KTH Sound&Vibration BOOK

Sound and Vibration n

und Source

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This material is protected by copyright and cannot be used or reproduced in any form without the written permission from: The Marcus Wallenberg Laboratory, KTH, SE-100 44 Stockholm, SWEDEN

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This course material was developed with support from the European Commission Tempus Program JEP-31018-2003 based on the Swedish book “Ljud och Vibrationer” used at KTH.

Chapters 1-12 are translated from “Ljud och Vibrationer” by Hans Bodén, Ulf Carlsson, Ragnar Glav, Hans-Peter Wallin, and Mats Åbom by Robert Hildebrand Chapters 13, 15 and 16 are written by Hans Bodén Chapter 14 is written by Ulf Carlsson and Hans Boden Chapter 17 is written by Mats Åbom Edited by Hans Bodén and Tamer Elnady

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CONTENTS

1 INTRODUCTION 1.1 The field of Sound and Vibration 1.2 Job Market for Sound and Vibration Engineers 1.3 Development 1.4 Principles, examples and countermeasures 2 FUNDAMENTAL CONCEPTS 2.1 Fundamental and applied mechanics 2.2 Definitions of sound and vibration fields 2.3 Peak value, mean value RMS-value and power 2.4 Longitudinal waves in gases and liquids 2.4.1 Longitudinal plane waves 2.4.2 Spherical waves 2.5 Diffraction 2.6 Models in room acoustics 2.6.1 Geometrical acoustics 2.7 Waves in solid media 2.8 Frequency analysis of sound 2.8.1 Time and frequency domain 2.9 Levels and DECIBEL 2.10 Filters 2.10.1 Band pass filters 2.10.2 Octave and third octave filters 2.11 Summation of sound fields, interference 2.12 Summation of frequency components 2.13 Important formulas 3 INFLUENCE OF SOUND AND VIBRATION ON MAN AND EQUIPMENT 3.1 The ear and hearing 3.1.1 The ear’s function 3.1.2 Measures of hearing 3.1.3 Measures of noise 3.1.4 Speech and masking 3.1.5 The influence of noise on man 3.1.6 Hearing injuries 3.1.7 Hearing protection 3.1.8 Sound quality 3.2 Effects of shock nd vibration 3.2.1 Machinery and vehicle vibrations

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3.2.2 Effects on man 3.3 Standards 3.3.1 ISO 3740 , ISO 3747, 3.3.2 ISO 2631-1, ISO 2631-2 3.3.3 ISO 5349 3.3.4 ISO 8662 3.3.5 ISO 4866 3.4 Regulations and recommendations 3.4.1 Machines 3.4.2 Vehicles 3.4.3 Work environment 3.4.4 Buildings 3.4.5 External noise 3.5 Important formulas 4 SIGNAL ANALYSIS AND MEASUREMENT TECHNIQUES 4.1 Complex numbers and rotating vectors 4.2 Fourier methods in sound and vibration 4.2.1 Fourier series 4.2.2 Fourier transforms 4.2.3 Parseval’s relationships 4.3 Measurement systems for sound and vibration 4.3.1 The measurement chain 4.3.2 Microphones 4.3.3 Accelerometers 4.3.4 Mounting of accelerometers 4.3.5 Calibration of transducers and measurement systems 4.4 Important formulas 5 VIBRATIONS OF SIMPLE MECHANICAL SYSTEMS 5.1 Mechanical power 5.2 Linear systems 5.2.1 One degree of freedom systems 5.2.2 Two degree of freedom systems 5.2.3 Multi degree of freedom systems 5.2.4 Frequency response functions 5.2.5 Damping 5.2.6 Mechanical-electrical circuits 6 THE WAVE EQUATION AND ITS SOLUTIONS IN GASES AND LIQUIDS 6.1 The wave equation in a source-free medium 6.1.1 Equation of continuity 6.1.2 Equation of motion 6.1.3 The thermodynamic equation of state 6.1.4 The homogenous linearised wave equation 6.2 Solutions to the wave equation

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6.2.1 General solution for free one-dimensional wave propagation 6.2.2 Harmonic solution for free one-dimensional wave propagation 6.2.3 Sound intensity for free one-dimensional wave propagation 6.2.4 Energy and energy density for free one-dimensional wave propagation 6.2.5 General solution for free spherical wave propagation 6.2.6 Harmonic solution for free spherical wave propagation 6.2.7 Sound intensity for free spherical wave propagation 6.3 Important formulas 7 REFLECTION TRANSMISSION AND STANDING WAVES 7.1 Reflection and transmission of plane waves 7.1.1 Normal incidence against a rigid boundary 7.1.2 Normal incidence at a boundary between two elastic half spaces 7.1.3 Plane wave propagation in three dimensional space 7.1.4 Non-normal incidence at a boundary between two elastic half spaces 7.1.5 Non-normal incidence at a boundary between a fluid and a solid 7.2 Eigen-frequencies and eigen-modes 7.2.1 Eigen-frequencies and eigen-modes in rooms 7.3 Important formulas 8 THE WAVE EQUATION AND ITS SOLUTIONS IN SOLIDS 8.1 Introduction 8.2 Wave propagation in infinite and semi-imfinite media 8.3 Quasi-longitudinal waves in beams 8.3.1 The wave equation for quasi-longitudinal waves in beams 8.3.2 Quasi-longitudinal waves in infinite beams 8.3.3 Quasi-longitudinal waves in finite beams 8.3.4 Reflection and transmission of quasi-longitudinal waves at area changes 8.3.5 Standing quasi-longitudinal waves in beams 8.4 Torsional waves in axles 8.4.1 The wave equation for torsional waves in straight cylindrical axles 8.4.2 Torsional waves in straight axles 8.5 Bending waves in beams and plates 8.5.1 The bending wave equation for beams and plates 8.5.2 Bending waves in an infinitely long beams 8.5.3 Bending waves in finite beams 8.5.4 Dispersion 8.5.5 Reflection and transmission at a boundary between two beams 8.5.6 Standing waves in beams 8.5.7 Standing waves in plates 8.6 Mechanical impedance and mobility 8.7 Damping in solid structures 8.7.1 Material damping

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8.7.2 Losses at boundaries 8.7.3 Losses in built up structures 8.7.4 Damping of beams and plates using absorbing material 8.7.5 Mathematical description of damping 8.7.6 Experimental determination of damping 8.8 Important formulas 9 ROOM ACOUSTICS 9.1 Energy methods 9.1.1 Energy balance for simple an coupled systems 9.1.2 Relationship between wave theory and energy based methods 9.2 Room acoustics 9.2.1 Sabine’s equation 9.2.2 Sound fields in rooms 9.2.3 Acoustic absorbers 9.2.4 Sound transmission through walls 9.3 Important formulas 10 SOUND GENERATION MECHANISMS 10.1 Monopoles 10.2 Dipoles 10.3 Quadropoles 10.3.1 Examples of quadropole sources 10.4 Influence of boundaries 10.4.1 Examples of hard and soft surfaces 10.5 Line sources 10.6 Sound radiation from vibrating structures 10.6.1 Infinite plane surfaces 10.6.2 Finite plates with bending vibrations 10.7 Point excited plates 10.8 Flow generated noise 10.8.1 Scaling laws for flow generated noise 10.8.2 Whistling 10.9 Important formulas 11 VIBRATION ISOLATION 11.1 Types of isolation 11.2 General about vibration isolation 11.3 Measures of vibration isolation 11.4 Prediction of vibration isolation 11.5 Models for prediction of vibration isolation 11.5.1 Rigid mass – ideal spring – rigid foundation 11.5.2 Flexible foundation 11.5.3 Wave propagation in the isolator 11.5.4 Non-rigid machine 11.5.5 General expression for vibration isolation

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11.6 Vibration isolation in practice 11.6.1 Design of vibration isolators 11.6.2 Methods for improving vibration isolation 11.6.3 Commercial vibration isolators 11.6.4 Dynamic stiffness 11.7 Important formulas 12 SOUND IN DUCTS 12.1 Principles for sound reduction in ducts 12.1.1 Insertion and transmission loss 12.1.2 Requirements on silencers 12.2 Sound propagation in ducts 12.2.1 The modified wave equation 12.3 Reactive silencers 12.3.1 Area changes 12.3.2 Expansion chambers 12.3.3 Side branch resonators 12.4 Electrical – acoustic circuits 12.4.1 Four-pole theory 12.5 Resistive silencers 12.6 Important formulas 13 INDUSTRIAL NOISE AND VIBRATION CONTROL 13.1 13.2 13.3

13.4

13.5 13.6

Motivation for industrial noise control Systematic approach to industrial noise control Noise control at the source 13.3.1 Noise generated by fluctuating forces in structures 13.3.2 Noise generated by fluid flow Noise control during the propagation path 13.4.1 Control of structure borne sound 13.4.2 Control of airborne borne sound Noise control at the receiver References

14 MACHINE CONDITION MONITORING 14.1 14.2 14.3 14.4

14.5

Introduction Basic ideas of machine monitoring Typical defects in gears and rolling bearings Vibrations of gears and bearings 14.4.1 Vibration characteristics of non-defective gears 14.4.2 Vibration characteristics of non-defective bearings 14.4.3 Vibrations of defective gears 14.4.4 Vibrations of defective bearings Monitoring methods 14.5.1 Early time domain methods

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14.6

14.7 14.8

14.9 14.10 14.11

14.5.2 Spectral methods 14.5.3 Cepstral methods 14.5.4 Envelope methods Machine condition indicators 14.6.1 RMS-value, peak-value and crest factor 14.6.2 Kurtosis 14.6.3 Defect severity index Residual time to failure estimation Measurement techniques 14.8.1 Instrumentation 14.8.2 Data acquisition 14.8.3 Signal filtering 14.8.4 Normalized order analysis User interface Signal processing tools References

15 VEHICLE NOISE AND VIBRATION CONTROL 15.1 15.2 15.3 15.4

15.5

15.6

Motivation for vehicle noise and vibration control Character of vehicle noise Measurement of exterior vehicle noise Vehicle noise sources 15.4.1 Engine noise 15.4.2 Exhaust and intake noise 15.4.3 Cooling system noise 15.4.4 Tyre-road noise 15.4.5 Aerodynamic noise Vehicle noise and vibration control 15.5.1 Engine noise control 15.5.2 Exhaust and intake noise control 15.5.3 Interior noise and vibration control References

16 NOISE AND VIBRATION IN PIPES AND DUCTS 16.1 Sound generation in pipes and ducts 16.1.1 Turbulent boundary layer generated sound 16.1.2 Sound generation by pipe discontinuities 16.1.3 Control valve sound generation 16.1.3.1 Classification of valves 16.1.3.2 Examples of valve types 16.1.3.3 Valve noise source mechanisms 16.2 Sound transmission in pipes 16.2.1 Fluid-borne sound 16.3.2 Structure-bone sound 16.3 Sound radiation from pipes

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16.3.1 Excitation by structure-borne sound 16.3.2 Excitation by fluid-borne sound 16.3.4 Radiation from pipe openings 16.4 Noise control techniques 16.4.1 Noise control at the source 16.4.2 Noise control during the propagation path 16.4.3 Control of structure borne sound 16.4.4 Reduction of sound radiation 17 SOUND GENERATION FROM FLUID MACHINES 17.1 Classification of fluid machines 17.2 Flow generated sound 17.2.1 The high Mach-number range 17.2.2 The case of liquids discontinuities 17.2.3 The character of the sound 17.3 Noise control

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Chapter 1: Introduction

CHAPTER ONE INTRODUCTION

Acoustics can be regarded as the science of sound and vibration. Sound today refers not only to those mechanical wave motions in air that give rise to sensations of hearing, but even to low-frequency (infrasonic) and high-frequency (ultrasonic) motions that cannot be sensed by hearing, as well as analogous wave motions in, for example, water (underwater acoustics). In solid materials, one speaks instead of vibrations or structure-borne sound. The phenomenon of hearing has fascinated mankind all through the ages. The mathematical theory of sound propagation can be said to have begun with Isaac Newton (1642 - 1727), whose work Principia (1686) contained a mechanical interpretation of sound as pressure pulses propagating in a medium. A more solid mathematical and physical groundwork of theory was provided by Euler (1707 – 1783), Lagrange (1736 – 1813), and d’Alembert (1717 – 1783). That development took place just as continuum mechanics and field theory began to take form, and the wave equation was formulated for functions of space and time. The modern theory of sound and vibrations is the product of the efforts of these mathematical physicists.

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Chapter 1: Introduction

NATURAL SCIENCES

TECHNICAL SCIENCES

LIFE SCIENCES

ART

Figure 1-1 Acoustics spans such diverse areas as biology, art, the natural sciences and technology. The proposed subdivision of the field (adjacent) covers, accordingly, many aspects and fields of science. The unshaded area describes the activity at a typical university sound and vibration laboratory. (Source: From R.B. Lindsay, who published the "acoustic wheel" in the Journal of the Acoustical Society of America 1964, vol 36)

1.1 THE FIELD OF SOUND AND VIBRATION In today’s well-developed, technological society, the number of systems that emit sound and vibrations is steadily increasing. Examples are machines, vehicles, and processes of all types, in which driving forces and engine power are constantly. Examples are machines, vehicles, and processes of all kinds, in which driving forces and engine power are continually being increased even as simultaneous efforts are made to hold down weight and materials usage. This implies the need for an ever more intensive research and development effort to identify and alleviate noise and vibration disturbances, and satisfy the needs of mankind for an acceptable environment. The fundamentals of Sound and Vibrations are part of the broader field of mechanics, with strong connections to classical mechanics, solid mechanics, and fluid dynamics. The subject of Sound and Vibrations encompasses the generation of sound and vibrations, the distribution and damping of vibrations, how sound propagates in a free field, and how it interacts with a closed space, as well as its effect on man and measurement equipment. Technical applications span an even wider field, from applied mathematics and mechanics, to electrical instrumentation and analog and digital signal processing theory, to machinery and building design. Several of the more important areas are worthy of particular mention:

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Chapter 1: Introduction

Machinery and Vehicle Acoustics deals with constructive measures to bring about machines, vehicles, and processes that are quieter and vibrate less; that requires knowledge of how sound and vibrations are generated, and how that generation relates to such physical parameters as flow velocities, masses, stiffnesses, losses, and geometry, for example. The most common reason that sound arises, in technical applications, is that a time-varying force excites vibrations of a mechanical structure, which then radiates sound. We can convince ourselves of that by knocking on a tabletop. The surrounding air is influenced by the vibrating structure, and responds with contractions and expansions. The mechanical form of energy we call sound has arisen. Analysis of the mechanisms of sound generation helps us to answer the question of why it sounds differently when we hit a tabletop with the soft part of the index finger, than it does when we hit it with the steel tip of a ball-point pen, and why it is louder when we knock the middle of the table with our knuckles than when we do the same thing at the edge of the table. The dull rumble of a Harley – Davidson motorcycle has become a defining characteristic – so important that the factory sought a patent on the sound to prevent competitors from plagiarizing it. That cash register click of a Mercedes door closing shut is something that other auto manufacturers strive for. The concept of Sound Quality is becoming ever more entrenched. Sound should convey a sense of product quality and reliability. The automobile industry has made pioneering efforts in this area and sound quality is given a high priority, along with such other important vehicle characteristics as road performance, safety and design. To describe sound, such subjective designators as “sharpness”, “rawness”, and “boxiness” are used. Several manufacturers have developed software packages that make it possible to modify the sound of products in order to mimic different design variations. The modified sounds are then judged by a “panel of listeners”. In the future, many consumer, industrial, and transportation products will be “sounddesigned”. Model

Technology

Vibrations

Force

Force Roughness Acoustic radiation

Mathematical structure Three stages Source

Response

Acoustic radiation

Examples: Roughness forces Gear tooth forces Mechanical imbalances Aerodynamic forces

Geometries Masses Stiffnesses Losses

Surface area Mode shape (oscillation pattern) Resonances Structure – air coupling

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Figure 1-2 Forces that cause vibrations appear in countless situations. It can be a matter of a roller in a bearing exhibiting outof-roundness, or perhaps small surface irregularities at the contact between a railway wheel and rail. The ability of the forces to induce vibrations depends on mechanical laws in which such parameters as the structure’s geometry, mass distribution, stiffness and losses come into play. The structure’s effectiveness as an acoustic radiator depends on the surface area, its oscillation pattern, and the coupling between the structure and the air at the surface of separation. Sound generation in connection with mechanical structures can be described in three stages: Source – Response - Radiation

Chapter 1: Introduction

N

N

L

p j = ∑ Zij vi Si = ∑ ∑ Zij Yki Fk Si i =1

i =1 k =1

Figure 1-3 In a machine as complex as a car, the three stages of sound generation occur in many parallel chains. The velocity v at a specific point, induced by a force F at another specific point, can be described by a so-called frequency response function Y. Similarly, other frequency response functions Z can describe the relationships between the velocities on the surfaces and the sound pressure p at an interior point. The frequency response functions can, moreover, be combined to Car Passenger describe all three stages in the Vibration Forces Sound propagation chain. By adding i l pressure the contributions from all significant forces, via the dominant radiating surfaces, the total sound pressure at a location of interest in the interior is obtained. The noise in the interior comes mainly from three sources: • • •

The driveline, i.e., the engine, transmission, and drive axles. The contact zone between the tires and the roadway. Airflow over the car body.

Sound passes from the source into the passenger compartment in two distinct ways: as structure-borne and as airborne sound. Structure-borne sound has essentially propagated in the form of vibrations from the source to the receiver, whereas air-borne sound had already radiated as sound, e.g., in the engine compartment, before its transmission into the passenger compartment (Sketch: Volvo Technology Report, nr 1 1988)

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Chapter 1: Introduction

Flow Acoustics is the study of the direct generation of sound in an elastic gaseous or liquid medium. The most common sound generation mechanisms are volume, force, and moment fluctuations in the medium, throughout which the elastic energy then spreads. Such phenomena are of great significance for noise from, for example, propellers, jet engines, fans, and vehicles, as well as for the propagation of sound in ducts. Figure 1-4 Volvo’s wind tunnel is used to find ways to reduce flow-induced sound. Mitigating such sound typically requires an even flow over the body, as well as fine tolerances and good fits, especially at doors and other seals. Airflow over projecting details can cause whistling. The small-scale turbulence around the car body gives a roaring type of noise, and if there are openings an airstream can sometimes create whistling sounds (Photos: Volvo Technology Report, nr 1 1988)

Room Acoustics, or Building Acoustics, deals with how sound fields are built up in various types of rooms and enclosed spaces, as well as how sound is transmitted through different types of structures, such as walls and systems of joists.

Figure 1-5 Heavy insulating surfaces effectively mitigate sound transmission from one enclosed space to another, as from, for example, the engine compartment to the cabin of a truck. For vehicles such as ships, cars, trains and aircraft, however, low weight is also a critical demand. Enormous efforts are therefore made in the transportation industry to bring about light, yet effective, designs for sound reduction. In the picture, the roof of a passenger car has been mounted in the opening between a sound reflective “reverberant room” at MWL, KTH (seen), and an adjacent sound absorbing ”anechoic room” (not seen). Using microphones and mathematical models for the sound interaction with the measurement rooms, the sound insulating properties of the car roof can be deduced (Photo: HP Wallin, MWL)

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Chapter 1: Introduction

Signal analysis is an important part of the science of sound and vibrations. The associated experimental methodology is primarily based on transducers, such as microphones and accelerometers, that convert sound and vibrations into equivalent electrical signals for further analysis. Until the 1960’s, most acoustic measurement instrumentation was analogue; then, digital computer technology arrived on a broad front. A numerical algorithm, FFT (Fast Fourier Transform), that transformed signals from the time domain (signal strength as a function of time) to the frequency domain (signal strength as a function of frequency), was a breakthrough in experimental methodology. Figure 1-6 To solve a noise problem, in which roof vibrations are suspected as the direct source, the car is driven on a chassis dynamometer.. The roof motion is registered as a function of time with the help of a motion sensor an accelerometer. Transforming the signal to the frequency domain, i.e., expressing its strength as a function of frequency, often greatly facilitates interpretation. Such a transformation is carried out by electrical filters or by a computer. That broad subject is called signal analysis. Its mathematical foundation is derived from mathematical statistics and from Fourier methods. In the case illustrated, a poorly made transmission is to blame. A shaft in the transmission, rotating at 1800 revolutions per minute, is bent or out-ofbalance, giving rise to a dominant vibration disturbance at 30 Hz. (Sketch: Brüel & Kjær, Structural Testing, Part 1)

Signal

Signal analysis

Time

Amplitude

Frequency

The Sound and Vibration subject area offers a wealth of opportunities for both the theoretically-inclined and the more practically / experimentally-oriented who are interested in applying themselves towards the improvement of machinery and vehicle designs, or towards quieter workplaces and societal environments.

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Chapter 1: Introduction

Train 100 meters away (100 km / h)

Normal conservation

Impact riveting

Quiet bedroom Weakest perceivable Sound

Eccentric press

Sound insulated lounge

Coarse grinding

Pain threshold

Near a jet plane starting up

Highest sound level that can be attained

Chainsaw

Manual grinding

Air-cooled electric motor 50 kW

Spray painting

Figure 1-7 The strength of sound is often given in dB. A change of a single dB is hardly noticeable. An increase of 8-10 dB is perceived as a doubling of the degree of disturbance. In order to adjust for variations in the sensitivity of the ear at different frequencies, corrected values are often used. The result is called Sound Level, and indicated in units of dB(A). (Diagram: ASF, ”Bullerbekämpning” [in Swedish], 1977, Illustrators: Anette Dünkelberg /Arne Karlsson)

1.2 JOB MARKET FOR SOUND AND VIBRATION ENGINEERS The task of minimizing sound and vibration in mechanical constructions falls primarily upon those who are responsible for their design and analysis, i.e., primarily engineers with mechanical, vehicle, and machinery backgrounds. Such issues have a great deal of relevance these days, and industry is therefore preoccupied with them. It is usually larger manufacturing companies that have the greatest need for engineers in this field. The vehicle industry employ many sound and vibration engineers but also other industrial sectors that manufacture products in which some type of energy conversion takes place (compressors, separators, turbines, fans, appliances, etc.) have a great need for this kind of knowledge. There are, moreover, many consulting firms in the field. Others that have a need for specialists in this area include national and local authorities that have regulatory responsibilities, as well as national research organizations that have the task of dproducing new knowledge.

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Chapter 1: Introduction

1.3 DEVELOPMENT

Figure 1-8 Swedish billboard demonstrating the marketing value of acoustic performance, describing the “world’s quietest dishwasher” as “Unbelievably unhearable”) (Photo: HP Wallin, MWL)

Classical acoustics was summarized by Lord Rayleigh in his fundamental treatise "The theory of sound" 1894-96. That classical theory serves as the basis for the modern science of sound and vibrations. Lord Rayleigh received the Nobel prize in physics in 1904 for his discoveries in the field of optics. The field of noise control engineering started to develop in in the 1930’s. Development on a larger scale came in thethe 1950’s in connection with for instance the effort to build quiet submarines and civil jet engine airplanes. The knowledge acquired spread to other areas of machinery acoustics. In Sweden Atlas Copco was one of the pioneers. They developed quiet pneumatic machinery and compressors, which had a great impact on the noise level along streets and at city squares. 18

Chapter 1: Introduction

Today, technical specifications on sound and vibration performance are made for a large share of the products that are delivered by manufacturing industry. The EU has brought about further demands on products to be sold in Europe, with respect to sound and vibration performance. Quality oriented manufacturers consider quiet, vibration-free products as a crucial quality characteristic and a marketable feature, so that all larger manufacturers tend to have departments responsible for sound and vibration performance. In the future, we will see improved and more easily used computer-based computational tools, and a continued development of cheap, personal computer-based systems for measurement and diagnostics. An example of the development of new technology is the so-called active sound and vibration control. The method is based on the use of, for example, microphones registering a disturbing sound field. Making use of automatic control methodologies, an “anti-noise” (phase inverted sound) is emitted that, under some conditions, can eliminate the disturbing field in an effective way. Commercial systems of that type exist already in, for example, ventilation systems and for automobile interiors and airplane cabins. In the future, we can expect corresponding systems on the vibration side as well. Today, vibration isolators, primarily of rubber, are inevitably used to prevent engine vibrations in an automobile from spreading into the car body. By installing parallel electrodynamic vibrators that generate reversed-phase forces, those disturbing forces that manage to pass through the isolators can be counteracted.

Mikrofon Microphone Högtalare Loudspeaker

Kontrollenhet Controller

Signal From motor Signal från motor

Varvtalsgivare Tachometer

Signal motor Signal From från motor

Figure 1-9 Cross section of the cabin of a turbo-prop SAAB 340 airplane with active noise control. The microphones register the sound at a number of points and transfer the electrical signals to the control unit. The signals can be treated to drive the speakers in such a way (”antiphase”) as to reduce the noise level in the cabin. The tachometer signals from the engines help to fine tune the automatic control in the control unit. (Source: Ny Teknik)

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Chapter 1: Introduction

1.4 PRINCIPLES, EXAMPLES AND COUNTERMEASURES The following pages give some examples of principles and applications in the field of sound and vibrations. On the left side, the underlying principles are presented, and on the facing right side, examples and countermeasures. Source: Asf, Bullerbekämpning (in Swedish), 1977.

Rapidness of a process determines the high frequency content The faster that force, pressure, or velocity changes occur, the higher the frequency content of the resulting sound. A fast ping pong ball gives a high frequency pop when it hits the table, while a slow handball bounces against the floor with a dull, low frequency sound. Principle Slow impact against the floor – low frequency sound

Rapid impact against the table – high frequency

Figure 1-10a (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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Chapter 1: Introduction

Example - Countermeasure In the primitive tooth configuration, the teeth slap together so that the forces between them rise and fall rapidly. The high-pitched tones are then strong. The more effort is made to modify the tooth configuration, the more softly they can be made to mesh together. Finally, the forces can be made to rise and fall again slowly. The high-pitched tones are then no longer so dominant. Since the peak force Force on a tooth

Angular gear

Sound level

Time Angular gear teeth

Force on a tooth

Rounded gear teeth Rounded gear teeth

Time Low-pitched tones

Figure 1-10b (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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High-pitched

Chapter 1: Introduction

Make vibrating surfaces as small as possible An object with small surfaces can vibrate very strongly without radiating a great deal of noise. The lower the frequency of the disturbing tones, the greater the surface area before it becomes a disturbing noise source. Since there is practically always a risk of vibrations when dealing with machinery, the shells and housings used should be as small as possible. Principle The electric shaver vibrations are transmitted into the large glass shelf and the resulting noise level is high.

The vibrations are no longer transmitted and the noise diminishes.

Figure 1-11a (Sketches: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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Chapter 1: Introduction

Example: The hydraulic aggregate was a powerful Instrument panel

noise source. Since the wall vibrations of the oil tank were damped by the oil itself, most of the noise was radiated by the instrument

Motor

panel.

Pump

Countermeasure: The panel was separated from the aggregate, reducing the radiating surface area, and thereby even the noise level.

Instrument panel moved to the wall Oil tank

Figure 1-11b (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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Chapter 1: Introduction

Densely perforated sheet emits little noise

Large vibrating shells cannot always be avoided. They give off a lot of noise. The reason for their sound radiation is that the vibrating sheet pumps the air to and from, like the piston of a pump. If the sheet is perforated, then it “leaks” and the pumping effect is weakened. The reduction in sound radiation is many times greater than the mere reduction in surface area. Alternatives to perforated sheet are nets and gratings. Principle

Unperforated sheet metal

Sound level

Perforated sheet metal

Figure 1-12a (Sketch: Asf, Bullerbekämpning, 1977 Illustrator: Claes Folkesson).

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Chapter 1: Introduction

Shield over drive belt and flywheel, of unperforated sheet

Perforated sheet

Example: The shield over the flywheel and drive belt constituted a strong noise source. The shield was fabricated of unperforated sheet metal.

Countermeasure: A new shield was fabricated from perforated sheet and wire netting. One of the noise sources of the press was thereby eliminated.

Figure 1-12b (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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Wire mesh netting

Chapter 1: Introduction

Wind tone is removed by a changed profile or a small disturbing element When sound flows past an object at certain speeds, a strong pure tone called a Strouhal tone can arise. By extending the dimension of the object in the direction of the airflow, with a “tail” for example, or by disturbing the regularity of the object profile, the tone can be prevented. In a duct, a resonance can amplify a Strouhal tone so much that the duct can be damaged. Principle

Air Flow

Regular pattern of vortices gives a strong tone

Small disturbing objects

Length extension

Irregular vortex pattern

Irregular vortex pattern

Figure 1-13a (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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Chapter 1: Introduction

Wind

Sheet iron spiral Chimney

Example: The shield over the flywheel and drive belt constituted a strong noise source. It was fabricated of unperforated sheet metal. Countermeasure: A new shield was fabricated from perforated sheet and wire netting. One of the noise sources of the press was thereby eliminated.

Figure 1-13b (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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Chapter 1: Introduction

Pure tones can be cancelled with sound in anti-phase

When the sound only consists of a single tone, or several such within a narrow frequency band, it can be completely or partially cancelled out in an interference muffler. It consists of a branched duct in which sound propagates through two separate paths that subsequently recombine with different time delays. In its simplest variation, shown in the figure, the path difference L1-L2 determines the frequencies at which sound reduction occurs. The time-delayed sound behaves as if in anti-phase.

Principle

L1

L2 Figure 1-14a (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

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Chapter 1: Introduction

Branches of varying lengths Interference muffler

Inlet

Example: When the frequency of the disturbing tone, or the temperature of the gas, varies in time, the effective frequency band of the muffler can be widened by a variation of the path length difference through multiple paths. The improvement obtained at the nominal frequency is, however, somewhat less than in the preceding variation. The interference damper is suitable for use

Outlet

Figure 1-14b (Sketch: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

29

Chapter 1: Introduction

The ASU Sound & Vibration Lab. At the Faculty of Engineering, Ain Shams University The Sound and Vibration Laboratory at the Faculty of Engineering, Ain Shams University was established in 2004 by a European Union Grant (JEP-31018-2003) within the TEMPUS Program. This is the first laboratory in Egypt specialized in sound and vibration teaching and research. The establishment of the laboratory and the development of the courses were done in cooperation with: - The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL) at the Royal Institute of Technology (KTH) in Sweden. - Institute of Sound and Vibration Research (ISVR) at the University of Southampton in United Kingdom. Our mission is to produce a new generation of Egyptian engineers and researchers in the field of sound and vibration. ASU-SVL welcomes research institutions, industry and others to cooperate with our staff and use the laboratory facilities. The laboratory hosts the Acoustical Society of Egypt, whose mission is to group all engineers, researchers and students in Egypt interested in the field of sound and vibration. ASU-SVL has joined the X3-Noise Aircraft External Noise Network in Europe acting as the regional focal point for MEDA Countries in this network. X3-Noise is funded within the Sixth Framework Programme. The activities in the lab are divided into three main domains: education, research and consultancy. The lab is equipped with the state-of-the art sound and vibration measurement equipment including different types of analyzers, transducers, calibrators, and accessories. There is an anechoic room (80 m3), a reverberation room (80 m3), and an acoustic flow facility. There is a teaching lab of 13 computers working as data acquisition systems to be used in lab exercises in different courses.

30

Chapter 2: Fundamental Concepts

CHAPTER TWO FUNDAMENTAL CONCEPTS

The aim of this chapter is, partly, to give an overview of the subject area of sound and vibrations, before the more detailed descriptions that follow, and partly to present and define a number of important concepts early enough that technically interesting problems may be treated in parallel with the main presentation. In the introduction, we shall relate the subject of sound and vibration (sometimes expressed as "vibroacoustics") to the mechanical sciences as a whole, describe what types of mechanical elastic waves can arise, as well as the conditions for their existence, and define important quantities which specify sound and vibration fields. In the section “Diffraction”, we will examine the conditions under which a sound field can bend around different objects. Using models from the subject of room acoustics, we will make a short preliminary survey of different methods for analyzing how sound propagation and sound fields interact with various types of rooms. Sound and vibration disturbances can only be tracked as functions of time, but are often more effectively analyzed and characterized as functions of frequency. Important concepts, such as the time and frequency domains, are defined, and filtering and frequency analysis are described.

31

Chapter 2: Fundamental Concepts 2.1 BASIC AND APPLIED MECHANICS The science of sound and vibration is a part of applied mechanics. This can be subdivided as in Figure 2-1.

S tatics

P article m echanics

V ibroacoustics

R igid body m echanics

Fluid M echanics S trength of M aterials A pplied m echanics

Fundam ental M echanics

Figure 2-1 Proposed subdivision of mechanics.

In statics, systems are examined in equilibrium states, i.e., the object considered is neither accelerated nor decelerated, and often has, in practice, no velocity. In particle mechanics, one studies the motion of the center of gravity of an object. The object / particle has three degrees of freedom, i.e., translations which can be described in a Cartesian coordinate system. In order to describe rigid body movement, six degrees of freedom are needed: three translations and three rotations. In applied mechanics one studies the behavior of solid, liquid and gaseous media, including deformation, wave propagation, tension, and fracture, when these media are subjected to physical agents of various types, such as forces and temperature shifts. In strength of materials one studies strain and deformation properties with the objective of properly dimensioning a construction for good strength and stability. In fluid mechanics one describes various aspects of the motions of, above all, fluids and gases. A typical problem from this field would be to improve the lift of an airplane wing, for example, or reduce the head loss (pressure drop) across a ventilation duct. Vibroacoustics covers everything from the basics of how sound and vibrations are generated and how they propagate to the question of how vibrations in a solid structure, a combustion engine for example, give rise to acoustic radiation. The division of the fields of mechanics as in Figure 2-1 is a mere simplification which can be adjusted and supplemented in many ways. A model from particle mechanics can, for example, be applied in vibroacoustics to explain the appearance of a certain tone when we blow over the mouth of a bottle, as in Figure 2-2. Force

Mass

Spring

32

Figure 2-2 The tone that arises when we blow over the bottle opening can be determined by methods from particle mechanics. The phenomenon is called a Helmholz resonator. It was already employed in the amphitheaters of ancient Greece where clay pots were used as resonators, see chapter 10. (Sketch (far left):Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

Chapter 2: Fundamental Concepts 2.2 DEFINITION OF SOUND AND VIBRATION FIELDS Sound and vibration waves are mechanical elastic waves, and thus the conditions for their existence are that the medium possess mass and elasticity. If a mass particle is displaced from its equilibrium position, the elastic forces will seek to return it to its original position. The particle influences the surrounding particles and, in this way, a disturbance propagates in the medium. Sound waves transport relatively little mechanical energy; thus, as is familiar to all Figure 2-3 A disturbance in water on a pond gives rise from everyday life, speech is not a to a radial wave pattern. These water (or gravitation) particularly exhausting activity. A speaking waves are nevertheless driven, in contrast to sound and person sends out only a few thousandths of vibration waves, by a balance between the water’s a Watt. Nobody is ever going to be able to inertia and gravitational effects. (Photo: Klas Persson) cook potatoes by yelling at them. This means, on the other hand, that machines driven at high power levels have almost unlimited capacities to excite acoustic fields; small flaws can convert a share of the available mechanical power into acoustic disturbances. In vibroacoustics there are two classes of waves: longitudinal waves and transverse waves. Longitudinal waves have a particle motion which is parallel to the direction of wave propagation, see Figure 2-4a, while transverse waves have a particle motion which is perpendicular to the direction of propagation as in Figure 2-4b. These shall be described in further detail in later chapters.

Particle displacement

Wave Propagation

Wave Propagation

Particle Displacement

Figure 2-4b The hand shaking the rope generates a transverse wave in which particles oscillate perpendicular to the direction of wave propagation.

Figure 2-4a A loudspeaker generates longitudinal waves in which particles oscillate parallel to the direction of wave propagation.

G In a gas or a fluid, at a certain point in space r and time t, an acoustic wave can be G described by the acoustical field quantities sound pressure p (r , t ) [Pa] and particle G G velocity u (r , t ) [m/s]. Sound pressure variations are normally very small deviations of the ambient pressure, and the particle velocity is small compared to the speed of sound.

pt

Figure 2-5 An acoustic field ordinarily implies only small disturbances. Normal speech at a distance of several meters, for example, gives a sound pressure p of a few hundredths Pa superimposed upon atmospheric pressure p0, which is about 105 Pa.

Sound Pressure, p

Atmospheric Pressure p0 5 ~ 10 Pa

t

33

Chapter 2: Fundamental Concepts 2.3 PEAK VALUE, MEAN VALUE, RMS-VALUE, AND POWER Before going any further, we recall some formulas from electrical engineering which are used to characterize signals, for example sound pressure p(t). A harmonic signal is a signal which can be described by a sine or cosine function as p(t) = pˆ sin(ω t + ϕ), where

(2-1)

p(t) is the instantaneous, time-dependent sound pressure, pˆ is the peak value or simply the amplitude,

ω = 2πf is the angular frequency, ϕ is the phase angle. The time-averaged mean value of a signal, marked by an overbar, is p=

1 T

T

∫ p(t )dt

,

(2-2)

0

where T [s]is the time over which the average value is determined. The root mean square or RMS value is marked by ~ p and is defined according to ~ p=

1 T

T

∫p

2

(t )dt .

(2-3)

0

The rms-value is a very important, and oft-used, form, since it gives information about the time average of the signal power content. p (t)

T

Figure 2-6 The amplitude characteristics of a harmonic signal, of sound pressure for example, can be described in various ways. The period T is the time required for one complete oscillation.

p^ ~ p t ppeak-peak= 2 ^p

The relation between the peak value and the rms-value for an arbitrary signal is called the peak factor or crest factor TF = pˆ ~ p . (2-4) For a harmonic signal, the following relationship applies between the rms-value and the peak value: ~ p = pˆ

2 .

34

(2-5)

Chapter 2: Fundamental Concepts Exercise 2-1 Show that equation 2-5 is valid for a harmonic signal.

n W

Figure 2-7 The acoustic power W over an area S in an acoustic field is the product of the sound pressure, particle velocity, and area.

S

Sound Source

The instantaneous mechanical power is the product of the instantaneous force and the instantaneous velocity. In acoustics, consequently, the instantaneous acoustic power G G W (r , t ) [W], which is transported through a surface S with a normal vector n , as in Figure 2-7, is G G G G G (2-6) W (r , t ) = ∫ p (r , t )u (r , t )n dS . S

G In acoustics, we are typically interested in the time average of power quantities W (r ) , see chapter 3.

2.4 LONGITUDINAL WAVES IN GASES AND FLUIDS In gases and fluids, shear stresses are, as a rule, small enough to be neglected. Consequently, only longitudinal mechanical elastic waves can exist in such media. Longitudinal waves are characterized by a particle velocity which is parallel to the direction of wave propagation (see Figure 2-4a). Two cases are considered here for longitudinal waves: plane waves and spherical waves. More complicated wave fields can be constructed from these simple special cases. 2.4.1

Longitudinal plane waves

Longitudinal plane waves are characterized by the condition that points with the same acoustical state, i.e., the same sound pressure and particle velocity, form parallel planes. Figure 2-8 shows the acoustic field of an infinite duct with a harmonically (i.e., sinusoidally) oscillating piston at one end. Harmonic vibrational velocity v(t) = v.sin(2π ft)

Direction of Propagation

λ

35

Figure 2-8 A harmonically oscillating piston in an infinitely long duct gives rise to a plane longitudinal acoustic wave that propagates in the duct. The frequency of the wave is the same as the piston oscillating frequency. The quantity 2π f is the so-called angular frequency, designated by ω. The distance λ between two planes in the medium with the same acoustic state is the wavelength in the medium.

Chapter 2: Fundamental Concepts When the piston begins to move forward in the cylinder, it compresses the air before it. That compression propagates at a speed completely independent of the particle velocity, hereafter referred to as the disturbance propagation speed c [m/s]. After a half period, the piston moves in the opposite direction and creates an expansion in the medium, which also moves at the speed c through the medium. After period T [s], the disturbance will have propagated the distance λ [m], so that the following elementary relation applies

λ = cT = c f

(2-7)

c = fλ ,

(2-8)

or where

c is the disturbance’s propagation speed or the sound speed, f is (the disturbance’s) frequency, λ is the wavelength.

In air at normal pressure and temperature, c ≈ 340 m/s. For the case of a free plane longitudinal wave, i.e., wave propagation without reflections, there is a very simple relationship between the field quantities sound pressure p and particle velocity ux as well as the time average of the sound power W . Sound pressure and particle velocity are always in phase, i.e., they attain their respective maxima and minima simultaneously. The relation between them can be expressed p( x, t ) = ρ 0 cu x ( x, t ) ,

where

(2-9)

ρ0 [kg/m3] is the density in the undisturbed medium (in air at normal pressure and temperature ρ0 ≈ 1.21 kg/m3) and c is the sound speed.

The time-averaged sound power W becomes, according to (2-2) T

W =

1 W (t )dt . T∫

(2-10)

0

Putting (2-6) and (2-9) in (2-10) gives

W =

1 T

T

∫ 0

p 2 ( x, t ) S dt , ρ0c

(2-11)

which, in accordance with (2-3), gives W =~ p 2 S ρ0c .

(2-12)

G If we study the sound power per unit area, the so-called sound intensity I [W/m2], the time-average of the x-component is obtained as

Ix =W S = ~ p 2 ρ0c .

(2-13)

Thus, for a free plane wave, the time average of the sound intensity is proportional to the square of the rms-value of the sound pressure. 36

Chapter 2: Fundamental Concepts In linear vibroacoustics, in more general terms, the time-averaged sound power is proportional to the square of the rms-amplitude of the relevant field quantity, as W =C ~ p2∝~ p2 .

(2-14)

This important relation implies that if the constant of proportionality C is known, we can determine the time-averaged sound power from a measurement of the rms sound pressure alone, using a pressure-sensitive microphone. That is usually considerably simpler than determining the particle velocity and making use of (2-6) directly. For a free plane wave, C = S/ρ0c according to (2-12). Note that if the losses in the medium are neglected, and since the wave doesn’t spread through an expanding volume, then time-averaged quantities are independent of distance to the source, i.e., time-averaged sound intensity and sound pressure amplitude are independent of spatial position. 2.4.2

Spherical and cylindrical waves

In the preceding section on plane waves, we were able to show that when the wave does not suffer losses, certain quantities remain independent of the distance to the source. If the source is an arbitrarily placed sphere and all points on its surface oscillate radially at the same amplitude and phase, or if the source radius a is small compared to the sound wavelength, then the source will produce spherical waves, as shown in Figure 2-9.

a

λ

λ λ λ Figure 2-9 A spherical source which oscillates at the same amplitude and phase over its entire surface area gives rise to spherical wave propagation.

The mechanical power W emitted into the medium by the pulsating sphere spreads over an ever-expanding spherical area (fig 2-10); thus, the time-averaged sound intensity is I r = W 4πr 2 .

(2-15)

Figure

r 2r 3r

37

2-10 In spherical wave propagation, sound power is divided over an ever-increasing area. The intensity decreases to one fourth its original value for a doubling of the distance to the source, and to one ninth when the distance is tripled.

Chapter 2: Fundamental Concepts Examples of real phenomena giving rise to spherical wave propagation are loudspeakers and the outlets of exhaust pipes, provided that the dimensions of the source are small compared to the wavelength, i.e., at sufficiently low frequencies as implied by (2-8). As is evident from Figure 2-9, the curvature of the wave fronts decreases with increasing radius. For engineering purposes, the waves can be considered (locally) plane for radii r > λ / 3, and (2-13) is then applicable even for this spherical wave case. The rms sound pressure can then be expressed in the form

~ p = ρ 0 c W 4π r 2 .

(2-16)

If the source is, instead, and infinitely long cylinder, the entire surface of which oscillates with a uniform phase and amplitude, then cylindrical waves arise; see Figure 2-11.

3r Figure 2-11 An infinitely long cylindrical source oscillating with uniform phase and amplitude over its entire surface gives rise to cylindrical waves.

2r r

S 2S 3S

The mechanical power, often given in units of power per unit length in the case of a line source, is distributed over a cylindrical area, so that the sound intensity can be expressed

I r = W ' 2π r ,

(2-17)

where W ' [W/m] is the sound power per unit length. Examples of sound sources that can be regarded as cylindrical are electrical distribution cables, pipes, ducts, transport belts at breweries, and heavily trafficked roads. A pre-requisite is that the distance from the source be small compared to the length of the source. 2.5

DIFFRACTION

Diffraction takes place in all types of wave propagation. Water waves are not noticeably affected by the presence of a thin mooring post in the water. The waves roll on as if the post did not exist. On the other hand, we know that behind a break wall in a protected harbor, or behind a narrow spit of land jutting into the water, a shadow zone free of waves develops. The relationship between the size of the hindrance and the wavelength determines how the wave motion is bent about it. Visible light has an approximate wavelength of 10-7 m, whereas typical human speech has a wavelength of about 1 m. For this reason, we can hear someone speaking from behind a pole without being able to see her at the same time. Thus, wave motions with wavelengths large in relation to the obstacle are little influenced by it and spread behind it. If, instead, the wavelength is small in relation to the obstacle, then a shadow zone develops. 38

Chapter 2: Fundamental Concepts

Exercise 2-2 Go to an open area, without any reflecting objects, such as buildings and the like, in the vicinity. Ask a companion to stand about 5 meters away, facing away from you, and take turns voicing an extended (i) ooooooo........ (“oo” as in “food”, not “foot”) (ii) ssssssss........

Note: the oo-sound has a frequency of about 250 Hz and the s-sound is a hiss with a frequency around 6000 Hz. Compare the wavelengths with the diameter of the head. These diffraction phenomena, as well as the persistence of free plane waves as planar and of free spherical waves as spherical, can be explained with the help of Huygen’s principle: "Every point on a wave front can be described as the center of a secondary wave field, a so called elementary wave. The new position [a moment later] of the wave front is the tangent to the set of these elementary waves ", see Figure 2-12.

a)

b)

Figure 2-12 Huygen’s principle can be used to show that free plane waves remain plane, and spherical waves remain spherical as they propagate. In a), the tangent to the so-called elementary waves is seen to build a plane wave front, and in b) the corresponding principle for spherical waves is shown.

In Figure 2-13a, it is seen that at such low frequencies that the wavelength is large in relation to the opening in the wall, Huygen’s principle implies a spherical wave propagation from the opening. If the frequency is high, and the wavelength thereby small in relation to the opening, as in Figure 2-13b, then the wave propagates as a beam with shadow zones on each side of it. Shadow zone

Spherical propagation

λ

λ

λ a)

λ b)

Shadow zone

39

Figure 2-13 a) shows sound transmission through a hole, small in comparison to the sound wavelength. From the opening, spherical wave propagation occurs. b) shows the case of hole that is large with respect to the wavelength. In the middle, the wave passes relatively unhindered, while shadow zones are built up at the sides.

Chapter 2: Fundamental Concepts Exercise 2-3 Noise from roads and railways is an enormous problem. A common attempt at a solution is to install noise walls or barriers. How should noise barriers be placed?

a) Is it in the high or the low frequency range that noise barriers can be expected to provide the greatest benefit? b) Is the greatest effect obtained from the barrier when the source is near or far from it? c) Where should the receiver be located with respect to the barrier, to receive the greatest benefit? 2.6

Figure 2-14 For highways with average vehicle speeds exceeding 70 km/h, the dominant noise is that generated at the contacts between tires and the road surface.

ROOM ACOUSTICS MODELS

In many situations, sound interacts with a closed space such as rooms in dwellings, concert halls, or industrial facilities, but even automobile or railway car interiors, or ship or airplane cabins. Knowledge of sound propagation and sound fields in such spaces is therefore an essential part of acoustics. When sound from a source reaches one of the room’s bounding surfaces, a share of the sound power is reflected back into the room and a share absorbed by the wall. In room acoustics, three distinct methods or models are used to describe sound propagation and the sound fields that arise. At low enough frequencies that the wavelength of sound is of the same order of magnitude as the dimensions of the room, wave theoretical room acoustics is a powerful tool. It turns out that when some of the room’s dimensions are whole multiples of half the wavelength, the incident and reflected waves interact such as to bring about standing wave fields, which then dominate the sound field in the room. The particular frequencies at which that occurs are called eigenfrequencies, or resonance frequencies in ordinary speech. The wave pattern, with its characteristic sound maxima and minima, is called an eigenmode, or simply mode; see Figure 2-15. Figure 2-15 When a dimension of a closed space, an airplane cabin or a vehicle interior for example, is a whole multiple of the half wavelength, then constructive interference arises between incident and reflected waves. A standing wave pattern with nodes and anti-nodes results. The frequencies at which that happens are called eigenfrequencies, and the standing wave patterns are called eigenmodes. (Sketch: Brüel & Kjær, Technical Review).

40

Chapter 2: Fundamental Concepts

Direct wave Reflected wave Sound source Observation point

Figure 2-16 Near a source, direct unreflected sound dominates; such sound comprises the so called direct, or free field in which the power from the source is distributed over an ever expanding volume, resulting in a halving of sound pressure for a doubling of distance, according to relation (2-16) between sound power and sound pressure for spherical waves. Further away, sound that has been reflected at least once dominates; that sound constitutes the so-called reverberant field. With a stricter demand, that sound pressure be constant and independent of distance to the source, sound must arrive at an observation point from all directions, with random phase; it then fulfills the definition of an (ideal) diffuse field.

At higher frequencies, the eigenfrequencies are so tightly spaced that, for practical reasons, we must choose another mathematical description. In chapter 8, we will study statistical energy methods for so called diffuse fields, see Figure 2-16. In practice, the demands for such a field are seldom fulfilled. The actual field is then called a reverberant field. The third method, geometrical room acoustics, is described in the next section. 2.6.1

Reflections and geometrical acoustics

A free wave, incident upon a reflecting surface with irregularities much smaller than a wavelength λ, changes its direction, i.e., is reflected, in a predictable way. In Figure 2-17, the wave fronts of the incident and reflected waves are marked. It is convenient, at times, to indicate a wave by an arrow. The arrow is perpendicular to the wave fronts and points in the direction that the wave propagates. Using Huygen’s principle, we can show that: (i)

Against a plane surface, a plane wave is reflected as a plane wave and a spherical wave as a spherical wave.

(ii)

The direction of the incident wave, the normal to the reflecting surface, and the direction of the reflected wave, all lie in the same plane.

(iii)

The angle of incidence θi is equal to the angle of reflection θr, see Figure 2-17.

n

θi Incident plane wave

Wave fronts

λ

θr λ

Reflected plane wave

Wave fronts

41

Figure

2-17 With the help of Huygen’s principle, we can show that a plane wave, incident on a reflecting surface, is reflected as a plane wave, and that the angle of incidence θi is equal to the angle of reflection θr.

Chapter 2: Fundamental Concepts If the reflecting surface is rigid, the sound pressure amplitudes and phases of the incident and reflected waves are equal the point of reflection. An observer is influenced by both the direct sound from the source and the reflected sound. We can describe the sound field above the reflecting surface by a superimposition of the direct wave field direct from the actual source with the direct field of an identical, imagined mirror source, placed at the same distance from the surface, but on the opposite side; see fig 2-18. Wave fronts

Source Incident spherical wave d

Observer

Reflected spherical wave

Rigid surface d

Figure 2-18 A spherical wave incident upon a reflecting surface is reflected as a spherical wave. If the surface is rigid, the reflected wave can be regarded as equivalent to a direct field from an imagined mirror source, identical to the actual source, at the same distance from the surface, but on the opposite side.

Mirror source

In geometrical acoustics, a sound wave is represented by an arrow in the direction of wave propagation, just as a light wave is represented in geometrical optics. We can regard a wave as a beam originating from the source. The method can be used to explain how a parabolic microphone functions; see Figure 2-19. Geometrical acoustics is primarily applied in the design of auditoriums. The purpose is to bring about an even distribution of sound, without focuses and shadow zones; see Figure 2-20. The limitation of the method is that typically only the first, or possibly even the second, reflection can be studied before it becomes impractical to follow the “sound trail”. Roof

Balcony Stage Parquet

42

Incident waves

Microphone

Parabola Figure 2-19 In a parabolic microphone, the reflected waves are focused into a focal point, at which the microphone is located, to maximize the amplification. Figure 2-20 In geometrical acoustics, sound waves are regarded as beams. The method is often used in the design of large musical auditoriums. The limitation is that it is typically only possible to follow the first few reflections of the sound waves. (Source: Brüel & Kjær, Measurements in Building Acoustics.)

Chapter 2: Fundamental Concepts 2.7

WAVE TYPES IN SOLID MEDIA

Solid media can sustain both normal and shear stresses, and thereby resist both volume and shape changes. That implies that in solids, in contrast to gases and liquids, not only longitudinal waves, but even transverse waves, can exist. Transverse and longitudinal waves can also, in combination with each other, build special types of waves, e.g., bending waves. Here, we will acquaint ourselves with some important special cases. Figure 2-21a illustrates a longitudinal wave and Figure 2-21b a transverse wave. When the medium is infinite in the direction of wave propagation, so that no reflections occur, we speak of free wave propagation.

Particle motion Partikelrörelse

Utbredningsriktning Propagation direction

Particle motion Partikelrörelse

Utbredningsriktning Propagation direction

a) b) Figure 2-21 Important wave types in solid media. a) Longitudinal wave: particle motion is parallel to the direction of disturbance propagation. b) Transverse wave: particle motion is perpendicular to the direction of disturbance propagation.

In a bounded medium, reflections occur at the boundaries. At certain frequencies, the incident and reflected waves interfere, or interact, such that standing wave patterns arise. These frequencies are called, as in section 2.6, eigenfrequencies or resonance frequencies, and the oscillation pattern is called an eigenmode or mode. Bending oscillations

Figure 2-22 When a tuning fork is stricken, bending oscillations take place in the legs, essentially in the first eigenmode. That disturbance excites a longitudinal oscillation at the same frequency. That latter, in its turn, excites bending waves in the tabletop. As is well known, it isn’t until that large table surface has been put into contact with the tuning fork, and begun vibrating itself, that we begin to hear the tuning fork well.

Tuning fork

Table-top

Longitudinal oscillations

Bending oscillations

43

Chapter 2: Fundamental Concepts First eigenmode

Exercise 2-4 Hold a long measuring stick, extended about 1.5 m from your grasp, and shake it transversally at about 1 Hz. It will now (hopefully) be oscillating in its first eigenmode. Increase the frequency until you find the second eigenmode. Observe the antinodes, where there is a large amount of bowing, i.e., deformation, and the nodes, at which there is none.

2.8

Second eigenmode

Node AntiNode

CHARACTERIZATION OF SOUND ACCORDING TO FREQUENCY

Sound can be characterized by its frequency content. For mankind, audible sound is that which falls in the 20 – 20 000 Hz range. For frequencies lower that 20 Hz, we speak of infrasound, and over 20 000 Hz of ultrasound. Most acoustic phenomena are frequencydependent. It is well known that different sound sources have different character, see Figure 2-23. Slowly changing and “soft” events mainly generate low frequency sound, while short and fast events mainly generate high frequency sound. We can be convinced of that by slapping the soft part of the thumb against the tabletop, and then do the same with a ballpoint pen. As is evident from Figure 2-23, the audible region spans over a range of wavelengths from 17 m to 17 mm, which is of fundamental significance for the types of noise control measures that can be undertaken. Analysis of the frequency content of a sound is therefore of great importance in the field of sound and vibration.

44

Chapter 2: Fundamental Concepts Audible Hörbartsound ljud

Infrasound Infraljud

Ultrasound Ultraljud

λ

340 340

17 17

1

10 20

3,4 3.4

0,34 0.34

100

1000

-3 17 .10 -3 17*10

Våglängd [m] [m] Wavelength

10000 20000

Frekvens [Hz] Frequency [Hz]

Sound Sources: Ljudkällor Wind noise Vindbuller Ship machinery Fartygsmaskiner Cars, Bilar,Trains, tåg etc etc. Smelting oven Smältugn Skärande bearbetning Cutting tools

Sawing Sågning Air blasting Luftblåsning Echo Ekolod

Audible range Hörområden

Mankind Människa Hund Dogs Fladdermus Bats

Figure 2-23 Classification of sound according to frequency, and the relation between frequency and wavelength in air with a sound speed c = 340 m/s. Frequency ranges of various sound generation mechanisms. (Source: Asf, Bullerbekämpning, 1977.)

2.8.1

Time and frequency domains

Figure 2-24a shows a harmonic signal of frequency f0 as a function of time. It can also be described as a function of frequency, as in Figure 2-24b. All power in the signal is concentrated at the frequency f0. We have thus introduced the, for vibroacoustics fundamental, concepts of the time and frequency domains. These are equivalent concepts, and we can choose to describe a signal in either of these domains. We can only record a signal as a function of time. It can thereafter be transformed into the frequency domain by frequency analysis, i.e., by methods from Fourier analysis. Signal

Amplitude

T=1/f0

Time

a)

f0

Frequency

b)

Figure 2-24 Harmonic signal in a) the time domain, and b) the frequency domain.

45

Chapter 2: Fundamental Concepts Jean Baptiste Fourier (1768-1830) demonstrated more than a century ago that all periodic signals can be described by a summation of harmonic signals. We can illustrate that by means of the three-dimensional Figure 2-25. A periodic signal, e.g., a sound pressure p(t) that repeats with period T, can be written p(t ) = p(t + nT ) , n = 1, 2, 3,...

(2-18)

Periodic signals are very common in engineering applications. They can be sound pressures or vibrations arising from, for example, rotating machines whose sound and vibration generation is repeated in an identical fashion in cycle after cycle. In Figure 2-25, a signal is shown (heavy line) together with its harmonic components (thin line), at frequencies fn. Every periodic signal can be expressed as the sum of harmonic signals with a so-called Fourier series (see chapter 3) according to N

p(t ) = ∑ pˆ n cos(2π nf 0 t + ϕ n ) ,

(2-19)

n =1

where

pˆ n is the peak value of the n-th harmonic component, f0 = 1/T and nf0 are the frequencies of the n-th harmonic component, ϕ n is the phase angle.

Amplitude

0

0

T/3 Time [s]

3/T

2T/3 T

1/T

4/T

Figure 2-25 A periodic signal can be described in both the time and the frequency domains. In the time domain, the signal is represented by its variation in time, and in the frequency domain by the amplitude of the harmonic signals it is built up of. Here, the phase angles ϕn are zero for all frequency components.

2/T Frequency [Hz]

The description of a signal in the frequency domain is usually called a (frequency) spectrum and each line, frequency component, is called a tone or a spectral line. In order to obtain a complete description, the phase angle of each frequency component is also required. Typically, however, we are only interested in the amplitudes of the frequency components. The frequency spectrum is very useful when the signal is to be analyzed, e.g., it provides the opportunity to distinguish every frequency component, even in cases in which a single frequency is dominant in the signal.

46

Chapter 2: Fundamental Concepts Example 2-1 With today’s technology, it is possible to discern spectral or frequency components with amplitudes that are a 100 000 - th those of other frequency components present. A rotating machine always gives rise to a strong frequency component corresponding to the rotational speed. Assume that we try to discern weaker signals, such as those of a roller bearing that is beginning to wear out. We are thus trying to find very small frequency components in the presence of a very strong one. That is considerably easier in the frequency than in the time domain.

We can illustrate the foregoing by considering a simple compound signal consisting of two harmonic components, one with frequency f0 = 1/ T and another with a lower amplitude and double the frequency, i.e., 2f0 = 2/ T, as shown in Figure 2-26. Signal

0 a)

0.01

Amplitude

0.02

0.03

0.04

0.05 Time [s]

0 b)

50

100

150 200 Frequency [Hz]

Figure 2-26 a) A signal in the time domain (heavy line) consisting of two harmonic components (thin lines) with periods T and T/2. b) Corresponding signal showing two components in the frequency domain.

The more complex signal in Figure 2-27 gives rise to more spectral components in the frequency domain. Signal

0 a)

0.05 0.1 0.15 0.2 0.25 0.3 Time [s]

Ampl

0 b)

20

40

60

80 100 Frequency [Hz]

Figure 2-27 a) A gear that transfers a constant moment and alternates between having two and three teeth meshing at a time, gives rise to a square wave type of signal in the time domain. b) In the frequency domain, many high frequency components are needed to describe the rapid variations in the time domain.

47

Chapter 2: Fundamental Concepts If the signal is not periodic, but rather transient, as in Figure 2-28, or stochastic as in Figure 2-29, a continuous distribution is instead obtained in the frequency domain. The transformation from the time to the frequency domain, for transient signals, can be described with the help of a Fourier transform; see chapter 3. Signal

Amp

Tid [s] [s] Time

a)

b)

Frekvens [Hz] Frequency [Hz]

Figure 2-28 A transient excitation, e.g., a hammer blow against a plate structure, brings about a time-decaying signal. In the frequency domain, a continuous spectrum is obtained, i.e., the spectral components are infinitely dense. If the vibrations in the structure are dominated by an eigenfrequency, then graphs such as the one given above are obtained.

Amp

Signal

a)

Tid [s] [s] Time

b)

Frekvens [Hz] Frequency [Hz]

Figure 2-29 A purely stochastic process, such as a downpour of rain on a car roof, gives a frequency spectrum of constant amplitude, a so-called white noise. Another example is the noise caused by the turbulent boundary layer around the fuselage of an airplane.

2.9

LEVELS AND DECIBELS

In example 2-1, we found that it is possible, with modern digital technology, to discern spectral components with amplitudes less than 1/100 000 of those of other spectral components. With that in mind, how can we show the entirety of that spectrum, on a monitor screen for example? If the strong component is 100 mm tall, then the weak one is only 0.001 mm tall in a linear scale, and therefore not even visible. If we want to see the entire spectrum simultaneously, then the amplitude scale must be adapted in some way. The solution is to use a logarithmic scale, which compresses the high amplitudes and expands the low ones. 48

Chapter 2: Fundamental Concepts At the beginning of the 1920’s, it had become practical to carry out routine sound measurements. An acoustic group at Bell Systems in the U.S. introduced a measurement quantity that they called a “sensation unit”, which was based on a logarithmic scale with base 10. The unit Bel (after Alexander Graham Bell), defined as the base-10 logarithm of the quotient between two acoustic power values, eventually proved to be impractically large. Today, the unit commonly used is, instead, a tenth of a Bel. The logarithmic unit deciBel better reflects mankind’s sense of hearing, as we shall see in chapter 3, providing an ideal gradiation of the relevant range of values, neither too coarse nor too fine nor with unwieldy numbers. One deciBel (1 dB) corresponds both to the measurement precision that can typically be obtained in acoustic measurements and to the amount of change that a human can discern in ideal circumstances. The base-10 logar-ithm will henceforth be designated by log. Since the argument of the logarithm, i.e., the ratio of the power value of interest to some chosen reference power value, is dimension-less, then logarithmic quantities are called levels. The Sound Power Level LW indicates the acoustic power with respect to an internationally accepted reference of 10-12 W, as LW = 10 ⋅log

where

W , W ref

(2-20)

W is the time-averaged sound power,

W ref = 10 −12 W is the reference value of sound power. Sound Power Level Acoustic power [W]

-12

L [dB] ref 10 W W

100 000 000

200

1 000 000

180

10 000

160

100

Object

Acoustic power [W]

Saturn rocket

50 000 000

Four jetplanes

50 000

140

Large orchestra

10

1

120

Scream

1

0.01

100

0.000 1

80

0.000 001

60

0.000 000 01

40

0.000 000 000 1

20

0.000 000 000 001

Typical speech

Whispering

20 . 10 -6

10

-9

0

Figure 2-30 The sound power and Sound Power Level for a number of typical sound sources. Notice that the large linear span is effectively compressed by the logarithmic scale. (Source: Brüel &Kjær, Acoustic Noise Measurements.)

49

Chapter 2: Fundamental Concepts Example 2-2 The motor of a propeller plane generates a sound power of 0.1 W. Determine the Sound Power Level. Solution Using the given values in formula (2-20) gives LW = 10 ⋅log

10 −1 10 −12

= 10⋅ log 1011 = 11 ⋅10⋅ log 10 = 110 dB .

The Sound Intensity Level LI is defined as L I = 10 ⋅log

where

I I ref

,

(2-21)

I is the absolute value of the time average of the sound intensity, I ref = 10 −12 W/m2 is the reference value of sound intensity.

Example 2-3 If the Sound Intensity Level in an airplane is 60 dB, what is the sound intensity? Solution Rewriting formula (2-21) gives log

L = I . 10 I ref I

Algebraic manipulation, using properties of logarithms, gives I = I ref 10 LI / 10 . Finally, entering the given values yields I = 10 −12 ⋅ 10 60 / 10 = 10 −6 W/m2. In linear vibroacoustics, as indicated in formula (2-14), time-averaged power values are proportional to the squared rms-amplitudes of the field variables (e.g., pressure, particle velocity). Thus, to calculate logarithmic levels from the field variables, it is these squared rms-amplitudes that must be used. The Sound Pressure Level Lp (or SPL) is defined as L p = 10 ⋅log

where

~ p2 2 p ref

,

(2-22)

~ p is the rms-amplitude of the sound pressure, p ref = 2 ⋅ 10 −5 Pa is the reference value of sound pressure.

The reference value of sound pressure approximately corresponds to the lowest sound pressure that a young person with normal hearing can perceive at 1000 Hz; see chapter 3.

50

Chapter 2: Fundamental Concepts Example 2-4 The Sound Pressure Level (SPL) a meter away from a person speaking is about 60 dB. Determine the sound pressure at that SPL. Solution 2 According to (2-22), L p = 10 ⋅log ~ p 2 p ref = 20 ⋅log ~ p p ref , L / 20 ~ p = p ref 10 p .

so that

Using the values given above in this relation yields ~ p = 2 ⋅ 10 −5 ⋅ 10 60 / 20 = 2 ⋅ 10 −2 Pa. The Vibration Velocity Level Lv is defined as Lv = 10 ⋅log

where

v~ 2 2 v ref

,

(2-23)

v~ is the rms-amplitude vibration velocity, vref = 10-9 m/s is the reference level of vibration velocity.

The Force Level LF is defined as L F = 10 ⋅log

~ F2 2 Fref

,

where

~ F is the rms-amplitude of the force, Fref = 10-6 N is the reference value of force.

2.10

FILTERS

(2-24)

Frequency analysis implies the study of a signal’s distribution along the frequency axis. Such analysis has traditionally been carried out mathematically or by means of analog electrical filters constructed of conventional electrical components. With today’s digital technology, there are two methods that are primarily used: the Fast Fourier Transform (FFT), and digital filtering. Both make use of digitized measurement values. Each type of filter is named after its affect on the signal’s frequency spectrum (see Figure 2-31): (i) (ii) (iii) (iv)

Low pass filter. High pass filter Band pass filter Band stop filter

51

Chapter 2: Fundamental Concepts (i)

A

(ii)

A

Amplification (iii) Förstärkning

A

Förstärkning 1 Amplification f 1

Förstärkning Amplification

f

f 1

Amplitude Amplitud

f

f 1

Förstärkning Amplification

f (iv)

A

f

Frequency frekvens

Figure 1-31 Different filters influence on a noise signal’s frequency spectrum when the signal passes through them: (i) Low pass filter (ii) High pass filter (iii) Band pass filter (iv) Band stop filter (Source: Brüel & Kjær, course material.)

f

The filter type that is most common is the low pass filter. Such filters are often used at the input to a measurement system to filter away frequency components higher than those to be analyzed. These removed components would otherwise introduce errors during the digitization process by contaminating the low frequency components (“aliasing”). A filter 2.10.1

Band pass filters

An ideal band pass filter, such as the one in Figure 2-32a, suppresses components at all frequencies except those that lie within the bandwidth B (i.e., “passes” those in B). In practice, however, the edges of the band have a certain slope, as shown in Figure 2-32b, which implies that the frequency components immediately outside of the pass band are not completely eliminated. A common way to define the upper fu and lower fl frequency limits of the band is to indicate the frequencies at which the signal is reduced by 3 dB. Amplification [dB] Band width,

Amplification [dB]

B=fu-fl

0

0 -3

a)

fl

fu

Frequency b)

fl

fu

Frequency

Figure 2-32 a) Ideal band pass filter with infinitely steep cutoffs. b) Real filters have imperfect cutoffs. The upper and lower bounding frequencies are then defined by the frequencies at which the filter reduces the signal by 3 dB.

52

Chapter 2: Fundamental Concepts Band pass filters are named according to how the bandwidth varies along the frequency axis. Filters with bandwidths that do not vary along the frequency axis are called constant absolute bandwidth (CAB) filters; see Figure 2-33a. A filter with a bandwidth proportional to its center frequency, fc, is called a constant relative bandwidth (CRB) filter; see Figure 2-33b. Amplification Förstärkning[dB] [dB]

B =100 Hz

0

0

1k

2k

3k

4k

5k

6k

7k

8k 9k 10k Linjär Linearfrekvensskala frequency scale

Figure 2-33a CAB filter, with a bandwidth that does not vary along the frequency axis; it is typically presented with a linear frequency axis.

1

Amplification Förstärkning[dB] [dB]

BB==((66 22 − 16 62 2) fm) f m

0

8

16 31.5 31,5

63

125 250

500

1k

2k

4k

8k

16k

Linear frequency scale Logaritmisk frekvensskala Figure 2-33b CRB filter, with a bandwidth that is a certain percentage of the center frequency fc; it is typically presented with a logarithmic scale. Because of the logarithmic scale, the stacks in the figure do not get wider, moving to the right along the axis. The example in the figure is called a third octave band filter and has a band width that is about 23% of the center frequency.

2.10.2

Third-octave and octave band filters

Third-octave and octave band filters are CRB filters very widely used in the field of sound and vibrations. Center frequencies are standardized, and listed in table 2-2. Both types of filters are named with band numbers, as in table 2-2, or more often by their center frequencies, fc. As is evident from table 2-2, each octave band spans three third-octave bands, which explains the name of this category of filters.

53

Chapter 2: Fundamental Concepts Table 2-1 Definition of third-octave and octave band filters.

Octave band filter fl = f

Lower frequency limit

fl = f

2

c

fu = 2 f c

Upper frequency limit

B = f u − fl =

Bandwidth

Third-octave band filter

fc =

Center frequency

(

fu =

2 −1

)

2 fc

B = fu − fl = fc =

fl fu

c 6

(

6

62 2 fc

)

2 − 1 6 2 fc fl fu

Table 2-2 Standardized center frequencies and upper and lower frequency limits of third-octave and octave band filters. Shading indicates octave bands.

Band no.

Center frequency fc [Hz]

3rd-octave band filter fl – fu [Hz]

1

1.25

2

1.6

Octave band filter fl – fu [Hz]

Band no.

Center frequency fc [Hz]

3rd-octave band filter fl – fu [Hz]

1.12 - 1.41

23

200

178 - 224

1.41 - 1.78

24

250

224 - 282

3

2

1.78 - 2.24 1.41 - 2.82

25

315

282 - 355

4

2.5

2.24 - 2.82

26

400

355 - 447

5

3.15

2.82 - 3.55

27

500

447 - 562

6

4

3.55 - 4.47 2.82 - 5.62

28

630

562 - 708

7

5

4.47 - 5.62

29

800

708 - 891

8

6.3

5.62 - 7.08

30

1000

891 - 1120

9

8

7.08 - 8.91 5.62 - 11.2

31

1250

1120 - 1410

10

10

8.91 - 11.2

32

1600

1410 - 1780

11

12.5

11.2 - 14.1

33

2000

1780 - 2240

12

16

14.1 - 17.8 11.2 - 22.4

34

2500

2240 - 2820

13

20

17.8 - 22.4

35

3150

2820 - 3550

14

25

22.4 - 28.2

36

4000

3550 - 4470

15

31.5

28.2 - 35.5 22.4 - 44.7

37

5000

4470 - 5620

16

40

35.5 - 44.7

38

6300

5620 - 7080

17

50

44.7 - 56.2

39

8000

7080 - 8910

18

63

56.2 - 70.8 44.7 - 89.1

40

10000

8910 - 11200

19

80

70.8 - 89.1

41

12500

11200 - 14100

20

100

89.1 - 112

42

16000

14100 - 17800

21

125

112 - 141

43

20000

17800 - 22400

22

160

141 - 178

89.1 - 178

54

Octave band filter fl – fu [Hz]

178 - 355

355 - 708

708 - 1410

1410 - 2820

2820 - 5620

5620 - 11200

11200 - 22400

Chapter 2: Fundamental Concepts 2.11

ADDITION OF SOUND FIELDS, INTERFERENCE

The sound pressure level a meter away from a person speaking is about 60 dB. Two persons speaking together, and in fact talking at the same moment, do not bring about a sound pressure level of 120 dB, however. Logarithmic sound pressure levels cannot be added together in that way. In linear vibroacoustics, we can, according to the underlying mathematics, add together the solutions obtained from linear wave equations. The resulting wave motion at a point of interest can therefore be calculated by adding up the contributions from the individual sound waves at that point. For scalar quantities, such as sound pressure, this would be scalar addition, and for vector quantities it would be vector addition. For sound pressure, for example, the fundamental addition rule applies, that the total sound pressure ptot(t) is the sum of the individual waves’ sound pressures pn(t); i.e., p tot (t ) =

N

∑ p n (t )

.

(2-25)

n =1

In order to determine the SPL of the resulting total sound pressure, we must, in accordance with formula (2-22), use the rms sound pressure. To determine the rms value, we first consider the case of two separate sound waves acting at the same point. The total sound pressure’s squared rms value is, according to equation (2-3), 1 2 ~ ptot = T

T

∫

2 ptot (t )dt =

0

2 =~ p12 + ~ p 22 + T

1 T

T

∫ [ p1 (t ) + p2 (t )]

2

dt =

0

T

(2-26)

∫ p1 (t ) p2 (t )dt. 0

Considering the third term in (2-26), we can identify three cases: (i) Uncorrelated sources. In most circumstances in which noise is coming from more than one source, as from several machines in a workshop for example, the sources can be assumed to be statistically uncorrelated , so that the third term in equation (2-26) goes to zero. Hence, 2 ~ p tot =~ p12 + ~ p 22 .

(2-27)

Additional uncorrelated sound pressures can then be added to this result. The addition rule for uncorrelated sources is therefore N

2 ~ p tot =∑~ p n2 .

(2-28)

n =1

For uncorrelated sources, the squared rms sound pressures can evidently be added together. A corresponding addition rule for the SPL is derived from definition (2-22), from which we have L 10 ~ p 2 = p 2 10 p . (2-29) ref

55

Chapter 2: Fundamental Concepts p 22 , with Sound Pressure Levels Lp1 and If two uncorrelated sources give rise to ~ p12 and ~ Lp2, respectively, then, from (2-22) and (2-28), L 10 ⎞ 2 2 ⎛ L p1 10 ~ =~ + 10 p2 p tot p12 + ~ p 22 = p ref ⎜10 ⎟ , ⎝ ⎠

(2-30)

L 10 L 10 ⎞ + 10 p2 L ptot = 10⋅ log⎛⎜10 p1 ⎟ . ⎝ ⎠

(2-31)

so that

For levels, the addition rule which therefore applies is L ptot = 10 ⋅log

N

∑ 10

L pn 10

.

(2-32)

n =1

Example 2-5 A sound source causes a sound pressure level Lp1 at a certain point. What increase in SPL is provided by a second source, equal in strength to, but uncorrelated to, the first? Solution Formula (2-32) gives L ptot = 10 ⋅log(10

L p1 10

+ 10

L p1 10

) = 10 ⋅log(10

L p1 10

⋅ 2) = L p1 + 10 ⋅log 2 = L p1 + 3 dB .

With two equally strong, but uncorrelated, sources, the level is therefore increased 3 dB. Example 2-6 Elimination of the background level. A typical problem is that the SPL due to a machine must be determined, while other noise sources present cannot be shut down. The solution can therefore be to first measure the background level Lpb without the machine in question operating; then, start the machine and measure the total level Lptot and, from that, back calculate the sound pressure level Lpm due to the machine by itself.

a) Derive a formula for that purpose. b) Determine L p m when L ptot = 90 dB and L pb = 83 dB . Solution a) Assume uncorrelated sources. Formula (2-28) gives 2 2 2 ~ p tot =~ p m2 + ~ p b2 , ~ pm =~ p tot −~ p b2 L from which L pm = 10 ⋅log⎛⎜10 ptot ⎝

(

b) L p m = 10 ⋅log 10 − 10 9

8.3

10

− 10

L pb 10 ⎞

⎟. ⎠

) = 89.0 ≈ 89 dB. 56

Chapter 2: Fundamental Concepts (ii) Identical sources. This case is, of course, not so common in noise control situations. Two closely spaced loudspeakers driven with the same signal can be considered identical. In that case, p1(t) = p2 (t) and from (2-26) we receive ~ p2 = 4~ p2 . (2-33) 1

tot

The total Sound Pressure Level is then L ptot = 10 ⋅log

p12 4~ 2 p ref

= L p1 + 6 dB.

(2-34)

So, the Sound Pressure Level increases by 6 dB. (iii) Identical, oppositely phased sources p tot = 0 , and Lptot → - ∞. That In this case, p1 (t) = - p2 (t), so that equation (2-26) yields ~ is called destructive interference, and is a practical means to reduce noise in many cases. One example is so called active noise control, in which a phase-inverted antinoise is emitted and interferes with the original noise. In practice, 20-30 dB reductions are obtainable. 2.12

ADDITION OF FREQUENCY COMPONENTS

Section 2.10 discussed different ways to describe a signal’s distribution in the frequency dimension or in frequency bands. Narrow bands give detailed information on the distribution of energy, with relatively low amplitudes in each band. Figure 2-34 shows the same spectrum presented with different bandwidths. L p [dB] Octave band Third octave band

Narrow band frequency (log) [Hz]

Figure 2-34 The same sound spectrum presented in narrow band, third octave bands, and octave bands. The bigger the bandwidth, the more frequency components that contribute to any band, giving higher levels. The logarithmic frequency axis causes the CRB filters to have the same apparent width per band over the entire spectrum, while CAB filters, with their fixed band width over the entire spectrum, appear to grow more dense as frequency increases.

Summation of the sound pressures of individual frequency components is carried out in the same way as for the summation of sound pressures from multiple sources; from (2-28) and (2-32), N

2 ~ p tot =∑~ p n2 n =1

57

(2-35)

Chapter 2: Fundamental Concepts N

L ptot = 10 ⋅log ∑ 10

and

L pn 10

,

(2-36)

n =1

where the index n stands for individual frequencies or frequency bands, instead of distinct sources. The proof for each of these formulas is from Parseval’s relations for periodic and non-periodic functions, which in turn comes from Fourier analysis; see chapter 3. Example 2-7 The 1000 Hz octave band includes the 800, 1000, and 1250 Hz third-octave bands. Determine the octave band level, if the third-octave band levels are 79, 86 and 84 dB, respectively. Solution Formula (2-36) gives Lp = 10⋅log(107.9 +108.6 + 108.4) = 88.6 ≈ 89 dB.

2.14

IMPORTANT RELATIONS

PEAK VALUE, AVERAGE VALUE, RMS AMPLITUDE AND POWER Time average of a quantity 1 T

T

1 T

T

p=

where

∫ p(t )dt ,

(2-2)

0

T is the averaging time.

RMS amplitude of a quantity ~ p=

∫p

2

(t )dt .

(2-3)

0

LONGITUDINAL WAVES IN GASES AND LIQUIDS Longitudinal plane waves

Relation between sound speed, frequency and wavelength. c = fλ .

(2-8)

Time averaged sound intensity Ix =

~ W p2 . = S ρ0c

(2-13)

Spherical and cylindrical waves

Sound intensity of spherical waves Ir =

W 4πr 2

58

.

(2-15)

Chapter 2: Fundamental Concepts Sound intensity of cylindrical waves Ir =

W' . 2πr

(2-17)

LEVELS AND DECIBELS Sound Power Level

where

I , I ref

(2-21)

L p = 10⋅ log

~ p2 2 p ref

,

(2-22)

Lv = 10⋅ log

v~ 2 2 v ref

,

(2-23)

v ref = 10 −9 m/s is the reference level of vibration velocity.

Force level

where

L I = 10 ⋅log

p ref = 2 ⋅10 −5 Pa is the reference value of sound pressure.

Vibration velocity level

where

(2-20)

I ref = 10 −12 W/m2 is the reference value of sound intensity.

Sound Pressure Level

where

W , W ref

W ref = 10 −12 W is the reference value of sound power.

Sound intensity level

where

LW = 10 ⋅log

L F = 10 ⋅log

~ F2 2 Fref

,

(2-24)

Fref = 10 −6 N is the reference level of force.

ADDITION OF SOUND FIELDS, INTERFERENCE Fundamental addition rule for sound pressure N

p tot (t ) = ∑ p n (t ) .

(2-25)

n =1

Addition rule for uncorrelated sources N

2 ~ p tot =∑~ p n2 ,

(2-28)

n =1

N

L ptot = 10 ⋅log ∑ 10 n =1

59

L pn 10

.

(2-32)

Chapter 2: Fundamental Concepts ADDITION OF FREQUENCY COMPONENTS N

2 ~ p tot =∑~ p n2 ,

(2-35)

n =1

N

L ptot = 10 ⋅log ∑ 10 n =1

60

L pn 10

.

(2-36)

Chapter 3: Influence of Sound and Vibration on man and equipment

CHAPTER THREE INFLUENCE OF SOUND AND VIBRATION ON MAN AND EQUIPMENT

This chapter begins with a description of the function of the ear, and how we perceive sound, as a function of its frequency content and strength. That description serves as a foundation, as we then define the indices that are needed to depict how disturbing noise is. One well known such index is Sound Level, given in dB(A), which is intended to serve as a measure of the degree of disturbance. Sound affects not only our hearing, but even the entire body, and psychological disturbances can be more obvious than physical ones at times. Our hearing was developed to best serve our ancestors tens of thousands of years ago; they were in those times huntergatherers. Our nerve and gland systems have not changed, and the extra doses of hormones, blood sugar, and blood fats that gave our ancestors courage to fight or strength to flee, merely increase our stress level today. How a product sounds, its sound quality, is one of the characteristics that distinguishes it from competing products. Sound design of products will be an important tool of competition in the future. Vibrations appear in many situations. We experience them in the home, during transports of different types, and in professional life. Sometimes we even generate vibrations intentionally. In vibratory feeding systems, objects are induced to move forward along a vibrating path. Ultrasonic cleaners are used for sterilization. Vibrating boring machinery is used to bore in rock. In vibration testing, components and entire finished products are exposed to high vibration amplitudes to evaluate their ability to function in their service environment.

61

Chapter 3: Influence of Sound and Vibration on man and equipment Nevertheless, vibrations are usually unwanted and harmful. Vibrating production machinery degrades production tolerances and surface finish. An unbalanced turbine can bring about serious fatigue problems leading to breakdowns. Vibrations of hand-held machines can cause blood circulation problems in the hands, the so-called white finger syndrome. Low frequency vibrations in the earth’s crust, earthquakes, can demolish entire cities. Noise and vibrations are annoying, and parliament and authorities pass laws, regulations, norms and guidelines within their respective areas of responsibility. Regulations are more detailed than laws, and must be followed; otherwise the authorities may intervene. Norms and guidelines carry weight when an individual citizen, for example, appeals the decision of a municipal authority to one of the administrative courts that handles the type of issue in question. Verification by measurement is needed to ensure that regulations issued by authorities are fulfilled. To guarantee that such measurements are performed correctly, and that companies declaring the noise and vibration levels generated by their products compete in minimizing such levels rather than in devising misleading measurement techniques, both national and international standards committees have come to agreement on measurement standards in different areas. Standards from the International Organization for Standardization are designated ISO. The European Union, EU, has its own standards designated as EN-standards. These are often in agreement with corresponding ISO-standards. Laws, regulations, norms and guidelines are constantly modified.

Always use the original, in its latest update Short summaries of the type that are presented in this chapter are not only incomplete, but run the risk of becoming out of date as well.

3.1

THE EAR AND HEARING

3.1.1

The ear’s function

Anatomically, the ear is subdivided into three parts: the outer ear, middle ear, and inner ear; see Figure 3-1. The outer ear consists of the pinna, i.e., the visible part of the ear, and the auditory canal. The eardrum separates the outer from the middle ear. The middle ear is an air-filled cavity that obtains its oxygen supply from the eustachian tube, originating in the throat. In the middle ear, there are three auditory bones, or ossicles: the hammer, the anvil, and the stirrup, which connect the eardrum to the oval window of the cochlea, the boundary between the middle and the inner ear. The inner ear consists of the cochlea and the semicircular canals (organ of balance).

62

Chapter 3: Influence of Sound and Vibration on man and equipment

The main function of the pinna is to, like a funnel, receive sound waves and channel them into the Middle ear auditory canal, as well as Pinna enhance our sound localization Semicircular Auditory nerve abilities by virtue of its shape. canals The most important clues for sound localization are intensity and time differences in the sound wave that reaches the ear. A sound wave from the right reaches the right ear before it Cochlea reaches the left, and is also stronger in the right ear than the left. These clues cannot, Eustachian Auditory however, explain our ability to tube canal distinguish between sound from Stirrup before us and sound from behind Ear drum Anvil us, or between sound from above and sound from below. That Hammer information comes from the Figure3-1 Schematic of the ear’s anatomy (Source: Brüel & Kjær, shape of the pinna, which course material.) modifies the incoming sound. The auditory canal leads sound waves in towards the middle ear. Certain frequency regions are enhanced and others weakened, depending on the direction from which they are coming. Together, the pinna and the auditory canal cause resonances that increase the sound pressure at the eardrum, above all in the 2 - 7 kHz region; see Figure 3-2. Outer ear

Inner ear

Acoustic amplification [dB]

20 15 10 5 Figure 3-2 Acoustic amplification caused by reflections around the head and shoulders, in the pinna and the auditory canal. (Source: Pickles, 1988.)

0 -5 -10 200

500

63

1000 2000 Frequency [Hz]

5000

10000

Chapter 3: Influence of Sound and Vibration on man and equipment

In the eardrum, acoustic pressure variations are converted into mechanical vibrations, which are then amplified in the middle ear and transferred to the cochlea. Amplification is mainly brought about by a pressure jump due to the much larger area of the eardrum than the contact area of the stirrup against the oval window. The middle ear functions as a socalled impedance transformer, coupling air (low impedance) to the liquid of the cochlea (high impedance). If that transformation mechanism did not take place, then the majority of the incident sound would be reflected back into the auditory canal. There are two muscles in the middle ear. One is fixed to the hammer near the eardrum, and the other is fixed to the stirrup. These muscles firm up the chain of auditory bones. When the muscles are tensed, the stiffness of the ossicle chain is increased, which reduces the transmission of low frequency sound. The stirrup muscle tenses as a reflex against strong sound (about 75 – 95 dB above the weakest sound a person can perceive), which suggests that it serves to protect the inner ear from injury due to noise. The reflex is, however, too slow to protect against impulse sound (strong sound, short in duration, discussed further in section 3.1.6) and the muscles tire quickly due to high frequency noise. The eustachian tube is responsible for supplying air to the middle ear and equalizes static pressure differences between the middle ear and the surrounding air. That occurs when the eustachian tubes open to the throat, as when we swallow or yawn for example. If a static pressure difference nevertheless arises, the eardrum will bow out towards the lowest pressure, i.e., towards either the auditory canal or the middle ear. The increased tension in the eardrum that results causes pain and can, in extreme cases, result in bursting of the eardrum. The cochlea is a rolled-up tube that, is divided along its length into three channels; see Figure 3-3. The upper and lower channels, which are in mutual communication, are filled with a liquid rich in sodium ions, while the middle is filled with a liquid rich in potassium ions. The hearing organ with sensory cells, hair cells, is located on the basilar membrane, which divides the two lower channels. Stirrup

Oval window

Figure

Round window

Basilar membrane

3-3 Schematic of cochlea “unrolled”.

the

When the stirrup moves against the oval window of the cochlea, a pressure wave emitted into the upper channel of the cochlea. The wave then moves back along the lower channel towards the round window at the base of the cochlea, the membrane separating the lower channel from the middle ear. At the round window, pressure equalization can occur; thus, the round window moves in antiphase with respect to the oval window. Pressure changes in the channels also affect the basilar membrane. The membrane varies in stiffness and

64

Chapter 3: Influence of Sound and Vibration on man and equipment width along the length of the cochlea, from stiff and narrow at the base to soft and wide at the apex. Because of these stiffness differences, the mode excited in the cochlea has a maximum at a point that depends on the frequency; see Figure 3-4. High frequency sound has a maximum near the base of the cochlea, while low frequency sound has its maximum near the apex. Thus, a frequency separation of the incoming sound is effected, and the hair cells at different positions along the cochlea react to different frequencies. That implies that a loss of hair cells at the base of the cochlea results in high frequency hearing loss, while a loss of hair cells near the apex results in low frequency hearing loss. Sound wave

Cochlea ”unrolled”

0.6 0.3

Excitation frequency [kHz]

Amplification

Distance from the oval window [mm] Figure 3-4 The graph shows the mode shapes (oscillation patterns) of the basilar membrane at various excitation frequencies. The higher the excitation frequency, the nearer the oval window the maximum amplitude is to be found. (Picture: Brüel & Kjær, course material.)

When the membrane moves, the hair cells are stimulated, and conversion to a nerve signal takes place by means of a sequence of bioelectrical processes based on the differing ion content of the three channels. A difference in potential arises in the hair cells, and that potential difference produces an electrical impulse in the nerve fibers, which are connected to the hair cells. These nerve fibers combine in the auditory nerve, which carries the signal to the hearing centers in the brain, where it is interpreted as sound. 3.1.2 Measure of hearing The human ear has a wide working range. A young person with normal hearing hears sound in the frequency range 20 - 20 000 Hz, and in the 0 - 130 dB range of sound pressure levels. With advancing age, it is above all the ability to hear weak, high frequency sound that diminishes, as discussed further in section 3.1.6. In many different situations, we must be able to measure hearing or quantify the impression made by a sound. It is also necessary in order to be able to undertake countermeasures in a noisy workplace or to be able to rehabilitate a hearing injury. Our subjective experience of the strength of sound is not in exact agreement with the physically measured sound pressure. The frequency, among other things, affects our perception of sound strength. Figure 3-5 presents a group of curves that connect tones of different frequencies, but with the same Loudness, i.e., tones that are perceived to be equal in strength. The concept of loudness is defined as the sound pressure level a sinusoidal tone at 1000 Hz would have, in order to give the same subjective impression of strength as

65

Chapter 3: Influence of Sound and Vibration on man and equipment the sound to be assessed. The subjective impression of strength presupposes a person with normal hearing. The unit of loudness is the phone. 120 110 Phones phon

110 100

90

90

B-weighting is based on B -vägning baseras thepå 7070 phone phoncurve kurvan

80

80

L p , [ dB ]

C -vägning baseras C-weighting is based kurvan phone curve on på the90 90phon

100

70

70

60

60

A -vägning baseras A-weighting is based kurvan onpå the4040phon phone curve

50

50

40

40

30

30

20

20

Threshhold of hearing for a Hörtröskeln för person with normal hearing en normalhörande person

10

10 0 -10 31,5 20 31.5

63

125

250

500

1000

2000

4000

8000 12500

Frekvens, [Hz] [ ] Frequency Figure 3-5 Isophone curves. Along a curve, the loudness level is constant, i.e., it is subjectively experienced as equally strong. The lowermost, dashed curve, is the threshold of hearing for a normally-hearing person. The curves were measured in an acoustically treated room with loudspeakers, and using both ears. Both tones marked in the diagram, at 63 Hz and 1000 Hz, therefore have a loudness of 60 phones. Their respective sound pressure levels, on the other hand, are 75 dB and 60 dB.

The Threshold of hearing is defined as the lowest sound pressure level that induces any sensation of hearing. Even the threshold of hearing varies with frequency as is evident from the lower dashed curve in Figure 3-5. Therefore, frequency-specific sound, usually consisting of sinusoidal tones, is used to determine an individual’s hearing threshold. In cases of impaired hearing, the degree of impairment is indicated relative to a standardized hearing threshold, derived from the averaged hearing thresholds of young people with normal hearing. The most common method to graphically depict these relative hearing impairments is an audiogram, in which the statistical normal hearing level is represented as a zero level, and hearing deficiency is indicated in dB HL (Hearing Level) downwards in the diagram; see Figure 3-6. The region of normal hearing is shaded.

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Chapter 3: Influence of Sound and Vibration on man and equipment

-10 0 10 20 30 40 50 60 70 80 90 100 110 120

125

250

Frequency [Hz] 8000 -10 0 EARING 10 20 30 40 50 60 70 80 90 100 110 120

500 1000 2000 4000

NORMAL H

Hearing loss [dB HL] Figure 3-6 Audiogram, with the region of normal hearing shaded.

3.1.3

Measures of noise

What we regard as noisy varies from individual to individual. By noise, we usually mean unwanted sound in the audible region. The strength of sound is measured by a sound level meter, that, in its simplest form, gives the SPL in dB; see section 1.13. The SPL does not, however, take account of the nonlinearity of our perception with respect to frequency, as reflected in the concept of loudness. To better reflect the human perception of sound, sound level meters contain filters, so-called weighting filters, that amplify the microphone signal different amounts at different frequencies; see Figure 3-7. Amplification [dB] + 20 + 10 0 - 10 - 20 - 30 - 40 - 50 - 60 - 70 10

100

1 000 Frequency [Hz]

67

10 000

Figure 3-7 A, B, C and D-weighting curves. A-weighting is the most common. Under 1000 Hz, the amplication is negative, implying that these frequencies are damped to compensate for the lower sensitivity of mankind to low frequency sound; see Figure 3-5. (Source: Brüel & Kjær, kursmaterial.)

Chapter 3: Influence of Sound and Vibration on man and equipment Table 3-1 A-, B- and C-weighting for third-octave and octave bands. The octave bands are given in bold.

Frequency [Hz] 25 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000 10000 12500 16000 20000

A-weighting [dB] -44.7 -39.4 -34.6 -30.2 -26.2 -22.5 -19.1 -16.1 -13.4 -10.9 -8.6 -6.6 -4.8 -3.2 -1.9 -0.8 0 +0.6 +1.0 +1.2 +1.3 +1.2 +1.0 +0.5 -0.1 -1.1 -2.5 -4.3 -6.6 -9.3

B-weighting [dB] -20.4 -17.1 -14.2 -11.6 -9.3 -7.4 -5.6 -4.2 -3.0 -2.0 -1.3 -0.8 -0.5 -0.3 -0.1 0 0 0 0 -0.1 -0.2 -0.4 -0.7 -1.2 -1.9 -2.9 -4.3 -6.1 -8.4 -11.1

C-weighting [dB] -4.4 -3.0 -2.0 -1.3 -0.8 -0.5 -0.3 -0.2 -0.1 0 0 0 0 0 0 0 0 0 -0.1 -0.2 -0.3 -0.5 -0.8 -1.3 -2.0 -3.0 -4.4 -6.2 -8.5 -11.2

A-, B- and Cweighting are taken from the 40, 70, and 90 phone curves in Figure 3-5, respective-ly; see table 3-1. Originally, the thought was that Aweighting would be used at low, Bweighting at intermediate, and Cweighting at high SPL, thereby adjusting the measurement results to our perception of sound as it varies in both frequency and strength. Today, however, Aweighting is most often used, although C-weighting is at times applied, particularly in connection with impulsive sound. Dweighting is

primarily used in measuring aircraft noise. A sound pressure level that is measured or determined with weighting filters, is called a Sound Level. Assume that the measured sound level with an A-weighting filter is 75 dB. That is written LA = 75 dB(A). The sound level in dB(A) can be calculated from third-octave and octave band filters as N

L A = 10⋅ log ∑ 10

( L pn + ΔAn ) / 10

[dB(A)],

(3-1)

n =1

where

Lpn [dB] is the third-octave or octave band level in band n, ΔAn [dB] is A-weighting in band n.

Example 3-1 Determine, from the given octave band

f [Hz] Lpn [dB]

125 90 68

250 96

500 92

1k 90

2k 85

4k 86

8k 81

Chapter 3: Influence of Sound and Vibration on man and equipment levels, Lpn, the sound level in dB(A) Solution A-weighting is given in table 3-1, which, from (3-1), gives

ΔAn [dB] Lpn +ΔAn

-16.1 73.9

-8.6 87.4

-3.2 88.8

0 90

1.2 86.2

1.0 87.0

-1.1 79.7

[dB]

LA = 10 ⋅ log(107.39 + 108.74 + 108.88 + 109 + 108.62 + 108.7 + 107.97 ) ≈ 95 dB(A). Equivalent sound pressure level is a form of average sound pressure level during a given period of time. It is defined as the constant sound pressure level that represents the same total sound energy as an actual time varying sound pressure level during a given time period, eight hours for example. This measure is used to characterize a time-varying noise and create a measure of the disturbance or destructive influence of the sound. The length of the measurement period should always be given. Leq ,T = 10⋅ log(

where

1 T

T

p 2 (t )

0

2 p ref

∫

dt ) [dB],

(3-2)

Leq,T is the equivalent sound pressure level during time period T, p(t) is the instantaneous sound pressure, = 2 ⋅ 10 −5 Pa, is the pref T is the length of the measurement period.

reference

sound

pressure,

With the help of the definition of sound pressure level, the expression can be written Leq ,T = 10⋅ log(

1 T L p (t ) / 10 10 dt ) [dB], T ∫0

(3-3)

where Lp(t) is the instantaneous sound pressure level, or, if an A-weighted quantity is intended L Aeq,T = 10⋅ log(

where

1 T L A (t ) / 10 10 dt ) [dB(A)], T ∫0

(3-4)

LA(t) is the instantaneous A-weighted sound level.

Equivalent sound level can be registered with an integrating sound level meter or a dosimeter.. It can also be calculated starting from sound level measurements, and with simplifying assumptions on the sound variation during the measurement period. The definition implies that a strong, short-duration sound makes a big contribution to the equivalent sound level. A constant sound level of 100 dB(A) during 15 min corresponds to a constant sound level of 85 dB(A) over 8 hours, i.e., a workday. Table 3-2 shows equivalent values of sound level and exposure time, each corresponding to LA = 85 dB(A) over 8 hours. An increase of the sound level by 3 dB(A) corresponds to a halving of the exposure time for the same equivalent sound level. 69

Chapter 3: Influence of Sound and Vibration on man and equipment In many cases, that type of broadband analysis of the sound level does not give a sufficiently detailed picture of the strength of a sound. The sound pressure level is therefore often measured in different frequency bands (octave, third-octave, or narrow band) to identify variations with frequency; see section 1.10. Table 3-2 Equivalent exposures, i.e., values of sound level, LA, and exposure time, T, giving the same total sound energy.

Sound level LA [dB(A)] 82 85 88 91 94 97 100 103 106 109 112 115

3.1.4

Exposure time T 16 h 8h 4h 2h 1h 30 min 15 min 7.5 min 3.8 min 1.9 min 1 min 0.5 min

Speech and masking

Perhaps the most important function of our hearing is to understand speech. With a normal voice level, the sound level one meter away from a speaker in a quiet environment is 60 65 dB(A). With a raised voice it is about 75 dB(A), and with a very loud voice about 85 dB(A). Speech can therefore be said to vary in strength as well as frequency; Figure 3-8 shows the frequency and strength distribution at a distance of 1 meter from a speaker, in an audiogram. The area in the figure constitutes a so-called speech banana. With the help of the speech banana, it is possible to get an idea of what speech sounds a person with a certain hearing impairment can be expected to understand. Vowels are seen to fall in the lower frequency areas, and are relatively strong, while unvoiced consonants, such as “s”, “sh”, and “f”, for example, are high in frequency, and relatively weaker. Vowels carry energy, while consonants carry the most linguistic information. Our ability to comprehend speech depends on many factors. The character of the speech is one such factor. The speech signal normally contains a large excess of information with respect to its linguistic content. That excess is called redundancy. Every language has limits on different linguistic levels. After a certain linguistic sound, only certain other linguistic sounds are allowed to follow. After a certain word, only certain other words can grammatically follow. These limitations give rise to the redundancy. It is that redundancy that permits a person with normal hearing to follow a conversation in a very noisy room, or a speech signal that is severely distorted, as in a bad telephone connection, for example. Listeners take advantage of all of their knowledge of the language, the speaker, and the subject of conversation. A coherent conversation has high redundancy; a solitary word removed from its context has low redundancy. 70

Chapter 3: Influence of Sound and Vibration on man and equipment

125

-10 0 10 20 30 40 50 60 70 80 90 100 110 120

250

500 1000

A C B

2000

Frequency [Hz] 4000 8000 -10 0 10 20 30 40 D 50 60 70 80 90 100 110 120

Hearing Loss [dB HL]

Figure 3-8 The primary region of the speech spectrum. Area A is the area for fundamental tones of the human voice, B is the main area for vowels, C for voiced consonants, and D for unvoiced consonants. From the figure, we see that the region for vowels, B, goes from 300 Hz to 3000 Hz in frequency, and from 37 dB to 65 dB in the hearing loss dimension. A hearing loss of 65 dB in the 300 - 3000 Hz region therefore implies an inability to comprehend some vowels, and in practice, hardly any other speech sounds either.

The background noise naturally influences our ability to comprehend speech. One of the most obvious effects of noise is that it reduces the possibilities for conversation. This is called masking, and implies that a sound reduces the hearing perception of another sound. We often mean, by masking, that the masking sound makes the masked sound inaudible. Generally, it can be said that a signal is most easily masked by sound with similar or identical frequency content. The phenomenon of masking is asymmetric; narrowband sound masks more upwards than downwards in frequency, as seen in Figure 3-9. That means that strong bass sounds are generally more disruptive of speech comprehension than treble sounds with the same strength. Lp [dB] 100

Hearing threshhold with noise

Noise 80 60 40 20 0

Hearing threshhold without noise

10

100 1000 Frequency [Hz]

10000

71

Figure 3-9 Masking. The plot shows how a narrowband noise centered at 1000 Hz influences the hearing threshhold of sinusoidal tones. The lower, dashed curve shows the hearing threshhold without masking noise, and the upper curve shows the the hearing threshhold with the masking noise. Note that the masking effect is greater at frequencies higher than the center frequency of the noise than for lower frequencies. (Source: Brüel & Kjær,

Chapter 3: Influence of Sound and Vibration on man and equipment Vision is an aid for speech comprehension, especially in a loud environment. Despite that a very limited share of speech sounds can be visually deduced, lip reading can nevertheless provide the extra information needed in a difficult listening situation. 3.1.5

The influence of noise on man

Man is influenced by noise in many different ways, and different individuals are influenced to different extents. That applies to both the disturbing effects of noise and to the risk of suffering a hearing injury. A number of physiological effects result from noise exposure, such as contraction of blood vessels, dilation of the pupil, and effects on breathing. Noise reduces attention and can therefore degrade work performance. Even during sleep, our hearing monitors the environment, and sleep disturbances can occur due to noise. Sudden, unexpected, or unknown sounds stimulate the body’s defense mechanisms. This means that, among other things, blood pressure rises, muscle tension and heart frequency both increase, and blood vessels in the skin constrict. 3.1.6

Hearing injuries

Hearing injuries can be of many different types. A way to coarsely, but usefully, categorize hearing injuries is to speak of where the injury is located. One then speaks of hearing injuries of the conduction variety, or of the sensory-neural variety. By a conduction injury, it is meant that something hinders the sound conduction function of the ear somewhere between the pinna and the cochlea. Among the variants that can be named are absence of the auditory canal from birth, larger holes in the eardrum, inflammations in the middle ear, interruption of the chain of auditory bones, or fusing of the stirrup to the oval window. A pure conduction injury can give a maximum hearing loss of 60 dB. Conduction problems cause a weakening, but generally not a distortion, of sound. Hearing losses caused by conduction injuries can, as a rule, be successfully rehabilitated by surgery or with a hearing aid. A sensory-neural hearing injury affects the function of the inner ear or the auditory nerve. Usually, the hair cells in the cochlea are injured or completely absent. Common reasons are advancing age, inherited conditions, noise injury, infections, and tumors. Sensory-neural hearing loss often results in both a weakening and a distortion of sound. Such hearing loss is difficult to rehabilitate. If hair cells are completely absent, a hearing aid is of no help; in principle, such a device simply amplifies the incident sound signal. The indispensable connection to the auditory nerve is lacking in that case. With advancing age, hearing deteriorates. Figure 3-10 shows normal hearing for two age groups. It is primarily the higher frequencies that are impacted by age, and comparing to the speech banana in Figure 3-8 shows that it is the weak, high frequency, unvoiced consonants that are lost first, while the stronger vowels are usually heard well.

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Chapter 3: Influence of Sound and Vibration on man and equipment

125

250

500

1000

-10 0 10 20 30 40 50 60 70 80 90 100 110 120

2000

Frequency [Hz] 4000 8000 -10 0 man 50 yrs 10 20 30 man 70 yrs 40 50 60 70 80 90 100 110 120

Hearing Loss [dB HL]

Figure 3-10 Hearing loss due to age. The median value for men that have not been subjected to dangerous noise levels. Two age groups: 50 and 70 years.(Source: ISO 7029, 1986.)

Noise injuries are a common reason for hearing loss in our society. Men are affected more often than women. It has been debated as to whether that is due to a physiological gender difference, or merely due to greater exposure of men to noise than women, in their work, military service, and recreational activities. Powerful, but short lasting, noise often causes a temporary hearing loss. Hearing returns, as a rule, after a period without noise exposure, but the hair cells can nevertheless have suffered small but irreparable injury. If the noise exposure is repeated enough times, many small such injuries can result in damage that gives a measurable hearing loss. When the ear is exposed to long-lasting and powerful noise, the hair cells in the cochlea are damaged and a permanent hearing loss is the consequence; see Figure 3-11. Research indicates that noise containing impulsive sound is considerably more damaging to hearing than a constant sound with the same equivalent sound level. The explanation for that can be that the stirrup muscle doesn’t have time to tense itself, stiffening the chain of auditory bones, as discussed in section 3.1.1. Impulsive sounds are usually not perceived as especially strong, because of their short durations, but they are all the more damaging.

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Chapter 3: Influence of Sound and Vibration on man and equipment

a)

Figure 3-11 Hair cells photographed with a sweep electron microscope. The photos show three rows of outer hair cells. a) Normal,uninjured hair cells. b) Hair cells injured by noise. (Source: Göran Bredberg, Hörselkliniken, Södersjukhuset.)

b) An audiogram, depicting hearing loss caused by noise exposure, is shown in Figure 3-12. Noise injuries give rise to hearing loss curves with the characteristic feature that the degradation is most severe in the 4 - 6 kHz frequency range. The dip in the curve there is deepened and widened as the injury progresses. Both age-induced hearing loss and noise injuries are of the sensory-neural type of hearing impairment. It is the hair cells of the cochlea that are damaged or absent, and such conditions cannot be cured.

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Chapter 3: Influence of Sound and Vibration on man and equipment

-10 0 10 20 30 40 50 60 70 80 90 100 110 120

3.1.7

Frequency [Hz] 500 1000 2000 4000 8000 -10 0 10 20 30 40 50 60 70 80 90 100 110 120 Hearing Loss [dB HL] 125

250

Figure 3-12 Example of an audiogram of a noise injury. Note the large hearing loss at 4 000 Hz.

Hearing protection

The first step to take at a noisy workplace is to try to reduce noise emission from the noise source itself. Failing sufficient improvement there, measures to improve the acoustics of the facility are the next best alternatives. If the noise level nevertheless remains dangerously high, then use of hearing protection should be adopted. There are two types of hearing protection: earmuffs (see Figure 3-13), and earplugs. Earmuffs enclose the entire outer ear, whereas earplugs are placed in the auditory canal. The latter are of two types: disposable and reusable. There are different models of both earmuffs and earplugs, with varying acoustic properties. Noise measurements at a workplace serve as a basis for the selection of proper hearing protection. Information on the SPL in different frequency bands is important, since different types of hearing protection each have their own frequency dependencies. The protection selected must sufficiently attenuate sound so that the SPL in the auditory canal does not exceed risk levels for hearing damage. It must also fit well; earmuffs should be well-sealed against the head, and earplugs should be properly inserted. Hearing protection must be worn the entire time that one spends in the noisy environment. Even short interruptions drastically diminish the protection provided. It is also important that noise pauses, during which one is away from the noisy environment and can remove the hearing protection, are provided during the workday. Hearing protection must be maintained to ensure that the sound reduction provided does not diminish with time. The cuff (or ear cushion) of earmuffs should not be damaged, hardened, or dirty. Reusable earplugs must be kept clean; disposable ones must not be reused. In noisy workplaces, regular hearing tests of the employees should be carried out, in order to detect the onset of hearing problems as early as possible. 75

Chapter 3: Influence of Sound and Vibration on man and equipment

Obtain hearing protection that you really are satisfied with. Make sure that all parts are always working correctly.

Replaceable sweat pad

Cuff Foam layer

Dome liner

Dome

Headband

Figure

3-13 Earmuff-type hearing (Sketch: Arbetarskyddsnämnden, Buller och vibrationer ombord, Ill: Claes Folkesson.)

3.1.8

Sound quality

protection.

In the introduction, an overview of the concept of good sound quality was provided; this is product sound that exudes an impression of quality and reliability. The area, as such, is not new; automobile manufacturers have thought along those same lines for decades. The dilemma, however, has been the need to rely on listening and analyzing prototypes of car models, but without the benefit of reliable analysis tools to evaluate and simulate the effect of design modifications. All modifications had to be physically realized and tested, which is very time demanding. The computer-based analysis and simulation tools available today, however, make it possible to simulate the sound resulting from constructive design changes and play it back to listener panels of potential buyers. This implies that one can reduce the development time of new and improved products. Already today, and especially during the years to come, it will be possible to use the available analysis and simulation tools to listen to how a product will sound, even before a physical prototype exists, in the early planning stages. The purpose of sound quality work is not always to bring about the quietest possible product, but even to derive clear objectives for the product on the basis of the customer’s expectations, transform these into complementary objectives for the various component parts, and then design so as to attain the overall goal for the product sound. The procedure can be divided into four stages: (i) Analyze and understand the characteristic sound content of the design, and specify the objectives. (ii)

Couple the stated objectives to specific parts and functions of the design.

(iii)

Survey and subdivide the composed structure into an source – response – acoustic radiation system, as in figures 1-2 and 1-3 in the introduction.

(iv)

Predict the sound content of the product for different design variations, finding that giving the best sound quality.

76

Chapter 3: Influence of Sound and Vibration on man and equipment Predict the sound content of the product for different design variations, and evaluate their relative desirabilities. Items (i) to (iv) are more thoroughly described below. (i) Analyze and understand the characteristic sound content of the design, and specify the objectives. Typical consumers don’t describe a product’s sound in terms of decibels, phones, high frequency, and low frequency, but instead more subjective expressions, such as “sharp”, “dull” and “boxy”. The task is therefore to translate these subjective terms into objective and measurable quantities. The objective quantities should, as such, both be measurable and characterize the sound in a way relevant to the product at hand. They can be both conventional vibroacoustic concepts and other mechanical quantities such as rotational speed. In connection with electric motors that adjust seat position in cars, for instance, it is known that the rotational speed has a large bearing on the subjective experience of sound. That has little to do with man’s physical hearing, but is instead related to the “psychoacoustic” feeling: − Will this mechanism keep working once the warranty has expired? There are a number of methods to transform subjective impressions into objective quantities. The most common are “paired comparison” and “semantic difference analysis”. In the former, two alternative sounds are presented to the experimental subject, who is then asked which one is the most desirable. If there are many subjects, their responses are statistically treated. Finally, relations can be established between the subjective preferences and the measurable characteristics of the sound. The “semantic” difference technique demands that the panel evaluate the sound and rank it with respect to such subjective dimensions as “soft” and “hard”, “sporty” and “luxurious” and so forth. The panel’s evaluation can then be compared to the objective quantities. When the coupling between the subjective concepts and the objective quantities has been established, the objectives can then be formulated. Examples are: • Modification of the amplitudes of individual tones, in order to bring about a changed hearing threshold; compare to Figure 3-9. • Altered balance between low and high frequency tire noise, as heard from inside a car. • Elimination of the rotational speed variations of electric motors and fans. (ii) Couple the stated objectives to specific parts and functions of the design. In order to transform sound quality objectives into concrete countermeasures, the objectives derived from part (i) above must now be coupled to the designs of individual components and different operational states. A noise problem in which there are strong tonal components can, for example, be attributed to the driveline of a passenger car (engine – transmission – drive shaft – rear axle) for operation in a certain gear at a certain speed. (iii) Survey and subdivide the complete structure into an source – response –acoustic radiation system, as in figures 1-2 and 1-3 in the introduction.

77

Chapter 3: Influence of Sound and Vibration on man and equipment Consider the example of automotive interior noise, again. Beginning with the oscillator element, a first step might be to subdivide the sound into periodic signals from the engine and transmission, and the noise-like sound from the airflow around the car body and from the tires. After that, one might identify how much of the tire noise that comes from each particular wheel. As for the response element, the problem may be dominated by a local resonance in the transmission, for example. Finally, for the radiation element, it might be found that a resonance in the car roof is responsible for the majority of the radiated sound. (iv) Predict the sound content of the product for different design variations, finding that giving the best sound quality. Beginning with a computational model of the driveline, for example, we can predict the sound that would be generated by the proposed design. By then varying a parameter in the model, such as the engine speed for instance, we can hear when resonances or other undesirable sounds arise. If our starting point is a measurement rather than a model, then we could use a sound quality program to eliminate resonances or shift them to other engine speeds. In either case, we or the panel can then listen to the expected sound of the proposed product, before and after the proposed design modifications, and determine the solutions that are preferable. 3.2

EFFECTS OF VIBRATION AND SHOCK

3.2.1

Machinery and vehicle vibrations

Operation of machines and vehicles gives rise to forces. These forces, in turn, generate vibrations. In order to describe the vibrations, we need to know their amplitudes, frequencies, and sometimes even their mode shapes, i.e., the deformation pattern of the structure. In machines, the main sources of vibrations are often forces due to accelerations and retardations of masses. Examples are unbalanced shafts in rotating machinery, reciprocating motions in piston-based machines such as compressors and internal combustion engines, and reciprocating motions in sewing machines. In gears, the contact forces vary as the gear rotates, since the number of teeth in contact is not constant. Shocks and vibrations in gears are also caused by manufacturing variability (tolerances on the tooth geometry), surface roughness, and shaft misalignment. In electrical machines, such as motors and generators, the electromagnetic forces give rise to vibrations. In internal combustion engines, compressors, and other pneumatic and hydraulic machines, pressure variations in the medium are the significant vibration sources. How strong vibrations are acceptable? There are, of course, no definitive answers to that question. Many different factors could come into play, such as surface finish requirements in machining, requirements for assembly precision, or fatigue strength in the most extreme cases. In Figure 3-14, a method is proposed to divide machines into different classes, and to give criteria to judge them on the basis of vibration amplitude.

78

Chapter 3: Influence of Sound and Vibration on man and equipment Vibration velocity level -9

Lv [dB] rel. 10

m/s

Vibration velocity, rms v~ [mm/s]

153

45

149

28

145

18

141

11,2

137

7,1

133

4,5

129

2,8

125

1,8

121

1,12

117

0,71

113

0,45

109

0,28

105

0,18

Not allowed Not allowed Not allowed Not allowed Barely tolerable Barely tolerable Barely tolerable Barely tolerable

Allowed Allowed Good

Allowed Good

Allowed Good

Good Small machines, up to 15 kW

Large machines on stiff and heavy Middle-size machines, foundations with eigen15-75 kW or more, frequencies exceeding up to 300 kW on the machine’s rotatspecial foundations ional frequeny

Class I

Class II

Class III

Large machines with a rotational frequency exceeding the eigenfrequency of the foundation (e.g., turbo-machines)

Class IV

Figure 3-14 Proposed division and evaluation of machinery vibrations with respect to machine type and vibration level. (Source: Brüel & Kjær, information material.)

3.2.2

Effects on man

The human body is influenced by vibrations at any frequency, if their amplitude is high enough. When we study the effects on man, both the purely physical and the psychological effects must be considered. The interesting aspects are: (i)

The characteristics of the human body when subjected to vibrations and shocks.

(ii)

Effects of disturbances. (These can be divided into physical, physiological and psychological criteria)

(iii)

Acceptable exposure amplitudes for different exposure times and frequencies.

In order to simplify evaluation, international evaluation criteria have been developed. They divide the disturbances into so-called whole body vibrations and hand-arm vibrations. In the case of whole-body vibrations, the disturbances are assessed according to three criteria: (i)

Health.

(ii)

Comfort and feeling.

(iii)

Motion sickness.

A lot of work has been carried out to survey and specify these criteria, but because the human body is very complex, and varies from individual to individual, it is a difficult task. Relatively little is known about what amplitudes cause human injuries, since it is difficult

79

Chapter 3: Influence of Sound and Vibration on man and equipment to do experiments, for ethical reasons among others. Most studies have therefore been on animals. Animals, however, have different anatomical constructions and sizes than humans, making it uncertain how the results can be extrapolated to man. When the question is comfort and feeling, however, it becomes easier to carry out human experimentation, but even in this case, the results vary, especially due to psychological factors. Most studies have focused on drivers in different transport modes, such as trucks and airplanes. The human body consists of a relatively hard skeleton of bone that is bound together by ligaments, tendons, muscles and other tissues. The soft, vital organs, such as the heart and lungs for example, are protected in the chest cavity or in the abdominal cavity. They are held in place individually by ligaments and membranes, and can therefore undergo relative motions. The combination of soft tissue and bone, and softly fastened inner organs, with masses varying form very small up to a few kilos, constitutes a system that can react very differently at different disturbing frequencies. Low frequencies, under 1 Hz, occur in many transportation situations. These low frequency vibrations affect our sense of balance and bring about effects we usually call motion sickness. From a purely mechanical perspective, the entire human body follows these low frequency oscillations as a single unit; i.e., roughly speaking, a standing human body can be regarded as a rigid body for vertical motions up to Head (axial motion) (About 25 Hz) about 2 Hz. In the rest of the low Eye, interocular frequency region, 2 - 100 Hz in structures (30 – 80 Hz) rough terms, the body can, for some purposes, at low Partial shoulder amplitudes, be described as a particle model that describes at (4 – 5 Hz) Chest cavity the individual organs’ and body (About 60 Hz) Lung volume parts’ lowest eigenfrequencies; Hand Underarm see Figure 3-15. For whole(16 - 30 Hz) Abdomen body vibrations, the vibrations (4 – 8 Hz) Diaphragm are typically transmitted via the (Axially 10-12 floor for a standing person, and Hand Sitting person via a seat and seatback for a (50 – 200 Hz) sitting person. The physical Leg effects of transverse vibration, (Vary from 2 Hz to 20 Hz depending on position) (perpendicular to the backbone), differ from the physical effects of axial excitation (parallel to the backbone). One of the most important subsystems that can Standing person be put in motion, either for a standing or a sitting person, Figure 3-15 In the low frequency 2 - 100 Hz region, the is the trunk and abdomen, with body can, for some purposes, and for low resonances in the 3-12 Hz region. amplitudes, be regarded as a particle system. The individual organs and body parts have eigenFor resonances in the shoulders, frequencies, the approximate values of which are indicated in the figure. (Sketch: Brüel & Kjær, informationsmaterial.)

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Chapter 3: Influence of Sound and Vibration on man and equipment abdomen, and hips, see figures 3-16 and 3-17. These low frequency resonances make it very difficult to devise effective vibration isolation elements, for use in a driver’s seat for example, since physical principles dictate that they have even lower eigenfrequencies; they would then risk having a swaying feel. Other important resonances in the body are the head-throat-shoulder system at about 25 Hz, and the eye at about 30-80 Hz.

Amplification factor

Hip / platform Shoulders / platform Head / platform Platform

Waist / platform Bent knees

Frequency [Hz]

Amplification factor

Figure 3-16 Amplification and damping of vertical vibrations from a vibrating platform under the feet to different parts of the body, for a standing person. Eigenfrequencies from 3 to 8 Hz give amplified vibrations. Bent knees provide some attenuation. Under about 2 H, the entire body largely behaves as a rigid body. (Plot: Brüel & Kjær, Mechanical Vibration and Shock Measurements, data from Dieckmann/Radke.)

Head / shoulder

Shoulder / platform

Figure 3-17 Amplification or attenuation of vertical vibrations from a vibrating platform to different parts of the body, for a sitting person. (Plot: Brüel & Kjær, Mechanical Vibration and Shock Measurements, data from Dieckmann.)

Head / platform

Frequency [Hz]

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Chapter 3: Influence of Sound and Vibration on man and equipment At frequencies above the lowest resonances, vibrations are relatively well-damped, assuming that no relatively undamped resonance is excited. Figure 3-18 shows that a tone of 50 Hz is attenuated about 30 dB from the feet to the head, and about 40 dB from the hand to the head.

Head

Figure 2-18 Attenuation of vibrations, at a frequency of 50 Hz, along the human body. The attenuation is given in dB, with the excitation position providing a reference amplitude. The excitation positions are A for the hands, and B for the feet. (Diagram: Brüel & Kjær, Mechanical Vibration and Shock Measurements, data from von Bekesy.)

Head

Vibration attenuation [dB]

Platform excitation

Vibration attenuation [dB]

In the mid-frequency region, above about 80 - 100 Hz, the particle model becomes an all the more inadequate means to describe vibrations in the human body. At these frequencies, it is necessary to regard the body as a continuous medium in which wave propagation of different wave types occurs .. The effects of the vibrations are also dependent on where in the body they occur, and on how they are directed. In the ultra sound region, above several hundred thousand Hertz, the vibratory energy mainly propagates as longitudinal waves. Since the wavelengths at these high frequencies are small in comparison to the dimensions of soft body parts, geometric acoustics methods are applicable (see chapter 2). Hand-arm vibration is the other large problem area. All hand-held machines transfer vibrations to the body via the hand-arm system; see Figure 3-19. Vibrations can arise because of unbalanced rotating elements or reciprocating motions in a machine. Sometimes, however, it is not the machine itself that brings about the strongest effects, but the forces from the work process. That applies to power saws, grinding machines, and such hitting devices as pneumatic boring machines and jackhammers used in road construction.

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Chapter 3: Influence of Sound and Vibration on man and equipment

a), b) excitation frequency 37 Hz.

c), d) excitation frequency 152 Hz.

Figure 3-19 Oscillation pattern in the arm-shoulder (a,c) and hand-arm (b,d) from vibration excited into the right hand (grasping the source). The visual pattern photographed is brought about by double-pulsed holography. In both cases, the excitation is perpendicular to the plane of the photograph, with an acceleration of 30 m/s2. The dark lines, so-called interference fringes, connect points undergoing the same motion. The interference fringes can be compared to isometric (equal altitude) lines on a map; as such, they show a vibration pattern (mode). The displacement difference between two adjacent interference fringes corresponds to about half a light wavelength, i.e., about 3·10-7 m. A large distance implies lower vibration amplitudes, and a short distance implies higher amplitudes. In a), the arm-shoulder system is excited at 37 Hz. The high-density interference pattern in the soft parts around the shoulder blades shows that the vibrations are transmitted effectively along the entire path from the hand grip via the lower and upper arm. In c), the arm-shoulder system is excited at 152 Hz. The sparse interference pattern on the upper arm shows that higher frequencies are more severely attenuated. b) and d) show excitation of the hand-arm system at 37 Hz and 152 Hz, respectively. The dense interference pattern in the elbow area at 37 Hz, as compared to 152 Hz, also demonstrates the more effective attenuation of high frequencies. (Hologram and photo: Lennart B M Svensson, Produktionsteknisk Mätteknik, KTH.)

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Chapter 3: Influence of Sound and Vibration on man and equipment Vibrations transmitted to a standing or sitting person can cause such problems as motion sickness, reduced comfort, and diminished working performance. Hand-arm vibrations can even cause physical injuries if the vibration amplitudes and exposure times are sufficient. It is often local injuries that occur in the hand-arm system, mainly to blood vessels, nerves, the skeleton, or joints. In the case of blood vessel damage, the walls of the thinnest blood vessels thicken, and the blood flow is reduced. The hands are, to a large extent, the body’s temperature regulator. If the body temperature gets too high, blood flow through the hands increases. If, on the other hand, the body temperature gets too low, the blood flow is reduced by contraction of blood vessels. With injured blood vessels that have thickened walls, however, the blood flow to such peripheral body parts as the fingers can be blocked off completely, and the affected body part whitens. That phenomenon has been extensively studied, as is often called white finger syndrome. The injury is characterized by loss of feeling, numbness, and pricking sensations in the fingers; see Figure 3-20. In the case of nerve injuries, feeling in the hand is reduced.

Figure 3-20 Vibrations transferred to the hand-arm system can increase the thickness of the walls of narrow blood vessels in the outer body parts, and constrict blood flow. In cases so severe that the blood supply to the fingers is completely cut off, the fingers whiten. That is called white finger syndrome. (Sketch: Brüel & Kjær, Human Vibration.)

The vibration exposure sufficient to cause injury is not completely known, neither with respect to the amplitude, the frequency content, or the exposure time. Despite difficulties and lack of quantitative information, the standard presented in section 3.3.3 does provide guidelines to judge the risk that blood flow disturbances of the white finger variety will occur.

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Chapter 3: Influence of Sound and Vibration on man and equipment 3.3

STANDARDS

In the following sections, a brief description is provided of international standards in the areas of acoustics, and vibrations and shock. 3.3.1 Standard ISO 3740 Acoustics – Determination of sound power levels of sound sources – Guidance for the use of fundamental standards and for working out machine-specific noise measurement methods Before we delve into the standard itself, let us first say something about its background. In chapter 2, we considered two possible ways to describe noise disturbances – sound pressure and sound power, or alternatively, their logarithmic equivalents, the sound pressure level and the sound power level. The sound pressure level is a useful quantity for describing the sound at a specific place, but not as useful for characterizing the acoustic behavior of a machine. This is because the sound pressure level depends on both the distance and the direction to the machine, as well as the environment in which it is placed. Environment, in an acoustics context, refers to the distance to reflecting surfaces, the room’s volume, and the sound absorption of the reflecting surfaces. The sound power level is, instead, a measure of the total radiated sound power from a machine, and is largely independent of the environment. The sound power and the corresponding sound power level are therefore preferable whenever sound data is to be provided for a certain machine. Based on sound power data, we can: (i)

Calculate the sound pressure level and sound level at a given point in a room, provided we know the distance from the source, the room’s volume, and its sound absorption properties.

(ii)

Compare sound data for machines of both the same and different types and sizes.

(iii)

Determine whether the machine fulfills agreed-upon specifications.

(iv)

Plan for noise control measures that will be needed after the machine is installed.

(v)

Sound power data can, moreover, provide a measure of performance of success of efforts to develop quieter products.

A complete description of a machine’s sound emission would consist of the total radiate sound power, and its directivity. The directivity is a measure of a how the machine’s acoustic radiation varies in different directions. Many sources, however, have very little directivity. That is especially true of sources that are small compared to the sound wavelength, which is often the case at low frequencies. ISO 3740 briefly describes the principles of the determination of sound power levels, and gives guidance on choosing the appropriate sound power measurement standard. They are summarized in table 3-3. Table 3-4 gives an overview of the precision of the various methods. If a particular measurement standard exists for the machine type of interest, that would naturally be used.

85

Chapter 3: Influence of Sound and Vibration on man and equipment Table 3-3 Standard ISO 3741-3747 on different methods to determine the sound power levels of machines and equipment. (Source: ISO 3740-3747 and Brüel & Kjær, Sound Power.) Interna- Class of Test Volume of tional Method environment sound standard source ISO 3741 Precision Reverberant room with specified data 3742 Should be < 1% of the 3743 Technical Special test room reverberant volume room

3744

3745

Indoors in a Largest large room, dimension or outdoors < 15m

Precision

3746

Survey

3747

Survey; only relative method

Anechoic or Should be < semi0.5% of the anechoic test room room volume No No special restrictions. environment Only limited by the available measurement space No special No a environment, limitations sound source not moveable

Type of noise

Continuous, broad band

Measureable sound power levels Third octave, octave

Continuous, pure tones, narrow band Continuous, Octave, broad band, A-weighted narrow band, pure tones Third octave, octave, A-weighted

Other information obtainable

A-weighted sound power level

Sound power levels with other weighting filters Directivity, sound pressure as a function of time, sound power levels with other weighting filters

All

Sound pressure as a function of time, sound power levels A-weighted with other weighting filters

Continuous, broad band, narrow band, pure tones

86

Sound power level in octave bands. Sound pressure level at specified locations

Chapter 3: Influence of Sound and Vibration on man and equipment Table 3-4 Uncertainty in the determination of sound power levels, expressed as the largest values of the standard deviation in dB. (Source: ISO 3740 and Brüel & Kjær, Acoustics Sound Measurement.) International standard ISO

Octave band [Hz]

125

250

500

Third octave band [Hz]

100160

200315

400630

3

2

1,5

5 3 1 1,5

3 2

2

3741 3742 3743 3744 3745 3746 3747

Anechoic room Semi-anechoic room

1 1,5

10004000 8005000

8000 630010000 3

-

1,5 0,5 1

3 2,5 1 1,5

2 2 -

For a source that generates discrete tones For a source that emits an evenly distributed noise over the frequency band of interest

Aweight

5 5 4

From table 3-3, it is evident that three classes of methods exist: precision; technical; and, survey. The precision methods specify rigorous requirements for the measurement environment, demanding access to expensive acoustic measurement laboratories, as shown in figures 3-21 and 3-22; on the other hand, the survey methods do not have such requirements. The standards ISO 3741-3743 are based on measurements in diffuse fields, and ISO 3744-3746 on measurements in free fields. The last, ISO 3747, is intended for machines that cannot be moved, and therefore also makes no special demands on the environment.

Figure 3-21 Reverberant room at MWL, KTH (Royal Inst Techn, Sweden). In a reverberant room designed for precision measurements, rigorous requirements are to be met. The floor, walls, and ceiling are to absorb, at most, 6% of the sound power of the incident waves. The volume of the room should fall in the range of about 200 to 300 m3 and its dimensions must have the proportions to ensure that the eigenfrequencies, as discussed in section 5.2, be evenly distributed along the frequency axis. The plexiglass panels that hang from the ceiling are intended to improve the diffuse sound field. Despite all of that, an ideal diffuse sound field is not attainable; thus, several microphone positions are used in practice, or, as in the picture, a rotating microphone boom is used to obtain a spatial average in the room. The measurement object on the floor is a calibrated sound power source of the fan type. (Photo: HP Wallin, MWL.)

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Chapter 3: Influence of Sound and Vibration on man and equipment

1,1m

Figure 3-22 Semi-anechoic room at ASU Sound and Vibration Lab., Ain Shams University in Egypt. In the anechoic measurement room, the same conditions must apply as for propagation in a free field. When the sound waves reach the floor, walls, and ceiling, they must be effectively absorbed. Less that 1% of the incident sound power is permitted to reflect back into the room. In order to be effective at low frequencies, and to be approved for precision measurements, the mineral wool wedges that are commonly used must be at least about a quarter wavelength long; see the detail view. For the room in the picture, that implies a length of about 1.1 m to obtain a lower frequency bound of 80 Hz. The room in the picture has a hard, reflecting floor, and is therefore a so-called semi-anechoic room. In that type of measurement room, the sound field obtained is equivalent to a free field over a hard reflecting plane; that corresponds, for example, to the situation of a vehicle traversing a reflecting roadway. A semi-anechoic measurement room, in combination with a chassis dynamometer, a kind of rolling highway, may therefore be found at any automobile manufacturer. (Photo: HP Wallin, MWL.)

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Chapter 3: Influence of Sound and Vibration on man and equipment 3.3.1.1 Standard ISO 3747 Acoustics – Determination of sound power levels – survey method using a reference sound source This standard describes a method based on relative measurements using a so-called calibrated sound power source.. The standard permits reflecting surfaces in the vicinity of the measurement object. Before becoming too specific about the contents of the standard, we begin by getting acquainted with some background. For a calibrated sound power source, there should be a calibration protocol that gives the sound power in third octave and octave bands, and with A-weighting included. Two types of calibrated sound power sources exist. Firstly, there are the electroacoustic sources that consist of a generator/amplifier part and a loudspeaker part. Secondly, there are fan-type sound sources that contain a radial fan driven by a powerful electric motor, as in Figure 3-21. The overdimensioned motor supplies a constant rotational velocity which, in turn, generates a stable sound spectrum. An important requirement for such reference sources is that, in addition to a constant sound power, they shoud also have a low directivity, i.e., they should radiate about the same in all directions. The standard does not make any demands on the measurement environment. That means that the sound waves recorded by the measurement microphone can be either direct sound from the source, or reflected sound; see Figure 3-23.

Reflected sound

Microphone Source

Direct sound

Figure 3-23 The standard for sound power determination ISO 3747 specifies no requirements for the measurement environment, which means that both direct and reflected sound can reach the microphone positions.

According to the preceding section, the sound power level is a measure of the radiated sound power from a machine, and essentially independent of the acoustic environment, whereas the sound pressure level depends on both the distance and the direction to the machine, and the environment in which it is placed. For a certain acoustic environment, i.e., a specific room, and a specific distance to the sound source the machine’s sound power level LW can be related to the resulting sound pressure level Lp at a point, using the so-called room correction K [dB], in the formula L p = LW + K .

(3-5)

The method for the sound power determination with a calibrated sound power source is based on determining the room correction K using the known sound power of that source. 89

Chapter 3: Influence of Sound and Vibration on man and equipment

The sound pressure level from the measurement object is measured first, after which it is replaced by the calibrated reference source, and the sound pressure level measured again at the same microphone positions. The sound power level LW of the measurement object can then be determined from (3-5), as

(

)

LW = L p + LWr − L pr ,

where

(3-6)

L p is the sound pressure level from the measurement object,

LWr is the sound power level of the reference source, L pr is the sound pressure level of the reference source.

Usually, the sound pressure level is measured at a number of points on a measurement surface around the measurement object. Then, the spatial average of the sound pressure levels L p is used in (3-6), as

L p = 10 ⋅log(

where

1 N

N

∑ 10

L pn 10

) [dB],

(3-7)

n =1

Lpn is the sound pressure level at point n.

Example 3-2 At five measurement points, the sound pressure levels 79, 82, 83, 79 and 81 dB are measured. Determine the spatial average. Solution

(3-7) gives L = 10 ⋅log ⎡ 1 (10 7.9 + 10 8.2 + 10 8.3 + 10 7.9 + 10 8.1 )⎤ = 81.09 ≈ 81 dB . p ⎢⎣ 5 ⎥⎦ ISO 3747 places no demands on the measurement environment, and is intended for measurement objects that cannot be moved to a measurement environment that gives a higher degree of precision. The overview given below is considerably shortened, and assumes a measurement object placed freely on a reflecting floor. Measurement object: Machines that cannot be moved, and their parts and components. Running condition is to be specified. Sound Character:

Stationary. Broad band, narrow band, discrete tones and combinations of these.

Quantities determined:

A-weighted sound power level.

Frequency region:

Octave bands with center frequencies from 125 Hz up to 8000 Hz.

Background levels:

The background level for each octave band is to be at least 3 dB below the corresponding sound pressure level when the 90

Chapter 3: Influence of Sound and Vibration on man and equipment

measurement object or reference source are operated. The correction for the background level is to be made if it is not at least 10 dB lower; see example 1-6. Placement of reference source:

The calibrated reference source can either be placed at the ordinary position of the measurement object, or at one or more points around the measurement object, depending on its size.

Microphone positions:

To find the microphone positions, a hypothetical parallelepiped reference surface, surrounding the measurement object and reference source positions, is first defined. The microphone positions are then placed on another, typically parallelepiped, surface, with its sides parallel the reference surface, and at a distance of 1 m from it. For measurement objects with horizontal dimensions of less than a meter, the microphones are placed in the middle of the five measurement surfaces, see Figure 3-24. Microphone position

Figure 3-24 The microphone positions on a measurement surface are at distances of 1 m from the reference surface. The reference surface surrounds the measurement object and the reference source.

3.3.2

Measurement surface

1m 1m 1m Reference surface

Reflecting floor

Standard ISO 2631-1 1997 Vibration and shock – Guidance for evaluating the effects of whole-body vibrations on man. Introduction

Machines and means of transport of all kinds subject man to vibrations that affect comfort, working ability, and, in the worst case, health. Despite the inherent difficulties due to the involved situations, and to some extent the lack of reliable knowledge, an international standard has nevertheless been agreed upon. The standard primarily seeks to simplify the assessment and comparison of measurement results concerned with whole-body vibrations. The standard is concerned with vibrations transferred to the whole body from the surface one stands, sits, or lays upon. It consists, presently, of two parts: Part 1 ISO 2631-1 General requirements

91

Chapter 3: Influence of Sound and Vibration on man and equipment

Pat 2

ISO 2631-2 Vibration and Shock - Measurement and guideline values for the assessment of comfort in buildings.

The standard includes directions on how to apply the results, and how and what should be measured. In the following two sections, the main features of parts 1 and 2 are briefly described. 2.3.2.1 ISO 2631-1 Part 1: General requirements Introduction:

The primary objective of part 1 is to specify methods to report measurement values used in evaluating whole body vibrations, with respect to: (i)

Health.

(ii)

Comfort and sensation.

(iii)

Motion sickness.

Part 1, by itself, does not contain any guideline values for evaluating the affects on health, comfort, sensation and motion sickness. It also does not treat the influence of the vibrations on working productivity or the possibility to carry out special work tasks, since those things are highly dependent on the individual in question, the work situation, and the actual work task. Motion sickness is not handled any further in this description, aside from the frequency bands of relevance given below, and the frequency curve in Figure 3-26. Frequency region: (i) (ii)

Health, comfort and sensation, Motion sickness,

0.5-80 Hz. 0.1-0.5 Hz.

Measurement directions and positions: The vibrations are to be measured in a coordinate system defined by Figure 3-25, and on the surface that transfers these to the human body:

(i)

Standing person – foot surface.

(ii)

Sitting person – the seat.

(iii)

Lying person – pelvis and back.

The standard states that, for sitting persons and evaluation with respect to comfort, the comfort can, in some cases, be affected by rotational vibrations about the coordinate axes, in the sitting surface, as well as vibrations in the foot support surface. Moreover, for lying persons, without a soft pillow under the head, vibrations under the head are also to be accounted for when evaluating either comfort or health. We will refrain from describing such cases in this more cursory description. If the strength of the vibrations varies over the support surface, then a spatial average, corresponding to that of section 3.3.1.1 is to be determined. 92

Chapter 3: Influence of Sound and Vibration on man and equipment

x

z

z y

x y

Lying down

Standing

Hand transferred vibrations

y

z x z

z

y Sitting

x Figure 3-25 Coordinate systems for measurement of the effects of vibrations on the human body. Hand transferred vibrations are treated in section 3.3.3. (Sketch: Arbetarskyddsnämden, Buller och vibrationer ombord, Ill: Claes Folkesson.)

Measurement Quantities: The vibration amplitudes are to be weighted by a weighting filter using a method analogous to that of determining the sound level in dB(A); see Figure 3-7.The weighting curves reflect the sensitivity of the body to whole-body vibrations in the vertical and horizontal directions. The measurement quantity is the rms acceleration a~w [m/s2], in the frequency band of interest. The index w indicates that the quantity is frequency weighted according to a~w =

N

∑ (Wn a~n ) 2 ,

(3-8)

n =1

where Wn is the weighting factor as a function of the frequency, according to Figure 3-26 and table 3-5. The weighting factor Wd,n is used in the x- and y-directions, and Wk,n is used in the z-direction. a~n [m/s2] is the rms acceleration in the n-th frequency band, e.g., measured in third-octave bands. N is the number of the frequency band.

93

Chapter 3: Influence of Sound and Vibration on man and equipment Weighting factor Wk,, Wd,, Wf 10

1 Wk 0.1 Wd 0.01 Wf 0.001

0.0001

0.00001 0.01

0.1

10

1

100

1000

Frequency f [Hz] Figure 3-26 The weighting factors Wk, Wd and Wf reflect the sensitivity of the human body to vibrations in the x-, y- and z-directions. The factor Wd is used in the x-and ydirections, Wk in the z-direction and Wf in the vertical direction, in connection with the assessment of motion sickness. In the standard, there are a total of six weighting curves for different directions and purposes. (Source: ISO 2631-1.) Table 3-5 The weighting curves Wk and Wd, in third octave bands. (Source: ISO 2631-1.)

Frequency [Hz] 0.5 0.63 0.8 1 1.25 1.6 2 2.5 3.15 4 5 6.3 8 10 12.5 16 20 25 31.5 40 50 63 80

Weighting factor Wk 0.418 0.459 0.477 0.482 0.484 0.494 0.531 0.631 0.804 0.967 1.039 1.054 1.036 0.988 0.902 0.768 0.636 0.513 0.405 0.314 0.246 0.186 0.132

94

Weighting factor Wd 0.853 0.944 0.992 1.011 1.008 0.968 0.890 0.776 0.642 0.512 0.409 0.323 0.253 0.212 0.161 0.125 0.1 0.08 0.0632 0.0494 0.0388 0.0295 0.0211

Chapter 3: Influence of Sound and Vibration on man and equipment

As discussed in section 2.3, the peak factor is the ratio of the peak value to the rms value of a signal. If the peak factor, calculated with the frequency weighted peak and rms values, does not exceed nine, then the signal’s rms value can be determined in the usual way described in section 2.3. If the vibrations contain shocks that are so strong that the peak factor exceeds 9, then another method that gives greater weight to the peaks should be used instead. In the assessment with respect to health effects, the standard is mainly concerned with sitting, because the effects on lying and standing persons are not sufficiently well known. After calculating the frequency weighted accelerations in the x-, y- and z-directions, from formula (3-8), the values in the x- and y-directions are multiplied by an direction factor K = 1.4. Then, the assessment is made using the highest value obtained. Example 3-3 In the driver’s seat of a work machine, an 8 Hz tone dominates in all three coordinate directions. The following rms-values have been measured: x-direction; a~x = 0.84 m/s2, y-direction; a~ = 0.71 m/s2, y

z-direction;

a~z = 0.48 m/s2.

Which value is to be used to make the assessment with respect to health effects? Solution Using the weighting filter from table 3-5, or from Figure 3-26, one obtains

Quantity Acceleration a~ [m/s2] Weighting factor W Direction factor K Weighted acceleration K ⋅ a~w [m/s2]

x-direction 0.84 W d = 0.253

y-direction 0.71 Wd = 0.253

z-direction 0.48 W k = 1.036

K = 1.4

K = 1.4

K = 1.0

0.30

0.25

0.50

The value in the z-direction is therefore the highest, and should be used in the assessment. For assessment with respect to comfort, the vector amplitude of the frequency-weighted acceleration, a~v , is determined from a~v = a~w2, x + a~w2, y + a~w2, z ,

(3-9)

where a~w, x , a~w, y and a~w, z are the frequency-weighted accelerations in the respective directions, each obtained from formula (3-8).

95

Chapter 3: Influence of Sound and Vibration on man and equipment For assessment with respect to the sensitivity threshold, i.e., the lowest value a normal person can detect, the assessment is to be made with respect to the highest value in any coordinate direction, using formula (3-8). Vibration type/ spectrum: The vibrations can be periodic, stochastic, transient or combinations of these, i.e., they can be of both narrow band character, with tonal components, or broad band with noisy character. They can be measured or evaluated with band pass filters of CAB or CRB-type; see section 2.10.1. The latter case includes, for example, a third octave band filter. Guideline values:

As mentioned at the outset, the standard itself does not contain any guideline values, but some of its appendices give guidance on evaluating the effects of vibration on health, comfort, sensation, and motion sickness. Below, a short summary of the first three is provided. Health: In the case of daily exposure for 8 hours, no health effects have been reported for K ⋅ a~w ≤ 0.43 m/s2. Moreover, for K ⋅ a~w ≥ 0.73 m/s2, injury is probable. Two methods are also provided to normalize exposure times less than 8 hours per day. The information given applies to completely healthy people, and must be used with extreme caution, as well as the understanding that it usually takes a number of years before any health effects can be observed. Comfort: Acceptable values for comfort vary according to many factors, such as exposure time, noise, expectations, and the actual activity one is engaged in, such as reading, writing, computer work, or consumption of food and drink. One of the appendices gives the following as probable reactions to the indicated vibration levels experienced by passengers in mass transportation. Table

3-6 Assessment of comfort (Source: ISO 2631-1.)

Acceleration, vector amplitude a~v < 0.315 m/s2 0.315 < a~v < 0.63 m/s2 0.5 < a~v < 1.0 m/s2 0.8 < a~v < 1.6 m/s2 1.25 < a~v < 2.5 m/s2 a~v > 2.0 m/s2

when

traveling

by

mass

transportation.

Degree of comfort Not uncomfortable A little uncomfortable Rather uncomfortable Uncomfortable Very uncomfortable Extremely uncomfortable

Sensitivity threshold: Half of a healthy, alert population can just barely sense a vertical acceleration with a peak value aˆ w = 0,015 m/s2, weighted with the weighting filter Wk.

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Chapter 3: Influence of Sound and Vibration on man and equipment

Example 3-4 On the floor aboard a vehicle, the following vertical and horizontal acceleration values have been measured, in third octave bands.

f [Hz] a~z [m/s2] a~ x = a~ y [m/s2]

2 0.50

2.5 0.70

3.15 0.40

4 0.50

5 0.40

0.45

0.60

0.35

0.45

0.30

Other frequencies are negligible in this context, and the peak factor is lower than nine. How will a passenger assess the vibration comfort? Solution: We apply the frequency-dependent weighting filter from table 3-5, and then calculate the frequency-weighted acceleration in the three coordinate directions, using formula (3-8),

f [Hz] Wk Wd

2 0.531 0.890

a~w, z =

2.5 0.631 0.776

3.15 0.804 0.642

4 0.967 0.512

5 1.039 0.409

2 (0.531 ⋅ 0.50) 2 + (0.631 ⋅ 0.70) 2 + ... + (1.039 ⋅ 0.40) 2 = 0.88 m/s ,

2 a~w, x = a~w, y = (0.890 ⋅ 0.45) 2 + (0.776 ⋅ 0.60) 2 + ... + (0.409 ⋅ 0.30) 2 = 0.70 m/s .

When evaluating with respect to comfort, the vector amplitude a~v of the acceleration, from formula (3-9) is determined. a~v =

2 0.70 2 + 0.70 2 + 0.88 2 = 1.32 m/s .

According to table 3-6, the assessment made is “uncomfortable” to “very uncomfortable”.

97

Chapter 3: Influence of Sound and Vibration on man and equipment 3.3.2.2 ISO 2631-2 Influence of vibration and shock on people in buildings Frequency range: 1 - 80 Hz. Measurement point & direction: The measurement point is chosen where the vibrations are greatest. Those are normally vertical vibrations in the middle of the floor with the greatest dimensions. Measurement Quantities:

Vibration type/ spectrum: Guidelines:

The vibration amplitudes are to be weighted by a weighting curve in the same fashion as for determining the sound level in dB(A); see Figure 3-7. The weighting curve reflects the sensitivity of the body to whole-body vibrations, and is therefore based, in principle, on the curves of that type given in Figure 3-26. The measurement quantity can either be the rms velocity v~w or the rms acceleration a~w . The index w indicates that the value is frequency-weighted. The measurement quantity can either be measured with a frequencyweighting filter, or calculated in a corresponding way, based on thirdoctave bands.

Either narrow or broad band vibrations can be treated. Table 3-7 Guideline values for the assessment of the effect of vibrations and shock on humans in buildings.

Weighted velocity, v~w [m/s] (0.4 - 1.0)·10-3 > 1·10-3

Moderate disturbance Probable disturbance

Weighted acceleration, a~w [m/s2] (14.4 - 36.0)·10-3 > 36·10-3

The guidelines are not intended to be applied to temporary activities such as building and installation work. The standard maintains that few people experience vibration levels below “moderate disturbance” as disturbing. The vibration levels that fall into the “moderate disturbance” category occasionally result in complaints. In the “probable disturbance category”, the vibrations are very noticable and often experienced as disturbing. If the frequency-weighted value is dominated by a particular frequency, which is often the case due to building resonances, then the frequency-weighted value can be replaced by the unweighted rms velocity at the dominant frequency; the degree of disturbance can be read from Figure 3-27.

98

Chapter 3: Influence of Sound and Vibration on man and equipment Speed v~ [m/s]

Probable disturbance

0.01

Moderate disturbance

0.001

0.0001 1

2

16 4 8 Frequency [Hz]

32

64

Threshold of sensitivity according to ISO 2631-1 1985

Figure 3-27 Threshold of sensitivity and regions of moderate disturbance and probable disturbance as functions of frequency for vibratons in buildings. The curves are based on the rms velocities of vibration without weighting. (Source: SS 460 48 61)

Vibration duration:

When the vibrations vary in time, the running conditions of the vibration source are to be recorded. For traffic-induced vibrations, established methods from mathematical statistics can be used.

Example 3-5 In a factory building, a floor resonance is measured to be at 31.5 Hz. The unweighted rms vibration velocity is 1.0 mm/s. Determine whether there is a risk of complaints. Solution Reading from Figure 3-27, it turns out that the value given falls on the boundary between the regions of moderate disturbance and probable disturbance. Complaints are possible. 3.3.3 Standard ISO 5349 Vibration and shock – Guidelines for the measurement and assessment of hand-transmitted vibration Frequency region: 5 - 1500 Hz. Measurement points & directions: The measurement results are to be given in a coordinate system in accordance with Figure 3-25. The measurements are to take place in three directions, in the immediate vicinity of the surfaces that transfer vibrations into the hand, i.e., at the hand grip when applicable. The assessment is based on components in the direction showing the highest measurement values.

99

Chapter 3: Influence of Sound and Vibration on man and equipment Measurement Quantity:

The rms acceleration in one or more directions is measured. The following options are called out: measurement in third-octave bands, 6.3 - 1250 Hz, (i) measurement in octave bands, 8 - 1000 Hz, (ii) (iii) measurement of frequency-weighted values from 5.6 - 1400 Hz. For the case of frequency weighting, a weighting filter from figure 3-28 is to be used. In the first two cases, the third-octave and octave band values are to be converted to frequency-weighted accelerations. The frequency-weighted acceleration is calculated from a~h, w =

N

∑ (K n a~h,n ) 2

,

(3-10)

n =1

where Kn is the weighting factor as a function of frequency, from Figure 3-28, and table 3-8. a~h,n [m/s2] is the rms acceleration within the n-th third-octave band or octave band. The index h stands for hand. N is the number of frequency bands. Vibration duration:

The assessment of the effects of hand-transmitted vibrations is based on a daily exposure time. The total daily exposure time is assumed, when making the assessment, to be 4 hours. To make comparisons to other exposure times possible, an equivalent-energy frequency weighted acceleration for a 4 hour exposure time is used. Then, if the total daily vibration exposure deviates from 4 hours, it can be normalized to the 4hour equivalent energy value, by the formula a~h, w,eq ( 4) =

1 T4

τ

∫ (a h,w (t )) ~

2

dt ,

(3-11)

0

where a~h, w,eq ( 4) is the equivalent-energy, frequency-weighted rms acceleration for a 4-hour exposure. a~h, w (t ) is the rms value as a function of time for the frequencyweighted acceleration. τ is the total daily work time in hours. T4 is four hours. If the exposure time T deviates from four hours, the corresponding equivalent-energy acceleration for four hours is calculated as a~h, w,eq ( 4) = T T4 a~h, w,eq (T ) [m/s2],

where

a~h, w,eq (T )

is

the

equivalent-energy

acceleration for a time period of T hours, T4 is four hours.

100

(3-12) frequency-weighted

Chapter 3: Influence of Sound and Vibration on man and equipment Table 3-8 Weighting factor Kn in third-octave bands. (Source: ISO 5349.) 1.2 Weighting factor, Kn

Frequency [Hz]

1

0.8

0.6

0.4

0.2

0 6.3 10

16

25

40 63 100 160 250 400 630 1000 Frequency [Hz]

Figure 3-28 The weighting factor Kn reflects the sensitivity of the human body to vibrations transferred to the hand-arm system. (Source: ISO 5349)

6.3 8 10 12.5 16 20 25 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250

Weighting factor Kn 1 1 1 1 1 0.8 0.63 0.5 0.4 0.3 0.25 0.2 0.16 0.125 0.1 0.08 0.063 0.05 0.04 0.03 0.025 0.02 0.016 0.0125

Example 3-6 For a handheld machine, a frequency-weighted equivalent-energy acceleration value of a~h, w,eq (5) = 15 m/s2 for a time interval T = 5 h is measured. Determine the corresponding

value for a time period of four hours. Solution (3-12) gives 2 a~h , w,eq ( 4 ) = 15 5 4 = 16.8 m/s .

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Chapter 3: Influence of Sound and Vibration on man and equipment Guidelines: The standard does not provide boundaries for safe exposure, but only guidelines. In an appendix that is not formally a part of the standard, information is given that makes it possible to estimate the probability of developing white finger syndrome, as a function of the frequency-weighted energy equivalent rms value for four hours of daily exposure, a~h, w,eq ( 4) ,

and the exposure time in years. The curves show the time that passes before signs of white finger syndrome begin to appear for 10, 20, 30, 40 and 50 % of the exposed persons. 25

exponerade individer

Exposure time in years before blood flow Exponeringstid i antal år innan blodflödesstörningar disturbances of the white finger variety appear av typ "vita fingrar" uppträder i procentuell andel av in a given percentage of the individuals

20

10

50 % 40 % 30 % 20 % 10 %

5

3

2

1

2

5

10

20

50

Ekvivalent frekvensvägd acceleration, a h,w,eq(4) , [ / ] Equivalenf frequency weghted acceleration, Figure 3-29 The relation between equivalent frequency-weighted acceleration and the exposed. In the proposed changes of1999, only the 10% -line is retained.(Source: ISO 5349.)

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Chapter 3: Influence of Sound and Vibration on man and equipment 3.3.4

ISO 8662 Hand machines – Handheld motor driven machines – Measurement vibrations in the hand grip

The standard describes the procedure for the measurement of vibrations in the hand grip of handheld, motor-driven machines. The purpose is to describe a laboratory method that, to the extent possible, gives the same results that would be obtained in real working conditions. ISO 8662 consists of the following parts. • Part 1: General requirements. This part contains general requirements for the measurement of vibrations in any hand-held machine. The other parts of the standard describe more specific procedures for particular kinds of machines. • Part 2: Chipping hammers and riveting hammers. • Part 3: Rock drills and rotary hammers. • Part 4: Grinding machines. • Part 5: Pavement breakers and hammers for construction work. • Part 6: Impact drills. • Part 7: Wrenches, screwdrivers, and nut runners. • Part 8: Polishers and orbital sanders. • Part 9: Rammers. • Part 10: Nibblers and shears. • Part 11: Fastener driving tools. • Part 12: Saws and filing machines. • Part 13: Die grinders. • Part 14: Stone-working tools and needle scalers. 3.3.5 Standard ISO 4866 Vibration and shock – Building vibrations Guidance for measurement of vibrations and assessment of its effects on buildings

Vibrations in a building can originate from many different sources. Examples of sources of building vibrations are earthquakes, rock blasting, wind, and machinery installation. Most of the injuries suffered by buildings are due to human activities resulting in vibrations in the 1 - 150 Hz range. Natural sources, such as earthquakes, primarily cause building damage at very low frequencies, typically 0.1 - 30 Hz. Wind excitation is commonly in the 0.1 - 2 Hz frequency range. The vibration amplitudes that such sources induce vary within a large range. The standard does not provide guideline values, but simply indicates typical amplitudes and frequencies from different types of sources; see table 3-9.

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Chapter 3: Influence of Sound and Vibration on man and equipment Table 3-9 Typical frequency ranges and vibration amplitudes of building vibrations caused by various excitation sources. (Source: ISO 4866.)

Source

Frequency range f [Hz]

Displacement ~ ξ [μm]

Velocity v~ [mm/s]

Traffic 1 - 80 1 - 200 0.2 - 50 road/rail Blasting 1 - 300 100-2500 0.2 - 500 groundborne Piling 1 - 100 10 - 50 0.2 - 50 groundborne Machines 1 - 300 10 - 1000 0.2 - 50 outdoor, groundborne Noise traffic, 10 - 250 1 - 1100 0.2 - 30 machines outdoors Machines 1 - 1000 1 - 100 0.2 - 30 Indoor 5 0.1 - 30 0.2 - 400 Earthquake 10 - 10 5 0.1 - 10 Wind 10 - 10 Noise 5 - 500 Indoor Symbol key: C = continuous process, T = transient process 3.4

Acceleration a~ [m/s2]

Character see symbol key

0.02 - 1

C/T

0.02 - 50

T

0.02 - 2

T

0.02 - 1

C/T

0.02 - 1

C

0.02 - 1

C/T

0.02 - 20

T T

REGULATIONS AND RECCOMENDATIONS

Regulations on noise and vibrations cover many areas of societal concern, but even so, such laws and regulations do not cover all relevant situations. The various authorities involved have limited areas of responsibility, and there are holes in important areas; e.g., for traffic noise, there are requirements for individual vehicles, but not for situations in which vehicles act in concert, i.e., for the areas surrounding highways The inner market of the EU came into being on January 1, 1995. The objective of the inner market is to create conditions of free passage over national borders of goods, services, capital, and people. To bring that about, the EU issues so-called directives, and the national authorities must adapt their respective laws and regulations to ensure that they are not in conflict with those. 3.4.1

Machines

The EU machine directive (89/392 EEC with amendment 91/368 EEC) is one of the more important directives, and deals with a large number of machinery types, from power saws to rock crushers. Certain machines are excepted from the directive, such as cranes, medical machines in direct contact with patients, amusement park machines, boilers, tanks, pressure vessels, fire arms, means of passenger and freight transport, police and military machines, etc. If a product is not specifically excepted as such, the manufacturer is then 104

Chapter 3: Influence of Sound and Vibration on man and equipment responsible to ensure compliance with the requirements of the directive, in order for the product to have unrestricted access to the EU’s inner market. The machinery directive has two objectives: (i)

To eliminate technical hindrances to trade.

(ii)

To incorporate provisions for health and safety in machinery operation.

The directive has many aspects. As far as noise and vibrations are concerned, there are general formulations, such as "machines are to be designed and produced to minimize noise and vibrations to the lowest possible levels, with regard to technical advances and the availability of noise and vibration countermeasures, above all at the source. " It also mandates that technical documentation contain the following information: (i)

Noise. • If the equivalent A-weighted sound level LA,eq is lower than 70 dB(A), that is to be indicated. Example: LA,eq < 70 dB(A). • If LA,eq is greater than 70 dB(A), the actual level is to be specified: Example: LA,eq = 80 dB(A). • If LA,eq is greater than 85 dB(A), then the A-weighted sound power level is to be given as well. Example: LA,eq = 96 dB(A), LW,A =104 dB(A). • If the instantaneous C-weighted level, LC, exceeds 130 dB(C), that level is to be indicated. Example: LC = 135 dB(C).

The noise levels given above refer to the operator position. The running condition of the machine, as well as the measurement method, are to be given. The simplest approach is to use an EN-standard for the specific type of machine, or, if that doesn’t exist, a standard from the International Standards Organization, ISO. (ii)

Vibrations. Hand-arm vibrations: • If the frequency-weighted rms acceleration a~w from section 3.3.3 is less than 2.5 m/s2, that is to be indicated. Example: a~w < 2.5 m/s2. • If a~w is greater than 2.5 m/s2, the actual value is to be indicated. Example: a~w = 2.8 m/s2. Whole body vibrations: • If the frequency-weighted rms acceleration a~w , based on the relevant standard, is less than 0.5 m/s2, that is to be indicated. Example: a~w < 0.5 m/s2. • If a~w is greater than 0.5 m/s2, the actual value is to be indicated. Example: a~w = 1.2 m/s2.

The manufacturer is to issue an assurance of EU-compliance, if the machine fulfills the requirements, and may then mark the machine with the CE-logo shown in Figure 3-30. For excepted machines, there are other directives that apply.

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Chapter 3: Influence of Sound and Vibration on man and equipment

Figure 3-30 CE-logo applied to a machine, which serves as an assurance of compliance with the EU Machinery directive. Guarantees free access to the EU inner market.

3.4.2

Vehicles

3.4.2.1 Motor vehicles

EU directives (70/157 EEC & 96/20 EC);proscribes the highest permissible sound levels for different categories of vehicles. Table 3-10 Highest allowed sound levels of motor vehicles, as proscribed by the EU directives 70/157 EEC, with amendment 96/20 EC.

Vehicle category

Passenger car Bus or truck with a total weight up to 3.5 tons with a total weight up to 2 tons with a total weight over 2 tons, but < 3.5 tons Bus with a total weight over 3.5 tons with a motor power 150 kW with a motor power of 150 kW or higher Truck with a total weight over 3.5 tons with a motor power less than 75 kW with a motor power betw 75 kW and 150 kW with a motor power over 150 kW Motorcycle (depending on cylinder volume) Moped

Highest allowed sound level, dB(A) EU directives 70/157 EEC & 96/20 EC 74

76 77 78 80 77 78 80

The measurements are, primarily, to be performed in accordance with an international standard, ISO 362. The standard proscribes a specific test path, and that the vehicle is to drive by a microphone at a distance of 7.5 m, as illustrated in Figure 3-31. For motorcycles and mopeds, there are special measurement instructions.

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Chapter 3: Influence of Sound and Vibration on man and equipment

7.5 m 7.5 m

1.2 m

Figure 3-31 Measurements of external vehicle (pass-by) noise are to be carried out in accordance with an international standard, ISO 362, which states that the vehicle is to drive by a microphone set-up at 7.5 m distance. From the beginning of the test path, A-A, until its end, B-B, the vehicle is to accelerate at full throttle from a certain specified speed. No hindrances to sound are permitted within a radius of 50 m. (Picture: SCANIA, Fordonsakustik och buller.)

3.4.2.2 Airplanes

For aircraft, environmental compliance certificates are issued, and guarantee that the type of airplane in question meets the requirements of the international organization ICAO, International Civil Aviation Organization. The certification is issued in connection with the so-called type-certification. The member countries of the ICAO incorporate its regulations in their respective national requirements for airplane certification. Airplanes with take-off weights exceeding 9000 kg are permitted certain maximum noise values, depending on the weight and number of motors. The three points used in certification pertain to noise data for take-off, side-line, and landing. The take-off value is established at two points. One of those is 6500 m from the “break-release”-point and directly under the flight path. The “break-release”-point is the point at which the airplane releases the brakes and begins the take-off. The other take-off value is called “sideline”, and is measured where the highest value is obtained, at a distance of 650 m for socalled chapter 2-airplanes and 450 m for chapter 3-airplanes. The landing value is measured at a point 2000 m before the touch-down point, directly beneath the flight path. The socalled chapters correspond to the sharpening of environmental standards by the ICAO. Chapter 3-airplanes, i.e., the airplanes that only meet the requirements at 650 m distance, will be prohibited from flying early in the 2000-decade. No new registrations of that type will be permitted in the EU. Noise data is measured linearly in third-octave bands, and corrected for, among other things, duration and differences in the levels in each third octave band, before summation. The result is given in units of EPNdB, Effective Perceived Noise.

107

Chapter 3: Influence of Sound and Vibration on man and equipment For airplanes with take-off weights under 9000 kg, only a maximum value during take-off is proscribed. The value is indicated as a sound level in dB(A). 3.4.3

Workplace environment

3.4.3.1 Workplaces, with certain exceptions

EU has issued the directive 86/188 EEC "On the protection of workers from risks due to exposure to noise while working ". It is a so-called minimal directive, which means that the member countries are permitted to issue stricter requirements. Sweden has taken advantage of that option in the proclamation AFS 1992:10 Noise, issued by the labor protection administration. That proclamation dictates, among other things, that if specific limits on noise exposure are exceeded, the reasons are to be investigated, and action is to be taken; see table 3-11. Table 3-11 Maximum allowed noise levels, as specified by the Swedish Labor Protection Administration proclamation AFS 1992:10 Noise.

Equivalent A-weighted sound level in a typical work day

LA,eq = 85 dB(A)

Maximum A-weighted sound level (excepting impulsive sound)

LA,max = 115 dB(A)

C-weighted peak value of impulsive sound

LC,peak = 140 dB(C)

The workplace safety law, on which the proclamation named above is based, applies to all workplaces with certain exceptions, e.g., aboard ships. 3.4.3.2 Noise aboard ships

Ships are required to comply with the UN’s International Maritime Organization IMO Resolution A: 468 (XXII) Code of Noise Levels Onboard Ships. That recommendation is applied voluntarily by the member states, and contains: • Maximum allowed sound levels in different spaces, such as the machine room, the navigation spaces, cabins, etc. • Limiting values for noise exposure. The limit for the equivalent A-weighted level, LA,eq, is 80 dB(A) for a period of 24 hours, which corresponds to 85 dB(A) over 8 hours. • Requirements on acoustic insulation between cabins.

108

Chapter 3: Influence of Sound and Vibration on man and equipment 3.4.4

Dwellings

There are no EU regulations on the noise in dwellings but there are national regulations. As an example information about Swedish regulations for noise in dwellings will be given. The Swedish health protection law regulates the application of measures to prevent or eliminate sanitary nuisances. A sanitary nuisance is any kind of disturbance that can be injurious to human health, and which is neither minor nor temporary. The assessment of the measures to be undertaken against sanitary nuisances is carried out by the municipal authority responsible for applying the health protection law. 3.4.4.1 New buildings

For newly built dwellings and premises, the Housing Authority (Boverket) building regulation BFS 1998:38 contains directives and general direction on noise from installations inside and outside of buildings, as well as on noise from road traffic. The rules promulgated proscribe, among other things, requirements on insulation against airborne sound, maximum sound levels from footfalls in stairways and rooms above, maximum sound levels from such installed equipment as elevators, etc., and maximum reverberation times. The reverberation time describes how sound reflective a room or space is, as described in detail in chapter 7. Moreover, maximum traffic-induced sound levels, indoors and outdoors, are specified. The rules for dwellings make reference to the Swedish standard SS 02 5267 Building acoustics – sound classification of spaces in buildings dwellings. The objective of the standard is to simplify the work of the building personnel to improve the acoustic quality of buildings, and to classify buildings for the sake of consumers. The requirements are divided in to four sound classes: •

Sound class A:

Very good acoustic conditions.

•

Sound class B:

Minimum requirements on good living environment; the tenants might still be disturbed, however.

•

Sound class C:

Meets the requirements of Swedish authorities.

•

Sound class D:

Used when sound class C cannot be met by renovation work; for example, a 100 year-old stone house.

Table 3-12 gives examples of the highest allowed sound levels of different disturbance sources and spaces.

109

Chapter 3: Influence of Sound and Vibration on man and equipment Table 3-12 Highest allowed equivalent and maximum sound levels from installations and traffic, indoors and outdoors, at dwellings and other types of premises (Source: BFS 1998:38.)

Room or Space

Dwellings/ Installations Kitchen Bedrooms

Dwellings/traffic Kitchen Bedrooms

Equivalent sound level (Sound of long duration) LA,eq [dB(A)] LC,eq [dB(C)] For installations, the length of the measurement period is equivalent to the duration of the disturbance; for traffic, it is one day

Highest sound level (Sound of short duration) LA [dB(A)]

LA,eq = 35 dB(A) LA,eq = 30 dB(A) Additionally, for bedrooms, LC,eq = 50 dB(C)

LA = 40 dB(A) LA = 35 dB(A)

LA,eq = 35 dB(A) LA,eq = 30 dB(A)

LA = 45 dB(A) May be exceeded up to five times per day 2200-0600.

Outside at least half of the bedrooms of an apartment

LA,eq = 54 dB(A)

In at least one outdoor space/balcony connected to the apartment Other premises/installations Places of instruction (class-rooms), sleep or rest Other premises/traffic Health care premises, day care and recreation centers, places of instruction (classrooms)

LA,eq = 54 dB(A)

Offices

LA,eq = 40 dB(A)

LA,eq = 30 dB(A)

LA = 35 dB(A)

LA,eq = 30 dB(A)

LA = 45 dB(A) May be exceeded up to five times per day 2200-0600.

110

Chapter 3: Influence of Sound and Vibration on man and equipment 3.4.4.2 Existing buildings

The Social Administration (Socialstyrelsen) published, in 1996, as part of its collected works, a document titled " Noise indoors and high noise levels, SOSFS 1996:7 (M), General guidance.” That is intended to be used as a tool to help municipalities and others in their work to alleviate different types of societal noise problems occurring indoors in dwellings, and premises for instruction and health care, meeting places, etc. The guidance is intended to suggest when a societal noise problem constitutes a sanitary nuisance in accordance with the health protection law. In some cases, it is not possible to alleviate a sound problem to the extent recommended by that guidance. According to the health protection law, that type of situation is to be handled on a case-bycases basis, with an investigation into whether it is reasonable to demand full compliance. That investigation is to take into account both economic and technical aspects. This does not apply, however, to noise from air, road, or railway traffic. Table 3-13 gives guideline values for indoor noise. Table 3-14 makes recommendations for assessing low frequency noise indoors, and, finally, table 3-15 makes recommendations for assessing sanitary nuisances at high sound levels. Table 3-13 Guideline values for assessing sanitary nuisances, with respect to indoor noise. (Source: SOSFS 1996:7.)

Maximum Equivalent

Level LA,max = 35 – 45 dB(A) LA,eq = 30 dB(A)

Comments: The higher guideline value for maximum noise (45 dB(A)) is intended as a protection against sleeping difficulties, waking, etc. To be regarded as a sanitary nuisance, it is sufficient that the guideline value is exceeded a few times, during a night for example. The lower guideline value for maximum noise (35 dB(A)) is used in assessing whether a disturbance, in certain cases, can be regarded as a risk of sanitary nuisance. It should, in that case, be a recurrent phenomenon. The guideline value for equivalent noise (30 dB(A)) refers to the time period at which the disturbing activity takes place. It seeks to protect against sleeping disturbances, speech masking, and perceived disturbance. Table 3-14 Recommendations for assessing the sanitary nuisance, for equivalent low frequency noise indoors. (Source: SOSFS 1996:7.)

Thirdoctave band f [Hz]

Equivalent Third octave bands level Leq [dB]

Comments: The recommendations can help to assess whether a perceived disturbance of equivalent low frequency noise is a risk of sanitary nuisance.

31.5 40 50 63 80 100 125 160 200

56 49 43 41.5 40 38 36 34 32

3-15 Recommendations for assessing sanitary nuisances at high sound levels. (Source: SOSFS 1996:7.) Comments: The recommendations on the boundary at which high sound levels constitute a risk of sanitary nuisance are applied to discotheques, concerts, etc. Both indoors and outdoors.

Table

111

Maximum Equivalent

Level LA,max = 115 dB(A) LA,eq = 100 dB(A)

Chapter 3: Influence of Sound and Vibration on man and equipment 3.4.5

External noise

3.4.5.1 External industry noise

For external industry noise there are no common EU regulations but as an example information about Swedish national regulations are given in this section. The Swedish Nature Conservation Authority (Naturvårdsverket) has issued, in RR 1978:5 External industry noise – general guidance, guideline values for external industry noise. These are based on equivalent sound levels in dB(A) outdoors and in free field conditions, i.e., no hindrances or buildings are to be located in the immediate vicinity of the measurement point; see table 3-16. Table 3-16 The Nature Conservation Authority guidelines on external industry noise. The figures without parentheses apply to newly established industrial sites, and with parentheses to already-existing industrial sites. (Source: RR 1978:5.)

Highest sound level in dB(A)

Equivalent sound level in dB(A) Area

Instantaneou s nighttime sound, 22-07

Daytime hours 07-18

Weekdays 1822, Sundays and holidays 07-18

Workplaces without noisy activities

60, (65)

55, (60)

50, (55)

-

Dwellings and recreational areas adjacent to dwellings, and educational and health buildings

50, (55)

45, (50)

40, (45)

55, (55)

Recreational areas and mobile recreation where exposure to nature is important

40, (45)

35, (40)

35, (40)

50, (50)

Nighttime hours 22-07

If the noise contains audible tones or impact noise, the requirements are made 5 dB(A) stricter. The rules also specify the measurement methodology, and that the noise is to be measured in the free field, at a 1.5 meter elevation, under certain permissible atmospheric conditions and wind speeds, etc.

112

Chapter 3: Influence of Sound and Vibration on man and equipment 3.5

IMPORTANT RELATIONS

THE EAR AND HEARING Measures of sound Table

3-17 A-, Band C-weighting The octave bands are indicated in bold print.

Frequency [Hz]

A-weighting [dB]

B-weighting [dB]

25 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000 10000 12500 16000 20000

-44.7 -39.4 -34.6 -30.2 -26.2 -22.5 -19.1 -16.1 -13.4 -10.9 -8.6 -6.6 -4.8 -3.2 -1.9 -0.8 0 +0.6 +1.0 +1.2 +1.3 +1.2 +1.0 +.5 -0.1 -1.1 -2.5 -4.3 -6.6 -9.3

-20.4 -17.1 -14.2 -11.6 -9.3 -7.4 -5.6 -4.2 -3.0 -2.0 -1.3 -0.8 -0.5 -0.3 -0.1 0 0 0 0 -0.1 -0.2 -0.4 -0.7 -1.2 -1.9 -2.9 -4.3 -6.1 -8.4 -11.1 N

Sound level

L A = 10⋅ log ∑ 10

for

third-octave

and

octave

bands.

Cweighting [dB] -4.4 -3.0 -2.0 -1.3 -0.8 -0.5 -0.3 -0.2 -0.1 0 0 0 0 0 0 0 0 0 -0.1 -0.2 -0.3 -0.5 -0.8 -1.3 -2.0 -3.0 -4.4 -6.2 -8.5 -11.2

( L pn + ΔAn ) / 10

[dB(A)]

(3-1)

n =1

Equivalent sound pressure level Leq,T = 10⋅ log(

1 T

T

p 2 (t )

0

2 p ref

∫

dt ) [dB]

(3-2)

T

Leq,T = 10 ⋅log(

1 L (t ) / 10 10 p dt ) [dB] T

∫ 0

113

(3-3)

Chapter 3: Influence of Sound and Vibration on man and equipment

114

CHAPTER FOUR

MATHEMATICAL METHODS AND VIBRATIONS OF SIMPLE MECHANICAL SYSTEMS

This chapter introduces mathematical methods commonly applied to the analysis of acoustics and vibrations problems. Firstly, it covers rotating vectors, or phasors, and the description of physical quantities using complex numbers; these topics are then applied to the determination of mechanical power and the study of linear oscillations. 4.1

COMPLEX NUMBERS AND ROTATING VECTORS

Using Fourier analysis, complicated, time-dependent events that exactly repeat, or are periodic, can be decomposed into their harmonic components. Doing so amounts to approximating the time-dependent function describing the event by the sum of a set of sine and cosine functions. In more involved cases, the resulting mathematical expressions may be very complicated. As a mathematical tool to simplify the computations in such cases, one may make use of rotating vectors or phasors; see Figure 4-1. If the projection on the x-axis is called the real part and the projection on the y-axis the imaginary part, we can describe the function with the aid of a complex, rotating vector according to e iωt = cos(ωt ) + i sin(ωt ) .

(4-1)

That relation is usually called Euler’s identity. A summary of formulas pertaining to complex numbers is given in Appendix A. 115

Chapter 4: Mathematical methods A time-dependent function v(t) = A cosωt + B sinωt can, with the help of (4-1), be written in the form v(t) = ((A - iB) eiωt + (A + iB) e-iωt ) / 2. Thus, a complete description of the event can be obtained by using two rotating vectors with equal, but oppositely directed, rotational velocities: +ωt and -ωt. Because the factor preceding e-iωt is the complex conjugate of the factor preceding eiωt, it suffices to use only one of them in calculations, and results for the other case can always be found by complex conjugation. Thus, in computations, one uses a complex-valued function v(t) = ((A - iB) eiωt = A eiωt, in which complex quantities are indicated by bold print. The actual time function can always be recovered by taking the real part of the final result, e.g., F(t) = Re(F(t)). y sin ωt

ωt

ωt

x Figure 4-1 Rotating vectors are used as aids in the description of harmonic functions. The projection on the x-axis, of a component with a circular frequency of ω, gives the function cos(ωt), and the projection on the y-axis is sin(ωt).

cos ωt

ωt

Example 4-1 A harmonically-varying force has a time-dependence of F(t) = Fˆ cosωt. The complex force can therefore be written in the form F(t) = Fˆ eiωt = Fˆ (cosωt + i sinωt), where F(t) = Re( Fˆ iωt). Example 4-2 A harmonically-varying force has a time dependence of F(t) = Fˆ1 cosωt - Fˆ2 sin ωt. The complex force can therefore be written in the form F (t ) = ( Fˆ1 + iFˆ2 )e iωt = ( Fˆ1 + iFˆ2 )(cos(ωt ) + i sin(ωt )) = = ( Fˆ1 cos(ωt ) − Fˆ2 sin(ωt )) + i ( Fˆ1 sin(ωt ) + Fˆ2 cos(ωt )),

where F(t) = Re( Fˆ eiωt).

116

Chapter 4: Mathematical methods Example 4-3 As an alternative to the treatment given in example 4-2, the force can be written as F (t ) = Fˆ1 cos(ωt ) − Fˆ2 sin(ωt ) = Fˆ cos(ωt + ϕ ) ,

where

Fˆ = Fˆ12 + Fˆ22 and ϕ = arctan( Fˆ2 Fˆ1 ) . The complex force can therefore be written as F (t ) = Fˆ e iωt +iϕ = Fˆ e iωt ,

where Fˆ is a complex amplitude Fˆ = Fˆe iϕ and F (t ) = Re(F (t ) ) . Example 4-4 A harmonic vibration with a peak velocity amplitude of vˆ , circular frequency ω and phase angle ϕv at time t = 0 can be written, using the complex vector approach, as v (t ) = vˆe i (ωt +ϕ v ) .

Determine the amplitude and phase of the velocity, acceleration, and displacement, and sketch these in the complex plane. Solution Velocity: Amplitude: Phase t = 0: t = t:

vˆ . ϕv . ωt + ϕv.

Acceleration: d d a(t ) = v(t ) = vˆe i (ωt +ϕ v ) = dt dt = iω vˆe i (ωt +ϕ v ) = iω v =

⎧ = ⎨e iπ ⎩ i

2

= cos

π

π 2

+ i sin

Im

π

= e 2 ωvˆe i (ωt +ϕ v ) = ωvˆe

⎫ = i⎬ = 2 ⎭

ωv

π

i (ωt +ϕ v + ) 2 .

a(t) = ωvˆei(ωt +ϕv +π 2) v

Transition from velocity to acceleration therefore implies multiplication by iω. Amplitude: Phase t = t:

ωvˆ . ωt + ϕv + π/2,

ω t + ϕv +π/2 ωv −ωv

117

v(t) = vˆei(ω t +ϕv ) v

ω t + ϕv v

Re

Chapter 4: Mathematical methods (Phase angle π/2). Displacement:

Im

x(t ) = ∫ v(t )dt = ∫ vˆe i (ωt +ϕ v ) dt = =

1 iω

vˆe i (ωt +ϕ v ) =

{

= − i = −e = e − iπ / 2

iπ / 2

v iω

=e

= −i

− iπ / 2

1

ω

v

vˆe i (ωt +ϕ v ) =

}=

 i (ω t + ϕ v ) v( t ) = ve

v

vˆ i (ωt +ϕ v ) vˆ i (ωt +ϕ v −π / 2) e = e .

ω

ω t + ϕv

ω

v

Transition from velocity to displacement implies, therefore, division by iω. Amplitude: vˆ ω . Phase t = t: ωt + ϕv - π/2, (Phase shift -π/2).

Example 4-5 The complex conjugate (see appendix A) of the force can be used for calculating both the force amplitude (modulus) and its real part. The complex conjugate of F is indicated by F*. A harmonically-varying force, and its complex conjugate, are shown in the adjacent figure.

ω t + ϕ v - π/2

v ω x( t ) =

− v ω

Re v

ω

e i (ω t + ϕ v − π

Im

F = F e iω t ωt -ω t

Re

F * = F e − iω t

The modulus of a force is given by

F = FF * = Fˆe iωt Fˆe −iωt = Fˆ 2 = Fˆ . The real part of the force can be calculated from Re(F) =

2)

1 1 (F + F * ) = ( Fˆ cos(ωt ) + iFˆ sin(ωt ) + Fˆ cos(ωt ) − iFˆ sin(ωt )) = 2 2 1 = (2 Fˆ cos(ωt )) = Fˆ cos(ωt ). 2

118

Chapter 4: Mathematical methods 4.2

MECHANICAL POWER

A harmonically-varying force and velocity have respective time dependencies as follow in (4-2) and (4-3): F (t ) = Fˆ1 cos(ωt ) + Fˆ2 sin(ωt ) = Fˆ cos(ωt + ϕ F ) = Re( Fˆe i (ωt +ϕ F ) ) ;

(4-2)

v(t ) = vˆ1 cos(ωt ) + vˆ 2 sin(ωt ) = vˆ cos(ωt + ϕ v ) = Re(vˆe i (ωt +ϕ v ) ) .

(4-3)

From fundamental mechanics, the instantaneous mechanical power can be calculated from W (t ) = F (t )v(t ) = Re(F (t )) Re( v (t )) = =

1 1 (F(t ) + F(t ) * ) ( v (t ) + v(t ) * ) = 2 2

= (F (t ) v (t ) + F(t ) * v(t ) * + F(t ) v(t ) * + F(t ) * v(t )) / 4,

(4-4)

which can be expressed in the form W (t ) = (Re(F(t ) v (t )) + Re(F(t ) v(t ) * )) / 2 = = Re( Fˆvˆe i 2ωt +iϕ F +iϕ v ) / 2 + Re( Fˆvˆe iϕ F −iϕ v ) / 2 = = Fˆvˆ(cos(2ωt + ϕ F + ϕ v ) + cos(ϕ F − ϕ v )) / 2 .

(4-5)

Typically, it is the time average of the power that is of interest, and which corresponds to the power that is fed into a mechanical system. The time average of the first term, above, is zero, so that the time-averaged power can be expressed as T

W =

1 1 W (t )dt = Fˆvˆ cos(ϕ F − ϕ v ) , T 2

∫

(4-6)

0

where the overbar indicates time-averaging. Thus, W=

1 1 Re(Fv * ) = Re(F * v) . 2 2

(4-7)

From (4-6), it is evident that maximal power is delivered when force and velocity are in phase, i.e., ϕF = ϕv , whereas no power, at all, is delivered when the phase shift between those quantities is 90º. In that latter case, one can speak of the force being “90º ahead” of the velocity, ϕF - ϕv = 90º, or “90º behind” the velocity, ϕv - ϕF = 90º.

119

Chapter 4: Mathematical methods 4.3

LINEAR SYSTEMS

Often in Vibrations and Acoustics, as well as in other fields for that matter, our interest resides in the calculation of what effect a certain physical quantity, called the input signal, has on another physical quantity, called the output signal; see Figure 4-2. An example is that of calculating what vibration velocity v(t) is obtained in a structure when it is excited by a given force F(t). That problem can often be solved by making use of the theory of linear time- invariant systems. Figure 4-2 A linear time-invariant system describes the relationship between an input signal and an output signal. For example, the input signal could be a velocity v(t), and the output signal a force F(t), or the input signal an acoustic pressure p(t) and the output signal an acoustic particle velocity u’(t).

F(t),v(t) p(t),u(t) Input Signal

Linear time-invariant system

F’(t),v’(t) p’(t),u’(t) Output Signal

From a purely mathematical standpoint, a linear system is defined as one in which the relationship between the input and output signals can be described by a linear differential equation. If the coefficients are, moreover, independent of time, i.e., constant, then the system is also time invariant. A linear system has several important features. The superposition principle implies that if the input signal a(t) gives rise to an output signal b(t), and the input signal c(t) gives rise to an output signal d(t), then the input signal a(t)+c(t) yields the output signal b(t)+d(t). The homogeneity principle states that if the input signal a(t) is multiplied by a constant α, then the output signal is α b(t). A linear system is also frequency-conserving, in the sense that only those frequency components that exist in the input signal can exist in the output signal. Example 4-6 The figure below, from the introduction, shows an example in which the forces that excite an automobile are inputs to a number of linear systems, the outputs from which are vibration velocities at various points in the structure. The vibration velocities are then, in turn, inputs to a number of linear systems, the outputs from which are sound pressures at various points in the passenger compartment. By adding up the contributions from all of the significant excitation forces, the total sound pressures at points of interest in the passenger compartment can be found. In this example, the linear system is described in the frequency domain by so-called frequency response functions, to be described in detail in section 4.3.4. The engine is fixed to the chassis via vibration isolators. If the force F1 that influences the chassis can be cut in half, then, for a linear system, all vibration velocities v1 – vN caused by the force F1 are also halved. In turn, the sound pressures p1 – pN, which are brought about by the velocities v1 – vN, are halved as well. If we make the simplifying assumption that all of the forces acting on the car are uncorrelated, and that those forces are the only sound sources acting, it implies that if all of them can be reduced by 5 dB, then both the linear sound pressure level Lp and the A-weighted sound pressure level LA, are reduced by 5 dB

120

Chapter 4: Mathematical methods

Zji

Yik

Body Forces

Vibration velocities

Passenger compartment

Sound Pressure (Picture: Volvo Technology Report, nr 1 1988.)

In this chapter, linear oscillations in mechanical systems are considered, i.e., oscillations in systems for which there is a linear relation between an exciting force and the resulting motion, as described by displacements, velocities, and accelerations. Linearity is normally applicable whenever the kinematic quantities can be regarded as small variations about an average value, implying that the relation between the input signal and the output signal can be described by linear differential equations with constant coefficients. 4.3.1

Single Degree of Freedom Systems

In basic mechanics, one studies single degree-of-freedom systems thoroughly. One might wonder why so much attention should be given to such a simple problem. The single degree-of-freedom system is so interesting to study because it gives us information on how a system’s characteristics are influenced by different quantities. Moreover, one can model more complex systems, provided that they have isolated resonances, as sums of simple single degree-of-freedom systems. Figure 4-3 shows a mechanical single degree-of-freedom (“sdof”) system consisting of a rigid mass m, a spring with spring rate κ, and a viscous damper with a damping coefficient dν. The spring and the viscous damper are located between the mass and the foundation, and are considered to be massless. That implies that the forces on the opposing endpoints of each are equal and oppositely directed, for both elements.

121

Chapter 4: Mathematical methods

F(t) x(t) m

κ

dv

Figure 4-3 Single Degree-of-Freedom System.

Newton’s Second Law gives the equation of motion of the system, in the form m

d 2 x(t ) dt

2

= Fx ( x(t ),

dx(t ) , t) . dt

(4-8)

Fx contains the spring force, the damper force, and the external exciting force Fx = −κx(t ) − d v

dx (t ) + F (t ) , dt

(4-9)

where m is mass of the body, κ is the spring constant, dv is the viscous damping coefficient, F(t) is the external excitation, x is the displacement of the mass, dx / dt its velocity, d 2x / dt2 its acceleration. These two equations lead to a second order linear differential equation with constant coefficients, d 2 x (t ) dt

2

+ 2δ

dx(t ) + ω 02 x(t ) = g (t ) , dt

(4-10)

in which the following simplifications have been incorporated:

ω0 = κ m ,

δ = d v 2m ,

g (t ) = F (t ) m .

(4-11)

where ω0 is the eigenfrequency of the system, and δ is the damping constant. The solution to the differential equation consists of both a homogeneous part xh(t) that corresponds to the homogeneous differential equation, i.e., with the right hand side equal to zero, and a particular solution xp(t) that corresponds to the non-homogeneous differential equation, i.e., with the right hand side non-zero. If there is damping in the system, then the free vibrations can be assumed to have been damped out after a number of periods, and only the forced vibrations remain. Thus, it is usually only the particular solution that is of interest, i.e., the second term in (4-12) x (t ) = x h (t ) + x p (t ) . Because the system is linear, its particular solution, when the exciting force is described by the rotating vector (4-13), represents an oscillation at the excitation frequency, but with a different phase and amplitude. A reasonable assumption for xp is given by (4-14),

122

Chapter 4: Mathematical methods g (t ) = gˆe iωt ,

x p (t ) = xˆ p e iϕ e iωt = xˆ p e iωt .

(4-13,14)

That assumed form, substituted into (4-10), provides the following result:

− ω 2 xˆ p e iωt + i 2ω δ xˆ p e iωt + ω 02 xˆ p e iωt = gˆe iωt .

(4-15)

The phase and magnitude of the complex amplitude xˆ p can now be determined to be gˆ

xˆ p =

(ω 02

− ω 2 ) + i 2δω

,

(4-16)

xˆ p = xˆ p e iϕ , xˆ p =

ϕ = arctan

(4-17)

gˆ (ω 02

− ω ) + (2δω )

2δω

ω − ω 02 2

2 2

,

(4-18)

2

+ nπ , n = 0, 1, 2, … .

(4-19)

From (4-11) and (4-16), it is apparent that forω « ω0, the stiffness κ determines the displacement. Thus, the low frequency response is stiffness-controlled. On the other hand, for ω » ω0, the mass m determines the displacement response; the high frequency response, therefore, is mass-controlled. Finally, for ω ≈ ω0, the value of the viscous damping coefficient dν is decisive for the displacement; the response at frequencies around the natural frequency is therefore said to be damping-controlled. The magnitude of the amplitude xˆ p varies with circular frequency ω. A normalized response, called the amplification factor φ, can be defined as

φ=

4.3.2

xˆ p (ω ) xˆ p (ω = 0)

⇒

φ=

1 (1 − (ω ω 0 ) 2 ) 2 + 4(δ ω 0 ) 2 (ω ω 0 ) 2

.

(4-20)

Two degree-of-freedom systems

The simple single degree-of-freedom system can be coupled to another of its kind, producing a mechanical system described by two coupled differential equations; to each mass, there is a corresponding equation of motion (see Figure 4-4). To specify the state of the system at any instant, we need to know time t dependence of both coordinates, x1 and x2, from which follows the designation two degree-of-freedom system.

123

Chapter 4: Mathematical methods

κ1

F1(t)

κ2

F2(t)

κ3

m2

m1 Figure 4-4 Two degree-offreedom system.

dv3

dv2

dv1

x1(t)

x2(t)

Newton’s second law for each mass gives m1

m2

d 2 x1 (t ) dt 2 d 2 x 2 (t ) dt

2

dx (t ) dx (t ) ⎞ ⎛ = F1x ⎜ x1 (t ), x 2 (t ), 1 , 2 , t ⎟ , dt dt ⎝ ⎠

(4-21)

dx (t ) dx (t ) ⎞ ⎛ = F2 x ⎜ x1 (t ), x 2 (t ), 1 , 2 , t ⎟ , dt dt ⎝ ⎠

(4-22)

F1x = −κ 1 x1 (t ) − κ 2 ( x1 (t ) − x 2 (t )) − dν 1

dx1 (t ) ⎛ dx (t ) dx (t ) ⎞ − dν 2 ⎜ 1 − 2 ⎟ + F1 (t ) , (4-23) dt dt ⎠ ⎝ dt

dx (t ) ⎛ dx (t ) dx (t ) ⎞ F2 x = κ 2 ( x1 (t ) − x 2 (t )) − κ 3 x 2 (t ) + dν 2 ⎜ 1 − 2 ⎟ − dν 3 2 + F2 (t ) . (4-24) dt dt ⎠ ⎝ dt

(4-21) - (4-24) give m1

d 2 x1 (t ) dt

2

+ dν 1

dx1 (t ) ⎛ dx (t ) dx (t ) ⎞ + dν 2 ⎜⎜ 1 − 2 ⎟⎟ + dt ⎠ dt ⎝ dt

+ κ 1 x1 (t ) + κ 2 ( x1 (t ) − x 2 (t ) ) = F1 (t ) , m2

d 2 x 2 (t ) dt

2

(4-25)

dx (t ) ⎛ dx (t ) dx (t ) ⎞ − dν 2 ⎜⎜ 1 − 2 ⎟⎟ + dν 3 2 − dt ⎠ dt ⎝ dt

−κ 2 ( x1 (t ) − x 2 (t )) + κ 3 x 2 (t ) = F2 (t ) .

(4-26)

Matrix and vector notation can be incorporated into (4-25) and (4-26), which is useful for generalizing to an arbitrary number of degrees-of-freedom. The matrix formulation even makes it possible to solve the system of differential equations using software that performs matrix computations. (4-25) and (4-26) are therefore expressed as G G G d 2x dx (4-27) [M ]⋅ 2 + [D]⋅ + [K ]⋅ xG = F dt dt where

124

Chapter 4: Mathematical methods

[M ] = ⎡⎢

m1

⎣0

0 ⎤ ⎡d + dν 2 , [D] = ⎢ ν 1 ⎥ m2 ⎦ ⎣ − dν 2

− dν 2

⎤ ⎥, dν 2 + dν 3 ⎦

−κ2 ⎤ κ1 + κ 2 , κ 2 + κ 3 ⎥⎦ ⎣ −κ2

[K ] = ⎡⎢

(4-28,29)

(4-30)

G ⎧ x (t ) ⎫ ⎧ F (t ) ⎫ G x (t ) = ⎨ 1 ⎬ and F (t ) = ⎨ 1 ⎬ . ⎩ x 2 (t )⎭ ⎩ F2 (t )⎭

(4-31,32)

Once again, let the excitation forces and the particular solutions be expressed by rotating vectors F1 (t ) = Fˆ1e iωt , F2 (t ) = Fˆ 2 e iωt , (4-33,34) x 1p (t ) = xˆ 1p e iωt , x 2p (t ) = xˆ 2p e iωt .

(4-35,36)

Putting (4-33,34,35,36) into (4-27) gives

{ }

{ } { }

{ }

(4-37) − ω 2 [M ]⋅ xˆ p + iω [D]⋅ xˆ p + [K ]⋅ xˆ p = Fˆ . G Solving to the homogeneous equations with the force vector F set equal to zero leads to the system’s eigenfrequencies. Setting, moreover, the damping matrix equal to zero, in order to obtain the undamped eigenfrequencies, the latter are found to be real. Damping, on the other hand, brings about complex-valued eigenfrequencies; the complex values contain information on both the undamped eigenfrequencies and the system damping. The eigenfrequencies ω1 and ω2 are given by the homogeneous equation − ω 2 [M ]⋅ {xˆ } + [K ]⋅ {xˆ } = {0} . (4-38) The condition for the existence of solutions to (4-38) is that the system determinant is identically zero, i.e.,

det(−ω 2 [M] + [K ] ) = 0

.

(4-39)

For a two degree-of-freedom system, (4-39) has two solutions corresponding to two eigenfrequencies. A system with n degrees-of-freedom has n eigenfrequencies. The eigenfrequencies of the two degree-of-freedom system are

ω1,2 =

κ1 + κ 2 2m1

κ +κ3 + 2 ± 2m 2

(κ 1 + κ 2 )2 (κ 2 + κ 3 )2 4m12

+

4m 22

κ 2 − κ 1κ 2 − κ 1κ 3 − κ 2κ 3 . + 2 2m1 m 2 (4-40)

From linear algebra, it is known that there is an eigenvector corresponding to each eigenvalue (eigenfrequency). These eigenvectors are mutually independent (orthogonal), and contain information on how the system oscillates in the vicinity of their respective

125

Chapter 4: Mathematical methods

eigenfrequencies. Specifically, they describe the system’s mode shapes, as noted in section 1.7. The mode shapes, x1 and x2, are obtained by substituting the eigenfrequencies, i.e., the solutions of (4-39), into (4-38), yielding − ω12 [M ]⋅ {xˆ 1 } + [K ]⋅ {xˆ 1 } = {0} ,

(4-41)

− ω 22 [M ]⋅ {xˆ 2 } + [K ]⋅ {xˆ 2 } = {0} .

(4-42)

Example 4-7 A method that provides a respectable amount of isolation of a vibrating machine (see chapter 9) is to use a so-called double-elastic mounting; see Figure 4-5.

Fstör m2 x2

κ2

Mass

dv2 m1

κ1

dv1

x1

Base Figure 4-5 A double-layered elastic mounting can provide considerable isolation of a vibrating machine.

Suppose that the vibrating machine is represented by a point mass m2, and that its vibrations are generated by a harmonic excitation force Fexc with a circular frequency ω . To reduce the resulting vibrations in the foundation, the machine is to be isolated by incorporating a spring – mass – spring system between the machine and the foundation, as illustrated in Figure 4-5. The parameters of the model are as follows: m1 = 100 kg, m2 = 500 kg, κ1 = 5⋅106 N/m, κ2 = 1⋅106 N/m, dv1 = 100 kg/s and dv2 = 200 kg/s. a)

Determine, for the isolated system, the (i) undamped eigenfrequencies. Which frequencies, i.e., vibration frequencies generated by the machine, is the mounted machine sensitive to? (ii) mode shapes.

b) In order to quantify the effect of vibration isolation on the foundation, it is common to compare the force on the foundation with and without the isolation system present.

126

Chapter 4: Mathematical methods

Determine (i) (ii) (iii)

the force Fwithout on the foundation without the isolation system in place; the force Fwith on the foundation with the isolation system in place; the ratio γ = Fwithout / Fwith. Sketch, in a frequency diagram, that ratio calculated using three different choices of the spring constant κ2: 1⋅106 N/m; 5⋅106 N/m; and, 1⋅107 N/m. In which frequency band is the vibration isolation effective?

Solution a) In the first task, the undamped ([D] = 0) vibration isolation system’s eigenfrequencies, and corresponding mode shapes, are determined. The undamped system’s eigenfrequencies can be calculated with the aid of (i) formula (4-40), with the third spring constant κ3 set equal to 0. Thus, ⎧ 60343 ⎧245.6 rad/s ≈⎨ ⎩ 1657 ⎩ 40.7

ω1, 2 = " ≈ ⎨

⎧39.1 Hz. f1, 2 ≈ ⎨ ⎩6.48

i.e.,

The system’s undamped mode shapes are the solutions to the homogeneous (ii) system, with the circular frequency ω set to each of the eigenfrequencies in turn. The undamped ([D] = 0) homogeneous system of equations of motion is, with the assumed harmonic solution forms, exactly as given in (4-38). That, with values entered, becomes ⎧⎪ 6 ⋅10 6 xˆ 1 − 1 ⋅10 6 xˆ 2 − 100ω n2 xˆ 1 = 0 , n = 1, 2, ⎨ ⎪⎩− 1 ⋅10 6 xˆ 1 + 1 ⋅10 6 xˆ 2 − 500ω n2 xˆ 2 = 0

in which x1 and x2 (see Figure 4-5) indicate the coordinates of both masses. By multiplying the second of these equations by the factor −

1 ⋅10 6 − 100ω n2 6 ⋅10 6

,

and substituting in one of the two numerical values for the eigenfrequency, it becomes identical to the first equation. That implies that both equations are linearly dependent, in complete agreement with the theory. A linearly dependent system, with two unknowns, has an infinite number of solutions along a straight line in the x1- x2- plane. In order to solve the system, the equation of that line must be determined. Set the amplitude of x1 to α in the second equation of the set, and solve for the amplitude of x2; that is found to be xˆ 2 =

1 ⋅10 6 1 ⋅10 6 − 500ω n2

⋅α .

If the amplitude of x1 has the value α, then that of x2 must have the value given by the formula above. Then, the eigenvector corresponding to eigenfrequency ωn has the form

127

Chapter 4: Mathematical methods 1 ⎫ ⎧ 6 ⎪ ⎪ 1 10 ⋅ ψ n =α ⋅⎨ ⎬, ⎪1 ⋅10 6 − 500ω 2 ⎪ n ⎭ ⎩ G

where α is an arbitrary constant. Thus, the eigenvector is a vector with a specified direction, but arbitrary length. The physical interpretation of the eigenvector’s direction is the ratio between the amplitudes of motion of the two masses in a resonant oscillation. Putting in the eigenfrequencies as described, with α set to 1 for the sake of convenience, provides an eigenvector or mode form for each,

ωn = ω1 :

ωn = ω2 :

⎧

1 ⎫ ⎬, ⎩− 0.034⎭

G

ψ1 = ⎨

G

⎧1⎫ ⎬. ⎩5.8⎭

ψ2 = ⎨

The interpretation of the first eigenvector, for example, is that if the system is excited by an excitation frequency near the first eigenfrequency, the system vibrates resonantly with the amplitude of the second mass 0.034 times that of the first. The minus sign, moreover, indicates that the masses move in opposite phase, i.e., in mutually opposing directions. b) In the second part of the problem, the influence of the vibration isolation system on the foundation is studied; in other words, the force on the foundation, with and without isolation in place, is to be compared. (i)

Without isolators, the entire excitation force is transmitted to the foundation, i.e.,

Fˆwithout = Fˆexc . (ii) With the vibration isolation system, the force on the foundation can be determined from the spring-damper relation for the spring-damper element between mass 1 and the foundation, i.e.,

Fˆ with e iω t = κ 1 xˆ 1e iω t + iωd v1 xˆ 1e iω t = {subst.} = (5 ⋅ 10 6 + iω100)xˆ 1e iω t , where the displacement amplitude of mass 1, in the particular excitation case in which the machine excitation is Fexc, should be used. That displacement amplitude (of mass 1) can, in turn, be calculated by solving the non-homogeneous system of equations (4-37) with the relevant values substituted in, and with the common factor eiω t cancelled out; thus,

⎧⎪− 100ω 2 xˆ p1 + iω 300xˆ p1 + 6 ⋅ 10 6 xˆ p1 − iω 200xˆ p 2 − 1 ⋅ 10 6 xˆ p 2 = Fˆexc ⎨ 6 2 6 ⎪⎩ − iω 200xˆ p1 − 1 ⋅ 10 xˆ p1 − 500ω xˆ p 2 + iω 200xˆ p 2 + 1 ⋅ 10 xˆ p 2 = 0 From the second of these two equations, the displacement amplitude of mass 1 is solved in terms of that for mass 2. Putting that into the first equation, the amplitude of mass 2 can then be expressed in terms of the excitation force amplitude, as

128

Chapter 4: Mathematical methods

xˆ p 2 =

1 ⋅ 10 6 + i 200ω ⋅ Fˆexc . (1 ⋅ 10 6 + i 200ω − 500ω 2 )(6 ⋅ 10 6 + i300ω − 100ω 2 ) − (1 ⋅ 10 6 + i 200ω ) 2

The force on the foundation, with the isolation system in place, is therefore

Fˆ with =

(5 ⋅ 10 6 + i100ω )(1 ⋅ 10 6 + i 200ω ) ⋅ Fˆexc , (1 ⋅ 10 6 + i 200ω − 500ω 2 )(6 ⋅ 10 6 + i300ω − 100ω 2 ) − (1 ⋅ 10 6 + i 200ω ) 2

which is apparently a function of the frequency of the exciting force. The ratio γ between the force on the foundation without and with the isolation, (iii) using results from (i) and (ii) above, is γ (ω ) =

Fˆ without (1 ⋅ 10 6 + i 200ω − 500ω 2 )(6 ⋅ 10 6 + i300ω − 100ω 2 ) − (1 ⋅ 10 6 + i 200ω ) 2 . = (5 ⋅ 10 6 + i100ω )(1 ⋅ 10 6 + i 200ω ) Fˆ with

When the magnitude of the ratio is larger than one, the isolator is effective, i.e., it isolates the foundation from the vibrations of the machine. If the ratio is less that one, the incorporation of the isolator is actually counterproductive, i.e., the force on the foundation is larger than it was without the isolator in place. When a vibration isolator is to be designed, it is a matter of choosing an isolator with the kind of properties that cause the ratio to be greater than 1 for all strong frequencies in the spectrum of the excitation force. In the figure, below, the ratio is plotted for three different Fˆutan Fˆmed values of the spring 4 10 constant κ2. From the figure, it is 3 κ 2 = 5 ⋅ 106 N/m 10 evident that the location of the 2 κ 2 = 1 ⋅ 106 N/m eigenfrequencies is 10 important to the isolation behavior. 1 10 At the eigenκ 2 = 10 ⋅ 106 N/m frequencies, the 0 isolation has a deep 10 minimum, at which -1 it is negative-valued, 10 i.e., the force on the Område där the vibrationsisoleringen foundation has Area where vibration isolation -2 ökar kraften mot underlaget 10 increases the forceon the foundation decreased after incorporation of the -3 vibration isolation 10 0 10 20 30 40 50 60 70 80 90 100 system. Frekvens [Hz] This example shows that even for a two degree-of-freedom system, solution by hand calculation is a challenge. A numerical method of computation would often be a suitable approach.

129

Formatted: Font

Chapter 4: Mathematical methods

4.3.3

System with an arbitrary number of degrees-of-freedom

The results from the two degree-of-freedom system can be generalized to a system with an arbitrary number of masses cascaded, i.e., coupled in series, as in Figure 4-6.

F1(t)

κ1

κ2

Fn(t)

F2(t) • • •

m2

m1

mn

dv2

dv1

x1(t)

κn+1

x2(t)

dvn+1 xn(t)

Figure 4-6 System with n cascaded masses.

The equations of motion become m1

d 2 x1 (t ) dt

2

+ dν 1

dx1 (t ) ⎛ dx (t ) dx (t ) ⎞ + dν 2 ⎜⎜ 1 − 2 ⎟⎟ + dt dt ⎠ ⎝ dt

+κ 1 x1 (t ) + κ 2 ( x1 (t ) − x 2 (t ) ) = F1 (t ) ,

m2

d 2 x 2 (t ) dt

2

⎛ dx (t ) dx (t ) ⎞ ⎛ dx (t ) dx (t ) ⎞ − dν 2 ⎜⎜ 1 − 2 ⎟⎟ + dν 3 ⎜⎜ 2 − 3 ⎟⎟ − dt ⎠ dt ⎠ ⎝ dt ⎝ dt

−κ 2 ( x1 (t ) − x 2 (t )) + κ 3 ( x 2 (t ) − x3 (t )) = F2 (t ),

m n −1

d 2 x n −1 (t ) dt

2

(4-44)

(t ) dx (t ) ⎞ ⎛ dx (t ) dx (t ) ⎞ ⎛ dx − dνn −1 ⎜⎜ n − 2 − n −1 ⎟⎟ + dνn ⎜⎜ n −1 − n ⎟⎟ − dt dt ⎠ dt ⎠ ⎝ dt ⎝

−κ n −1 ( x n −2 (t ) − x n −1 (t )) + κ n ( x n −1 (t ) − x n (t )) = Fn −1 (t ),

mn

(4-43)

d 2 x n (t ) dt

2

+ dνn +1

(4-45)

dx n (t ) ⎛ dx (t ) dx (t ) ⎞ + dνn ⎜⎜ n − n −1 ⎟⎟ + dt dt ⎠ ⎝ dt

+ κ n +1 x n (t ) + κ n ( x n (t ) − x n−1 (t ) ) = Fn (t ).

130

(4-46)

Chapter 4: Mathematical methods

The mass matrix, damping matrix, and stiffness matrix, respectively, become ⎡m1 ⎢0 [M ] = ⎢⎢ # ⎢ ⎣0 ⎡dν 1 + dν 2 ⎢ −d ν2 ⎢ ⎢ 0 [D] = ⎢ ⋅ ⎢ ⎢ ⋅ ⎢ ⋅ ⎣⎢ ⎡κ 1 + κ 2 ⎢ −κ 2 ⎢ ⎢ 0 [K ] = ⎢ ⎢ ⋅ ⎢ ⋅ ⎢ ⎢⎣ ⋅

0 ⎤ # ⎥⎥ , % 0 ⎥ ⎥ 0 mn ⎦

0

"

m2

0

0

"

(4-47)

− dν 2

0

⋅

⋅

dν 2 + dν 3 − dν 3 ⋅

− dν 3

0

⋅

• •

• •

⋅ •

⋅

0

− dνn −1

dνn −1 + dνn

⋅

⋅

0

− dνn

−κ2

0

⋅

⋅

κ2 +κ3 −κ3

−κ3

0

⋅

⋅

• •

• •

⋅ •

⋅

0

⋅

⋅

⋅

⋅

⎤ ⎥ ⎥ ⎥ ⋅ ⎥, ⋅ ⎥ −κn ⎥ ⎥ κ n + κ n+1 ⎥⎦ ⋅

− κ n −1 κ n −1 + κ n −κn

0

⎤ ⎥ ⋅ ⎥ ⎥ ⋅ ⎥ ,(4-48) ⋅ ⎥ − dνn ⎥ ⎥ dνn + dνn+1 ⎦⎥

(4-49)

where non-zero elements not shown in the equations are marked with a •, and zero-valued elements are marked with a ⋅. One can even allow masses to be coupled in parallel, as in Figure 4-7.

κ2 κ1

F1(t)

F4(t)

m2 d v2

m1 d v1

κ4

F2(t)

κ3

F3(t)

x2(t)

κ5

d v4

κ6 m4 d v6

m3 x1(t)

x4(t) d v5

d v3

x3(t)

Figure 4-7 System with parallel coupling.

131

Chapter 4: Mathematical methods

The equations of motion become d 2 x1 (t )

m1

dt

2

+ dν 1

dx1 (t ) ⎛ dx (t ) dx (t ) ⎞ + dν 2 ⎜⎜ 1 − 2 ⎟⎟ + dt dt ⎠ ⎝ dt

⎛ dx (t ) dx (t ) ⎞ + dν 3 ⎜⎜ 1 − 3 ⎟⎟ + κ 1 x1 (t ) + κ 2 ( x1 (t ) − x 2 (t ) ) + dt ⎠ ⎝ dt + κ 3 ( x1 (t ) − x 3 (t ) ) = F1 (t ),

m2

d 2 x 2 (t ) 2

dt

(4-50)

⎛ dx (t ) dx (t ) ⎞ ⎛ dx (t ) dx (t ) ⎞ − dν 2 ⎜⎜ 1 − 2 ⎟⎟ + dν 4 ⎜⎜ 2 − 4 ⎟⎟ − dt ⎠ dt ⎠ ⎝ dt ⎝ dt

−κ 2 ( x1 (t ) − x 2 (t )) + κ 4 ( x 2 (t ) − x 4 (t )) = F2 (t ), m3

d 2 x 3 (t ) dt 2

(4-51)

⎛ dx (t ) dx (t ) ⎞ ⎛ dx (t ) dx (t ) ⎞ − dν 3 ⎜⎜ 1 − 3 ⎟⎟ + dν 5 ⎜⎜ 3 − 4 ⎟⎟ − dt ⎠ dt ⎠ ⎝ dt ⎝ dt

−κ 3 ( x1 (t ) − x 3 (t )) + κ 5 ( x 3 (t ) − x 4 (t )) = F3 (t ), m4

d 2 x 4 (t ) dt

2

+ dν 6

(4-52)

dx 4 (t ) ⎛ dx (t ) dx (t ) ⎞ + dν 4 ⎜⎜ 4 − 2 ⎟⎟ + dt dt ⎠ ⎝ dt

⎛ dx (t ) dx (t ) ⎞ + dν 5 ⎜⎜ 4 − 3 ⎟⎟ + κ 6 x 5 (t ) + κ 4 ( x 4 (t ) − x 2 (t ) ) + dt ⎠ ⎝ dt + κ 5 ( x 4 (t ) − x3 (t ) ) = F4 (t ).

(4-53)

The mass matrix, damping matrix and stiffness matrix, respectively, become ⎡m1 ⎢0 [M ] = ⎢⎢ 0 ⎢ ⎣0 ⎡ dν 1 + dν 2 + dν 3 ⎢ − dν 2 [D] = ⎢⎢ − dν 3 ⎢ 0 ⎣

0

0

m2

0

0

m3

0

0

0 ⎤ 0 ⎥⎥ , 0 ⎥ ⎥ m4 ⎦

− dν 2

− dν 3

dν 2 + dν 4 0

0 dν 3 + dν 5

− dν 4

− dν 5

132

(4-54)

⎤ ⎥ − dν 4 ⎥, ⎥ − dν 5 ⎥ dν 4 + dν 5 + dν 6 ⎦ 0

(4-55)

Chapter 4: Mathematical methods ⎡κ 1 + κ 2 + κ 3 ⎢ −κ 2 [K ] = ⎢⎢ −κ 3 ⎢ 0 ⎣

−κ 2

−κ 3

κ 2 +κ 4 0

0 κ3 +κ 5

−κ 4

−κ 5

⎤ ⎥ −κ 4 ⎥. ⎥ −κ 5 ⎥ κ 4 +κ 5 +κ 6 ⎦ 0

(4-56)

The general principle for generating these matrices, for systems in which the directions of forces and velocities are defined as in figures 4-6 and 4-7, can be summarized in the following way: (i)

the mass matrix is diagonal.

(ii)

a diagonal element in the stiffness or damping matrix is the sum of the spring rates or damping coefficients, respectively, of all springs / dampers connected to the mass indicated by the row number of the element.

(iii)

an off-diagonal element at a specific row and column position in the stiffness or damping matrix has the opposite (negative) of the value of the spring rate or damping coefficient, respectively, for the connection between the mass indicated by the row number and that indicated by the column number.

4.3.4

Frequency response functions

A frequency response function is defined as the relation between an output signal Y(ω) from a linear system, expressed as a function of the circular frequency ω, and the corresponding input signal X(ω), H (ω ) = Y(ω ) X(ω ) . (4-57) In other words, it can be interpreted as the proportionality constant in the linear relation between the complex input and output amplitudes. It is one of the most important quantities used in the analysis of sound and vibration problems. If the input signal is a force on a structure, knowledge of the frequency response function permits the computation of the resulting vibration at different points in the structure; if the input signal is a pressure at a point in a ventilation duct, it permits the calculation of the sound pressure at the outlet. Examples of some frequency response functions, commonly used in vibrations and acoustics, are given in table 4-1.

133

Chapter 4: Mathematical methods

Table 4-1 Examples of frequency response functions commonly used in vibrations and acoustics.

Quantity

Input quantities

Relation

Dynamic flexibility or Receptance H(ω)

Displacement x(ω) Force F(ω)

H (ω ) = x(ω ) F (ω ) (4-58)

Mobility or mechanical admittance Y(ω)

Velocity v(ω) Force F(ω )

Y(ω ) = v(ω ) F(ω ) (4-59)

Accelerance A(ω)

Acceleration a(ω) Force F(ω)

A(ω ) = a(ω ) F(ω ) (4-60)

Dynamic stiffness κ(ω)

Displacement x(ω) Force F(ω)

κ(ω ) = F(ω ) x(ω ) (4-61)

Mechanical impedance Z(ω)

Velocity v(ω) Force F(ω)

Z(ω ) = F(ω ) v(ω ) (4-62)

Acoustic impedance Z(ω)

Acoustic volume flow rate Q(ω) Acoustic pressure p(ω)

Z(ω ) = p(ω ) Q(ω ) (4-63)

Specific impedance Z(ω)

Acoustic particle velocity u(ω) Acoustic pressure p(ω)

Z(ω ) = p(ω ) u(ω ) (4-64)

In section 4.3.1, the motion of a single degree-of-freedom system was considered. The dynamic flexibility (4-58), i.e., the relation between the displacement x(ω) and the force F(ω), for the single degree-of-freedom system can, as in (4-16), be described by H (ω ) =

1κ x(ω ) 1 = = . 2 2 F (ω ) − ω m + iω 2mδ + κ 1 − (ω ω 0 ) + i 2 (ωδ ω 02 )

(4-65)

The frequency response functions can be presented graphically in a number of different ways. One possibility is to divide them up into real and imaginary parts H (ω ) = Re(H (ω )) + i Im(H (ω )) .

(4-66)

For the single degree-of-freedom system considered, Re(H (ω )) =

(1 − (ω ω 0 ) 2 ) κ , (1 − (ω ω 0 ) 2 ) 2 + ( 2 ωδ ω 02 ) 2

134

(4-67)

Chapter 4: Mathematical methods

Im(H (ω )) = −

2 ωδ (κω 02 ) (1 − (ω ω 0 ) 2 ) 2 + ( 2 ωδ ω 02 ) 2

.

(4-68)

Figure 4-8 shows graphs of the real and imaginary parts of H(ω), normalized by 1/κ and with δ as a parameter. For δ = ω0, the system is critically damped, for δ < ω0, it is weakly, or subcritically, damped, and for δ > ω0 it is strongly, or supercritically, damped.

κRe(H(ω)) δ=0

4

δ = 0,1ω0 δ = 0,3ω0

3 2

δ = 0,7ω0

1 0

δ = ω0 δ = 2ω0

-1 -2 -3 -4 0

a)

1

ω /ω0

2

3

κIm(H(ω)) 0

δ = 2ω0 δ = ω0 δ = 0,7ω0

-1

-2

δ = 0,3ω0

-3

-4

δ = 0,1ω0 -5

b)

0

1

ω /ω0

2

3

Figure 4-8 Dynamic flexibility of a single degree-of-freedom system divided into real and imaginary parts.

135

Chapter 4: Mathematical methods

Another possible representation of the frequency response function is in terms of its amplitude and phase angle H (ω ) =

1κ (1 − (ω ω 0 ) 2 ) 2 + (2 ωδ ω 02 ) 2

ϕ (ω ) = arctan

2 ωδ ω 0 2 (ω ω 0 ) 2 − 1

,

(4-69)

.

(4-70)

A figure in which the amplitude and phase curves are plotted is usually called a Bode diagram, see Figure 4-9.

κ H(ω )

5

δ=0

4

δ = 0,1ω0 δ = 0,3ω0

3

δ = 0,7ω0

2

δ = ω0 δ = 2ω0

1 0

0

1

ϕ [rad]

ω /ω0

2

δ=0 δ = 0,1ω0 δ = 0,3ω0 δ = 0,7ω0 δ = ω0

0

3

δ = 2ω0

− π 0

1

ω /ω0

2

3

Figure 4-9 Bode diagram of the dynamic flexibility of a single degree-of-freedom system, with separate plots of the amplitude and the phase angle.

136

Chapter 4: Mathematical methods

From Figure 4-9, it is apparent that the damping strongly influences the amplitude at a resonance, i.e., ω = ω0. The behavior well away from the resonance frequency is the same, regardless of whether or not there is damping in the system. By considering the magnitude of H(ω) in the vicinity of the resonance frequency, a measure of the losses can be inferred. A polar plot, in which the real part of H(ω) projects onto the x-axis and the imaginary part of H(ω) onto the y-axis, is called a Nyquist diagram, as in Figure 4-10.

κ Im[H(ω)]

Figure 4-10 Nyquist diagram of the dynamic flexibility of a single degree-offreedom system. The damping can be found using the diameter of the circle. A large circle corresponds to small damping and vice versa.

ω→∞

ω=0

0

ω

-1

δ = 2ω 0 δ = ω0 δ = 0,7ω 0 δ = 0,3ω 0 δ = 0,1ω 0

-2

-3

-4

-5

-3

4.3.5

-2

-1

0

κ Re[H(ω)]

1

2

3

Damping

For a single degree-of-freedom system, we have thus far only used the viscous damping coefficient dv or the damping constant δ to describe the losses. For structures, it is more common to use the so-called loss factor to describe the influence of different types of damping. In order to demonstrate how the loss factor is defined, it will be useful to first introduce energy quantities for a single degree-of-freedom system. Considering a sinusoidal displacement x = xˆ sin(ω t + ϕ ) , the kinetic and potential energies can be expressed, respectively, as 2

E kin =

m ⎛ dx ⎞ m 2 2 2 ⎜ ⎟ = ω xˆ cos (ωt + ϕ ) , 2 ⎝ dt ⎠ 2

E pot =

κ 2

x2 =

κ 2

xˆ 2 sin 2 (ωt + ϕ ) .

137

(4-71)

(4-72)

Chapter 4: Mathematical methods

The energy dissipated (i.e., spent or “lost”) in one period, can be expressed as E dis = ∫ Fd dx = ∫ d v

T

T

2

dx ⎛ dx ⎞ dx = ∫ d v ⎜ ⎟ dt =d v xˆ 2ω 2 ∫ cos 2 (ωt + ϕ ) dt = dt ⎝ dt ⎠ 0

0

= d v xˆ 2ω 2 T 2 = d v xˆ 2ωπ ,

(4-73)

in which we have used the relation ωT = 2π. The lost factor is defined as the dissipated energy per radian, divided by the maximum potential energy, E dis 2π ω dv ωδ = =2 . κ max( E pot ) ω02

η=

(4-74)1

For the case of the harmonic excitation of a single degree-of-freedom system, x p (t ) = xˆ e iωt ,

F(t ) = Fˆ e iωt ,

(4-75)

substitution into (4-10) yields ( −ω 2 m + iω d v + κ ) xˆ = Fˆ .

(4-76)

( −ω 2 m + κ (1 + iη )) xˆ = Fˆ .

(4-77)

Then, from (4-74),

As such, losses can be incorporated by defining a complex spring constant κ = κ (1 + iη )

4.3.6

.

(4-78)

Analogue mechanical - electrical circuits

When analyzing the differential equations (4-8), (4-9), and (4-10), which describe the single degree-of-freedom system, similarities to the equations used to describe electrical circuits can be observed. Those similarities are used to define so-called analogue mechanical-electrical circuits. It is even possible to define analogue acoustic-electrical circuits, but we limit the discussion to just mechanical systems at the moment. Assume that electric potential, or voltage, U and current I in an electrical circuit have a harmonic timedependence. U(t ) = Uˆe iω t , I (t ) = Iˆe iω t +ϕ .

(4-79)

In order to describe active circuit elements, i.e., those that provide power to the circuit, two ideal source models are used: an ideal current source, and an ideal voltage source; see 1 For resonant systems with small loss factors, it can be shown that max( E ) ≈ E , where E is the total energy pot of the system. For the dissipated energy, Edis = WdisT applies. Putting these expressions into (4-74) gives the following approximate expression for the dissipated power, Wdis = ηωE .

138

Chapter 4: Mathematical methods

Figure 4-11. An ideal voltage source delivers a constant voltage, regardless of the circuit to which it is connected, and an ideal current source delivers a constant current, regardless of the circuit to which it is connected. I

I

U

U

Figure 4-11 An ideal voltage source and an ideal current source.

U

I

The complex ratio of the voltage to the current is called the impedance; see Figure 4-12, Z=

U = R + iX , I

(4-80)

where R is the resistance and X is the reactance. I

U

Z

Figure 4-12 General passive circuit element impedance Z.

Impedance

The most common passive circuit elements are resistors, R, inductors, L, and capacitors, C, shown in Figure 4-13.

I

I

R

U

a) Resistance

I

U

C

b) Capacitance

U

L

c) Inductance

Figure 4-13 Three common passive circuit elements are the resistor, the capacitor and the inductor. Characteristic for passive circuit elements is that they do not supply energy to the system.

139

Formatted: Engl

Chapter 4: Mathematical methods

For the resistor, U (t ) = RI (t ) ⇒ U = RI .

(4-81)

For the capacitor, U (t ) =

1 1 I (t )dt ⇒ U = I. C∫ iω C

(4-82)

For the inductor, U (t ) = L

dI (t ) ⇒ U = iω LI . dt

(4-83)

That approach to writing the relation between circuit elements, in complex form, gives simple computational formulas for the determination of the voltage and current in circuits. It is usually called the iω-method. In the complex plane, a representation as in Figure 4-14 is obtained. Im I U = iω L I U = RI

o

90 Figure 4-14 Relation between current and voltage for a resistor, capacitor and inductor, presented in the complex plane.

Re

Formatted: Engl U=

I iω C

For the resistor, the voltage and current are in phase. For the capacitor, the voltage lags the current by 90°. For the inductor, the voltage is 90° ahead of the current. In the case of a single degree-of-freedom system with a harmonically-varying velocity v = vˆe iω t ,

(4-84)

the acceleration and displacement are also harmonic a=

∫

dv = iωvˆeiωt = iω v , dt

x = vdt =

1 iω

vˆeiωt =

1

(4-85)

v .

(4-86)

F = i ω mv .

(4-87)

iω

For a mass, Newton’s second law states F (t ) = m

dv(t ) , dt

By comparison to (4-83), it is clear that if the force corresponds to voltage, and the velocity to current, then the mass corresponds to inductance (m ⇔ L). A spring obeys

140

Chapter 4: Mathematical methods

∫

F=

F (t ) = κ v(t )dt ,

κ iω

v .

(4-88)

Comparing that to (4-82) shows that the spring rate is analogous to the reciprocal of an electrical capacitance (κ ⇔ 1 / C). A viscous damper, by definition, follows the law F = dv v .

F (t ) = d v v(t ) ,

(4-89)

Comparison to (4-81) reveals that the viscous damping coefficient is analogous to an electrical resistance (dv ⇔ R). Table 4-2 Summary of mechanical-electrical circuit equivalents, for passive circuit elements.

Component Mass Spring

Equation F = iωm v F = vκ / iω F = dv v

Viscous damper

Equivalence m↔ L κ ↔1 C dv ↔ R

Example 4-8 To isolate a machine of mass m = 700 kg, it is mounted upon springs with a combined spring rate κ = 2⋅106 N/m. Due to an imbalance, the machine is subjected to a force F of 50 N, at a frequency of 15 Hz. We would like to calculate the force acting on the foundation, if it can be regarded as completely rigid, i.e., the point impedance ZU as defined in (4-62) is infinitely large. The problem can be solved by setting up an equivalent mechanical-electrical circuit.

v

F

iωm

κ iω

vU

v

ZU = ∞

F

F(t) x(t) m

κ

vu(t)

iωm

κ iω

FU

A rigid mass must have the same velocity at the input as at the output, while the forces at those two points differ. A mass is therefore coupled in-series to the circuit. An ideal massless spring, on the other hand, has the same force at its input and output, while the velocities differ. Thus, it is coupled in-parallel to the circuit, i.e., between the relevant voltage (force) and ground. Because the spring is coupled to a rigid foundation with zerovelocity (vU ), the branch that comes after the spring, in-parallel, must have an infinite

141

Chapter 4: Mathematical methods impedance, and can therefore be eliminated. The force Fu acting on the foundation is then obtained by voltage splitting, as FU = F

κ iω 1 κ , =F =F 2 i ωm + κ i ω κ −ω m 1 − ω 2 ω 02

where ω 02 = κ m . With the input data entered, one obtains

FU = Fˆe iϕ t = 50e i 2π 15t

1

1 − (2π ⋅ 15) ⋅ 700 2 ⋅ 10 2

142

6

= −23.7e −i 94.2t N.

Chapter 4: Mathematical methods

4.5

IMPORTANT RELATIONS

MECHANICAL POWER The time average of the mechanical power W=

1 1 Re(F v * ) = Re(F * v ) . 2 2

(4-7)

LINEAR SYSTEMS Single degree-of-freedom system Differential equation of motion

d 2 x(t ) dt

ω0 =

where

In the frequency plane

2

+ 2δ

κ

dx(t ) + ω 02 x(t ) = g (t ) dt

δ =

m xˆ p =

dv 2m

g (t ) =

(4-10)

F (t ) m

gˆ (ω 02

− ω 2 ) + i 2δω

(4-11)

(4-16)

g (t ) = gˆe iω t

(4-13)

x p (t ) = xˆ p e iωt

(4-14)

det(−ω 2 [M] + [K ]) = 0

(4-39)

where

Two degree-of-freedom system

Frequency response function Dynamic flexibility H (ω ) =

1κ 1 x(ω ) = = 2 2 F (ω ) − ω m + iω 2mδ + κ 1 − (ω ω 0 ) + i 2 ωδ ω 0 2

143

(4-65)

Chapter 4: Mathematical methods

Damping Loss Factor

Complex spring constant

η=

ωdv E dis 2π ωδ = =2 κ max( E pot ) ω02 κ = κ (1 + iη) .

(4-74)

(4-78)

Analogue mechanical-electrical circuits Mechanical force and velocity correspond to electrical voltage and current, respectively For a mass,

F = iωmv

(4-87)

so that the mass corresponds to an inductance (m ↔ L ) . For a spring,

F = vκ iω

(4-88)

so that the spring constant corresponds to the inverse of a capacitance (κ ↔ 1 C ) For a viscous damper

F = dv v

so that the viscous damping coefficient corresponds to a resistance (d v ↔ R ) .

144

(4-89)

Chapter 5: Fourier Methods and Measurement Techniques

CHAPTER FIVE

FOURIER METHODS AND MEASUREMENT TECHNIQUES

This chapter introduces Fourier methods of analysis permitting the analysis of vibroacoustic problems in the frequency domain. The chapter is concluded with the important subject of measurement, which includes not only transducers, such as microphones and accelerometers, but also a brief description of digital measurement systems. 5.1

FOURIER METHODS IN VIBROACOUSTICS

5.1.1

Fourier series

Most machines emit periodic disturbances to the surroundings, either in the form of fluctuating forces, acting via the machine mounts, or in the form of sound. The reasons for these periodic disturbances can be, to name a few examples, the meshing of gear teeth, imbalances in rotating shafts, or periodic pressure fluctuations that arise in the cylinders of internal combustion engines due to the intake-exhaust cycles; see Figure 5-1a. The disturbances that arise in practice are, as mentioned, periodic, but not usually purely harmonic. They consist of a combination of different frequency components, as exemplified in Figure 5-1b. To analyze the problem in the frequency domain, a method is needed to divide up a measured signal into its harmonic components, so that they can be individually analyzed. For periodic signals, it is possible to use a Fourier series expansion.

145

Chapter 5: Fourier Methods and Measurement Techniques 5.1.1.1

Approximation of signals

As a first step in deriving a method to decompose a periodic signal into its harmonic components, we will study how to best approximate a signal a(t) with a signal b(t). We assume that the fitting of the two signals is to take place during the time interval 0 < t < T. To carry out the approximation in the simplest possible way, we multiply b(t) by a constant β that is varied to adjust the approximation as well as possible, a(t ) ≈ β b(t ) . The error in the our approximation then becomes

(5-1)

e(t ) = a (t ) − β b(t ) .

(5-2)

Next, the averaged squared error is computed over the entire time interval 0 < t < T,

1 T

ε=

T

∫ (a(t ) − βb(t )) dt . 2

(5-3)

0

We minimize ε with respect to β by differentiating, and setting the resulting derivative equal to zero, T

dε 1 = 2( βb 2 (t ) − a (t )b(t ))dt = 0 . dβ T

∫

(5-4)

0

from which T

β = ∫ a(t )b(t )dt 0

T

∫b

2

(t )dt .

(5-5)

0

We now introduce the useful concept of orthogonality. If the signals a(t) and b(t) are orthogonal, there is no connection between the two signals, and β equals zero. The orthogonality condition for the two signals a(t) and b(t) on the time interval 0 < t < T is therefore T

∫ a(t )b(t )dt = 0 .

(5-6)

0

The energy of a signal is proportional to the time integral of the signal squared; compare to formula (2-3). Thus, T

∫

E a = α a 2 (t )dt ,

(5-7)

0

in which α is a proportionality constant. Given a signal composed of two parts, v(t)=a(t)+b(t), its energy becomes Ev = α

T

T

T

T

0

0

0

0

2 2 2 ∫ (a(t ) + b(t )) dt = α ∫ a (t )dt + α ∫ b (t )dt + α ∫ 2a(t )b(t )dt .

If the signals are orthogonal, we get 146

(5-8)

Chapter 5: Fourier Methods and Measurement Techniques T

T

∫

∫

Ev = α a 2 (t )dt + α b 2 (t )dt = E a + Eb . 0

(5-9)1

0

Thus, the energy of the combined signal is the sum of the individual energies. If the signals are not orthogonal, equation (5-8) must be used. Possibly, the approximation of the signal a(t) given by (5-1) in combination with (5-5) gives an inadequate fit. To further reduce the error, we incorporate another signal c(t) with a proportionality constant γ , and write the approximation as a(t ) ≈ β b(t ) + γc(t ) .

(5-10)

e(t ) = a(t ) − β b(t ) − γ c(t ) ,

(5-11)

The error in that case becomes

and the mean squared error is

ε=

1 T

T

∫ (a(t ) − β b(t ) − γ c(t ))

2

dt .

(5-12)

0

We first minimize ε with respect to β T

∂ε 1 = 2( β b 2 (t ) + γ b(t )c(t ) − a (t )b(t ))dt = 0 . ∂β T

∫

(5-13)

0

If we can now choose b(t) and c(t) to be mutually orthogonal, the second term in (5-13) is eliminated, and we obtain T

β = ∫ a(t )b(t )dt 0

T

∫b

2

(t )dt ,

(5-14)

0

i.e., the same expression as in equation (5-5). If we minimize the error with respect to γ we obtain ∂ε 1 = ∂γ T

T

∫ 2(γc

2

(t ) + β b(t )c(t ) − a(t )c(t ))dt = 0 .

(5-15)

0

If b(t) and c(t) are orthogonal, we can solve for γ in the same way as before, T

γ = ∫ a(t )c(t )dt 0

T

∫c

2

(t )dt .

(5-16)

0

By comparison to (5-5), the result obtained is the same as what would have followed from the approximation a(t) = γ c(t). To improve upon that, we therefore only need to add 1 Compare to section 2.1.

147

Chapter 5: Fourier Methods and Measurement Techniques another orthogonal signal and minimize the mean squared error versus a(t), independently of the other signals included in the approximation. Examples of functions that are orthogonal are sin(nω0t) and cos(nω0t). They are orthogonal for all integer values of n over the time interval T = 2π / ω0. In the next section, we will take advantage of that orthogonality property to decompose a periodic signal into sine and cosine components. That gives the desired decomposition of the signal into its frequency components, given by fn = n / T. 5.1.1.2 Fourier series decomposition

Assume that we have a signal a(t) that is periodic, with period T, and which we wish to approximate with the help of sine and cosine functions, ∞

∞

n =1

n =1

a(t ) = β 0 + ∑ β n cos(nω o t ) + ∑ γ n sin( nω o t ) .

(5-17)

That is called a Fourier series decomposition of the signal a(t). The coefficients βn and γn can be calculated separately, and given by (5-5)

β0 =

T 2

−T 2

−T 2

∫ a(t )1dt

T 2

T 2

−T 2

−T 2

∫ a(t ) cos(nω 0 t )dt

βn =

T 2

2 ∫ 1 dt =

2 ∫ cos (nω 0 t )dt =

1 T

2 T

T 2

∫ a(t )dt ,

(5-18)

−T 2

T 2

∫ a(t ) cos(nω 0 t )dt ,

n = 1,2,3, … ,

−T 2

(5-19) T 2

γn =

∫ a(t ) sin(nω 0 t )dt

−T 2

T 2

2 ∫ sin (nω 0 t )dt = T 2

−T 2

T 2

∫ a(t ) sin(nω 0 t )dt ,

n = 1,2,3, … .

−T 2

(5-20) The interval of integration in (5-18), (5-19) and (5-20) is -T/2 to T/2, but could just as well have been 0 to T. The coefficient β0 represents the signal’s time average. It can also be shown that corresponding sine and cosine terms can be combined into a single cosine term with a phase angle ϕn. ∞

a(t ) = β 0 + ∑ δ n cos(nω 0 t − ϕ n ) ,

(5-21)

n =1

where

δ n = β n2 + γ 2n

ϕ n = arctan(γ n β n ) + mπ , m = 0, 1, 2, …, where

δ1 gives the signal’s amplitude for the first tone, or fundamental tone, δ2 gives the signal’s amplitude for the second tone, or the first overtone, δ3 gives the signal’s amplitude for the third tone, or second overtone, i.e., 148

Chapter 5: Fourier Methods and Measurement Techniques p [bar]

Exhust

6 Combustion

4

Ignition 2

0

0

200

400

a)

600

800

1000

Crankshaft angle φ [Degree]

1200

1400

L p [dB] 200

180

160

140

120

0

200

b) Figure

400 600 Frequency [Hz]

800

1000

5-1 Cylinder pressure in a single-cylinder, two-cycle engine running at 5500 rpm. a) The pressure as a function of the crankshaft angle φ. The pressure variation repeats itself periodically after 360° , i.e., after a complete cycle. The different phases are compression, ignition, combustion, exhaust, and intake. For a two-cycle engine, exhaust and intake occur simultaneously, by virtue of the inflowing gases pushing out the exhaust gases. b) The sound pressure level as a function of frequency. The periodic pressure variation, after Fourier series decomposition, results in a frequency spectrum with discrete frequency components For a running speed of 5500 rpm, the fundamental frequency is f0 = 5500/60 = 91.7 Hz.

In Figure 5-1, the Fourier series method is illustrated for the case of pressure in a twocycle engine. In Figure 5-1a, the pressure is given as a function of the crankshaft angle φ instead of time t. That can be done fairly simply, since there is a relation φ = ω t between these. In Figure 5-1b, the corresponding sound pressure level, Lp, is given as a function of the frequency. In order to be able to denote each frequency component as a complex, rotating vector, which, as demonstrated in section 4.1, results in simpler computations and a more compact symbolic expression, a complex Fourier series can be defined, a(t ) =

∞

∑ δ n e inω t 0

n = −∞

149

,

(5-22)

Chapter 5: Fourier Methods and Measurement Techniques where the complex coefficient δn can be determined from T 2

δn =

−inω t ∫ a(t )e 0 dt

−T 2

T 2

2 ∫ 1 dt =

−T 2

1 T

T 2

∫ a(t )e

−inω 0t

dt .

(5-23)

−T 2

A table summarizing certain useful signals’ complex Fourier coefficients is given in Appendix D. In (5-22), there are components with “negative frequencies”. That is because, as shown in section 4.1, a real quantity can be described with the aid of two oppositely rotating complex vectors, each of which is the complex conjugate of the other; see Figure 5-2. βn

-f

Real

δ n = β n + iγ n

γn βn δ n* = β n - iγ n γn

Imag

+f Figure 5-2 Three-dimensional description of the frequency spectrum of a periodic signal. Complex Fourier series contain components with both negative and positive frequencies. (Source: Brüel &Kjær, Frequency Analysis.)

150

Chapter 5: Fourier Methods and Measurement Techniques

Example 5-1 Consider a rectangular wave, as in the figure below.

a (t)

A

-T / 4

-3T / 4

T/ 4

3T / 4

t

-A

From (5-18)

β0 =

1 T

0

∫ − Adt +

−T 2

1 T

T 2

∫ Adt = − T [t ] −T 2 + T [t ] 0 A

A

0

0

T 2

=−

A A + =0 2 2

i.e., the time-averaged value is, as expected, equal to 0. Equation (5-19) yields

βn =

2 T

0

∫ − A cos(nω 0t )dt +

−T 2

2 T

T 2

∫ A cos(nω 0t )dt = 0 T 2

0

2 A ⎡ sin(nω 0 t ) ⎤ =− ⎥ ⎢ T ⎣ nω 0 ⎦ −T

2

2 A ⎡ sin(nω 0 t ) ⎤ + ⎥ ⎢ T ⎣ nω 0 ⎦ 0

=−

A A sin(nπ ) + sin( nπ ) = 0, nπ nπ

and (5-20) gives

γn =

2 T

0

∫ − A sin(nω 0 t )dt +

−T 2

0

2 A ⎡ cos(nω 0 t ) ⎤ = ⎥ ⎢ T ⎣ nω 0 ⎦ −T

2 T

T 2

∫ A sin(nω 0 t )dt = 0 T 2

2

2 A ⎡ cos(nω 0 t ) ⎤ + ⎥ ⎢− T ⎣ nω 0 ⎦ 0

=

⎧ 4A 2A ⎪ n = 1,3,5,... (1 − cos(nπ )) = ⎨ nπ nπ ⎪⎩ 0 n = 0,2,4,...

Thus, the Fourier series can be expressed as a(t ) =

4A ⎡ 1 1 ⎤ sin(ω 0 t ) + sin(3ω 0 t ) + sin(5ω 0 t ) + ...⎥ . ⎢ π ⎣ 3 5 ⎦

It only consists of the odd sine components. That only sine components appear in the Fourier series is because the rectangular wave is an odd function; odd functions can be decomposed into sine components, since the sine function itself is odd. By definition, a function is odd if it has the property that a(-x) = -a(x), and even if it has the property that a(-x)=a(x). If the rectangular wave is shifted T/4 to the left, making it symmetric about t=0, it becomes an even function instead, and be built up exclusively of cosine functions. How good an approximation of the original rectangular waveform that one obtains, in example 5-1, depends, naturally, on the number of terms included in the Fourier series 151

Chapter 5: Fourier Methods and Measurement Techniques decomposition. In Figure 5-3, the gradual improvement of the curve fit is evident as more and more terms are included in the series. 1 Com pon en t 2 Com pon en t s

8 Com pon en t s

1

a( t ) 0

-1

0

0.5

1.0 t/ T

1.5

2.0

32 Com pon en t s 1

a( t ) 0

-1

0

0.5

1.0 t/ T

1.5

2.0

Figure 5-3 Synthesis of a rectangular waveform with different numbers of terms in the Fourier series decomposition. The graph shows the gradual improvement of the Fourier series approximation when more Fourier components are included in the summation. In the intervals in which the original time function has continuous derivatives, the Fourier series approximation can be arbitrarily close to the original function with enough terms. For discontinuous points, such as those at t/T = 0, 0.5, 1.0, 1.5, etc., there is always a “ripple”, no matter how many terms are included in the approximation. That is called the Gibb’s phenomenon in mathematics. It can be shown that the size of the ripple is, at a maximum, about 18% of the size of the discontinuity.

152

Chapter 5: Fourier Methods and Measurement Techniques

Example 5-2 The nature of the periodic force applications that bring about sound and vibration determines how great are the problems that arise. The figure below shows Fourier series decompositions of a rectangular wave, a triangular wave, and a sine wave. LF [dB]

LF [dB]

0

Square Wave

0

Triangular Wave

-10

-10 t -20

-20

-30

-30

-40

-40

-50

t

1

3

5

7

9

11

13

15

17

19

-50

1

3

Frequency Component, n

5

7

9

11

13

15

17

19

Frequency Component, n

LF [dB]

The amplitude of the overtones decays more slowly for the rectangular wave (as Sine -10 1/n, where n refers to the n-th frequency component) than for the triangle wave -20 (decays as 1/n2), whereas a sine wave only t has a single frequency component. -30 Because the overtones often fall in the -40 more disturbing frequency bands, it is a good design principle to always make -50 9 11 13 15 17 19 1 3 5 7 force applications as soft and “sinusoidal” Frequency Component, n as possible. Another phenomenon that occurs in the case of periodic forcing is that the distance Δ f between the frequency components becomes larger, the shorter the period T ; i.e., Δ f = 1 / T. That is illustrated in the figure below, for a periodically repeated rectangular pulse. 0

LF [dB]

LF [dB] TP = 0.8 ms

0

TP = 0.8 ms

0

T = 0.05 s

T/5 = 0.01 s

-10

-10

-20

-20

-30

0

100

200

300

400

500

600

-30

0

Frequency [Hz]

100

200

300

400

500

600

Frequency [Hz]

That fact can be used to minimize the number of frequency components excited in sensitive frequency bands, if it is possible to change the period.

153

Chapter 5: Fourier Methods and Measurement Techniques 5.1.2

Fourier transform

Many machines or processes give rise to sound or vibrations which are not periodic, but rather random (stochastic) or transient; see section 2.8. Roughness of the contacting surfaces between a wheel and its path, for instance, or between meshing gear teeth, bring about randomly varying vibrations. Turbulence in flowing media gives rise to randomly varying sound. For non-periodic disturbances, a Fourier series decomposition cannot be made; instead, one must use a so-called Fourier transform. The Fourier transform can be developed from the complex Fourier series (522,23) of a periodic train of pulses, as shown in Figure 5-4.

αT

F(t)

Figure 5-4 Periodic train of pulses with amplitude 1, perod T and pulse width αT.

1

−T

0

T

t

Equation (5-23) provides the coefficients of the complex Fourier series, 1 δn = T

αT 2

∫

1 ⋅ e −inω0t dt =

(e

)

inω 0αT / 2

−αT 2

2i sin(αnω 0T / 2) − e −inω0αT / 2 = , inω 0T inω 0T

(5-24)

sin (αnπ ) . αnπ

(5-25)

but T = 2π /ω0 , which implies that

δn = α

The Fourier series of the pulse train becomes F (t ) =

∞

∑α

n = −∞

sin(αnπ ) inω 0t e . αnπ

(5-26)

The Fourier series coefficients for the cases α = 1 2 , 1 4 and 1 8 are shown in Figure 5-5. From that figure, it is clear that if the pulses are permitted to glide farther and farther apart in the time domain, T → ∞ and the pulse width α T is held constant, from which it follows that α → 0, then the spectral lines approach each other in the frequency domain, i.e., become infinitely dense.

154

Chapter 5: Fourier Methods and Measurement Techniques

1,0 0,8

δn

α = 1/ 2

0,6

α

0,4 0,2 0 -8

-4

0

4

8

Frequency Component , n

a) 1,0

α = 1/ 4

0,8

δn

α

0,6 0,4 0,2 0 -16

-8

0

8

16

Frequency Component , n

b) 1,0

α=

0,8

δn

α

1 8

0,6 0,4 0,2 0 -32

-16

0

16

32

Frequency Component , n

c)

Figure 5-5 Fourier series decomposition of a periodic pulse train with constant pulse width αT. For smaller α and increasing period T, the frequency components become all the more densely packed a) α = 1 / 2, b) α = 1 / 4 and c) α = 1 / 8.

To derive a relation for a non-periodic event, we therefore consider the limiting case of the period T becoming infinite. If we substitute in the expression for the Fourier coefficients (5-23) into the Fourier series (5-22), we then obtain 1 ∞ inω 0t F (t ) = e F (t )e −inω 0t dt . T n = −∞ −T 2

∑

T 2

∫

155

(5-27)

Chapter 5: Fourier Methods and Measurement Techniques To adapt that to the limiting case when the period T goes to infinity, the interchange ω0 → dω is made because ω0 = 2π / T, and nω0 transforms into a continuous variable ω, i.e., nω0 → ω . The step size in the summation becomes infinitesimally small, and the summation in (5-27) transforms to an integral

F (t ) =

1 2π

∞

∫

−∞

∞

e inω 0t ( ∫ F (t )e −inω 0t dt )dω .

(5-28)

−∞

The expression inside the parentheses is identified as the Fourier transform of the signal, ∞

F (ω ) =

∫ F (t )e

−iωt

dt ,

(5-29)

−∞

and the inverse Fourier transform is given by F (t ) =

1 2π

∞

∫ F(ω )e

iω t

dω .

(5-30)

−∞

Appendix C contains a table of the Fourier transforms of some useful signals. The Fourier transform is a complex quantity, which, in the case of F(t) representing a force, has the units N/Hz. In order for F(t) to be real, F(-ω ) = F*(ω ) must hold; compare example 4-5.

Example 5-3 Calculate the Fourier transform of a single force pulse, with pulse width Tp, as illustrated in the adjacent figure. Applying (5-29) yields F(t) Tp 2 −iωt ˆ F F(ω ) = Fe dt =

∫

−T p 2

=

Fˆ ⎛ iωT p 2 −iωT p 2 ⎞ −e ⎜e ⎟= ⎠ iω ⎝ sin ωT p 2 = FˆT p . ωT p 2

(

)

-Tp/2

Tp/2

t

The amplitude spectrum becomes F(ω ) = FˆT p

sin(ωT p / 2)

ωT p / 2

.

The Fourier transform is real, which implies that the phase spectrum is determined by the sign of sinω Tp. The Fourier transform’s amplitude spectrum and phase spectrum are shown in the figure below, in which a dimensionless frequency ωTp has been incorporated. Note that the transform of the rectangular pulse corresponds to the case in which α → 0 in section 4.4.2. The amplitude spectrum in the figure below is therefore the result obtained in Figure 5-5, in the limit as α → 0. The discrete spectrum for the pulse train has, in the case of a single pulse, transformed into a continuous spectrum.

156

Chapter 5: Fourier Methods and Measurement Techniques F(ω ) FˆT p

Fas ϕ [rad/s]

1,4 π

1,2 1,0 0,8 0,6

π/2

0,4

0

0,2 0 −8π −6π −4π

−2π

0

ωΤp

2π

6π

4π

8π −8π −6π −4π −2π

0

ωΤp

2π

4π

6π 8π

Example 5-4 By analysis of the transient forces in the time and frequency domains, respectively, a number of general conclusions, useful in machine and equipment design, can be drawn.

The smaller the impulse ( I = ∫ F (t )dt ), the lower the amplitude in the frequency domain. The figure below illustrates the effects of two different modifications to the impulse. Note that a dimensionless frequency f Tp is used. LF [dB]

LF [dB]

I=Ip/2

I=Ip 10

10

Tp/2

Tp 0

0

-10

-10 0

2

1

1

0

2

f Tp

f Tp

LF [dB] I=Ip/2

10

Tp 0

-10 0

1

2

f Tp

Increased duration or pulse width Tp in the time domain translates into a lowering of the cutoff frequency (the frequency at which the level has fallen 3 dB with respect to the maximum amplitude). By making the pulse longer (increasing duration), a lower frequency 157

Chapter 5: Fourier Methods and Measurement Techniques excitation is thereby obtained. That can exploited to shift the excitation into a frequency band which is less disturbing or in which the structure is not as effectively excited. The figure below illustrates that effect. L F [dB]

LF [dB]

T=T p 0

0

T 0 t

-10

t

-10

-20

-20

0

1

0

2

1

2

f Tp

f Tp L F [dB]

If the rise or fall time of the pulse is lengthened, the amplitude decays more rapidly with frequency above the cutoff frequency; see the adjacent figure. That can be exploited to reduce the high frequency content in the excitation. The same also applies to higher time derivatives. The more rounded and “soft” the excitation is in the time domain, the more rapidly the high frequency content decays.

T = 4 Tp

0

-10

t

-20

0

2

1

f Tp

LF [dB] 0 -2

Tp

-4 -6

Tp

-8 -10 0

0,5

f Tp

1

1,5

158

Chapter 5: Fourier Methods and Measurement Techniques

5.1.3

Parseval’s relations

Let F1(ω ) and F2(ω ) be Fourier transforms of the time functions F1(t ) and F2(t ). Then, it can be shown that ∞

∫

∞

F1 (t ) F2 (t )dt =

−∞

∫ F1 (ω )F2

*

(ω )

−∞

dω , 2π

(5-31)

which is called Parseval’s relation. To prove (5-31), put the inverse Fourier transform of F1(t ) into the left hand side of (5-31) ∞

∫

∞

F1 (t ) F2 (t )dt =

−∞

∫

∞

−∞ ∞

=

∫ F1 (ω )e

F2 (t )

iω t

−∞

dω dt = 2π

⎤ dω ⎡∞ F1 (ω ) ⎢ F2 (t )e iωt dt ⎥ = ⎥ 2π ⎢−∞ −∞ ⎦ ⎣

∫

∫

∞

=

∫ F1 (ω )F2

*

(ω )

−∞

dω . 2π

Parseval’s relation can be used to calculate the mean square value of a quantity from a measured frequency spectrum. The mean square value is given by (2-3), in which we let T → ∞ because we do not have a periodic function, ⎛ ~ ⎜1 F 2 = lim ⎜ T →∞ ⎜ T ⎝

⎞ ⎟ F 2 (t )dt ⎟ . ⎟ 2 ⎠

T 2

∫

−T

(5-32)

Parseval’s relation yields ∞

∞

−∞

−∞

2 ∫ F1 (t )dt =

* ∫ F1 (ω )F1 (ω )

dω = 2π

∞

∫ F1 (ω )

−∞

2

dω . 2π

(5-33)

Putting (5-33) into (5-32), and using the relation F(ω ) = F*(-ω ), yields ⎛ ~ ⎜1 F 2 = lim ⎜ T →∞ ⎜ T ⎝

⎛ Ω2 ⎞ ⎞ ⎛2Ω2 ⎞ ⎜1 2 ⎟ 2 dω ⎟ 2 dω ⎟ ⎜ ω F(t ) dt ⎟ = lim ⎜ F (ω ) F ( ) = . lim 2π ⎟⎟ Ω→∞ ⎜ Ω 2π ⎟ ⎟ Ω →∞ ⎜ Ω − Ω 2 2 0 ⎝ ⎠ ⎝ ⎠ ⎠ (5-34)

T 2

∫

−T

∫

∫

159

Chapter 5: Fourier Methods and Measurement Techniques

Parseval’s relation for periodic signals can also be derived, as T 2

∫

F1 (t ) F2 (t )dt = T

−T 2

∞

∑ d1n d *2n

(5-35)

n = −∞

in which d1n and d2n are the Fourier series coefficients of F1(t ) and F2(t ), respectively. Using (5-22), ∞ ⎛ ∞ ⎞ ⎜ d1n e inωt ∑ d *2 m e −imωt ⎟ dt = ∑ ⎜ ⎟ m = −∞ ⎠ 2 ⎝ n = −∞

T 2

T 2

−T 2

−T

∫ F1 (t ) F2 (t )dt = =

∞

∞

T 2

n = −∞

m = −∞

−T 2

∑ d1n ∑ d *2m ∫ e i(n−m)ωt dt,

T 2

yet because

∫

∫e

i ( n − m)ωt

−T 2 T 2

then it follows that

⎧T för n = m , dt = ⎨ ⎩ 0 för n ≠ m

∫ F1 (t ) F2 (t )dt = T

−T 2

∞

∑ d1n d *2n

.

n = −∞

(5-35) can be used to calculate the mean squared value of a periodic signal from its Fourier components, as indicated by formula (2-35) ~ 1 F2 = T

T 2

∫

F 2 (t )dt =

∞

∑ dn

2

.

(5-36)

n = −∞

−T 2

If the summation is only carried out over the “positive” frequencies, n = 0, 1, 2, … , then ~ 2 Fn2 = 2 d n must be used as the mean square value of the n-th frequency component. That follows from d − n = d *n , from which d − n = d n . That also provides the basis for computing the third octave band spectrum, for example, from a narrow band spectrum, or the octave band spectrum from a third-octave band spectrum.

160

Chapter 5: Fourier Methods and Measurement Techniques

Example 5-5 Measurement of the sound pressure level has been carried out in the third octave bands with center frequencies 800 Hz, 1000 Hz and 1250 Hz, from which the results given in the table below were obtained.

f [Hz] Lp [dB]

800 73.4

1000 69.8

1250 72.1

We now wish to calculate the sound pressure level for the octave band with the center frequency 1000 Hz. Solution Calculate, firstly, the mean squared value of the sound pressure in the third octave bands, using formula (2-29).

Then, sum up the mean squared values in accordance with Parseval’s relation, 2 ~ p oct p12 + ~ p 22 + ~ p32 . =~

f [Hz] 2 ~ p [Pa2]

800 8.75⋅10-3

1000 3.82⋅10-3

1250 6.48⋅10-3

Calculate the sound pressure level as L p = 10 ⋅log( ~ p oct p ref ) . 2

2

2 ~ p oct =~ p12 + ~ p 22 + ~ p32 = 8.75 ⋅ 10 −3 + 3.82 ⋅ 10 −3 + 6.48 ⋅ 10 −3 = 1.91 ⋅ 10 −2 , 2 2 L p = 10log( ~ p oct p ref ) = 10 ⋅ log(1.91 ⋅ 10 −2 4 ⋅ 10 −10 ) = 76.8 dB.

5.2

MEASUREMENT SYSTEMS FOR SOUND AND VIBRATIONS

The nature of sound and vibrations to be measured can vary widely. Sound can be “noisy” (roar or hiss-like), like that from a heavily trafficked highway, while vibrations of a machine are often dominated by the rotational frequency and its multiples. A machine under constant loading gives off a stationary noise, while the noise at an airport tends to be intermittent. Moreover, the purpose of measurements varies. If it is merely a question of a noise disturbance survey in an industrial facility, then relatively simple, single-channel instruments are used; see figure 5-6. If the purpose, on the other hand, is to determine the mode shapes of a large structure, such as an airplane fuselage, then a larger measurement system with two or more channels is required.

161

Chapter 5: Fourier Methods and Measurement Techniques Figure 5-6 The most common portable instrument for measurement and analysis of sound is a sound level meter, which can be found in various makes and models. It measures the total sound pressure level throughout the audible band, but is often also equipped with octave band and third-octave band filters for frequency band analysis. To imitate the sensitivity of human hearing to sound with different frequency contents, so called A, B, and C weighting filters can be used; the measurement quantities obtained are called Sound Levels in dB(A), dB(B) and dB(C), and are discussed in chapter 2. Power (rms) amplitudes, as from (1-3), can normally be displayed for a selected integration time T. Standardized settings are slow, fast, impulse, and peak, with decreasing integration time in the order given, and faster updating of the measurement quantities in that order. Many modern digital sound level meters are of the integrating type, meaning that they can calculate time-averaged values over extended periods, such as hours or days. (Photo: Brüel & Kjær.)

Figure 5-7 In vibroacoustics, many quantities are defined in Swedish and international standards and measurement procedures. The intent of standardization is to, among other things, facilitate the comparison of measurement results from different sources and product characteristics from different manufacturers. (Sketch: Brüel & Kjær, informationsmaterial.)

Enimportant viktig princip An principle: Never begin measurements without first clearly defining the purpose and – poorly planned measurements Påbörja aldrig en documenting mätning innan the syftecircumstances och mätsituation är kartlagda - dåligt planlagda can seldom or never be properly interpreted later. mätningar kan sällan eller aldrig utvärderas i efterhand.

162

Chapter 5: Fourier Methods and Measurement Techniques

5.2.1

The measurement chain

The measurement systems that are marketed today are primarily digital, i.e., sound pressure and vibrations are converted into digital values for later treatment in more or less advanced signal processors. Ever since the beginning of the 1920’s, when analog measurement began to be practical, a number of methods and measurement quantities have been developed. These are, today, well established and difficult to dispense with. Thus, while digital technology offers ever more sophisticated possibilities, measurement systems are nevertheless often adapted to be able to compare measurement results with those obtained in the past using analog technology. Digital measurement systems have a more complicated structure than analog ones. A flow chart for a digital, single channel system is shown in figure 5-8. S o u n d o r v ib ra tion sign al

T ra n sd u cer

E lectrica l sig n a l

S ig n al co n d ition in g

S ig n al

Im p ed an ce m atch in g

A ccelero m eter T im e

A m plification

M icro ph o ne

A n a log filter

A n a log -d igita l co n v ersio n

A m p lificatio n

S ign al

F req u en cy a n a lysis

D igital

1

filtering

F ast F o u rier F req u en cy

T im e

T ransform

L ow p ass filtering

M easu rem en t d a ta p rocessin g

G ra p h A m p litu d e

E xam ple: A v erag in g R M S am p litu d e d eterm in atio n A -, B -, C -w eig h ting F req uency

Figure 5-8 Flow chart of a digital, single-channel measurement system. For multi-channel systems, the analog and digital conversions are synchronously controlled for simultaneous sampling of the measurement signals. That improves the precision in computing quantities that depend on several channels, such as frequency response functions; see chapter 3.

5.2.2

Transducers

The types of transducers that are most commonly used in vibroacoustics are microphones to measure sound pressure, accelerometers to measure accelerations of solid structures, and force transducers to measure forces on solid structures. The principles behind force transducers are not described here, but are very similar to those for accelerometers. 163

Chapter 5: Fourier Methods and Measurement Techniques

Transducers convert measured quantities, such as sound pressure, into equivalent electrical signals. A harmonically varying sound pressure induces a harmonic electrical signal at the same frequency. For a certain microphone, the output voltage U(t) [V] due to a certain sound pressure p(t) is U (t ) = C p(t ) ,

(5-37)

where the proportionality constant C should be independent of amplitude and frequency, to the extent possible. A number of characteristics are common to all types of transducers: Sensitivity:

Indicates the ratio of electrical output to mechanical input. Example: A microphone’s sensitivity is given in mV/Pa.

Frequency band: Indicates the upper and lower frequency limits, between which the transducer sensitivity varies within a given (small) tolerance range. Dynamic range: Indicates the upper and lower amplitude limits between which the transducer sensitivity varies within a given (small) tolerance range. The dynamic range is commonly given in dB with respect to a reference value. The lower dynamic boundary is often determined by the transducer’s electrical noise, and the upper boundary by when the transducer is loaded beyond its mechanical linear region. 5.2.3

Microphones

For acoustic measurements with high demands on the precision, condenser microphones are used. Condenser microphones consist of a thin metal membrane called a diaphragm, separated from an opposing “backplate” electrode by an air gap (figure 5-38). The diaphragm and backplate constitute the electrodes of a condenser which is polarized by an electrical charge. When the diaphragm vibrates, due to the sound pressure, the capacitance of the condenser varies and an electrical output signal is generated. That electrical output signal is proportional to the sound pressure. As we established in earlier sections, a sound field is affected by reflection and diffraction when it is disturbed by the presence of objects. When the sound frequency is so high that the wavelength approaches the size of the microphone, then the microphone itself will influence the sound field. A small microphone has a higher upper frequency limit, but a lower sensitivity. Measurement microphones are therefore made in various sizes, and with built-in corrections for different types of sound fields. Microphone sizes are given in inches; typical sizes are 1", 1 2 ", 1 4 " and 1 8 ". A 1 2 "-microphone is standard in most measurement situations; see table 1-3 for details. relief vent Hål förPressure statisk tryckutjämning

Diaphragm

Insulator Isolator

Shield Membran Diaphragm Motelektrod Backplate electrode

Hylsa Casing

164

Backplate electrode

Cavity Insulator Output terminal

Chapter 5: Fourier Methods and Measurement Techniques Figure 5-38 The electrode charges can be brought about in two ways. For externally polarized microphones, an external voltage is applied across the diaphragm and backplate. Pre-polarized microphones are charged by means of a thin electrical material that is placed on the backplate. That solution is usually more expensive, but nevertheless preferred for portable instruments, since it avoids the complications inherent in requiring external electrical voltage. (Source: Brüel & Kjær, Measurement Microphones)

The different types of microphones are: Free field:

Free field microphones are intended for use in direct fields and should therefore be directed towards the sound source. They have built-in corrections that compensate the microphones influence on the sound field.

Pressure:

Pressure microphones are mainly intended for calibrations in small cavities and for mounting positions flush with walls and the like. These do not compensate for their own affect on the sound field. They measure the actual sound pressure on the microphone diaphragm.

Diffuse field: Diffuse field microphones are used in diffuse sound fields, i.e., they should have a flat sensitivity curve as a function of frequency for sound that falls in from all directions.

The microphone capsule is directly connected to a pre-amplifier. Its main task is to convert the microphone’s high output impedance to a low one, permitting connection to long cables or to a measurement system with relatively low input impedance. Table 5-1 Data for some types and sizes of measurement microphones. Microphone Diameter Type of sound field

B&K 4145 1" (25.4 mm) Fee field

2.6 Hz - 18 kHz Frequency range (±2 dB) 50 Sensitivity [mV/Pa] Dynamic range [dB] 11 - 146 (with recommended Preamp)

B&K 4165 ½" (12.7 mm)

2.6 Hz - 20 kHz 50

B&K 4135 ¼" (6.35 mm) Free and diffuse field 4 Hz - 100 kHz 4

B&K 4138 1/8" (3.2 mm) Pressure and diffuse field 6.5 Hz - 140 kHz 1

15 - 146

36 - 164

55 - 168

Free field

165

Chapter 5: Fourier Methods and Measurement Techniques

5.2.4

Accelerometers

Piezoelectric accelerometers are the most commonly used vibration transducers. In some measurement situationsError! Bookmark not defined., however, other types may be preferred, such as optical and inductive transducers or strain gauges. The construction of a piezoelectric accelerometer of the compression type can be seen in figure 5-39.

Pre-loaded fjäder spring Förspänd

Figure 5-39 Fundamental construction of a piezoelectric accelerometer of the compression variety. The active elements are the piezoelectric discs, on which a mass is resting. The mass is preloaded by a spring and the entire arrangement is enclosed in a metal capsule on a stable mounting plate.

Seismic Rörlig mass massa Electrical Elektrisk + Output signal utsignal

Piezoelectric element Piezoelektriska element

_

Baseplate / mounting plate Bas/monteringsplatta Vibrating object Vibrerande objekt Threaded stud Fästskruv

When the accelerometer, which is firmly mounted to the measurement object, is subjected to vibrations along its axis of symmetry, the mass gives rise to a force that varies as the acceleration varies. That force deforms the piezoelectric discs, which then produce, because of their piezoelectric properties, a charge on the surfaces of the discs proportional to the force, and thereby to the acceleration as well. The piezoelectric effect can be obtained for either compression or shear of the piezoelectric material. Simplified models are shown in figure 5-40.

F + + + + -

q

+

+ + + + + + -

F

-

F

a)

F

q

+

Figure 5-40 The piezoelectric materials usually are artificially polarized ceramics that give rise to a charge, q, when subjected to a) compression, or b) shear. The accelerometers that are based on the shear principle can be made less sensitive to other types of deformations, as for example, those caused by temperature variations. (Source: Brüel & Kjær, Piezoelectric Accelerometers and Vibration Preamplifiers)

b)

In selecting suitable accelerometers, there are primarily three characteristics of interest: sensitivity, given in charge per unit acceleration [pC/ms-2]; internal resonance frequency; and the accelerometer’s total mass. The sensitivity and resonance frequency are strongly dependent on the mass; see figure 5-41. Accelerometers are therefore available with masses ranging from one gram, for very high vibration levels (shocks), up to 500 grams, intended for very low levels.

166

Chapter 5: Fourier Methods and Measurement Techniques Sensitivity

Accelerometer with a large mass

Useful frequency range ~fr /3

Accelerometer with a small mass

fr

fr

Frequency (log)

Figure 5-41 The sensitivity of an accelerometer depends primarily on the seismic mass and the properties of the piezoelectric material. A larger accelerometer normally has a greater sensitivity. The seismic mass and the piezoelectric element constitute a mass-spring system with a resonance frequency fr at which the sensitivity increases dramatically. As a rule of thumb, it can be said that the useful frequency range is below fr /3.

Table 5-2 Data for some types and sizes of accelerometers. Accelerometer B&K 8306 500 Mass [g] Compression Type 4500 Internal resonance frequency [Hz] 0.06 - 1250 Frequency range [Hz] 1000 Sensitivity [pC/ms-2]

B&K 4370 54 Shear 18000 0.2 - 6000 10

B&K 4367 13 Shear 32000 0.2 – 10600 2

B&K 4344 2 Compression 70000 1 - 21000 0.25

Piezoelectric accelerometers are connected to preamplifiers, the primary purpose of which is to match the high output impedance of the accelerometer to the low input impedance of the measurement system. Typically, a charge amplifier is used; despite its name, it does not amplify charge, but rather gives an output voltage proportional to the accelerometer’s charge. The particular advantage of a charge amplifier is that the length of the cable between the accelerometer and the amplifier can be varied without altering the sensitivity. More advanced charge amplifiers even have other functions built in: (i)

Amplification of the signal to a level suited to the measurement system.

(ii)

Integration of the acceleration signal to velocity and displacement. Usually, velocity signals are preferred , especially for acoustic measurements.

(iii)

Low and high pass filters for damping of frequency components outside of the band to be analyzed, as, for example, around internal resonances and resonances due to the fastening to the measurement object.

It can be an advantage for some applications to use accelerometers with built-in preamplifiers. Such accelerometers are normally less sensitive to electrically induced noise in cables and thereby permit the use of long and inexpensive cables.

167

Chapter 5: Fourier Methods and Measurement Techniques

5.2.5

Mounting of accelerometers

There are three aspects, above all, that must be considered when mounting accelerometers to measurement objects: (i) (ii) (iii)

Measurement direction and placement; see figure 5-42. Mass loading of the measurement object; see figure 5-43. Attachment to the measurement object; see figure 5-44.

Figure 5-42 Measurement direction and placement. The accelerometer should be placed so that the measurement direction coincides with its axis of maximum sensitivity. Most often, the circumstances dictate what would be a useful placement of the accelerometer on the measurement object. If we want to check the condition of a roller bearing, a fundamental principle is to place the transducer as close to the bearing as possible. Position A is better than B for axial vibrations, and C is better than D for radial vibrations. (Sketch: Brüel & Kjær, course material)

A Axis of principal sensitivity (100%)

B

C

Transverse sensitivity ( ρ 2 c 2 . An example of this case would be sound in water, incident upon air. In this case, R < 0 and the sound pressure amplitudes have differing signs., i.e., they are 180˚ phase-shifted. That implies that, at the boundary, an incident positive sound pressure is reflected as negative.

207

Chapter 7: Reflection, transmission and standing waves If ρ1c1 / ρ2c2 → ∞, it follows that R → -1. The sound pressure amplitude of the reflected wave has approximately the same magnitude as that of the incident wave, but is negative-valued. The resulting sound pressure in the immediate vicinity of the boundary asymptotically approaches zero as R goes to -1. Thus far, we have assumed real specific impedances for the two media. Typically, however, medium 2 either has losses, or is limited in its extent. If so, then to describe the reflection against medium 2, we can incorporate such characteristics into a complex specific impedance Z at the boundary, by analogy to (4-115) p p Z= G G= u ⋅ n u⊥

,

(7-23)

G G in which p is the sound pressure at the surface, u is the surface velocity vector, and n is a unit vector directed into medium 2, normal to the surface. In order to handle cases of reflection against solid media, (7-23) can be written as p p Z= G G = v⋅n v⊥

,

(7-24)

G G where v is the velocity vector at the surface, and n is a unit vector directed into the medium, normal to the surface. The impedances given by (7-23) and (7-24) are generally complex, i.e., they have both real and imaginary parts in accordance with (4-115). Applying (7-23) and the boundary conditions (7-14) and (7-15), the impedance of medium 2 is Z2 =

p i ( x = 0, t ) + p r ( x = 0, t ) . u i ( x = 0, t ) + u r ( x = 0, t )

(7-25)

Deriving a reflection coefficient as before, we find that it is complex-valued, pˆ e iδ r Z 2 − ρ1c1 , R = Re iδ r = r = pˆ i Z 2 + ρ1c1

(7-26)

i.e., there is a phase difference δr between the amplitudes that can take on values other than 0 and π radians, which were the only possibilities for the case of real wave impedances. Medium 1

Medium 2

Wr Wt Wi

Figure 7-4 Symbolic picture of acoustic reflection and transmission at a boundary.

x

208

Chapter 7: Reflection, transmission and standing waves When considering incident sound from medium 1, the sound energy that is transmitted into medium 2 can be regarded as absorbed by medium 2. A very central concept in acoustics that describes the absorbing ability of a medium or a boundary is the absorption factor α.. The absorption factor is defined as

α=

W t Wi − W r W = = 1− r Wi Wi Wi

,

(7-27)

where Wi is the incident sound pressure, Wr is the reflected sound pressure and Wt is the transmitted sound power. Because the power can be expressed as W = Ix S, where S is the area and I is the intensity, which can be expressed as I = ~ p 2 / ρ c according to (4-83), 0

x

x

the absorption factor can be expressed as

α = 1+

7.1.3

I x,r I x ,i

= 1−

pˆ r2 pˆ i2

= 1− R

2

.

(7-28)

Propagation of plane waves in a three-dimensional space

Before analyzing the oblique incidence of a wave against a boundary, we consider how a wave can be described when its direction of propagation doesn’t coincide with a coordinate axis. For sound propagation in the positive x-direction in a Cartesian coordinate system, (4-69) implies that p( x ′, t ) = pˆ e i (ωt − kx′) ,

(7-29)

where the ´ (prime) symbol is used to distinguish that coordinate system from coming systems. W av e fron ts

y´ Figure 7-5 Plane wave propagation in the positive x’-direction. The wave fronts are surfaces joining points with identical phase.

G

λ

ey'

G

ex'

x´

To describe multi-dimensi onal propagation, an unprimed coordinate system is introduced. In that system, for simplicity, we begin by studying the propagation in the xy-plane, in order to then generalize to three dimensions. The primed system has been rotated through an angle ϕ1 about the z-axis relative to the unprimed, as shown in figure 7-6.

209

Chapter 7: Reflection, transmission and standing waves

Wave fronts y

y´ Figure 7-6 Plane wave propagation described in two coordinate systems. One has been rotated through an angle ϕ1 about the z-axis.

Ge´

y

x´

Ge G y e´

Gr

x

Ge

ϕ1

x

x

λ

In a so-called orthogonal transformation, the description can be transformed from the G primed to the unprimed system. The position vector r to a point on the x´-axis is indicated in the respective coordinate systems as G G G G (7-30) r = x ′e ′x = xe x + ye y i.e.,

G G G G x ′ = xe ′x ⋅ e x + ye ′x ⋅ e y ,

(7-31)

G G G G where ei ⋅ e j = cos(ei , e j ) in the transformation theory are usually called transformation G G coefficients, and are cosines of the angles between the base vectors ei and e j . The

expression (7-31) can also be stated in the form x ′ = x cos ϕ1 + y cos(90 D − ϕ1 ) = x cos ϕ1 + y sin ϕ1 ,

(7-32)

and (7-29) transforms in the unprimed system to p( x, y, t ) = pˆ e i (ωt − kx cos ϕ1 − ky sin ϕ1 ) . (7-33) G To further generalize the discussion, a unit vector n is introduced to designate the direction of propagation; it is expressed the respective coordinate systems as G G G G n = e ′x = n x e x + n y e y . (7-34)

G G G G From (7-34), applying the orthogonality relations e x ⋅ e x = 1 and e x ⋅ e y = 0 , it follows

that G G G G n x = e ′x ⋅ e x = cos(e ′x , e x ) = cos ϕ1 ,

(7-35)

G G G G n y = e ′x ⋅ e y = cos(e ′x , e y ) = sin ϕ1 .

(7-36)

The wave number vector is defined as

G G k =k ⋅n,

210

(7-37)

Chapter 7: Reflection, transmission and standing waves

G with a magnitude k = ω /c, and a direction n identical to the direction of propagation; it can be expressed as G G G G G k = k (n x e x + n y e y ) = k cos ϕ1 e x + k sin ϕ1 e y . (7-38) Thus, the components of the wave number vector, i.e., its x and y-axis projections, are

k x = k cos ϕ1 , k y = k sin ϕ1 ,

(7-39) (7-40)

respectively, and we conclude that the most general form of the solution becomes G

or in component form

G G p( r , t ) = pˆ e i (ωt − k ⋅r ) ,

(7-41)

G i (ωt − k x x − k y y ) p( r , t ) = pˆ e .

(7-42)

In three dimensions, it follows by analogous logic that G

G G i (ωt − k x x − k y y − k z z ) r p(r , t ) = pˆ e i (ωt − k ⋅ ) = pˆ e

where

,

G G k ⋅ r = constant,

(7-43) (7-44)

constitute surfaces of constant phase. Entering (7-43) into the wave equation (4-43) ∂2 p ∂x 2

provides the condition

+

∂2 p ∂y 2

+

∂2 p ∂z 2

=

1 ∂2 p

(7-45)

c 2 ∂t 2

G ω k = k = = kx2 + k y2 + k z2 c

.

(7-46)

That condition is an important relation that will be utilized in the discussion that follows.

7.1.4

Oblique incidence on a boundary between two fluid media

In order to analyze what happens when a plane acoustic wave with a certain angle of incidence θi reaches the bounding surface between two fluid media, it is necessary to supplement the types of boundary conditions used up to this point. These boundary conditions, which require continuity of pressure and particle velocity across the boundary surface, are supplemented with the condition that the incident, reflected, and transmitted waves have the same periodicity along the boundary surface, i.e., the plane x = 0 in figure 7-7.

211

Comment [UC1] (4-43)

Chapter 7: Reflection, transmission and standing waves

Medium 1

Medium 2

Z1 =ρ1c1

y

pr

λr

θr λi

θi

Z2 =ρ2c2

λr

y p

t

⇒

θt

λi

λi

x

λ t λt

λr

pi

θi θr

θt x

λt

Figure 7-7 Oblique incidence against a boundary surface between two fluid media.

From that condition, illustrated in figure 7-7, the projected wavelengths are equal, i.e.,

λi λt λr . = = sin θ i sin θ r sin θ t

(7-47)

On the side from which the incident wave arrives, the incident and reflected waves traverse the same medium, λi = λr, which yields θi = θr, i.e., the angle of reflection is equal to the angle of incidence. Equation (7-47) can be rewritten with the help of the relation c = f λ. Since the frequency is the same, f = c1/λ1 = c2/λ2 , and (7-47) can therefore be expressed as c1 c = 2 sin θ i sin θ t

.

(7-48)

That relation is called Snell’s law, and is also known from optics. From Snell’s law as a point of departure, two special cases are considered; (i)

c1 > c 2 .

Examples of this case are incident waves in water reflecting off an air-water interface. Snell’s law (7-48) dictates that the transmission angle satisfies sin θ t =

c2 sin θ i . c1

(7-49)

The case c1 > c2 yields θt < θi, i.e., the transmitted wave is redirected in closer to the normal, as shown in figure 7-8a. With the condition c1 > c2, θt has a maximum for grazing incidence, i.e., for θi = 90˚, as

212

Chapter 7: Reflection, transmission and standing waves

θ t max = arcsin

c2 . c1

(7-50)

The relationship is illustrated in figure 7-8b. a)

b) Medium 1

c1 > c 2

y

Medium 1

Medium 2

y

c1 > c 2

p

Possible angles Möjliga ofutbredningspropagation vinklar för for the transmitterad transmitted wave våg

p

r

t

p

θr θ

Medium 2

t

θ t max = arcsin

θt x

θi

i

pi

c2 c1

x

p

i

Figure 7-8 a) The transmitted wave bends is deflected closer to the normal. b) The illustration of potential angles of propagation for the transmitted wave.

(ii)

c1 < c 2 .

An example of this case would be sound in air, incident on an air-water interface. According to Snell’s law (7-48), θt > θi, i.e., the transmitted wave is deflected away from the normal; see figure 7-9a. For an increasing angle of incidence θi, we reach a boundary case, an angle of incidence θic, where θt = 90˚ and we have a transmitted wave that grazes the boundary surface. The angle of incidenceθic is given by θ ic = arcsin(c1 c 2 ) .

(7-51)

For θi > θic, a total reflection of the incident wave occurs. Thus, a transmitted wave only exists for angles of incidence for which θi ≤ θic. The relation is evident in figure 7-9b.

213

Chapter 7: Reflection, transmission and standing waves

a)

b) Medium 1

p

y

c1 < c 2

Medium 2

Medium 1

p

r

p

t

t

θt = 90°

θt θi

Medium 2

y

c1 < c 2

x

p

θi c

i

x

pi

Incident waves in this are totally reflected

Medium 1

Figure 7-9 a) The transmitted wave deflects away from the normal. b) Illustration of possible angles of incidence such that a transmitted wave exists.

We limit the analysis to apply to the case in which a transmitted wave exists. The following assumption can then be made for the sound pressure: p i ( x, y, t ) = pˆ i e p r ( x, y, t ) = pˆ r e p t ( x, y, t ) = pˆ t e

i (ωt − k1x x − k1 y y )

i (ωt + k1x x − k1 y y )

i (ωt − k2 x x − k2 y y )

= Rpˆ i e = Tpˆ i e

,

(7-52)

i (ωt + k1x x − k1 y y )

i (ωt − k 2 x x − k 2 y y )

,

(7-53)

.

(7-54)

Here, the pairs k1x, k1y and k2x , k2y represent components of the wave number vectors in each medium. R is the reflection coefficient, defined in (7-19),

R = pˆ r pˆ i

(7-55)

and T is the transmission coefficient, as defined in (7-21),

T = pˆ t pˆ i .

(7-56)

It turns out that both R and T are real, since the specific impedances of the media are; thus, to avoid unnecessary complication, they are not expressed as complex. Boundary conditions analogous to those for normal incidence can be set up. Continuity of pressure at x = 0 gives

p i ( x = 0, y, t ) + p r ( x = 0, y, t ) = p t ( x = 0, y, t ) .

(7-57)

Putting the pressure in the incident wave expression (7-52), in the reflected wave expression (7-53), and in the transmitted wave expression (7-54) yields

pˆ i e

i (ωt − k1 y y )

+ pˆ r e

i (ωt − k1 y y )

where Snell’s law, (7-48), finally yields

214

= pˆ t e

i (ωt − k2 y y )

,

(7-58)

Chapter 7: Reflection, transmission and standing waves pˆ i + pˆ r = pˆ t .

(7-59)

The boundary condition for continuity of particle velocity normal to the boundary surface is

u i ( x = 0, y, t ) cos θ i + u r ( x = 0, y, t ) cos θ i = u t ( x = 0, y, t ) cos θ t .

(7-60)

For plane waves, according to (4-76) and (4-77),

u = ±p ρ 0 c

(7-61)

and, using k1y = k2y, as above, then (7-60) can be expressed as pˆ i pˆ t pˆ cos θ i − r cos θ i = cos θ t . ρ1c1 ρ1c1 ρ 2 c2

(7-62)

From (7-59) and (7-62), the reflection coefficient R takes the form R=

pˆ r ρ 2 c 2 cos θ i − ρ1c1 cos θ t = pˆ i ρ 2 c 2 cos θ i + ρ1c1 cos θ t

(7-63)

and the transmission coefficient T according to T=

2 ρ 2 c 2 cos θ i pˆ t = pˆ i ρ 2 c 2 cos θ i + ρ1c1 cos θ t

.

(7-64)

In contrast to the derivation in the next section, we have taken account of wave propagation in medium 2 in this case. The surface is said to have a distributed response.

7.1.5

Oblique incidence from a fluid against a solid medium

The case of a plane wave reflecting obliquely against a solid medium is more complex than the corresponding reflection at the boundary surface between two fluids. The reason is that the solid medium, as opposed to the fluid, can support shear stresses. The general case of a stiff elastic medium, such as concrete for example, must account for the propagation of both longitudinal compress ional waves (such as those we treat in this chapter) and shear or transverse waves, in which the particle motion is normal to the direction of propagation. We now consider a special case, a so-called locally-reacting surface. To treat a reflection at a locally-reacting surface, every point on it is regarded as completely isolated from all other such points. For such a surface, the impedance becomes p Z= v , ⊥

(7-65)

independently of the angle of incidence θi of the sound. The velocity v⊥ of any point on the surface depends exclusively on the sound pressure p acting on that point. If we excite a point on the surface, no other points on the surface move and the acoustic disturbances in the locally-reacting medium only propagate perpendicular to the surface.

215

Chapter 7: Reflection, transmission and standing waves In summary, a locally-reacting surface can be regarded as such that (7-65) applies to every point with a determined, usually frequency-dependent, value of Z2, independently of the character of the acoustic field; see figure 7-10.

Medium 1 Z1 =ρ1c1

y

Medium 2 Z2

pr pt x

θi

Figure 7-10 Oblique incidence against a locally-reacting medium.

pi

The point impedance of the locally reacting surface, as described by (7-65), must ordinarily be assumed complex. That implies that, in deriving the reflection coefficient, the phase shifts of the reflected and transmitted waves must be accounted for, in contrast to the treatment in section 7.1.2. Continuity of pressure, as stated in (7-14), implies that pˆ i + pˆ r e iδ r = pˆ t e iδ t ,

(7-66)

and continuity of particle velocity normal to the boundary surface, as expressed in (7-17) and (7-65), that pˆ i pˆ e iδ t pˆ . (7-67) cos θ i − r e iδ r cos θ i = t ρ1c1 ρ1c1 Z2 Eliminating pˆ t eiδ t , yields the reflection coefficient R = Re iδ r =

pˆ r e iδ r Z 2 cos θ i − ρ1c1 = pˆ i Z 2 cos θ i + ρ1c1

.

(7-68)

The ideal locally-reacting surface, as described above, is a model, i.e., a means to simplify the description of a complicated reality. Below, we discuss the circumstances under which real media can, to a reasonable degree of precision, be described as locally-reacting. (i)

Anisotropic medium.

An anisotropic structure, e.g., a perforated structure as in figure 7-11a, or a honeycomb structure as in figure 7-11b, can effectively block particle motion parallel to the surface, while permitting motion perpendicular to it. That type of structure is used as a sound absorbent. The physical principle is that energy from the incident acoustic field is converted to heat by the viscous forces associated with the particle motion in and out of the cavities.

216

Chapter 7: Reflection, transmission and standing waves

a)

b) Figure 7-11 Sound absorbent in the form of a) perforated panel, and b) honeycomb structure.

(ii)

Medium with significant losses.

Medium 2 has large losses, i.e., the acoustic disturbances are strongly damped out. Examples are acoustic absorbents, such as various types of mineral wools, as well as earth surfaces with markedly porous character. (iii)

Medium with significant compliance.

We return to section 7.1.4, in which oblique incidence between two fluid media is treated, and especially focus on the case in which c2 « c1. In that case, there is a considerable deflection towards the normal, as in figure 7-8a. The transmission angle θt is relatively independent of θi , and the situation approaches that which applies to a locally-reacting surface. If the specific impedance of medium 2, i.e., ρ2c2 for a surface with a distributed response, and Z2 for a locally-reacting surface, is finite, then R approaches –1, since θi approaches 90° in formulas (7-63) and (7-68). That implies that the amplitude of the reflected wave is unchanged, but phase-shifted by 180°. That situation has important consequences for, among other things, sound propagation along absorbing walls and ceilings; see example 7-1.

Example 7-1

"Bad sound" due to the destructive interference resulting from grazing incidence?

7.2 EIGENFREQUENCIES AND EIGENMODES IN A THREE-DIMENSIONAL SPACE With the aid of wave-theoretical argumentation, this section will study how characteristic sound fields, standing waves, can be built up in such three-dimensional cavities as dwellings, lecture halls, and vehicle cabins. The method is based upon using solutions of the wave equation to set up mathematical statements of the boundary conditions that apply at the bounding surfaces of the room, such as floors, walls, and ceilings. The difficulty resides in determining these boundary conditions for irregular spaces, such as churches or rooms with bulky contents; as a result, exact analytical solutions only prove possible for 217

Chapter 7: Reflection, transmission and standing waves especially simple geometries, such as spherical, cylindrical, and parallelepipedic rooms. Despite the practical limitations, the theory is nevertheless essential to the understanding of MANY acoustic phenomena. Some pertinent questions of interest are: (i)

Why does a speaker sound one way in a certain room, and another way elsewhere?

(ii)

Why is it that there are very distinct maxima and minima in the sound field of an airplane cabin?

(iii)

Where, in a workspace, should a machine be located in order to minimize the resulting acoustic disturbance?

7.2.1

Parallelepipedic room

Let us consider a wave motion in a parallelepipedic room with dimensions as indicated in figure 7-12.

y

z

Figure 7-12 Parallelepipedic room with the dimensions lx, ly and lz.

lz

ly x

lx

The medium is loss-free and the walls are infinitely rigid. The boundary condition at all bounding surfaces is that the particle velocity normal to the surface be zero. For plane wave propagation in a three-dimensional space, then in general, according (7-43), G i (ωt ± k x x ± k y y ± k z z ) p(r , t ) = pˆ e ,

(7-69)

where the minus sign stands for propagation in the respective coordinate axis’ positive direction, and the plus sign for propagation in the corresponding negative direction. Because all possible combinations of propagation directions may occur, the sound field in the room will consist of eight waves, i.e., G −i ( k x x + k y y + k z z ) i(k x+ k y +k z ) p(r , t ) = ( pˆ 1e + pˆ 2 e x y z + + pˆ 3 e + pˆ 6 e

−i ( k x x + k y y − k z z )

i(k x x −k y y + k z z )

+ pˆ 4 e

+ pˆ 7 e

i(k x x+ k y y −k z z )

i(k x x −k y y −k z z )

+ pˆ 5 e

+ pˆ 8 e

−i ( k x x − k y y + k z z )

−i ( k x x − k y y − k z z )

+

) e iω t .

(7-70)

Putting (7-70) into the wave equation (6-43), expressed in Cartesian coordinates, yields the condition that the components kx, ky and kz of the wave number must satisfy, as in (7-46),

218

Chapter 7: Reflection, transmission and standing waves G ω k = k = = k x2 + k y2 + k z2 c

.

(7-71)

Practice Exercise 7-1 Show, by putting (7-70) into the wave equation (4-43), that the relation given above holds. Because the walls are rigid, the particle velocity perpendicular to the surface must be zero. Thus, the boundary condition is ux= 0 at x = 0 and x = lx,

uy= 0 at y = 0 and y = ly,

(7-72)

uz= 0 at z = 0 and z = lz. According to section 7.1.1, reflection at an infinite rigid surface occurs without change in amplitude or phase, and from that it follows that

pˆ 1 = pˆ 2 = pˆ 3 = pˆ 4 = pˆ 5 = pˆ 6 = pˆ 7 = pˆ 8 .

(7-73)

Setting pˆ 1 , pˆ 2 ,..., pˆ 8 to pˆ 8 and developing the exponential terms into the form eiθ = cosθ + isinθ, yields G (7-74) p(r , t ) = pˆ cos(k x x) cos(k y y ) cos(k z z )e iωt . The particle velocity can be determined with the help of the equation of motion (4-25) G ∂u ρ0 + ∇p = 0 . (7-75) ∂t G In component form, u x (r , t ) is obtained as G 1 u x (r , t ) = −

∂p

1

dt = − ρ 0 ∫ ∂x ρ 0 iω

∂p . ∂x

(7-76)

Putting the pressure (7-74) into the expression for particle velocity in the x-direction (776), k G u x (r , t ) = pˆ x sin( k x x ) cos( k y y ) cos( k z z ) e iω t . iωρ 0

(7-77)

The other velocity components are found by the same approach to be ky G sin( k y y ) cos( k x x ) cos( k z z ) e iω t , u y (r , t ) = pˆ iωρ 0

(7-78)

k G u z (r , t ) = pˆ z sin( k z z ) cos( k x x ) cos( k y y ) e iω t . iωρ 0

(7-79)

219

Chapter 7: Reflection, transmission and standing waves From (7-77), the boundary condition at x = 0 is clearly satisfied. Moreover, to satisfy that at x = lx, the condition sinkxlx = 0 is required; from that, the components of the wave number vector become k x = n xπ l x

,

(7-80)

satisfied for any integer-valued nx, i.e., nx = 0, 1, 2, ... . In a corresponding wave, the other components are k y = n yπ l y

,

(7-81)

k z = n zπ l z

,

(7-82)

where ny and nz, in the same fashion, can take on the values 0, 1, 2, ... . Putting (7-80), (781) and (7-82) into (7-71), and using the relation 2π f = ω = kc, the eigenfrequencies or eigenvalues of the parallelepipedic room are found to be c f (n x , n y , n z ) = 2

⎛ nx ⎜ ⎜l ⎝ x

2

⎞ ⎛⎜ n y ⎟ + ⎟ ⎜l ⎠ ⎝ y

2

⎞ ⎟ + ⎛⎜ n z ⎜l ⎟ ⎝ z ⎠

⎞ ⎟⎟ ⎠

2

.

(7-83)

Thus, the room has an eigenfrequency and a corresponding eigenmode for every combination of the indices nx, ny and nz. The composite sound pressure from all eigenmodes is obtained as p( x, y, z , t ) = ∑ pˆ i cos(k xi x) cos(k yi y ) cos(k zi z ) e iωit

,

(7-84)

i

where i stands for the summation over all excited eigenmodes with their respective unique combinations of indices nx, ny and nz, and where pˆ i is the amplitude of the individual mode form.

Example 7-2 A so-called reverberant room is a type of acoustic measurement room in which a minimal absorption by walls, floors, and ceilings is sought, as well as a uniform distribution of eigenfrequencies along the frequency axis. Table 7-1 lists all of the eigenfrequencies under 100 Hz calculated for such a room at the Marcus Wallenberg Laboratory for Sound and Vibration Research, MWL, KTH. The room has the dimensions lx = 7.86 m, ly = 6.21 m, lz = 5.05 m. For the sound speed c, a value of 340 m/s has been assumed.

220

Chapter 7: Reflection, transmission and standing waves Table 7-1 The calculated eigenfrequencies under 100 Hz of a reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m at MWL, KTH.

nx ny nz

f [Hz]

nx ny nz

f [Hz]

nx ny nz

f [Hz]

nx ny nz

f [Hz]

100 010 001 110 101 200 011 111 210 020 201

21.6 27.4 33.7 34.9 40.0 43.3 43.4 48.5 51.2 54.7 54.8

120 211 021 300 002 121 220 310 102 012 301

58.9 61.3 64.3 64.9 67.3 67.8 69.8 70.4 70.7 72.7 73.1

112 221 311 202 030 212 320 130 400 022 031

75.8 77.5 78.1 80.0 82.1 84.6 84.9 84.9 86.5 86.8 88.7

122 410 321 131 230 401 302 411 222 312 231

89.4 90.7 91.3 91.4 92.8 92.8 93.5 96.8 97.0 97.4 98.7

Evidently, the lowest eigenfrequency 21.6 Hz is obtained for nx = 1, ny = nz = 0. That implies a standing wave between those walls which stand furthest apart, i.e., lx = 7.86 m. A quick calculation shows that that distance corresponds to a half wavelength at 21.6 Hz. At low frequencies, the eigenfrequencies are relatively sparse, but they become all the more densely spaced as the frequency rises. The eigenfrequencies (0,0,1) at 33.7 Hz and (1,1,0) at 34.9 Hz are only spaced 1.2 Hz apart. Above that, there is a jump of 5.1 Hz to the next eigenfrequency (1,0,1) at 40.0 Hz. Eigenmodes can be divided into three categories: (i)

Axial eigenmodes.

Two of the indices nx, ny and nz are zero. From equation (7-84), it is apparent that the sound pressure only varies in one coordinate direction, and is constant in the other two. The particle velocity, moreover, is parallel to a coordinate axis. Consequently, there are absorption losses only at the walls at which the particle velocity is normal to the surface. Figure 7-13a provides a symbolic depiction of the spatial dependence of the normalized sound pressure, in the form of a graph of the eigenmode (2,0,0), which has the form p( x) 2π x . = cos pˆ lx

(7-85)

In figure 7-13b, the graph of the normalized particle velocity is presented; the expression is u x ( x) 2π = sin x . (7-86) uˆ x lx Note that where the sound pressure has an antinode (relative maximum), the particle velocity has a node, and vice versa.

221

Chapter 7: Reflection, transmission and standing waves 1 u( x) u^

maximum

p(x) ^ p

1

maximum

node

node 0

0

-1 x=0

a)

x = lx

-1 x=0

x = lx

b)

Figure 7-13 a) Symbolic picture of the spatical dependences of a) the normalized sound pressure and b) the normalized particle velocity, for the (2,0,0) mode. Compare these to figure 7-2.

(ii)

Tangential eigenmodes.

One of the indices nx, ny and nz is zero. The particle velocity is parallel to a pair of the walls and has components in two of the coordinate axes directions. Figure 7-14a illustrates the plane wave fronts and the propagation directions for the mode form (1,1,0). Figure 714b shows a symbolic picture of the sound pressure contours for the same eigenmode.

y

x Figure 7-14a Illustration of the plane wave fronts and directions of propagation for the tangential mode form (1,1,0). (Source: Brüel & Kjær, Architectural Acoustics.)

222

Chapter 7: Reflection, transmission and standing waves

y 1,0

p/pmax 0,8 0,6 0,4

0,4 0,2

0,2

0

0,8

1,0

0,6

0

0

0,2

0,4

0,4

0,6 1,0

1,0

0,2

0 0

0,8

0,8

0,6

x

Figure 7-14b Symbolic picture of the normalized sound pressure contours for the tangential mode form (1,1,0). (Source: Brüel & Kjær, Architectural Acoustics.)

(iii)

Oblique eigenmodes.

All of the indices nx, ny and nz are non-zero. The particle velocity has components in all three coordinate axes directions, resulting in reflections from all six walls. That circumstance is decisive for how acoustic absorbents shall be mounted in a room to prevent the development of strong standing wave field, so-called resonances. Figure 7-15 shows a symbolic picture of the sound pressure contours of the eigenmode (1,2,1). The situation we have studied above, i.e., the determination of z mode shapes of sound pressure and particle velocity is the socalled homogeneous problem of free oscillations, in which we have only taken account of initial and boundary conditions. The boundary condition was that the walls be loss-free and rigid, and the initial condition just that the oscillation had y already started at t = 0, after which no further excitation or damping took place; i.e., the x medium was source-free and lossless. Figure 7-15 Symbolic picture of the normalized sound pressure In reality, it is normally contours for the oblique eigenmode (1,2,1). (Picture: Brüel & Kjær, Technical Reveiw) the so-called particular problem with forced oscillations that is of

223

Chapter 7: Reflection, transmission and standing waves interest, i.e., the room is continually supplied sound power from a source. With boundary conditions as given above, the sound pressures and particle velocities become infinite, because the walls (and the medium) are assumed loss-free. That such does not, in fact, occur, is because even apparently rigid walls absorb some fraction, or some percentage, of the incident sound field. These losses are, however, balanced in stationary sound fields by the energy supplied by the sound source. Bearing that in mind, we can only use the derived relations for sound pressure and particle velocity to draw qualitative conclusions. When a sound source excites a room at certain frequencies, it is only those frequencies that exist in the room. Nevertheless, the sound field is largely built up of the eigenfrequencies and mode shapes that coincide with, or fall in the vicinity of, the frequencies sent out by the sound source. How strongly a particular mode is excited also depends on where in the room the sound source is placed, in relation to the nodes and antinodes of the eigenmode. Provided that the excitation is from a constant velocity source, i.e., a source that has a certain, fixed vibration velocity which is independent of where it is located in the room, the greatest excitation of an eigenmode occurs if the source is located at a particle velocity node. As is evident from figure 7-13, a particle velocity node corresponds to a sound pressure anti-node. Sketching the sound pressure contours of different eigenmodes in accordance with figures 7-13a and 7-14b, it turns out that every eigenmode has a sound pressure antinode in the corner of the room. Moreover, it turns out that only one eighth of all modes have a non-zero sound pressure in the geometric midpoint of the room. A sound source placed in a corner is, therefore, optimally-placed if it is desired to maximally excite the eigenfrequencies of a room. Similarly, if a microphone is placed in a corner, it registers the maximum sound pressure for every excited eigenmode. _________________________________________________________________________

Example 7-3 In order to investigate how the eigenmodes of a room “color” the sound at a certain listener position, given a loudspeaker at another given location, one can proceed as follows. In a reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m, a loudspeaker has been placed in one of the corners. A microphone is placed in the diagonally opposite corner. The loudspeaker sends out a noise of relatively constant amplitude in a frequency band of interest, while the sound pressure level is measured by the microphone. The result is shown in figure 7-16.

224

Chapter 7: Reflection, transmission and standing waves Lp [dB] 100

(1,1,1)

90 (1,0,1)

80

(2,0,0)

(1,1,0)

(0,1,1)

70 (0,0,1) (0,1,0)

(1,0,0)

60 50 40 20

25

30

35

40

Frekvens [Hz]

45

50

Figure 7-16 Sound pressure levels measured in a corner of a reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m. The sound source, a loudspeaker, was placed in a diagonally opposite corner (Compare to the calculated eigenfrequencies in table 7-1, example 7-2).

The indices of the eigenmodes corresponding to each peak in the curve are indicated above them. _________________________________________________________________________ It is often of great interest to know how many eigenfrequencies fall within a certain frequency band, such as a third-octave band for instance. Starting with expression (7-83), we can see that the frequency spacing of purely axial modes in the x, y, or z-directions is f (n x = 1, n y = n z = 0) =

c , 2l x

(7-87)

f ( n y = 1, n x = n z = 0) =

c , 2l y

(7-88)

f ( n z = 1, n x = n y = 0) =

c . 2l z

(7-89)

These axial eigenfrequencies may be marked out in a frequency space with the coordinate axes fx, fy and fz as in figure 7-17. From the Pythagorean theorem and relations (7-87), (788) and (7-89), it turns out that every eigenfrequency f (nx, ny, nz) is represented in that frequency space and that the distance from the point f (nx, ny, nz), corresponding to the (nx, ny, nz)- mode, to the origin is a measure of the eigenfrequency for that specific mode shape. All eigenmodes at or below a certain frequency f are included in the eighth-sphere in the frequency space lying between the positive fx-, fy-, fz -axes and the spherical surface with radius f.

225

Chapter 7: Reflection, transmission and standing waves

fz

f c/2l z fy Figure 7-17 Schematic picture of discrete eigenfrequencies f(nx, ny, nz) in a frequency space.

c/2l x c/2l y

fx

The total number of eigenmodes falling below a certain frequency f can be determined by counting the discrete points in the frequency space developed above. That being a timeconsuming task, it is more practical to try to develop a formula that provides the same information. To each point representing an eigenmode, according to (7-87), (7-88) and (7-89), there is a corresponding cube with a volume V=

c c c 2l x 2l y 2l z

in the frequency space. The number of eigenmodes N with an eigenfrequency less than f is obtained by dividing the volume of an eighth-sphere (of radius f ) with the volume that a single mode represents, from which 1 4π f 3 4π f 3 N≈ 8 3 = lxl ylz . c c c 3c3 2l x 2l y 2l z

(7-90)

Formula (7-90) is, however, incomplete. Certain eigenmodes are underrepresented. That applies to the tangential modes with a zero-value index, and which therefore lie on one of the three planes that contain a pair of coordinate axes. It also applies to the axial eigenmodes with two of the indices equal to zero, and which therefore lie along one of the frequency axes. Correcting for those, the total number of eigenmodes with an eigenfrequency lower than f is N≈

4π f 3 π f2 f V+ S+ L 3 2 c 8 3c 4c

226

,

(7-91)

Chapter 7: Reflection, transmission and standing waves where

V = l x l y l z is the volume of the room, S = 2(l x l y + l x l z + l y l z ) is the bounding surface area of the room, and L = 4(l x + l y + l z ) is the perimeter of the room.

In rather large rooms at high frequencies, the first term in (7-91) dominates. It can also be shown that as f → ∞, (7-91) also applies to rooms of forms other than parallelepipedic.

Example 7-4 Determine, with the aid of (7-91), the number of modes under 100 Hz for the reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m, which was treated in example 7-2. Compare the calculated number with the actual number that can be counted from table 7-1. Solution Setting V = 246.49 m3, S = 239.73 m2, L = 76.48 m and c = 340 m/s, then N = 26.27 + 16.29 + 2.81 = 45.37, which is in good agreement with the actual counted result of 44, from table 7-1. Differentiating equation (7-91) with respect to frequency,

πf dN 4π f 2 1 ≈ V+ S+ L , 3 2 df 8c 2c c from which it is evident that the number of eigenmodes ΔN in a frequency band B = fu – fl, centered around f, is ⎛ 4π f 2 π f 1 ⎞ V+ S + L ⎟B . ΔN ≈ ⎜ (7-92) ⎜ c3 8c ⎟⎠ 2c 2 ⎝ It should be noted that relations (7-91) and (7-92) are only approximate, and that the precision increases with the frequency and the bandwidth. _________________________________________________________________________ Example 7-5 Determine, for the reverberant room of example 7-2, the number of eigenmodes in the 80 Hz third-octave band. It spans the range of frequencies from 80 6 2 = 71.3 Hz up to

80 ⋅ 6 2 = 89.8 Hz . Solution Putting the values from example 7-4, as well as B = 18.5 Hz , into equation (7-92), yields Δ N =14.67. That is in good agreement with the 14 that can be counted from table 7-1. The number of modes in the third-octave band around 1000 Hz is, for the reverberant room of example 7-4, over 18000, which implies more than 78 eigenfrequencies per Hz. Thus, at medium to high frequencies, the eigenfrequencies are so densely spaced that it becomes completely impractical to consider individual eigenfrequencies and modes. Instead, it is fruitful to switch to energy methods, or so-called statistical room acoustics. See chapter 7.

227

Chapter 7: Reflection, transmission and standing waves

7.3

IMPORTANT RELATIONS

Normal incidence against a boundary between two elastic media Boundary conditions at the surface Continuity of pressure p i ( x = 0, t ) + p r ( x = 0, t ) = p t ( x = 0, t )

(7-14)

Continuity of particle velocity u i ( x = 0, t ) + u r ( x = 0, t ) = u t ( x = 0, t ) R=

Reflection coefficient

pˆ r pˆ i

(7-15) (7-19)

ρ1c1 ρ c − ρ1c1 ρ 2 c2 R= 2 2 = ρ c ρ 2 c 2 + ρ1c1 1+ 1 1 ρ 2 c2

(7-20)

pˆ T= t pˆ i

(7-21)

1−

Transmission coefficient

T=

2ρ 2 c2 = ρ 2 c 2 + ρ1c1

2 1+

ρ1c1 ρ 2 c2

(7-22)

Complex impedances In general,

p p Z= G G= u ⋅ n u⊥

(7-23)

Solid media

p p Z= G G = v⋅n v⊥

(7-24)

Reflection coefficient

Absorption factor

R = Re iδ r =

α=

pˆ r e iδ r Z 2 − ρ1c1 = pˆ i Z 2 + ρ1c1

Wt Wi − Wr W = =1− r Wi Wi Wi

228

(7-26)

(7-27)

Chapter 7: Reflection, transmission and standing waves

α = 1+

I x ,r I x,i

2

⎛ pˆ ⎞ 2 = 1 − ⎜⎜ r ⎟⎟ = 1 − R ⎝ pˆ i ⎠

(7-28)

Propagation of plane waves in a three dimensional space G

Sound pressure

G G i (ωt − k x x − k y y − k z z ) r p(r , t ) = pˆ e i (ωt − k ⋅ ) = pˆ e

(7-43)

G ω k = k = = k x2 + k y2 + k z2 c

Wave number vector

(7-46)

Oblique incidence against a boundary surface between two fluid media c1 c = 2 sin θ i sin θ t

Snell’s law

Reflection coefficient

R=

(7-48)

pˆ r ρ 2 c 2 cos θ i − ρ1c1 cos θ t = pˆ i ρ 2 c 2 cos θ i + ρ1c1 cos θ t

(7-63)

pˆ t 2 ρ 2 c 2 cos θ i = pˆ i ρ 2 c 2 cos θ i + ρ1c1 cos θ t

(7-64)

Transmission coefficient T=

Oblique incidence from a fluid against a solid medium Reflection coefficient R = Re iδ r =

pˆ r e iδ r Z 2 cos θ i − ρ1c1 = Z 2 cos θ i + ρ1c1 pˆ i

(7-68)

Eigenfrequencies and eigenmodes in a three-dimensional space c Eigenfrequencies f (n x , n y , n z ) = 2

⎛ nx ⎜ ⎜l ⎝ x

2

2 ⎞ ⎛⎜ n y ⎞⎟ ⎛ nz ⎟ + ⎟ ⎜ l ⎟ + ⎜⎜ l ⎝ z ⎠ ⎝ y ⎠

⎞ ⎟⎟ ⎠

2

(7-83)

Eigenmodes, sound pressure p ( x, y , z , t ) =

∑ pˆ i cos( k xi x ) cos( k yi y ) cos( k zi z ) eiω t i

229

(7-84)

Chapter 7: Reflection, transmission and standing waves Components of the wave number vector kx =

ky = kz =

n xπ lx

(7-80)

n yπ

(7-81)

ly n zπ lz

(7-82)

Number of eigenfrequencies under f [Hz] N≈

4π f 3c

3

3

V+

π f2 4c

2

S+

f L 8c

(7-91)

Number of eigenfrequencies in frequency band B ⎛ 4π f 2 π f 1 ΔN ≈ ⎜ V+ S+ 2 ⎜ c3 8 c 2c ⎝

230

⎞ L ⎟B ⎟ ⎠

(7-92)

CHAPTER EIGHT

WAVE EQUATIONS AND THEIR SOLUTIONS IN SOLID MEDIA

Vibrations in technical systems are, for various reasons, attracting ever more attention. The direct consequences of vibrations, such as durability issues and comfort problems, are well known. There are also indirect consequences of vibrations, such as their impact on function, manufactured product quality, and sound generation. The study of sound is closely connected to the study of vibrations. There are several reasons for that. First of all, vibrations caused by fluctuating forces acting on solid materials are very important sources of acoustic energy. Secondly, vibrations in solid structures are very important carriers of acoustic energy. Thirdly, vibrations of surfaces that bound air volumes generate and radiate sound into the air. Finally, the most effective approaches to mitigating noise are usually those nearest to the source of the noise. A strong foundation of knowledge of vibrations in solid media and structures is, for those reasons, extraordinarily important to both practical and theoretical noise control. This section imparts fundamental knowledge of oscillations and vibrations in solid media. The emphasis of the material is on those parts which are essential to the acoustician. The depiction begins with wave propagation in infinite media. After that, the effects of limits on the spatial extent of the structure are considered. In that context, the important topic of standing waves is also dealt with. Finally, a treatment of losses in solid materials, and their mathematical representation, is provided.

231

Chapter 8: Wave equations and their solutions in solid media 6.1

INTRODUCTION

Vibrations in structures are of great significance in many important technical problems. Comfort and performance of vehicles and vessels are considerably reduced by vibrations. Most noise problems one encounters in technical systems contain a link in which the acoustic disturbances are transmitted in the form of structural vibration; see figure 8-1. Many destructive accidents in technical systems are, directly or indirectly, caused by fatigue failure of parts subjected to vibrations. The functioning of most machines is negatively impacted by vibrations. If, for example, the tool in a machining operation vibrates, the cutting precision is degraded. To o find a rational and effective solution to these problems, it is necessary to be able to analyze the vibrations of a structure. That demands, in turn, fundamental knowledge of wave propagation in solid media.

Ω ut

Ω in

Figure 8-1 Acoustic energy is often transferred in the form of vibrations in solid media. Often, the source mechanism is that of time dependent contact forces between moving elements of the structure. In the case illustrated, fluctuating forces are generated in the contact zone of the meshing gear teeth. The resulting vibrations spread through the axles and bearings to the rest of the machine. Any attached large, easily excited plates of sheet metal radiate sound strongly.

Because of the great technical significance of vibrations, a number of special disciplines for the analysis of vibrations have arisen in the recent past. In experimental modal analysis, experimentally-determined characteristics of a structure are used to construct a mathematical model of the structure’s vibration behavior. Today, many technical products undergo vibration testing. The product is subjected to a predefined dynamic loading program, using a shaker. Afterwards, possible changes in the properties of the structure are analyzed. By measuring the vibrations at certain, well-selected points, one seeks by machinery monitoring to determine the condition of the measured machine. The idea is that a change in the machine’s condition reveals itself by a change in the machine’s vibrations. If, for example, bearing faults were to develop, certain characteristic frequency

232

Chapter 8: Wave equations and their solutions in solid media components might appear in the vibration spectrum. That information can be used to determine whether or not the machine should be shut down for inspection or service. 8.2

WAVE PROPAGATION IN INFINITE AND SEMI-INFINITE MEDIA

The difference between a fluid and a solid medium is essentially the ability of the latter to support shear stresses. If the medium is regarded as built up of layers, one upon another, there is, in a solid medium, a resistance to sliding of the layers relative to each other. As a result, a solid medium exhibits more types of wave motion than a liquid medium does. In an unbounded solid medium, two types of waves can propagate independently of one another: longitudinal waves and transverse waves. The longitudinal wave is identical to that which exists in gases and liquids; see chapter 4. In longitudinal, or compressive waves, the particle motions are parallel to the direction the wave propagates; see figure 8-2. In transverse, or shear waves, on the other hand, the particle motions are confined to a plane perpendicular to the wave propagation; see figure 8-3. The total displacement of a point in the medium is the sum of both longitudinal and transverse wave contributions.

a)

b)

Figure 8-2 a) Deformation of a volume element, and b) Particle motions, in a longitudinal wave. The longitudinal wave is characterized by pure compression of a volume element.

a)

b)

Figure 8-3 a) Deformation of a volume element, and b) particle motion, in a transverse wave. The transverse, or shear, wave is characterized by the shearing of a volume element, without change of volume.

If the medium has boundaries, additional wave types can arise. In semi-infinite media, there are, for instance, so-called Rayleigh waves. The Rayleigh wave is an example of a surface wave. Surface waves are characterized by an amplitude that decays with distance from the surface; see figure 8-4.

233

Chapter 8: Wave equations and their solutions in solid media

Propagation Direction

Figure 8-4 Particle motions in a transverse surface wave. The wave propagates to the right in the figure. The seismic waves of greatest import in an earthquake, for example, are of this type. Another example is the velocity field in the air adjacent to a plate vibrating harmonically in bending, at a frequency below the so-called coincidence frequency; see chapter 8.

All waves in solid media consist of a longitudinal and a harmonic part. Even though the particle displacement in a bending wave is largely normal (transverse) to the direction of wave propagation, there is also a longitudinal component. Similarly, a quasi-longitudinal wave in a bar has a transverse component. In practice, all elements of structures are limited in their extents. The waves in them will therefore experience reflections and transmissions at boundaries, and will interfere with other waves. The remainder of this chapter therefore concentrates on structures that can be used to describe and understand wave propagation in different types of vehicles and machines. 8.3

QUASI-LONGITUDINALWAVES IN BARS

Beams are elements of many structures. At frequencies that are not too high, say below 510 kHz, vibrations are transmitted in a beam primarily by longitudinal waves, torsional waves, and bending waves. When the beam is subjected to an axial disturbance that is not too high in frequency, a quasi-longitudinal wave is generated, and propagates along the beam axis. The beam is normally called a bar when discussing that wave type1. In a bar subjected to an external harmonic force directed along the bar axis, a wave is generated in synchrony to the force variations, with alternating tensile and compressive regions; the wave propagates along the axis, away from the point of force application. As a tensile region passes through a volume element, the element is extended in the axial direction, and contracted in the transverse direction. If, on the other hand, a compressive region passes, the element is contracted in the axial direction, and extended in the transverse. Such a wave is called a quasi-longitudinal wave; see figure 8-5. A quasi-longitudinal wave is not a pure longitudinal wave. It consists of both longitudinal and transverse components. If the wavelength λL of the oscillations is much greater than the bar thickness d, as in figure 8-5, the longitudinal component of the deformation is considerably larger than the transverse. For a circular cylindrical steel bar with a 5 cm radius, the ratio of the transverse to the longitudinal deformation is 0.01 at an 1 Although they may be geometrically identical, or all 3 wave types may even exist in the same structural element, the convention is to refer to a thin, extended 1D medium as a “beam” in the context of bending waves, a “bar” in the context of quasi-longitudinal waves, and a “shaft” in the context of torsional waves.

234

Chapter 8: Wave equations and their solutions in solid media excitation frequency of 1 kHz. In many technical applications, quasi-longitudinal waves are therefore treated as purely longitudinal waves.

λL U n d efo rm ed b eam

cL

d

cL

C o m p re s s io n

E x p a n s io n

Figure 8-5 Deformation in a quasi-longitudinal wave. In practice, the deformation in the radial direction is considerably less than that in the axial direction. Compared to bending waves, the sound radiation from quasi-longitudinal waves is therefore negligible.

Because the displacement perpendicular to the surface of the bar is relatively small, the direct sound radiation from a quasi-longitudinal wave is also relatively small. Despite that, a quasi-longitudinal wave can, indirectly, be very significant for the acoustic properties of a structure. Specifically, when the bar is part of a large composite structure, the quasilongitudinal wave can transfer acoustic energy to an effective radiator, such as a plate vibrating in bending, for instance. 8.3.1

Wave equation for quasi-longitudinal waves in a bar

We now construct the wave equation for quasi-longitudinal waves in a bar. To begin with, we study the relation between the bar’s deformation, and the internal forces and stresses acting in the bar. Consider a straight prismatic bar. Assume that a longitudinal wave propagates in the positive x-direction. Let ξ ( x,t) indicate the displacement of a particle at position x and time t. Assume that the length of a bar element at rest is Δx. Referring to figure 8-6, its length Δξ when subjected to a normal stress σ ( x,t) is given by ⎛ ⎞ ∂ξ ∂ 2ξ ∂ξ . Δξ( x, t ) = ξ( x + Δx, t ) − ξ( x, t ) = ⎜ ξ( x, t ) + Δx + 2 (Δx) 2 + ...⎟ − ξ( x, t ) ≈ Δx ⎜ ⎟ ∂ x ∂x ∂x ⎝ ⎠

The derivative of the displacement ξ with respect to position x is called the strain ε, i.e., ε ( x, t ) =

∂ξ . ∂x

(8-1)

The deformation, i.e., the strain of the element, is induced by the stress σ acting on the element. The stress can, in turn, be related to the external forces acting on the bar. Thus, we need a relation between the stress and the strain. That so-called constitutive relation is provided by the well-known Hooke’s law: “Stress is directly proportional to strain”,

235

Chapter 8: Wave equations and their solutions in solid media

σ = Eε = E

∂ξ , ∂x

(8-2)

where σ and ε are measured on a bar (test piece) subjected to axial load. The so-called elasticity or E-modulus, also called Young’s modulus, is defined by that relation, based on quasi-longitudinal deformation in a bar. The definition also includes a sign convention for σ. The stress is defined positive for positive strains, i.e., when the element extends, and negative when the element contracts2. Thus, stresses are positive when directed out from the element, i.e., when the element is subject to pulling forces; see figure 8-6. ξ ( x, t) ξ (x + Δx, t)

σ ( x + Δ x, t )

σ ( x, t )

σ =0

Δx

x

x + Δx

ξ ( x + Δ x, t ) − ξ ( x, t ) + Δ x

x

Figure 8-6 Displacements and stresses in longitudinal deformation.

By setting up the equation of motion for a small element Δx of the bar, we obtain yet another relation between the displacement ξ and the stress σ. Figure 8-7 shows an element cut out from the bar, with a length Δx and a cross sectional area S. The equation of motion for the element, in the x-direction, is given by (8-3): q ( x, t ) SΔx

σ ( x, t )S

Δ x/2

σ ( x + Δ x , t )S

ξ ( x, t)

Δx

x

q ( x, t ) SΔx

x + Δx

ξ( x + Δx 2 , t)

x

ξ (x + Δx, t)

Figure 8-7 Bar, and bar element, with forces and stresses indicated.

2 Note that the opposite applies to fluids. In a fluid, the equivalent of stress, i.e., the pressure p, is positive when the volume element is compressed; see chapter 4 for more discussion.

236

Chapter 8: Wave equations and their solutions in solid media

ρSΔx

∂ 2 ξ( x + Δx 2 , t ) ∂t 2

= − σ ( x , t ) S + σ ( x + Δx , t ) S + q ( x , t ) S Δ x .

(8-3)

where the displacement of the center of mass is ξ(x + Δx/2,t) and where external forces per unit volume are indicated by q. Expanded in a Taylor series for small Δx,

ρSΔx

∂ ξ Δx ∂2 ⎛ ⎞ + "⎟ = ⎜ ξ ( x, t ) + 2 x ∂ 2 ⎠ ∂t ⎝

∂σ ⎞ ⎛ = − σ ( x, t ) S + ⎜ σ ( x, t ) + Δx + "⎟ S + q( x, t ) SΔx . ∂x ⎠ ⎝

(8-4)

When Δx approaches zero, terms that contain Δx to powers higher than 1 can be neglected. Dividing by SΔx provides the equation of motion ∂σ ∂ 2ξ . +q = ρ ∂x ∂t 2

(8-5)

The desired wave equation for quasi-longitudinal waves is, finally, obtained by eliminating, for example, the stress from equations (8-2) and (8-5), ∂ 2ξ ∂x

2

−

1 ∂ 2ξ c L2

∂t

2

+

q =0, E

c L2 = E ρ ,

(8-6)

where cL is the phase velocity of quasi-longitudinal waves in a bar. Note that the wave equation (8-6), with q = 0, has exactly the same form as the wave equation for fluid media, as derived in chapter 4. Example 8-1 Using formula (8-6) and appropriate material data, the wave speed for quasi-longitudinal waves can be determined for some common structural materials. Table 8-1 Material data and quasi-longitudinal wave speed for some common materials.

Material

Density ρ [kg/m3]

E-modulus E [N/m2]

Poisson’s ratio3 υ

Steel Aluminum Concrete Rubber

7800 2800 2300 1200

2 ⋅ 1011

0.3 0.3 0.2 0.5

7.3 ⋅ 10 2.6 ⋅ 1010 0.8 ⋅ 10 6 − 40 ⋅ 10 6 10

Quasilongitudinal wave speed cL [m/s] 5100 5100 3400 26 - 180

3 Poisson’s number υ and E are the constants that define a homogeneous isotropic linear elastic material.

237

Chapter 8: Wave equations and their solutions in solid media 8.3.2

Quasi-longitudinal waves in infinite bars

The solutions to the wave equation (8-6), in the absence of external forces, represent the range of possible quasi-longitudinal motions of an infinite bar. Two solutions of particular interest are d’Alembert’s solution (see chapter 4), ξ ( x, t ) = f ( x − c L t ) + g ( x + c L t ) ,

(8-7)

and the plane harmonic wave solution ξ( x, t ) = ξˆ + e i (ω t − k L x ) + ξˆ − e i (ω t + k L x ) ,

(8-8)

where kL is the quasi-longitudinal wave number. The first terms in these solutions describe waves that propagate in the positive x-direction. The plane harmonic wave solution is particularly useful. It can be used, with Fourier methodology as discussed in section 3.4, to build up more complicated solutions. A consequence of the d’Alembert solution is that a wave retains its form as it moves along the bar. Thus, losses of energy from the wave do not occur in systems for which the d’Alembert solution applies; energy losses would imply a change in the amplitude of the wave along its path. Moreover, the retention of form also implies that the d’Alembert solution only holds for systems for which the phase velocity does not depend on frequency. Systems with that property are called non-dispersive; see section 8.5.4. Example 8-2 A quasi-longitudinal wave propagates in a bar. The bar is made of steel, and has a crosssectional area of 0.5·10-2 m2. The velocity field vx along the bar axis is given by vˆ exp(i(ω t − kx)) with an amplitude of vˆ = 0.1·10-3 m/s. Calculate the bar’s stress field. Solution The relation (8-2) between displacement and stress is useful in this context. The displacement is calculated first by performing a time integration of the velocity. The time integration is carried out by dividing by iω. Putting that into (8-2) and differentiating with respect to x then leads to σ=

E ∂v x E E i (ω t − k L x ) ik L vˆe i (ω t − k L x ) = − =− vˆe . iω ∂x iω cL

After putting in numerical values (cL for steel is taken from example 8-1), the magnitude of the stress amplitude is then calculated to be σˆ ≈ 3.4·103 [N/m2]. That can be compared to the yield stress for steel, 200-300 MN/m2. 8.3.3

Quasi-longitudinal waves in finite bars

Assume that we have a source that generates a plane harmonic wave that propagates in the positive x-direction. If the bar is infinite, the wave propagates ever farther away from the source. If the bar is, on the other hand, finite, the wave will sooner or later reflect against the end. The mechanical properties at the end of the bar determine how the wave is reflected. Mathematically, these properties are formulated as boundary conditions. For

238

Chapter 8: Wave equations and their solutions in solid media example, an unloaded end is described by proscribing that the normal stress be zero at the end. A fixed (rigidly blocked) end is described by proscribing that the displacement be zero there. The reflection of a wave at a bar endpoint affects the wave in two ways: (i)

The reflected wave amplitude changes with respect to that of the incident wave;

(ii)

The reflected wave can undergo a phase shift relative to the incident wave.

That, by analogy to chapter 5, can be expressed mathematically by a reflection coefficient ξˆ r . ξˆ

R = Re iδ r =

(8-9)

i

The reflection coefficient is, as such, a practical reformulation of the boundary conditions when the field consists of incident and reflected plane harmonic waves. Example 8-3 Formulate a) the boundary condition, and b) the reflection coefficient for figure 8-8a. ξˆ i e i (ω t − k L x )

E,ρ,S

Sσ ( − Δ x , t )

x

κ

x F

i (ω t + k L x ) a) ξˆi R e

Δx

b)

Figure 8-8 a) Bar end coupled to an ideal spring. b) Cross-sectional loads at the bar end.

Solution a) The state at the end of the bar is determined by the properties of the spring. Cut out a bar element of width Δ x at the end, as in figure 8-8b. Assume an ideal massless spring with spring rate κ. In that case, the spring load is F = −κ ξ(0, t ) .

(8-10)

That force acts on the end of the bar as well. By setting up the equation of motion in the xdirection for the small bar element, the force F can be related to the bar’s normal stress

ρSΔ x

∂ 2 ξ(− Δ x 2 , t ) ∂t 2

= F − σ(− Δx, t ) S .

(8-11)

If we then allow Δ x to approach zero, we have σ(0, t ) = F S = −

κ S

ξ(0, t ) .

(8-12)

b) Assume that a plane harmonic wave is incident upon, and reflects from, the end of the bar. The total displacement field is then

239

Chapter 8: Wave equations and their solutions in solid media ξ( x, t ) = ξˆi e i (ω t − k L x ) + ξˆ r e i (ω t + k L x ) .

(8-13)

Formula (8-2) can be used to determine the corresponding total stress field σ ( x, t ) = E

∂ξ = E ((−ik L )ξˆi e i (ω t − k L x ) + (ik L )ξˆ r e i (ω t + k L x ) ) . ∂x

(8-14)

The boundary condition (8-12), above, gives

κ E ((−ik L )ξˆi + (ik L )ξˆ r ) = − (ξˆi + ξˆ r ) , S

(8-15)

κ

κ 1− 1− ˆξ iωS ρE ik L SE r = R= . = ˆ κ κ ξi 1+ 1+ ik L SE iωS ρE

i.e.,

(8-16)

Table 8-2 gives some more examples of different boundary conditions for quasilongitudinal waves in bars. Table 8-2 Boundary conditions and reflection coefficients for quasi-longitudinal waves in a bar with some idealized fastening conditions.

Description

x

Mass

σ(0, t ) = 0

R=1

ξ(0, t ) = 0

R = -1

x

Rigidly Fixed

Damper

Reflection coefficient

x

Free

Spring

Boundary Condition

σ(0, t ) = −

κ x

dv x m

κ S

ξ(0, t )

R=

d ∂ ξ(0, t ) σ(0, t ) = − v S ∂t

R=

m ∂ 2 ξ(0, t ) S ∂t 2

R=

σ(0, t ) = −

1 − κ iωS ρE 1 + κ iωS ρE 1 − d v S ρE 1 + d v S ρE 1 − iωm S ρE 1 + iωm S ρE

If the bar’s length is finite, the wave components are repeatedly reflected against the ends. The motion can then be calculated by specifying a displacement field in the bar in accordance with (8-8), and thereafter determining the amplitudes ξˆ and ξˆ by putting −

+

the solution form into the boundary conditions. Examples that illustrate that method are given in sections 8.3.5.2 and 8.3.5.3.

240

Chapter 8: Wave equations and their solutions in solid media 8.3.4

Reflection and transmission of quasi-longitudinal waves at area changes

Reflections of waves at changes along the transmission path can be used to reduce the transfer of mechanical energy in a structure. An example is a quasi-longitudinal wave that is incident upon a joint connecting two bars of differing material and/or cross section; see figure 8-9a. x

x

ξ i e i ( ωt − k1x )

σ 1 (−Δ x, t ) S1

ξ i Te i ( ωt − k2 x )

ξ i R e i ( ωt + k1x ) E1 , ρ 1 , S1

E2 , ρ 2 , S2

σ 2 ( Δ x, t ) S 2

2Δ x

a)

b)

Figure 8-9 a) Reflection and transmission of a quasi-longitudinal wave at the joint between two semi-infinite bars. b) Cross sectional areas at the joint.

We can analyze the effect of the joint on the wave propagation by considering an incident plane wave, as shown in figure 8-9a. That gives rise to a reflected and a transmitted plane wave. The total displacement field in the coupled beams is therefore given by ⎧⎪ξˆ e i (ω t − k1x ) + Rξˆi e i (ω t + k1 x ) , ξ ( x, t ) = ⎨ i ⎪⎩ Tξˆi e i (ω t − k2 x ) ,

x5

ψn

-

1.875

4.694

7.855

10.996

14.137

(n-1/2)π

cosd(knx) - σ 1n sind(knx)

-

3.142

6.283

9.425

12.566

15.708

nπ

sin(knx)

0

3.927

7.069

10.210

13.352

16.493

(n+1/4)π

coss(knx) - σ n3 sins(knx)

-

3.927

7.069

10.210

13.352

16.493

(n+1/4)π

cosd(knx) - σ n4 sind(knx)

-

4.730

7.853

10.996

14.137

17.279

(n+1/2)π

cosd(knx) - σ n5 sind(knx)

0

4.730

7.853

10.996

14.137

17.279

(n+1/2)π

coss(knx) - σ n6 sins(knx)

Notation: cosd(x) = cosh(x) - cos(x), coss(x) = cosh(x) + cos(x), sind(x) = sinh(x)-sin(x), sins(x) =sinh(x)+sin(x) and σ11 ≈ 0.734, σ 12 ≈ 1.018, σ 31 ≈ 0.999, σ 14 ≈ 1.000, ... , σ 13 ≈ 1.001, σ 23 ≈ 1.000, ... , σ 14 ≈ 1.001, σ 24 ≈ 1.000, ... , σ 15 ≈ 0.983, σ 25 ≈ 1.001, σ 35 ≈ 1.000, ... , σ 16 ≈ 0.983, σ 26 ≈ 1.001, σ 36 ≈ 1.000, ... .

Otherwise, σ nm ≈ 1.000. 270

Chapter 8: Wave equations and their solutions in solid media

Example 8-11 Railway rails are mounted on heavy concrete slabs. The slabs are placed in the track bed with a spacing a of about 0.65 m. Every slab mass acts as a hindrance to the propagation of bending waves, which arise when a train wheel rolls on the rail. At those specific frequencies for which the bending half-wavelength equals a whole number multiple of the slab spacing a, however, the bending waves can propagate, unhindered, great distances from the source. What are the first few (lowest) such frequencies? Solution Example 8-8 provides some data for Swedish track. The first so-called pass-frequency is determined by setting half the bending wavelength equal to the distance between slabs, and then solving the frequency from the dispersion relation (8-60a; doing so,

f = 2π (4a 2 ) EI ρS ≈ 450.7 a 2 ≈ 1070 Hz. A system intended to warn about the approach of a train might therefore be based on the detection of vibration signals in the region around 1000 Hz.

8.5.7 Standing waves in plates Finite plates exhibit resonant properties. Just as for beams in bending vibration, the eigenfrequencies and eigenmodes of a plate depend on the edge fastening conditions. Figure 831 shows the six lowest eigenmodes of a plate with simply-supported edges. (1,1)

(2,1)

(3,1)

(2,2)

(1,2)

(3,2)

Figure 8-31 The six first eigenmodes for a rectangular plate simply supported at all four edges. The eigenfrequencies increase from right to left in the figure. The mode shapes are typically indicated by (m,n) where m gives the number of displacement antinodes as projected along a short edge of the plate, and n gives the number of displacement antinodes as projected along a long edge of the plate.

Compared to the eigenmodes of a beam, those of the plate are complicated by the fact that they also depend on the ratio of the shorter to the longer edge lengths. It is therefore impossible to summarize all of the properties in a single table, like table 8-7 above. In

271

Chapter 8: Wave equations and their solutions in solid media various handbooks and tables, there are collections of tables summarizing the resonance frequencies and mode shapes for various edge mounting conditions, and various aspect ratios. Table 8-8, below, gives some examples, for three different sets of edge conditions. Table 8-8 Eigenfrequencies of rectangular plates with certain boundary conditions. The mode’s indices (m,n) represent the number of displacement antinodes projected along the short sides (m) and the long sides (n) of the plate, respectively; see figure 8-31. (Source: Based on R D Blevins, Formulas for Natural Frequency and Mode Shape. 1979. Van Nostrand Reinhold.)

Edge Conditions F F

F

F

b

a

F = Free S S

S

S

b

a

S = Simply Supported C C

C

C

a

C = Clamped (Fixed)

μmn (m,n)

a/b

b

0.4 2/3 1 1.5 2.5 0.4 2/3 1 1.5 2.5 0.4 2/3 1 1.5 2.5

3.46 (1,3) 8.95 (2,2) 13.5 (2,2) 20.1 (2,2) 21.6 (3,1)

5.29 (2,2) 9.60 (1,3) 19.8 (1,3) 21.6 (3,1) 33.0 (2,2)

9.62 (1,4) 20.7 (2,3) 24.4 (3,1) 46.6 (3,2) 60.1 (4,1)

11.4 (2,3) 22.4 (3,1) 35.0 (3,2) 50.3 (1,3) 71.5 (3,2)

18.8 (1,5) 25.9 (1,4) 35.0 (2,3) 58.2 (4,1) 117.5 (5,1)

11.4 (1,1) 14.3 (1,1) 19.7 (1,1) 32.1 (1,1) 71.6 (1,1) 23.6 (1,1) 27.0 (1,1) 36.0 (1,1) 60.8 (1,1) 147.8 (1,1)

16.2 (1,2) 27.4 (1,2) 49.4 (2,1) 61.7 (2,1) 101.2 (2,1) 27.8 (1,2) 41.7 (1,2) 73.4 (2,1) 93.7 (2,1) 173.9 (2,1)

24.1 (1,3) 43.9 (2,1) 49.4 (1,2) 98.7 (1,2) 150.5 (3,1) 35.4 (1,3) 66.1 (2,1) 73.4 (1,2) 148.8 (1,2) 221.5 (3,1)

35.1 (1,4) 49.4 (1,3) 79.0 (2,2) 111.0 (3,1) 219.6 (4,1) 46.7 (1,4) 66.6 (1,3) 108.3 (2,2) 149.7 (3,1) 291.9 (4,1)

41.1 (2,1) 57.0 (2,2) 98.7 (3,1) 128.3 (2,2) 256.6 (1,2) 61.6 (1,5) 79.8 (2,2) 131.6 (3,1) 179.7 (2,2) 384.7 (5,1)

The eigenfrequencies are calculated from the table parameters, geometric data and material data using the formula

f m, n =

μ mn Eh 2 . 2 2πa 12 ρ (1 − υ 2 )

Here, a is the length of a long edge, h the plate thickness, ρ the plate density, E the elastic modulus and υ Poisson’s ratio.

Example 8-12 Part of the boundary surface of a machine consists of steel sheet, which can be regarded as a rectangular, simply-supported (at all edges) plate. The thickness of the sheet is 1 mm, and its sides are 0.6 m and 0.4 m long. In order to avoid the excitation of resonances in the plate by the operation of the machine, the eigenfrequencies of the plate need to be known to the designer. a)

Determine the two lowest eigenfrequencies of the plate.

b)

How are the eigenfrequencies influenced by doubling the plate thickness?

Solution a) All information needed to solve the problem can be found in tables 8-1 and 8-8. From table 8-1, for steel: E = 2·1011 N/m2, ρ = 7800 kg/m3 and υ = 0.3. According to the problem statement, moreover, a = 0.6 m, a/b = 1.5 and h = 0.001 m. From table 8-8, for the

272

Chapter 8: Wave equations and their solutions in solid media simply-supported boundary conditions along the edges, and for the given parameters, the two lowest eigenfrequencies follow from

μ11 = 32.1 and μ21 = 61.7. and the deformation patterns for both of these modes are shown in figure 8-31. Putting these values into the formula for the eigenfrequencies beneath table 8-8, the lowest two are found to be f11 =

and

32,1

2 ⋅1011 ⋅ 0,0012

2π 0,6 2

12 ⋅ 7800(1 − 0,3 2 )

f 21 =

≈ 21,7 Hz

μ 21 61,7 ⋅ f11 = ⋅ 21,7 ≈ 41,8 Hz. μ11 32,1

b) According to the formula in table 8-8, the eigenfrequency is directly proportional to the plate thickness. When the plate thickness is doubled, the eigenfrequencies are doubled as well,

f11 ≈ 2 ⋅ 21.7 = 43.4 Hz and

8.6

f 21 ≈ 2 ⋅ 41.8 = 83.6 Hz.

MECHANICAL IMPEDANCE AND MOBILITY

The typical machine is a complicated, composite structure. The analysis of composite structures can be considerably simplified by taking advantage of the concepts of impedance or mobility. A mechanical impedance is a frequency response function, as discussed in chapter 3, that describes the relation between an exciting point force and the resulting velocity at a given point. The mechanical impedance and the mobility, are defined as (see table 3-1)

and

Z(ω, xF, xv ) = F(ω ) /v(ω ) ,

(8-68a)

Y(ω, xF, xv ) = v(ω ) /F(ω ),

(8-68b)

where xF is the position of the excited force, and xv is the point at which the resulting velocity is referred to. Thus, the mobility is the reciprocal of the impedance. Figure 8-32 exemplifies the concepts of impedance and mobility, for the case of a beam fixed at one end. Z = Fˆ / vˆ

Figure 8-32 Quantities that are included in the definitions of mechanical mobility and impedance.

xv

273

xF vˆ e iω t

Fˆ e iω t

x

Chapter 8: Wave equations and their solutions in solid media

Example 8-13 Determine the so-called driving point mobility for quasi-longitudinal waves in a bar fixed at one end, if the excitation and response points are both at the free end of the bar, as in figure 8-11. Solution We can use formula (8-39) from section 8.3.5.3, which gives the displacement field for that case. The velocity field is obtained by multiplying (8-39) by iω. The mobility is thereafter found by dividing (8-39) by F, Y(ω ) =

sin(ω ρ E x) vˆ x iωξˆ( x) iω sin( k L x) i = =− =− . k L ES cos(k L L) Fˆ Fˆ S ρE cos(ω ρ E L)

(8-69)

Figure 8-33 shows the mobility as a function of the excitation frequency.

lo g Y

0

5000

10000 F req uency[H z ]

15000

20000

Figure 8-33 Mobility for quasi-longitudinal waves in a bar fixed at one end; see figure 8-11. The peaks in the curve correspond to the eigenfrequencies.

Assume that the mobility of a structure is known. It is relatively simple to, using formula (8-68b), calculate the resulting velocity in the structure for different alternative excitation forces. If the mobilities of the parts of a composite structure are known, a system of equations can be specified, the solutions to which are the motions of the composite structure. Chapter 9 shows how the impedance concept can be used to simplify the analysis of vibration isolated machines. Table 8-9, below, provides the impedances of some common mechanical structures.

274

Chapter 8: Wave equations and their solutions in solid media

Table 8-9 Point impedances of some common infinite and semi-infinite structural elements.

Mass Spring Longitudinal wave in bar Bending wave in an endexcited, slender ½-∞ beam

Z = iωm

v

F

v

Z = κ iω

F

F

Z = S Eρ

v F

Z = 0.5 ρSc B (1 + i )

v

Bending wave in a slender ∞ beam

F

Edge excited thin ∞ plate

F

Z = 2 ρScB (1 + i )

v

Z = 3.5 D p ρh

v F

Thin ∞ plate excited far from the edges

v

Z = 8 D p ρh

Example 8-14 Assume that a small motor with a mass of 1 kg is so compliantly mounted that the affects of the surroundings can be neglected at frequencies above 40 Hz. Calculate the velocity if a 100 Hz harmonic force, with an amplitude of 0.5 N, excites the motor. Solution Because influences from the surroundings can be ignored at 100 Hz, the motor can be treated, as a rigid 1 kg mass, to a first approximation. From table 8-9, the impedance of a rigid mass can be used, so that v = Z −1F = F iωm = 0,5 (i 2π 100 ⋅ 1) ≈ −i ⋅ 8,0 ⋅ 10 −4 m/s.

8.7 LOSSES IN SOLID STRUCTURES The losses in a mechanical structure have, in a couple of cases, great significance for the vibrations. If, for example, a beam vibrating in bending is much longer than a wavelength, the losses reduce the amplitude of the bending wave considerably as it traverses the beam. In the same way, the losses have very important effects on resonant vibration fields; see figures 8-15 and 8-34. In resonant fields, the propagation path is long, because of the repeated reflections between the ends. Increased losses are therefore an effective way to diminish the reverberant part of a vibration field. It can be shown that the energy in the reverberant field is inversely proportional to the loss factor η E ∝1 η .

275

(8-70)

Chapter 8: Wave equations and their solutions in solid media

Resonance in the blade

Rubber Steel

Reinforcements Figure 8-34 The losses in the structure are of great significance to the amplitudes of resonant vibrations. While grinding the teeth of the circle saw blade, powerful resonance vibrations build up in the blade, causing a strong grating sound. A way to minimize the vibration field, and the resulting noise level, is to apply a rubber-clad disc to the saw blade. That will increase the losses, partly because of the high internal losses in rubber materials, and partly because of friction at the contact between the rubber and the saw blade. (Source: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

By losses, one typically means irreversible energy conversion from vibrational to other energy forms, e.g., thermal energy. Often, a structure’s losses are divided into two different components: internal losses, i.e., so-called material losses and boundary losses.

276

Chapter 8: Wave equations and their solutions in solid media

8.7.1

Material losses

Material losses include those caused by atomic and molecular process within the material itself. Examples of such processes are dislocations, changes in the crystal structure, changes in the molecular structure, and the excitation of oscillations at the molecular level. All of these processes are so-called relaxation phenomena, i.e., they taper off exponentially with time. The characteristic decay time, or relaxation time, can, depending on the phenomenon in question, vary from 10-9 s up to several hours. The relaxation time determines the frequency bands in which the phenomenon has the greatest significance; a short relaxation time implies a strong effect at high frequencies, and vice versa. Thus, it need not be the same dissipation mechanism that dominates in each frequency band. As a result, the loss factor tends to vary with the excitation frequency; see figure 8-35. 0,100 0,090 0,080 0,070 0,060 ♦

0,050

∗ ♦

η

0,040

∗ ♦

∗

♦

∗

0,030

0,020

10

100

Frequency [Hz]

1000

∗

+

♦

+

♦ ♦

10000

Figure 8-35 Experimentally determined loss factor for plexiglass. The half-bandwidth method was used to determine the loss factor; see section 8.7.6 for more details.

Material losses are, especially for many metals, relatively small. For a metal, the material loss factor η might typically be of the order 10-4 - 10-3. For a material with relatively high internal losses, e.g., certain plastics, the loss factor might fall in the range 10-2- 10-1. Generally, the internal losses in a material increase with increased shear deformation. Table 8-10 summarizes the typical loss factors of some common materials, and some common structures.

277

Chapter 8: Wave equations and their solutions in solid media

Table 8-10 Typical loss factors for some common materials and structures. (Source: E E Ungar, Damping of Panels, in, L L Beranek. Noise and Vibration Control, McGraw-Hill, 1971)

Material/Structure Steel Aluminum Concrete Tile Sand (dry) Gypsum Panel Wood Plywood Particle Board Plastics Carbon-fiber reinforced plastics Aluminum structure, riveted Building

8.7.2

Loss factor η 0.0002-0.0006 0.0001 0.01-0.05 0.01-0.02 0.12-0.6 0.0008-0.003 0.01 0.01-0.013 0.01-0.03 0.01-0.1 0.001 0.01-0.05 0.01

Boundary losses

Boundary losses include those losses that can be attributed to the boundary surfaces of a structure. An important example is surface friction. Surface friction comes about when two surfaces in contact slide relative to one another. In joints and hinges of various kinds, as well as such machine elements as linkages, bearings, and gears, that type of friction is significant. Acoustic radiation from vibrating surfaces to the surrounding media (e.g., air, or water) is another example of a kind of boundary losses. A sometimes important contribution to the total loss factor is that from so-called air pumping. Air pumping occurs in joints in which air is pressed in and out of the small spaces between mating surfaces.

8.7.3

Losses in composite structures

In composite structures, such as machines, vehicles, and buildings, the boundary losses are normally significantly higher than the material losses in the components. The reason for that is, of course, that the losses at the boundary surfaces between the structural elements and the surroundings are significantly higher. It is therefore often the case that a increase in the material losses does not reduce vibrations as much as expected.

8.7.4

Increase of losses in beams and plates by means of absorbers

In two special situations, the losses may be very effectively increased by combining ordinary metallic structural materials with materials that have higher internal losses – socalled absorbers. When the vibrations of parts of a structure, such as attached sheet metal, are predominantly reverberant, that approach may be promising. Another case would be when waves transit a relatively long, reflection-free path, such as a beam. In the first case, the dissipation influences the vibrational energy in the structural member. In the second case, it influences the energy transferred by the beam. 278

Chapter 8: Wave equations and their solutions in solid media To get the full benefit of the added material, it must be suitably placed. The dissipation is maximized when the shear deformation is as large as possible. Thus, the material should be placed in areas with large vibration amplitudes. Plates, shells and beams can be specially adapted to provide high losses. For example, a plate can be clad with a viscoelastic material that has high losses. In order to increase the shear deforma-tion, and by extension the losses, the shear layer can be sandwiched between two metallic sheets (MPM plate); see figure 8-36. Simplified methods to calculate the resulting loss factors of these structural elements can be found in the structural acoustics literature. V iscoelastic layer

M etal

M etal Figure 8-36 Plate of metal-plastic-metal type (MPM-plate), which has high losses This type of plate is designed so that the viscoelastic material mainly works in shear. That maximally exploits the material properties, and therefore reduces the layer thickness needed to obtain a desired loss factor.

8.7.5 Mathematical description of dissipation For strictly periodic processes, it can be shown that relaxation processes cause a phase shift, i.e., a time delay, between stress and strain processes. That implies, by the definition of mechanical work, that some work is performed by each stress-strain cycle. That work is the dissipated energy, i.e., vibrational energy irreversibly converted to other forms. The phase shift also implies that the effect of losses can be described in a mathematically convenient way. Specifically, a phase shift between stress and strain can be introduced by a complex-valued E-modulus (compare to section 3.3.5), E = E (1 + iη ) ,

(8-71)

where η is the material’s internal loss factor. That relation can be used as a definition of the loss factor. More commonly, however, an equivalent definition based on energy methods is used, as in formula (3-74). Via the E-modulus, even complex phase velocities and wave numbers can be defined. For bending waves in beams, for example, a truncated series development of equation (8-60a), gives an approximation valid for small η, i.e., kB = 4

ρS ρS ω =4 ω ≈ k B (1 − i η 4) . EI EI (1 + iη )

(8-72)

If that wave number is put into the plane wave solution, then equation (8-61) provides a physical interpretation of the complex wave number, ζˆ ( x) = ζˆ0 e −ik B x = ζˆ0 e − k B xη 4 e −ik B x ,

i.e., an exponentially decaying wave solution, see figure 8-37.

279

(8-73)

Chapter 8: Wave equations and their solutions in solid media ζˆ 0 e − k B xη / 4 e − ik B x 1,00 0,80 0,60 0,40 0,20 0,00 -0,20 -0,40 -0,60 -0,80 Distance x

Figure 8-37 Wave propagation in materials with internal losses. The amplitude of the bending waves decreases with distance from the source. The original, undamped wave amplitude is assumed to be ζ0.

An important consequence of losses is that the vibration amplitude is limited. If one supplies a constant quantity of vibrational energy per unit time to a structure, the structure’s vibrational energy increases successively to a certain level E0. The stationary level occurs when the energy supplied per unit time is balanced by the energy dissipated per unit time. If the energy supply is interrupted, the vibrational energy decays again to zero, as does the amplitude. Chapter 7 derives the result that the energy sustaining vibrations decays according to E (t ) = E 0 e −ηω t ,

(8-74)

where E0 is the initial energy value. The loss factor determines how quickly the structure can “suck away” the vibrational energy. A more general discussion of energy storage in systems is given in section 7.1.

Example 8-15 The loss factor of a long beam is 0.0005. How much is a bending wave’s amplitude damped per wavelength? Solution The decay can be studied with the help of the 1st factor in (8-73). After one wavelength, i.e., kB x = 2π, the amplitude is changed by a factor

e −2πη / 4 = e −0.0005π / 2 ≈ 0.9992 ,

280

Chapter 8: Wave equations and their solutions in solid media i.e., hardly 1 per thousandth. After 100 wavelengths, the amplitude has diminished to about 92 % of its initial value. If an absorber can raise the loss factor to, say, 0.1, then the amplitude would diminish to 85 % of its initial value.

8.7.6

Experimental determination of the loss factor

In many situations, it is important to be able to estimate the loss factor. In practice, theoretical predictions of the loss factor are impossible to make to an acceptable degree of precision. Thus, some kind of experimental method is required. Some such methods are briefly described here. (i)

If the stress and strain are measured simultaneously, the loss factor can be directly determined from the phase shift between them. That method is, due to high demands on the phase precision, mainly useful at low frequencies.

(ii)

By recording the decay of vibrational energy as a function of time, the loss factor can be determined from equation (8-74). That is the so-called reverberation time method. The reverberation time T60 is the time it takes for the energy of vibration to decrease by a factor of106, i.e., the velocity level must fall 60 dB; see figure 838. The loss factor is then found from the relation

η = 2.2 f T60 .

(8-75)

That method is very useful if η is to be determined in, for example, octave or third-octave bands.

Lv [dB] 100 80

60 dB

60 40

T60

20 0

1

2 Time [s] 3

4

5

6

Figure 8-38 The reverberation time T60 is the time it takes for the velocity level to fall by 60 dB. A vibration source, e.g., an electrodynamic shaker, excites vibrations in the structure. When the vibration velocity reaches stationary conditions, the energy supply i.e., the shaker, is interrupted, and the decay process recorded as a function of time. From that recorded signal, the reverberation time, and thereby the loss factor as well, can be determined; see equation (8-75).

(iii)

A narrow band version of method (ii) is performed by exciting the structure with a harmonic force. When the excitation is interrupted, the decay process is

281

Chapter 8: Wave equations and their solutions in solid media recorded. From the relative decrease in energy, i.e., the square of the amplitude, η can be determined. (iv)

Frequency response functions, such as mobility7, i.e., motion normalized by force, see chapter 3 and section 8.6, have peaks at the resonance frequencies. The width of a resonance peak is directly proportional to the loss factor at that frequency. That relation can be used in the so-called half-value bandwidth method in which the loss factor is determined using the formula,

η ( f = f res ) = Δ B f f res ,

(8-76)

where fres is the resonance frequency and ΔB f is the half-value bandwidth between the frequencies, on each side of the peak, at which the mobility has fallen by a factor of the square root of 2 relative to the resonance peak. The name half-value bandwidth comes from the fact that the bandwidth is determined from the points at which the energy is half of that at the resonance peak. By defining the so-called mobility level LY, Y2 LY = 10 ⋅ log , (8-77) 2 Yref where Y is the magnitude of the mobility and Yref = 1·10-3 m/Ns is the reference value for mobility, the loss factor is determined by a measurement of the mobility level. The halfvalue bandwidth is then determined by the frequency lines at which the mobility level has decayed by 3 dB with respect to the resonance peak; see figure 8-39. -80 -81 3 dB -82 LY [dB] -83 -84 -85

Δb f = fö - fu

-86 30

40

fu 50 Frequency [Hz]

fö

60

70

7 Other such frequency response functions, besides mobility, are the dynamic flexibility (receptance) and the accelerance.

282

Chapter 8: Wave equations and their solutions in solid media Figure 8-39 Calculation of the loss factor from the half-value bandwidth of a resonance peak. The half-value bandwidth is the distance along the frequency axis between the frequencies at which the mobility level has fallen by 3 dB relative to the maximum value at the resonance frequency.

This type of measurement can be quickly and easily performed using the FFT analyzers that have become readily available in this day and age; see figure 8-40. The main disadvantage is that the losses can only be measured at the resonance frequencies of the test object. Suspension wires

Beam

Accelerometer

Charge amplifiers

FFT-analyser

Force transducer F(t) Excitation hammer a(t)

PC Frequency

Figure 8-40 Instrumentation used in the measurement of the material losses in a test beam vibrating in bending. The test beam is excited by a light hammer blow. The exciting force is measured with a force transducer, and the resulting vibration velocity in the test beam with an accelerometer. These signals are recorded, and an FFT analyzer calculates the mobility function between the force and velocity. The mobility function may is then plotted, and from the plot, the loss factor can be found by applying the half-value bandwidth method to the resonance peaks seen in the mobility spectrum.

Example 8-16 From a measurement of the mobility of a structure, the half-value bandwidth of a resonance peak is found to be 23 Hz. The resonance frequency is at 1990 Hz. Determine the loss factor of the structure at 1990 Hz. Solution Putting the given values into formula (8-76) yields

283

η = 23 1990 ≈ 0.012 .

Chapter 8: Wave equations and their solutions in solid media

8.8

IMPORTANT RELATIONS

LONGITUDINAL WAVES IN BARS ε ( x, t ) =

Strain

∂ξ . ∂x

σ = Eε = E

Hooke’s law

(8-1)

∂ξ . ∂x

(8-2)

∂ 2ξ ∂σ =ρ . ∂x ∂t 2

Equation of motion

∂ 2ξ

Longitudinal wave equation

2

∂x

−

(8-5)

1 ∂ 2ξ

c L2

∂t

2

= 0 , c L2 =

E

ρ

.

Plane wave solution ξ( x, t ) = ξˆ + e i (ω t − k L x ) + ξˆ − e i (ω t + k L x ) .

(8-8)

ξˆ r = Re i δ r . ˆ ξ

(8-9)

E E ∂θ ε= γ = Gγ = Gr . 1+υ 2(1 + υ ) ∂x

(8-45)

R=

Reflection coefficient

(8-6)

i

TORSIONAL WAVES IN SHAFTS

Hooke’s law

τ=

Equation of motion

Torsional wave equation

J

∂ 2θ ∂t

2

∂ 2θ ∂x

2

=

−

∂M . ∂x

(8-46)

1 ∂ 2θ

cT2

∂t

2

= 0 , cT2 =

G

ρ

.

(8-48)

BENDING WAVES IN BEAMS AND PLATES Normal strain

ε( z ) = ε 0 + z ρ = ε 0 −

284

∂ 2ζ ∂x 2

z.

(8-51)

Chapter 8: Wave equations and their solutions in solid media

σ = Eε 0 − E

Normal stress

Cross-sectional moment

∂ 2ζ ∂x 2

M y = − EI b

∂T ∂ 2ζ + q ′ = ρS ∂x ∂t 2

Equations of motion

z = ε0 − E

Bending wave equation in a beam

∂ 2ζ ∂x 2

∂ 2ζ

z.

∂x 2

.

and −

(8-52)

(8-53) ∂M y ∂x

+T = 0 .

∂ 2ζ ∂ 2 ⎡ ∂ 2ζ ⎤ D + ρS = q′ . ⎢ ⎥ ∂x 2 ⎣⎢ ∂x 2 ⎥⎦ ∂t 2

Bending wave equation in plates ∇ 2 ( D p ∇ 2ζ ) + ρh

∂ 2ζ ∂ t2

= q ′′ .

(8-54,55)

(8-56)

(8-57)

Bending stiffness of a plate

Dp = Eh3/(12(1-υ2)).

(8-58)

Plane wave solution

ζ ( x, t ) = ζˆ e i (ωt − k B x ) .

(8-59)

Dispersion relation for bending waves General solution

k B4 = ω 2 ρS D .

ζ = ( Ae ik B x + Be −ik B x + Ce k B x + De − k B x )e iω t .

(8-60) (8-61)

Phase velocity in a beam

c f = ω 4 D ρS .

(8-65)

Group velocity in a beam

c g = 2 ω 4 D ρS .

(8-66)

Phase velocity in a plate

c f = ω 4 D p ρh .

(8-65a)

MECHANICAL IMPEDANCE AND MOBILITY Definition of impedance

Z(ω , x F , xv ) = F(ω ) v(ω ) .

(8-68a)

Definition of mobility

Y(ω , x F , xv ) = v (ω ) F(ω ) .

(8-68b)

285

Chapter 8: Wave equations and their solutions in solid media LOSSES IN SOLID STRUCTURES

Energy in the reverberant field Complex E-modulus Energy decay

E ∝1 η .

(8-70)

E = E (1 + iη ) .

(8-71)

E (t ) = E 0 e −ηωt .

(8-74)

η = 2.2 f T60 .

(8-75)

Experimental determination Reverberation time method Half-value bandwidth method

η ( f = f res ) = Δ B f f res .

286

(8-76)

CHAPTER NINE

ENERGY METHODS APPLIED TO ROOM ACOUSTICS

This chapter discusses suitable methods for the analysis of sound fields, or of sound transmission, at high enough frequencies that the wavelengths are small compared to typical distances traversed by the sound. Energy-based methods can be used for the analysis of vibrations and acoustics problems in such circumstances. After a short introduction to the subject area, including a bit of historical background, energy balance equations are treated for simple and for coupled acoustic systems. An equation which relates the reverberation time and the loss factor of an acoustic system is then derived. That result is specialized to the case of room acoustics and, with the help of the diffuse field and diffuse intensity concepts, an expression for the reverberation time of a room – Sabine's formula – is derived. We then discuss the measurement of the reverberation time and absorption of a room, and consider the structure of a sound field radiated by an acoustic source. Finally, acoustic absorbers are discussed, and an analysis is made of the sound transmission between two rooms. In that last context, the concepts of the transmission factor and the sound reduction index of partitions (walls) are studied, and some typical examples of insulating partitions are presented.

287

Chapter 9: Energy Methods Applied to Room Acoustics

9.1

OVERVIEW OF ENERGY METHODS

In the study of acoustic systems, we can choose between a number of different methods to describe acoustic fields; see section 1.6. For example, an exact description of the sound field as the sum of eigenfunctions (modes) may be used. For the case of a sound field in a room, as studied in chapter 5, the modal density grows quickly, however, with frequency; see equation (5-92). This implies that exact descriptions of the sound field ordinarily become unreasonably difficult at even low to moderate frequencies. This is a completely general conclusion and applies to all bounded (finite) acoustical systems with low to moderate damping, as, for instance, the case of bending waves on a steel plate. In order to make such an analysis possible at high frequencies, various energy-based methods have been developed instead. These are based on ignoring the wave character of the sound field, and, instead, on treating the field as the superposition of independent sound rays which can be regarded, locally, as propagating plane waves. These waves propagate in accordance with the laws of geometrical acoustics (compare to geometrical optics); i.e., they follow straight-line paths which can be bent if there is a variation in the medium's wave impedance, or which change direction when reflected by a bounding surface (wall of a room). In an energy-based analysis, the field is characterized by its total energy content, and sound rays by their local energy density or (alternatively) by their intensity. If there are several sources, these are treated as incoherent, which implies that the sources' contribution to the sound field at a certain point can be added on a power basis (see equation (1-28)). Because the wave character of the sound is not considered, energy-based methods cannot describe such phenomena as interference and diffraction. In order to classify different acoustic problems and assess whether an energy-based method is suitable, we can resort to a dimensionless number called Helmholtz' Number (He). That number is defined as He = kl

,

(9-1)

in which k = 2π/λ is the wave number and l a typical dimension (units of length) for the considered system. The wave number k is dependent on which type of mechanical wave is being considered. For sound in a room, k corresponds, of course, to the wave number for airborne sound, and for bending waves in a plate, for example, k should be interpreted as the bending wave number. Helmotz' number gives a measure of the size of the system as measured in sound wavelengths. In order for a standing wave (a mode) to arise in a system, it is necessary that the system be at least a half wavelength large in some direction. This means that if He is much bigger than π, we can expect a large number of modes in the system, so that energy-based methods are applicable. When He is about the same order of magnitude as π, the system's behavior is dominated by a relatively small number of modes and an exact description of the field is possible. The case in which He is much smaller than π is special, and means that the sound wavelength is much larger than the dimensions of the system, in which case it is no longer reasonable to speak in terms of wave propagation. In that low frequency region, the system's behavior can be modeled as an

288

Chapter 9: Energy Methods Applied to Room Acoustics equivalent discrete mechanical system consisting of masses, springs, and viscous dampers. This type of system is described in chapters 3, 9, and 10. In summary, three different frequency regions can be distinguished, for which we introduce the following designations: no-modes region, few-modes region, and manymodes region. A more precise definition of these terms, as well as a summary of the discussion content, is given in table 9-1. Table 9-1 Classification of finite acoustic systems with low or moderate damping (i.e., resonant systems), using the Helmholtz number.

Frequency Region

Character

No-Modes Region

He « π

Wave propagation is ignored. The system can be modeled as a discrete mechanical system.

Few-Modes Region

He ≈ π

A few eigenmodes dominate. A complete mathematical description of the field is possible.

He » π

A large number of eigenmodes control the behavior. Analysis is only practicable using energy-based methods.

Classification

Many-Modes Region

The classical application of energy methods in acoustics (in the analysis of bounded systems) is the field of room acoustics; it is this application which is discussed in this chapter. The original work in the room acoustic field was done by W C Sabine, W S Franklin, and G Jaeger, at the beginning of the 1900's. As far as systems without boundaries are concerned (infinite systems), and which are really not a part of the subject matter of this chapter, the classical application is geometrical acoustics. The earliest work in that area was also carried out at the beginning of the 1900's and is, when we don't have flow in the medium, analogous to the work done in the field of geometrical optics. The most important application is the study of sound propagation in the atmosphere or in the sea. Geometrical acoustics is also applied in room acoustics to analyze the initial phase of a sound field's development after a sound source is suddenly turned on, an analysis which is of interest for, among other things, the dimensioning of concert halls. The possibility of applying energy-based methods more generally, in solid structures for example, has received increasing attention during the last 20-30 years. Special methods in which a statistical approach is coupled to the energy-based description have been developed (SEA = Statistical Energy Analysis). The statistical approach handles the uncertainty, always present in real problems, as to the exact nature of boundary conditions, as well as to the values of other mechanical parameters (mass, stiffness, etc.).

289

Chapter 9: Energy Methods Applied to Room Acoustics Fundamental contributions to that field were made by, among others, R H Lyon, during the 1960's, so that today SEA is an accepted tool for the vibroacoustic analysis of arbitrary structures. An important application which stimulated the development of the original method was the need to be able to attack complicated vibroacoustic problems in the aerospace industry.

9.1.1

Balance of Energy in Simple and Coupled Acoustical Systems

Consider a bounded linear acoustic system into which we supply power via sources, on the one hand, and remove power by means of losses Wdis (dissipation) in the system, on the other hand. Assuming that the law of conservation of energy is applicable to acoustical energy, then the following balance equation can be written for the system dE = Win − W dis , dt

(9-2)

in which E(t) is the total acoustic energy in the system and W(t) refers to power. In the study of linear acoustic systems, the system's behavior is normally analyzed as a function of frequency. For energy-based analyses and applications in the many-modes region (see table 9-1), a number of adjacent frequency bands (octave and third-octave bands) are normally used. The quantities in equation (9-2), and in the other equations in this chapter, are therefore meant to refer to a specific frequency band. That implies that energy and power are calculated from the primary acoustic quantities (e.g., sound pressure), first after those latter have been band-passed filtered. The assumption of linearity also implies that the system dissipation can be characterized by means of a loss factor in accordance with the definition given in chapter 3; see the footnote to equation (3-74). Wdis = ηωE ,

(9-3)

in which η is the system’s loss factor. Putting equation (9-3) into (9-2) yields dE + ηωE = Win . dt

(9-4)

This first order differential equation describes how the system’s energy is built up after an external source has been turned on. A means of determining the loss factor, and hence even the damping, of an acoustic system, is to first excite the system using a source within a certain frequency band (e.g., white noise in an octave band). When the system has thereafter attained a stationary condition, we suddenly turn off the source and measure how the energy decays in time. Assume that we turn off the source at t = 0, and the system’s energy content in the stationary state is E0. For t > 0, equation (9-4) reduces to dE + ηωE = 0 . dt

(9-5)1

1 It should be noted that this equation describes a non-stationary (transient) event, whereas the definition of the loss factor (equation (3-74)) is based on a stationary condition. For small loss factors (≈ quasi-stationary events),

290

Chapter 9: Energy Methods Applied to Room Acoustics This equation has the solution E (t ) = E 0 e −ηωt , t > 0.

(9-6)

An event of the type described by equation (9-6) is called, in acoustics, a reverberant event. By studying the reverberant event, it is possible to experimentally determine the loss factor of an acoustic system as a function of frequency. From equation (9-6), we can define a characteristic time T that constitutes a measure of the duration of the reverberant event. In acoustics, T is normally defined as the time it takes for the energy in the system to decay to a value 10-6 (60 dB) times the original value E0; it is ordinarily called the reverberation time, and that particular definition goes back to the early work in the area of room acoustics by W C Sabine.

Example 9-1 Determine the relation between a system’s reverberation time T, and its loss factor η. Solution According to the definitions given above, the relation 10-6 = e-ηωT must apply, from which it follows that 6⋅ln 10 = ηωT , i.e., T = (6⋅ln 10)/ηω ≈ 13.82/ηω. _____________________________________________________________________ To round off this section, we will consider the energy balance for the case of two coupled acoustic systems (1 and 2). That result will later be applied to the study of sound transmission between two rooms. The point of departure for such an analysis is that we apply the energy balance equation (9-2) to the two systems. That gives

dE1 = W1,in − W1,dis , dt

(9-7)

dE 2 = W 2,in − W 2,dis . dt

(9-8)

Referring to figure 9-1, the terms on the right-hand side are now divided up into different contributions. The power input to system 1 can be written in the form W1,in = W11 + W21 ,

(9-9)

in which W11 is the power input into system 1 from a source in system 1, and W21 is the power input into system 1 from system 2. Moreover, the dissipation in system 1 can be expressed as

W1,dis = η1ωE1 + W12 ,

(9-10)

the distinction is unimportant. For cases in which the loss factors are large, however, equation (9-5) should instead be seen as a definition of the system’s loss factor, distinct from that given in equation (3-74).

291

Chapter 9: Energy Methods Applied to Room Acoustics in which W12 is the power input to system 2 from system 1. W22 W11 W21

η ω E1 1

System 1 E1

System 2 E2

η ω E2 2

W12

Figure 9-1 Two coupled acoustic systems: E indicates total energy and W power transport.

The first term in equation (9-10) represents the portion of the energy loss from system 1 that is completely lost, i.e., converted to thermal energy or radiated to the surroundings. In the same way, and using analogous symbols, we obtain for system 2 the relations

W2,in = W22 + W12 ,

(9-11)

W2,dis = η 2ωE 2 + W21 .

(9-12)

Putting equations (9-9) to (9-12) into (9-7) and (9-8) yields the differential equations dE1 + η1ωE1 + W12 = W11 + W21 , dt

(9-13)

dE 2 + η 2ωE 2 + W21 = W22 + W12 . dt

(9-14)

Equations (9-13) and (9-14) constitute a pair of coupled differential equations that describe how the energy builds up in the two coupled systems after a pair of external sources are turned on. 9.1.2

Relation between Wave Theory and Energy-Based Methods

The equations studied in the preceding section describe how the total acoustic energy of a system changes with time. In formulating these equations, no regard was given to the fact that the energy in the system is transported by waves that propagate, and are reflected, between the boundaries of the system. That aspect will now be given consideration, in order to better understand how a complete field description can, in the right circumstances, be simplified to an energy-based description. In that regard, we shall define the concept of a diffuse field and derive the relation between energy density and incident intensity for

292

Chapter 9: Energy Methods Applied to Room Acoustics such a field. These latter concepts are also necessary precursors to the derivation of the socalled Sabine’s formula in the next section. Our discussion and analysis in this section will be based on the case of a room with hard walls. The results obtained will have general validity and applicability to all types of acoustic systems in the many-modes region. We consider a sound field in the room that falls in the many-modes region, implying that it is built up of a large number of modes (standing waves). Assume that the field, in the immediate vicinity of some arbitrary point in the room, can be regarded as the superposition of propagating plane waves. As is evident from section 5.2, that is the case for a parallelepiped (rectangular prismatic) room. For an arbitrarily-shaped room, it is a reasonable assumption in the many-modes frequency region. Our assumption implies that the sound field, for a specific harmonic component, can be expressed as p=

∑

G G

pˆ n e i (ω t − kn ⋅r ) .

(9-15)

n

An energy-based analysis presupposes that the field can be regarded as built up of independent sound beams that may locally be considered as plane waves. The question is then under what circumstances the plane waves that build up the field in equation (9-15) can be regarded as independent. Since we are concerned with energy quantities, we determine pˆ 2 , which takes the form ˆ2

p =

∑ pne

2

G G −ikn ⋅r

=∑

n

m

G G G G m ⋅r −ik n ⋅r

∑ p ∗m p n e ik

.

(9-16)

n

Because the summation contains cross-terms m ≠ n, the waves are not independent, as we would desire. In order to obtain independent (uncorrelated) waves, a spatial averaging of the field is carried out. Assume that we average equation (9-16) over a sufficiently large volume; the cross terms will then disappear because l

1 ikx e dx ≈ 0 , l

∫ 0

for a sufficiently large value of l. Moreover, the small remaining contributions from the cross-terms have differing signs and tend to cancel each other out. The conclusion can therefore be drawn that, after averaging a sufficient volume (with a diameter of the same order of magnitude as a wavelength), one obtains pˆ 2 ≈ ∑ pˆ n2 ,

(9-17)

n

in which the particular form of brackets used indicates spatial averaging. Equation (9-17) means that the waves can be added in an energy sense and considered uncorrelated (compare section 1.11). In practice, our result implies that we must spatially average whenever we carry out measurements in connection with energy-based methods. Nevertheless, recall that we restricted our analysis to that of a single harmonic component

293

Chapter 9: Energy Methods Applied to Room Acoustics (pure tone). An alternative to spatial averaging, when broad-band sources are investigated, is averaging over frequency, i.e., taking measurements by frequency band. That is directly evident from equation (9-16), since the phase of the exponential function depends on the product kr. We turn now to the relation between energy density and the intensity impinging on a wall in a room in which the sound field is completely dominated by modes (standing waves). In acoustics, rooms that have that character, i.e., are bounded by hard walls with little sound absorption, are called reverberant rooms. The opposite is a room with completely absorbing walls – a so-called anechoic room; in the latter, there are no modes, but rather just a free field surrounding any source. Assume that we have a sound field that fulfills equation (9-17), and that, moreover, all plane waves incident on a point in the room have the same strength and are uniformly distributed over all possible angles of incidence. A sound field that fulfills these criteria, and in which all points in the room are equivalent in the sense that they have the same energy density, is called an ideal diffuse field. The rms sound pressure in such an ideal diffuse field is 1 ~ p d2 = ∑ pˆ n2 = (same strength of all waves ~ p02 ) = N~ p 02 , (9-18) 2 n where N is the number of plane waves incident on a point in the room. Since equation (918) is the sum of plane wave contributions, the energy density ε is obtained by dividing that equation by ρ c2 – see equation (4-99) – so that ε = ~ p 2 ρ c 2 . The power incident d

0

d

0

upon a surface S, belonging to the walls of the room, will now be calculated. We first regard a plane wave that is incident from a certain direction against S (see figure 9-2). The power that meets S is Wn =

~ p 02 S cos θ n , ρ0c

(9-19)

where θ n is the angle of incidence relative to the wall’s normal direction. The total incident power is obtained by adding up all the contributions from the diffuse field Wd = ∑ Wn′ = n′

~ p 02 S cosθ n′ ρ0c ∑ n′

.

(9-20)

The summation in equation (9-20) is only over the modes n´ that are incident on the wall. When the number of modes is large, we can approximate equation (9-20) by an integral over all space angles constituting a hemisphere adjacent to the wall.

294

Chapter 9: Energy Methods Applied to Room Acoustics

kn

k1 k2

Figure 9-2 Diffuse sound field incident upon a wall.

θn

Because the density of waves (number/space angle) is N /4π, then against every small space angle increment ΔΩ ,there are a number of waves NΔΩ /4π. For large N, one therefore obtains ~ p2 Wd ≈ 0 S ∫ cos θ NdΩ / 4π . (9-21) ρ0c That integral can be solved by first expressing the space angle increment, making use of

θ, in the form dΩ = 2πsinθ dθ, and then carrying out the integration over the interval 0 to π /2. The result is Wd ≈

N ~ p 02 S

.

4ρ 0c

(9-22)

With the aid of equation (9-18), we can re-express that result in the form Wd =

~ p d2 S ε cS = (or, alternatively, in terms of energy density) = d 4ρ0c 4

(9-23)

in which the equality applies in the limit as the number of modes becomes infinite. We define the diffuse intensity Id as the power, as expressed in (9-23), per unit area, incident on a boundary surface (wall). That provides us the following relation, which is fundamental to classical room acoustics (sometimes called statistical room acoustics): Id =

εdc 4

.

(9-24)

The relation we have derived by considering sound in a room is very general, and applies as an approximation for the many-modes region to all types of wave fields bounded in a three-dimensional (3-D) system. For example, that relation can also be found in the theory of electromagnetic black-body radiation, in which it gives the electromagnetic power incident on the walls of a cavity in thermal equilibrium. The result in equation (9-24)

295

Chapter 9: Energy Methods Applied to Room Acoustics depends, however, on the dimension number of the enclosed space. Repeating our derivation for the cases of two-dimensional and one-dimensional rooms, respectively, we obtain

Id =

and Id =

εd c , for 2-D π εdc 2

(9-25)

, for 1-D.

(9-26)

If equations (9-24) through (9-26) are to be applied to other types waves than airborne sound, it is important to note that the speed of sound is to be interpreted as that speed at which energy propagates, the so-called group velocity (see chapter 6). For sound waves in liquids and gases, that is normally the same as the ordinary sound speed (phase velocity), but for dispersive waves (i.e., those with a frequency-dependent phase velocity), the two wave speeds do differ; that is the case for bending waves, for example. 9.2

ROOM ACOUSTICS

In this section, we will make use of the results already obtained in this chapter to study sound fields in rooms. As mentioned in the introduction to the chapter, that is the classical application of energy-based methods in acoustics. The point of departure for the derivations that are made is a room with hard walls in the many-modes region. Additionally, all sound absorption is assumed to occur at reflections against the various surfaces in the room. 9.2.1

Sabine’s formula

Sabine’s formula provides a relation between the reverberation time T and the acoustic damping (absorption) of a room. In order to derive that formula, we make use of the relation between reverberation time T and the system’s loss factor (see example 9-1) T = (6 ⋅ ln 10)/ηω

.

(9-27)

In order to obtain the room’s loss factor, we use equation (9-3)

η=

Wdis . ωE

(9-28)

We assume that the sound field in the room can be regarded as an ideal diffuse field with energy density εd. The total acoustic energy in the room can then be written in the form E = Vεd, where V is the volume of the room. When a plane wave with intensity In reaches a surface of the room, a certain sound absorption is obtained. If the surface has the absorption factor α(θ ), then (for the definition of α, see equation (5-27)) Wn,dis = α n I n S cos θ n ,

296

(9-29)

Chapter 9: Energy Methods Applied to Room Acoustics where S is the surface area and the index n indicates the direction from which the corresponding wave is incident; see figure 9-2. The total absorbed power is obtained by a summation over all waves that are incident in the diffuse field. By analogous reasoning to that which led to equation (9-24), we can derive the following result for the dissipated power Wdis : Wdis =

ε d cS

π 2

∫ α (θ ) sin 2θ

4

dθ .

(9-30)

0

The average of the absorption factor defined by equation (9-30) is usually called the absorption factor for diffuse incidence, π 2

αd =

∫ α (θ ) sin(2θ )dθ

.

(9-31)

0

Note that, because sin(2θ ) has a maximum at θ = 45°, the appearance of α(θ) around that angle is of great significance in the computation of αd. The absorption factors for perpendicular (θ = 0°) and grazing incidence (θ = 90°) have no influence whatsoever on the value of αd. In the case of a room with several absorbing surfaces that differ in their respective absorption characteristics, the total absorbed power becomes Wdis =

εdc 4

∑ α d ,m S m

.

(9-32)

m

where the summation includes all surfaces m that contribute to the total absorption of the room as a whole. Sometimes, one even makes use of an average absorption factor for the room, defined as

αd =

∑ α d ,m S m m

S

,

(9-33)

where S = ∑ S m . m

Making use of equations (9-28) and (9-32), we can now calculate the loss factor for the entire room as c αd S . (9-34) η= 4ωV That result can, making use of equation (9-27), be expressed as a reverberation time

297

Chapter 9: Energy Methods Applied to Room Acoustics

T=

(24 ⋅ ln 10)V V = ( with c = 342 m/s) = 0.161 . αd S c αd S

(9-35)

That result is Sabine’s formula for the reverberation time of a room. The first derivation of it resembling ours just given was published in 1903 by W.S.Franklin. Sabine, himself, found the equation empirically after a series of experiments, around the year 1900. When the absorption characteristics of a room are to be described, one ordinarily reports either an average absorption factor, as in (9-33), or, alternatively, the so-called equivalent absorption area A. That area is defined as the total absorbing surface with α = 1, i.e., an acoustic "black hole", which would give the same absorption as the room considered. Mathematically, A can be computed from A = ∑ α d ,m S m ,

(9-36)

m

where the unit for A is usually called [m2 Sabine], or abbreviated to [m2S]. Table 9-2 Absorption data for different materials. (Source: M D Egan, Concepts in Architectural Acoustics, McGraw-Hill, 1972.)

Material Tile Concrete Plywood Window glass Draperies

Untreated Painted 1 cm Pressed thin against wall Thick, drawn up

Concrete floor with linoleum layer with thick mat Wood floor Ceiling

Absorption factor αd

Description

Gypsum slabs Plywood 1cm

125 Hz

250 Hz

500 Hz

1 kHz

2 kHz

4 kHz

0.03 0.36 0.10 0.28

0.03 0.44 0.05 0.22

0.03 0.31 0.06 0.17

0.04 0.29 0.07 0.09

0.05 0.39 0.09 0.10

0.07 0.25 0.08 0.11

0.35

0.25

0.18

0.12

0.07

0.04

0.03

0.04

0.11

0.17

0.24

0.35

0.14

0.35

0.55

0.72

0.70

0.65

0.01

0.01

0.02

0.02

0.02

0.02

0.02

0.03

0.03

0.03

0.03

0.02

0.02 0.15 0.29 0.28

0.06 0.11 0.10 0.22

0.14 0.10 0.05 0.17

0.37 0.07 0.04 0.09

0.66 0.06 0.07 0.10

0.65 0.07 0.09 0.11

Besides the surfaces of the room itself, additional absorption of acoustic energy is also obtained from any objects (e.g., furniture) and persons that may be present. These added absorbing entities are normally characterized by a supplemental equivalent absorption energy ΔA. Table 9-3 gives examples of the supplemental absorption energy provided by certain objects.

298

Chapter 9: Energy Methods Applied to Room Acoustics

Table 9-3 Added equivalent absorption area from several objects (Source: H Kuttruff, Room Acoustics, Applied Science, 1973.)

Object

Description

Standing human Student, incl seat Chair

With coat Without coat

125 Hz 0.17 0.12

added absorption area [m2S] 250 Hz 500 Hz 1 kHz 2 kHz 0.41 0.91 1.30 1.43 0.24 0.59 0.98 1.13

4 kHz 1.47 1.12

Sitting

0.20

0.28

0.31

0.37

0.41

0.42

Cushioned

0.55

0.86

0.83

0.87

0.90

0.87

9.2.1.1 Limitations of the Sabine-Franklins theory and later work There are a number of limitations of the theory that underlies Sabine’s formula. We will briefly discuss these, as well as touch upon the development of the field of statistical room acoustics after Sabine’s and Franklin’s classical work. One of the most important limitations of the theory is, of course, the assumption of an ideal diffuse field. That idealization has several parts, above all that the field consist of incoherent waves. As indicated in section 9.1.2, that condition is fulfilled in practice by taking spatial and frequency averages of our energy quantities. An ideal diffuse field should be, moreover, homogeneous and isotropic, i.e., all points in the room are to be equivalent, and all directions of propagation equally probable. That requires that the absorption be uniformly distributed throughout the room, so that certain directions of propagation do not become dominant. An example is a rectangular prismatic room in which a wall has a much higher absorption coefficient than all others, leading to a 2-D diffuse field more so than a 3-D one; in consequence, a longer reverberation time is obtained (compare equations (9-24) and (9-25)) than would be expected from a direct application of Sabine’s formula. In order that the assumption of homogeneity remain valid during the reverberation event, it must also not occur too rapidly. Since absorption occurs at surfaces in the room, there is a tendency for the energy density to be smaller in their immediate vicinity than elsewhere in the room. In order for the field to remain homogeneous, the typical time that it takes for sound energy to even itself out throughout the room must be shorter than the reverberation time. It can be shown that that limits the validity of Sabine’s theory to rooms in which ≤ 0.3. A long series of modified versions of the classical Sabine-Franklin theory of reverberant events have been proposed. For example, the work of R F Norris and C F Eyring was published around 1930. That work takes, as a starting point, a sound beam that propagates and is reflected against the surfaces of the room. The reverberant event is obtained as an average over all possible paths that the beam takes through the room. That approach is also the basis of most later work on energy-based methods for room acoustics. Despite the many advances made after the original development of Sabine’s theory, it nevertheless remains the standard approach normally applied in practice, in routine analysis of the reverberant behavior of a room. Sabine’s formula is also used in standardized measurements of the acoustic absorptive capacity of materials. Such measurements are performed in special reverberant rooms in such a way that Sabine’s theory can be assumed to apply. Measurements of this type must also take account of the

299

Chapter 9: Energy Methods Applied to Room Acoustics phenomenon of absorption (damping) within the medium itself. The effect of such damping is particularly important in rooms that have a large volume, and at high frequencies. Sabine’s formula, incorporating a correction for damping in the medium, is available from most acoustics handbooks, and from the ISO standards for reverberant room measurements; see section 2.3.1. 9.2.1.2 Measurement of reverberation time Finally, this section will conclude with a discussion of how reverberation time can be determined experimentally. The standard method of measuring reverberation is based on excitation by a broad-band noise source in a certain frequency band (third-octave or octave-band). Because energy density is proportional to the square of the sound pressure, we can record a reverberant event by measuring the time decay of the sound pressure level after a source is abruptly interrupted. As a step in computing the sound pressure level, we must determine the rms sound pressure. For sound fields with an energy content that varies in time, the rms value is calculated by means of a so-called moving average, as 1 ~ p 2 (t ) = Tav

t

∫p

2

(τ )dτ

,

t −Tav

where Tav must be much shorter than the reverberation time T. The traditional way to carry out the measurement is to record the sound pressure level on a print-out with a linear SPL scale. 10 dB

T = 4 .1 s

T = 5 .4 s

T = 4 .6 s

1 0 m m /s 500 H z

1000 H z

2000 H z

Figure9-3 Examples of reverberation curves measured in third-octave bands in the reverberation room at the Marcus Wallenberg Laboratory for Sound and Vibration Research, MWL, KTH.

If the recording paper unfurls at a constant rate, then an ideal reverberant event, in accordance with Sabine’s theory, would be drawn as a straight line. In practice, deviations from that ideal occur. In part, there are random fluctuations due to the finite averaging interval used to compute the rms value of the pressure; additionally, non-uniform distribution of absorption in the room results in some degree of curvature in the decay curve. An example of reverberation curves measured in a special reverberant room is shown in figure 9-3.

300

Chapter 9: Energy Methods Applied to Room Acoustics 9.2.2

Sound fields in rooms

We now analyze a stationary sound field in a room due to a source located within it. A stationary sound field can be interpreted as one with a time-invariant frequency spectrum. Within the framework of an energy-based model, we can divide up the sound field into two parts: the sound beams that have just departed from the source and not yet been reflected by any surfaces; and, the sound beams which have already been reflected one or more times by surfaces. The first part of the field, which is not as yet affected by the room, can be regarded as the same field which would be radiated in a free field environment without bounding surfaces; if the source is, however, in the vicinity of a boundary, e.g., in the middle of the floor or in a corner, then a free field is only obtained in those directions in towards the interior of the room. The part of the total sound field that has, as such, free field character, is usually called the direct field.. The other part of the field, consisting of reflected sound beams, is usually called the reverberant field. The two fields can be considered uncorrelated, so that the total sound field can be obtained by adding the respective energy densities of each. In a stationary situation, the power input to the reverberant field must equal the power lost in reflections at the surfaces of the room. We assume that the reverberant field is an ideal diffuse field, and can then calculate the power dissipated in the room using equations (9-32) and (9-33), which yield W dis =

Direct field

εdc αd S 4

.

(9-37)

Reverberant field

W dir

Figure 9-4 Sound field in a room: decomposition into a direct field and a reverberant field.

The power Wdis must correspond to the power provided by the direct field minus that which is lost in the first reflection, i.e., W dis = W dir (1 − α d ) ,

(9-38)

where Wdir is the power sent out by the source. Note that we have chosen to use the absorption factor for diffuse incidence in equation (9-38). That choice was not obvious, and implies, in fact, that we have assumed an isotropic source, and that the direct field spreads out so as to reach the surfaces of the room with a uniform distribution of angles of

301

Chapter 9: Energy Methods Applied to Room Acoustics incidence. For a strongly directional source, or a room with non-uniform distribution of absorbent, that assumption doesn’t hold, and another value of the absorption should be used instead. The value we selected corresponds, nevertheless, to the conventional assumption made in such analyses. The energy density of the direct field, at such distances that sound beams can be regarded as locally plane waves (i.e., in the far field – see (4-100) and (4-123)), is given by Γ W dir ε dir = , (9-39) 4πcr 2 where Γ is the directivity indexError! Bookmark not defined. (direction factor) of the source, which indicates how the sound beam varies in different directions, and r is the distance from a reference point at the source (origin) to an observation point. The directivity index is defined such that the integral of Γ over all space angles that point from the source into the interior of the room, is 4π. For a source that radiates equally in all directions (isotropic source), and is located in the middle of the room, Γ = 1; for the same source on the floor, Γ = 2; for that same source located at an edge between the floor and a wall, Γ = 4; and, for the same source located in a corner of the room, Γ = 8 (see figure 95).

Γ=2

Close to one reflecting surface

Γ=1 At a distance from all reflecting surfaces

Γ=4

Γ=8 At the intersection of three reflecting surfaces

At the intersection of two reflecting surfaces

Figure 9-5 Directivity index Γ (direction factor) for an isotropic source located at various positions in a room with perpendicular surfaces. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

From (9-37), (9-38) and (9-39), the total energy density of the room can be expressed as ΓWdir 4(1 − α d )W dir ε tot = ε dir + ε d = . (9-40) + c αd S 4πcr 2

302

Chapter 9: Energy Methods Applied to Room Acoustics Both energy densities in this equation are related to the rms sound pressure in accordance with the plane wave equation (4-99). Equation (9-40) gives, therefore, an rms sound pressure, from a source in the room, of ⎡ Γ 4(1 − α d ) ⎤ 2 2 ~ p tot p dir p d2 = ρ 0 cWdir ⎢ =~ +~ + ⎥ . 2 α d S ⎥⎦ ⎣⎢ 4πr

(9-41)

Alternatively, expressing the result in terms of sound pressure level in air, taking specific impedance to be ρ0c = 400 Pa s/m, then ⎡ Γ 4⎤ dir + ⎥ [dB] Ltot p = LW + 10 ⋅ log ⎢ 2 A′ ⎦ ⎣ 4πr

(9-42)

where A′ =

αd S (1 − α d )

,

is called the room constant. The distance from the source at which the direct field and the reverberant field are equally strong is called the echo radius. By equating the two terms in parentheses in equation (9-42), the echo radius is found to be ⎛ A′Γ ⎞ re = ⎜ ⎟ ⎝ 16π ⎠

12

.

(9-43)

Figure 9-6 illustrates the main features of the variation of sound pressure level with distance from a source in a room.

303

Chapter 9: Energy Methods Applied to Room Acoustics

Ldp + 10 Total sound pressure level

Ltot p

Ldp + 5

Ltot p [dB]

3 dB

Ldp Sound pressure level in Ldp reverberant field Sound pressure level in direct dir Lp field

Ldp − 5

Ldp − 10 0

1,0

2,0 Avstånd/Ekoradie Distance / Echo radius r/re

3,0

Figure 9-6 The distance-dependence of sound pressure level, with respect to the level in the reverberant field, according to equation (9-42). The distance to the source is measured in echo radii; see equation (9-43). Note that when the distance from the source is equal to the echo radius, the direct and reverberant fields contribute equally to the total sound pressure level, which therefore rises by 3 dB.

In practical applications of the results obtained above, A’ is usually interpreted as the equivalent absorption area A (see equation (9-36)), i.e., the effect of the factor (1-) is ordinarily ignored. In the context of Sabine’s theory, that is in fact reasonable, considering that the influence of that factor on A is proportional to 2, which should be insignificant in the frequency region in which the theory is valid, i.e., when ≤ 0.3 (see section 9.2.1.1). Finally, we note that the value of the sound power emitted by the source, Wdir, is dependent on the location of the source. As we will see in chapter 8, that is because the radiation impedance experienced by the source depends on its proximity to large (in terms of the Helmholtz number) reflecting surfaces. That also means that the sound power emitted by the source will depend on its location in the room, i.e., whether the source is placed on the floor or in a corner. That is an important consideration to be born in mind in practical applications of the results obtained above. 9.2.3

Acoustic absorbents

The most common measure of the acoustic behavior of a room is the reverberation time. Reverberation time alone, however, is rarely sufficient in and of itself. The distribution of the direct field, as well as reflections that might give rise to echo effects, must also be taken into consideration in, for instance, a lecture hall. Especially demanding venues, such as concert and opera houses, are usually designed with the aid of special computer

304

Chapter 9: Energy Methods Applied to Room Acoustics programs that, based on geometrical acoustics, predict the distribution of the sound field through the first few reflections. 2.5

T 60 , [s ]

2.0

Katolska kyrkor Catholic churches Konsertsalar Concert halls Konferensrum Conference halls Sound studios Studios

1.5

1.0

0.5

0 10

100

1000

10000

3

Volym[m ,[] ] Volume Figure 9-7 Optimal reverberation time in various locales, for the frequency range 500-1000 Hz, according to L.L. Doelle, Environmental Acoustics, McGraw-Hill, New York, 1972.

To perform well, locales of different types require different reverberation times, according to their respective functions. To bring about a desired reverberation time in a locale, while minimizing undesirable reflections (echoes), different kinds of acoustic absorbents are used. These absorbents are normally fixed to different reflecting surfaces, such as walls and the ceiling. In this section, we will concentrate on some basic kinds of absorbents, and discuss the mechanisms of sound absorption that they display. The absorption factor concept provides a measure of the performance of an absorbent material; for the special case of normal plane wave incidence, it is defined by equation (5-27). That definition can, moreover, be generalized to arbitrary sound fields without modification. In that case,

α =1−

Wr , Wi

(9-44)

where the symbols in the numerator and denominator of the second term indicate total reflected and total incident acoustic power, respectively, from/upon a given area S. It is important to note that the absorption factor, in the general case, depends on both the surface characteristics (including geometric form) and on the incident sound field. For the case of plane waves, and a sufficiently large (in terms of Helmholtz number) plane surface (the characteristics of which are identical at all points, and independent of direction, i.e., the surface is acoustically homogeneous and isotropic) the factor is only a function of the angle of incidence, which is specified by the angle θ between the incident wave normal and the surface normal. That assumption is standard in the derivation of formulas in room

305

Chapter 9: Energy Methods Applied to Room Acoustics acoustics, and in the practical application of the latter. For instance, we use that assumption in our derivation of the absorption factor αd for an ideal diffuse field (see equation (9-31)). 9.2.3.1 Porous absorbents

Porous absorbents consist of solid material with a degree of porosity (i.e., materials with cavities interconnected by channels) that permits air to be forced into the interior of the material matrix. Most often, porous absorbents are fibrous materials consisting of thin (220 μm) mineral or glass wool fibers, arranged in layers and with random fiber directions in planes parallel to the material surface. Alternatively, fibrous materials based on natural wood fibers are also available for building applications. For high-temperature applications, on the other hand, pressed metal wool might sometimes be preferred. When sound propagates in a porous fiber absorbent, acoustic energy dissipation results from the viscous forces arising as air is forced to flow through the small passages between the fibers. Additionally, heat transfer adds to the effect, to some extent; the temperature fluctuations inherent in a sound wave are evened out by contact with the fibers, which are better heat conductors than the air itself. That process is never completely reversible, implying that losses occur. Another mechanism at work, and which may be significant for sound absorption, is the coupling between sound and vibrations in the porous material’s solid matrix. That phenomenon is normally negligible for fibrous porous materials, except at low frequencies (under 300 Hz). For other types of porous materials (e.g., foam), however, the effect of the ”fluid-structure” coupling may be quite significant. The classical model of a porous absorbent, mainly applied to the fibrous type, is based on considering the absorbent to be an equivalent fluid. In its simplest form, the approach replaces the absorbent by a homogeneous fluid with viscous damping. That viscous damping can be incorporated as an extra term in the equation of motion, equation (4-26),

ρ 0 iω u x +

∂p + φu x = 0 , ∂x

(9-45)

in which we have assumed a harmonic signal and written the equation in complex form, and in which φ refers to the flow resistance [Pa s/m2]. Assuming a homogeneous and isotropic absorbent, the equations in the y and z directions are completely analogous. In practice, however, the flow resistance perpendicular to the absorbent surface is usually greater than that parallel to it. As a result, we have to specify a flow resistance for each coordinate direction φx, φy and φz. The flow resistance in our model could be frequencydependent and complex. Handbook data on fiber materials is usually limited to values of φ for the case of a time-invariant flow (i.e., the case α = 0). Such a measurement is made by measuring the pressure drop of a steady flow over a slab of absorbent material, as in figure 9-8. From equation (9-45), it is evident that the pressure, in that case, varies linearly in x, so that

φ=

1 ( p1 − p 2 ) , ux h

where h is the thickness of the slab.

306

(9-46)

Chapter 9: Energy Methods Applied to Room Acoustics

p

Figure 9-8 The principle for the measurement of the flow resistance φ of an absorbent material. By measuring the pressure drop p1 – p2 and the flow rate ux over the absorbent test sample, the flow resistance follows from equation (9-46).

p

1

2

x ux

Figure 9-9 shows two typical flow resistance (φ) curves for fiber-type absorbent materials. Because the standard unit of φ in the SI-system is very small in magnitude, it is customary to instead use the corresponding unit in the cgs (cm-gram-second) system. Since φ represents a wave resistance per unit length, and the cgs unit of wave resistance is called a Rayl, the cgs unit of φ is therefore a Rayl/cm. To convert to SI units, we note that 1 Rayl/cm = 103 Pa·s/m2. The absorption factor of a porous absorbent is ordinarily determined from measurements specified by ISO standards. Traditionally, the absorption is either measured for the case of perpendicular incidence, using standing waves (in a so-called Kundt’s tube), or it is measured in a reverberant room. By measuring the difference in the reverberation time between the room with, and the room without, the absorbent, and making use of Sabine’s formula, the diffuse field absorption factor can be inferred. Calculating the absorption directly is also possible in some cases. For fibrous absorbents, for example, one might use the equivalent fluid model described in connection with equation (9-45). A necessary condition for such an approach, however, is access to reliable material data on the absorbent, which is often difficult to come by in practice. The most common way to use a porous absorbent in room acoustics is to locate it in front of a hard wall, for the purpose of reducing the wall reflections. We now consider that approach, in order to determine how to choose the flow resistance and absorbent thickness, in order to obtain the best possible sound absorption. The discussion will be based on simulated data obtained from a semi-empirical equivalent fluid model developed by Delany and Bazley (1969).

307

Chapter 9: Energy Methods Applied to Room Acoustics Flow resistance φ, [Rayl/cm] 100

10

1 10

Density [kg/m3]

100

Figure 9-9 Typical values of the flow resistance φ perpendicular to the surface of glass wool absorbents (solid line) and rock wool (dashed line).

This model is described in acoustics handbooks, and requires, as a prerequisite, the measurement of the flow resistance φ. In figure 9-10, some typical results are provided. As is clear from the figure, “optimal” absorption properties are obtained if we choose a dimensionless flow resistance that satisfies

φ h / ρ0c ≈ 2 ,

(9-47)

where h is the absorbent thickness. The vicinity of the optimum is, nevertheless, quite flat, so that any value in the interval 1 - 3 would be almost as good. Values far outside of that interval, however, would give clearly inferior absorption performance.

308

Chapter 9: Energy Methods Applied to Room Acoustics Absorption factor 1,0 0,8

3 21

10

0,6

α⊥

0,1

0,4 0,2 0,0

h 0

0,5

1,0

1,5

2,0

2,5

kh

3,0

3,5

4,0

4,5

5,0

Figure 9-10 Calculated absorption factor for a plane wave with normal incidence against a fiber-type absorbent mounted in front of a hard (completely reflective) wall. The parameter in the diagram is the dimensionless flow resistance φ h/ρ0c.

Equation (9-47) shows that the thinner the absorbent that we select, the greater the flow resistance needed to obtain a good sound absorption. For the optimal case, we obtain α⊥ ≥ 0.9 for a Helmholtz number kh greater than about 1, where k is the wave number in air and α⊥ is the absorption factor for normal incidence. That result is usually expressed as a rule of thumb: that an (optimal) absorbent can be expected to provide good absorption if its thickness is about a quarter wavelength. All curves in figure 9-10 seem to approach an absorption factor α⊥ = 1 at high frequencies, which is equivalent to the reflection coefficient approaching zero. According to equation (5-68), this means that the impedance Zabs of the absorbent must approach the specific impedance of the surrounding air. In other words, at high frequencies, a sound wave passes directly into the absorbent without reflection, and is then gradually converted into thermal energy as it propagates further. Example 9-2 Assume that we wish to dimension an absorbent to be placed in front of a hard wall that is to give α⊥ ≥ 0.9 above 500Hz. Solution The condition that kh ≈ 1 at 500 Hz gives

h = c/ 2πf = {c = 340 m/s} =0.11 m, and we therefore choose h = 10 cm. Equation (9-47) now provides the optimal value of the flow resistance φ 2ρ c φ ≈ 0 = {ρ 0 c = 400 Pa ⋅ s/m} = 8.0 ⋅ 103 Pa ⋅ s/m 2 . h

309

Chapter 9: Energy Methods Applied to Room Acoustics If we choose rock wool as our absorbent, then figure 9-9 shows that the selected flow resistance corresponds to a density of about 45 kg/m3. In figure 9-11, below, the calculated absorption factor in that case (case I) is shown. For comparison, the figure also shows the absorption obtained if we only use an “optimal” absorbent with h = 5 cm (case II). As would be expected, that about doubles the frequency at which α⊥ ≥ 0.9. Note that, for that thickness, double the flow resistance is needed to attain the optimum case. A very similar curve is obtained in case III, in which we move the same absorbent as in case II to a position 5 cm away from the wall. The low frequency damping is hardly affected at all, but we do get some fluctuations at higher frequencies. If the absorbent is moved too far from the wall, say more than 4-5 times its thickness, then while the low frequency performance is certainly good, there can be unacceptably large fluctuations in the high frequency performance. That approach to improving the low frequency performance, for a given absorbent thickness (say 5 cm), is often applied in practice, e.g., by mounting the absorbent to bolts providing something of an air column. An example would be lecture halls and auditoria in which the ceiling absorption is enhanced by mounting absorbent panels a couple of decimeters below the inner ceiling. Absorption factor 1,0 I

0,8

α⊥

II

Fall I 10 cm III Fall II 5 cm

0,6

0,4

Fall III 5 + 5 cm

0,2 0

0

1000

2000

3000 4000 Frequency, [Hz]

5000

6000

Figure 9-11 Calculated absorption factor for c = 340 m/s, for the three cases described in the example.

In many applications, the porous absorber must be given some kind of a covering layer. The purpose of that is partly to prevent the loss of fibers to the surroundings, which could constitute a health risk, and partly to protect the absorbent from various external agents, such as humidity and gases with high flow rates. The cover layer normally consists of a thin plastic foil or the like, or perhaps perforated plate. A cover layer normally degrades the acoustic performance of the absorber, especially at high frequencies; that is because the mechanical role of the absorber is that of an equivalent mass, which makes it more difficult for sound to penetrate into the absorbent. Assume that a 20% reduction of the absorption factor, from 1 to 0.8, is permissible at high frequencies. Then, it can be shown that, for a cover layer with surface density m″ and with negligible bending stiffness, the reduction in absorption is inconsequential up to a frequency of

310

Chapter 9: Energy Methods Applied to Room Acoustics fg =

ρ0c . 2πm ′′

(9-48)

Example 9-3 Assume that we would like to cover an absorbent by a cover layer, without degrading the absorption characteristics at frequencies under 3000 Hz. What is the maximum permissible surface density? Solution Using equation (9-48), we find m” ≤ 22 g/m2.

For perforated plate cover layers, an inertia effect also occurs; that is attributable to the local acceleration of the fluid at the perforation holes. Moreover, the mechanical characteristics of the plate also have an effect. If the plate is compliant in bending, it provides an added inertia, which should be added to that of the holes. Formulas to calculate the added acoustic mass can be found in most acoustic handbooks; see, also, chapter 10.

Sound absorbing baffles

Traverse

Sound absorbing suspended roof

Figure 9-12 In hard environments, with long reverberation times, it may be necessary to increase the absorption by installing added absorbent in the form of, for example, porous absorbers. These are often installed as ceiling baffles, along walls, or in other places where they do not hinder the room’s normal activities. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

9.2.3.2 Resonant Absorbers

This kind of absorbent consists of acoustic systems that provide sound absorption in frequency bands around the vicinity of their resonance frequencies. These are primarily used to obtain sound absorption at low and mid frequencies, up to 500 - 600 Hz. The types

311

Chapter 9: Energy Methods Applied to Room Acoustics of absorbers considered up to this point have all been broad-band, with a lower frequency bound approximately corresponding to a quarter wavelength equal to the absorbent thickness. As a result, such absorbers are most suitable for high-frequency applications, above about 500-600 Hz. Porous absorbers that provide good low frequency absorption must be made unreasonably thick (e.g., at 100 Hz, a thickness λ /4 ≈ 75 cm), so that such an approach is only undertaken in very special cases, such as anechoic rooms in laboratories. As an introduction to the discussion of resonance absorbers, a plane surface with an impedance of Z is considered. We assume that that impedance is equivalent to a springmass system with damping, giving Z = iωm ′′ +

κ + Rd , iω

(9-49)

where m″ is the mass per unit area, κ is the spring constant per unit area, and Rd is a term that describes the system damping. Note that these parameters can depend on the details of the incident sound field. Otherwise, we have what is called a locally reacting surface; see section 5.1.5. At the resonance frequency ω0, defined as the frequency at which Im(Z) = 0, the impedance is only due to damping: Z = Rd. According to equation (5-68), the reflection coefficient at that frequency, for an incident plane wave, is R − ρ 0 c cos θ i R= d . (9-50) Rd + ρ 0 c cos θ i As is evident from that equation, we can obtain zero reflection, i.e., R = 0, for a certain angle of incidence, by choosing Rd = ρ0c / cosθ i. ´For the case of normal incidence, θ i = 0, in particular, Rd must be chosen so as to equal the specific impedance of the surrounding fluid. The most common way to bring about a resonant absorber is to create the direct acoustic equivalent of a mass-spring system, a so-called Helmholtz resonator; see chapter 10. Such a resonator consists of a constriction (mouth), which gives rise to an inertial effect, and an enclosed fluid volume, which corresponds to a spring; see figure 9-13. To obtain the desired damping, the resonator can be partially, or completely, filled with porous absorbent material. Figur 9-13 Absorbent consisting of a group Helmholtzresonators.

The most effective approach is to locate the absorbent in, or near, the constriction. A Helmholtz resonator is a single degree-of-freedom system, and only gives, therefore, a single resonance frequency about which sound absorption is to be expected. Another common way to realize resonant absorption is to place an absorbent behind a compliant plate (panel); figures 9-14 and 9-15 illustrate that type of system, commonly referred to as a panel absorber.

312

Chapter 9: Energy Methods Applied to Room Acoustics

Low frequency High frequency

Low frequency High frequency

Influence of the distance between Influence of the plate thickness stffeners

Low frequency

High frequency

Influence of the distance to the wall

Figure 9-14 By varying such design parameters as the distance between fastening points (i.e., the stiffness), the panel thickness, and distance from the wall, the effective frequency band of a panel absorber can be tuned to some extent. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

The lowest resonance frequency of this system occurs when the contained air Wi Wr volume acts as a spring, and the panel as a mass. At higher frequencies, panel standing waves build up between the 0 absorber wall and the panel, giving rise to a h number of resonance frequencies (compare the phenomenon of standing waves in cavities, section 5.2). wall x Normally, however, the systems behavior is merely optimized for the Figure 9-15 Panel absorbent mounted to a hard wall. The panel is assumed to be compliant in lowest resonance frequency. That bending, and has a surface density m''. resonance can be calculated if we know the spring constant κ of the enclosed volume of air. To calculate κ, consider a harmonic plane wave, with perpendicular incidence; from the equation of continuity (eq (4-11)), iω ρ + ρ 0

∂u x =0 , ∂x

(9-51)

in which we have assumed that the entire space between the panel and the wall is air-filled. If we assume low frequencies (kh « 1), then the particle velocity field in a standing wave, directly in front of a hard wall, varies linearly, while sound pressure, and thereby density as well, are largely independent of x; see figure 5-2. Integration of equation (9-51), at low frequencies, therefore yields iωhρ + ρ 0 (u x (h) − u x (0)) = 0 ,

313

Chapter 9: Energy Methods Applied to Room Acoustics where ρ is assumed constant (independent of x). The boundary condition at the plate gives ux(0) = vx, and, at the hard wall, ux(h) = 0. Substituting the relation between sound pressure and density (p = c2ρ, equations (4-31) and (4-44)), finally, gives p=

vx iω h ( ρ 0 c 2 )

.

From that equation, the spring constant κ (per unit surface area) is directly obtained as

κ=

ρ0c 2 h

.

(9-52)

For oblique incidence at an angle θ , analogous logic finds that κ changes by a factor of 1/cos2θ. With absorbent behind the panel, another (lower) speed of sound applies. If the entire space is filled with absorbent, the isothermal sound speed in air is a good approximation for c at low frequencies. That speed can be obtained by dividing the

γ , after which the fundamental resonance frequency of the panel absorber can be calculated according to

adiabatic sound speed by

f0 =

1 2π

κ m′′

=

1 2π

⎧ 60 ⎪ m′′h , adiabatic 2 ⎪⎪ ρ0c = {for air} ≈ ⎨ m′′h ⎪ 50 , isothermal ⎪ ⎪⎩ m′′h

,

in which normal temperature and pressure of air have been assumed, when putting in the density and sound speed. For a panel absorber which partially filled with absorbent, the resonance frequency falls somewhere between the two values given by equation (9-53). Figure 9-16 shows the calculated absorption factor, for a normally-incident plane wave, against a panel absorber filled with absorbent. The calculation has been carried out by assuming that the system’s impedance can be written as Z = iωm″ + Zabs , in which the absorbent’s impedance is obtained from the model used earlier in this section (Delany and Bazley(1969)). For comparison, we can calculate the resonance frequency of the system in figure 9-16, from equation (9-53) (isothermal case). That yields

f0 ≈

⎧m′′ = 3.0 kg/m 2 ⎫ 50 =⎨ ⎬ ≈ 130 Hz , m′′h ⎩ h = 0.05 m ⎭

which is somewhat lower than the peak observed in figure 9-16, lying around 135 Hz.

314

(9-5

Chapter 9: Energy Methods Applied to Room Acoustics Absorption factor 1,0 0,8 0,6

α⊥

0,4 0,2 0,0 50

100

150

250 200 Frequency [Hz]

300

350

400

Figure 9-16 Calculated absorption factor for a panel absorbent (see figure 9-15). Data: m'' = 3.0 kg/m2, h = 5.0 cm, φ = 3.0 ⋅103 Pa ⋅s/m2 .

Absorbing walls

Hard wall Wooden beam Porous board covered with plastic laminate

Engine test benches

Figure 9-17 In the low frequency region, porous absorbers are inadequate. Panel absorbers may then serve as practical alternatives to reduce high levels of noise in, for example, workshop environments. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

315

Chapter 9: Energy Methods Applied to Room Acoustics 9.2.4

Sound Transmission through Insulating Partitions

To round off the treatment of room acoustics, consideration is now given to the transmission of sound through insulating partitions used in or between rooms (shields and walls, respectively). The purpose of such a partition is to stop the propagation of sound, mainly by providing an impedance jump that reflects the sound incident upon it. To characterize the ability of an insulating partition to stop sound, the concept of the transmission factor τ and the sound reduction index R are used. The transmission factor of a partition with an area S is defined as

τ=

Wt Wi

,

(9-54)

in which the denominator contains the total incident acoustic power Wi, and the numerator the total transmitted acoustic power Wt. It is important to note that the transmission factor, in general, depends on the mechanical properties of the partition itself, and its geometric form, as well as the specific form of the incident sound field. For the special case of plane waves, and a sufficiently large (measured in Helmholtz numbers) plane partition surface (which is acoustically homogeneous and isotropic), the factor is only a function of the direction of the incident wave relative to the normal direction (the angle of incidence θ). That is the case we consider in this treatment. The transmission factor of most insulating partitions is in the range 10-6 - 10-2. It can be greater at low frequencies, however. The sound reduction index is defined as R = 10 ⋅ log

1

.

τ

(9-55)

A deeper understanding of the concepts of absorption and transmission factors can be obtained from a consideration of figure 9-18. The figure illustrates the power balance that must hold for a sound field incident upon an insulating partition. Absorber Figure 9-18 Power balance for a sound field incident upon an insulating partition consisting of absorbent attached to a plate.

Wr

Wt Wdis

Wi

Plate

Using the notation from the figure, the power balance can be expressed in the form Wi = Wr + Wt + Wdis ,

316

Chapter 9: Energy Methods Applied to Room Acoustics

where Wdis is the power lost to damping in the absorber or elsewhere. Using the definitions of α and τ , that can be written as

α =τ +δ,

(9-56)

where δ = Wdis / Wi is the dissipation factor. For an absorbent mounted to a plate (figure 918), and except in the few-modes region (see section 9.2.4.3, figure 9-23), it is normally the case that τ « δ , which implies that the absorption factor is not significantly influenced by the plate (i.e., the plate can be considered “hard”). The reverse is not true, however. The transmission factor can be significantly altered by covering a wall with an absorbent, if the absorption factor is nearly one. We now consider the sound transmission between two rooms (systems 1 and 2) separated by a wall. Our point of departure is a stationary sound field emitted by a source in one of the rooms, in room 1 for example. Making use of the energy balance equations for two coupled systems, equations (9-13) and (9-14),

η1ωE1 + W12 = W11 + W21 ,

(9-57)

η 2ωE 2 + W21 = W12 ;

(9-58)

see figure 9-19. The loss factors in the respective systems (rooms) can, if we assume ideal diffuse fields, be calculated from equation (9-34), η1 =

cA1 , 4ωV1

(9-59)

η2 =

cA2 , 4ωV2

(9-60)

where A1 and A2 , are the equivalent absorption areas of the rooms in accordance with equation (9-36).

Source Room 2 Room 1 Figure 9-19 Sound transmission between two rooms: W is sound power, ε energy density, V volume and S the area of the insulating partition. Compare to figure 9-1.

317

Chapter 9: Energy Methods Applied to Room Acoustics

In order to compute the power transmission between the rooms, we consider the diffuse fields incident upon the wall separating them, and make use of the definition of the transmission factor W12 = τ d I d ,1 S ,

(9-61)

W21 = τ d I d ,2 S .

(9-62)

in which Id signifies the diffuse intensity and τd the transmission factor for a diffuse field. That factor can be calculated by an approach analogous to that for αd in equation (9-31). Note that the transmission factor from room 1 to 2 is assumed identical to that from 2 to 1. It can be shown that that reciprocity principle for transmissions factors has general validity. By expressing the field’s energy (E = εV) and diffuse intensity (see equation (924)) as energy densities, and putting those into the equations given above, the following relation then it follows from equation 9-58 that

ε d ,1 ε d ,2

=1+

A2 . Sτ d

(9-63)

Because ε d = ~ p d2 ρ 0 c 2 , an expression for the difference in the sound pressure level between the rooms is obtained directly from equation (9-63) L p,1 − L p,2 = 10 ⋅ log(1 +

A2 ) . Sτ d

(9-64)

That result shows that the sound reduction between two rooms depends on both the partition’s sound reduction index and the absorption in the receiving room (room 2). It can be observed that, if the absorption in the receiving room is zero, the same sound level is attained in each room, regardless of how good the insulating wall is. Equation (9-64) is the basis of a technique for the experimental determination of the transmission factor. Assume that we have two reverberant rooms, coupled by a rectangular aperture in their separating wall. A wall element to be tested is then mounted in the aperture. By measuring the difference in the sound pressure level between the rooms, with a sound source in one of them (the sender room, room 1), the transmission factor can then be inferred from equation (9-64). A necessary precondition is, however, that we have first determined the receiver room’s (room 2’s) equivalent absorption area, by a measurement of the reverberation time and an application of Sabine’s formula. In practice, the unit-valued term in the parentheses of equation (9-64) is normally ignored in the application of that equation to the prediction of the sound reduction between two rooms. The equations can then be written in the form A L p,1 = L p,2 + R d + 10 ⋅ log 2 . (9-65) S That approximation usually results in an error of less than 1 dB, which is negligible in practice, especially considering that our idealized model (assuming ideal diffuse fields) is already approximate in most rooms.

318

Chapter 9: Energy Methods Applied to Room Acoustics 9.2.4.1 Sound reductions across composite partitions

Often, instead of being completely homogeneous, the area of the partition may in fact consist of distinct sub-areas (sub-elements), each of which differs markedly from adjacent areas in its insulating characteristics; for example, a wall with windows is such a composite partition. Additionally, a real partition may have openings or fissures, which, although they may be small in area, have transmission factors that approach 1 at high frequencies; consequently, even relatively small openings can drastically degrade the sound reduction index of the partition. In order to determine the sound reduction index of a composite partition, we assume that the incident sound power is uniformly distributed over the area of all sub-elements of the partition. That is the case for an ideal diffuse field, for instance. This assumption, together with our definition of the transmission factor (equation (9-54)), directly yields W τ= t = Wi

∑ Wt , n ∑ τ n ( S n n

Wi

=

S ) Wi

n

=

Wi

∑ τ n (S n

S) ,

(9-66)

n

where the index n specifies a sub-element, and S = ∑ S n . For a sound reduction index R, n

one obtains the expression R = 10 ⋅ log

S

∑ S n ⋅10 − R

n

10

.

(9-67)

n

Example 9-4 A hole is to be made in a 10 m2-wall, with a 40 dB sound reduction index. How large a hole is permissible, if the sound reduction index is not to be degraded by more than 10 dB? Solution Assume that the transmission factor of the hole is 1. That certainly applies at high enough frequencies that the in-plane dimensions of the hole are large, in terms of Helmholtz number. If we let x = Shole /S, then equation (9-67), upon substitution of the pertinent values, yields the expression 30 = −10 ⋅ log( x ⋅ 10 0 + (1 − x) ⋅ 10 −4 ) ,

the solution of which is x = 9⋅10-4 , i.e., Shole = 9⋅10-3 m2. That roughly corresponds to a square hole with 10 cm sides. 9.2.4.2 Flanking transmission

Flanking transmission is a collective designation for the contributions to the sound field in a receiving room made by all transmission paths save the direct one. Figure 9-20 illustrates the situation. Transmission between two rooms can take place by way of a number of different paths: the direct path 1; the indirect path 2; etc. The direct path is that described by the transmission factor given above, in equation (9-54); it corresponds to airborne

319

Chapter 9: Energy Methods Applied to Room Acoustics sound incident on the separating wall, which in turn vibrates, and thereby constitutes a sound radiator in the receiving room. In practice, however, there is always a certain amount of flanking transmission, and the actual sound reduction is therefore somewhat lower than that predicted by considering the direct path alone; that degraded sound reduction index is called the field reduction index. Figure 9-20 Transmission paths between two rooms. Path 1 is the direct transmission path, while paths 2 and 3 are examples of other, indirect, paths. The transfer of acoustic energy via the indirect paths is usually referred to as flanking transmission.

2 1 3

That field reduction index is usually defined by R ′ = 10 ⋅ log(∑ Wt ,m / Wi ) −1 = 10 ⋅ log(τ + ∑ τ m′ ) −1 ,

(9-68)

m ≠1

m

in which the index m specifies the transmission path and ′ = τm

Wt , m Wi

, m ≠ 1.

9.2.4.3 Simple wall

We now consider the sound reduction index of a homogeneous plate surrounded by a fluid (e.g., air). When a sound wave strikes one side of the plate, vibrations result. If it is a thin plate, its response consists of bending waves. Sound radiation by those bending waves, on the other side, is the mechanism by which the transmitted sound field then arises in the receiving room. In order to analyze the transmission, we assume that we have an infinite plate upon which a plane wave is incident. If we choose a coordinate system so that the incident wave lies in the x-y-plane (see figure 9-21), then the bending wave field in the plate satisfies (−m′′ω 2 + D p

∂4 ∂y 4

) v x = iω ((pi + p r ) − pt ) ,

(9-69)

where m” is the mass per unit area of the plate, Dp is the bending stiffness, and we have assumed that all fields have harmonic time-dependencies. Equation (9-69), with the substitution ξ = vx / iω, corresponds to the bending wave equation (6-57) found in section 6.5.1, except that displacement occurs in the x rather than the z-direction.

320

Chapter 9: Energy Methods Applied to Room Acoustics y Figure 9-21 Sound transmission through a homogeneous (infinite) thin plate of thickness h. The fluid on each side is assumed to have the same sound speed and specific impedance as that on the other. The incident plane wave excites a forced bending wave with phase velocity c/sinθ.

pt

pr

θ

θ x

θ pi

Forced bending wave

The sound fields in equation (9-69) are symbolized as in section 5.1.3. Accounting for Snell’s law, we obtain the following result for these fields where they contact the plate (i.e., at x = 0), pα = pˆ α e

i (ωt − k y y )

(9-70)

,

where α = i, r, or t. The particle velocity field vx must have the same y-dependence as the sound field, i.e., v x = vˆ x e

i (ω t − k y y )

.

(9-71)

Substituting equations (9-70) and (9-71) into (9-69) gives (iωm ′′ +

D p k y4 iω

) vˆ x = (pˆ i + pˆ r ) − pˆ t ,

(9-72)

where ky = k sinθ and the angle θ is the same for all three sound fields. To solve our problem, we need two additional equations. These are obtained by requiring that the normal velocities of all fields, at the plate, be equal. With the aid of equation (5-62),

and

pˆ cos θ , vˆ x = t ρ0c

(9-73)

pˆ i − pˆ r = pˆ t .

(9-74)

Eliminating vˆ x and pˆ r from equation (9-72), using equations (9-73) and (9-74), leads to the result that (iωm ′′ +

D p k y4 pˆ t cos θ = 2(pˆ i − pˆ t ) . ) iω ρ0c

From this equation, we can directly determine the transmission factor as (Note: the power transmission in the x-direction, of both the incident and the direct fields, contains the same factor cosθ / ρ0c, which therefore cancels),

321

Chapter 9: Energy Methods Applied to Room Acoustics

τ (θ ) =

2 pˆ t

pˆ i

2

1

= ⎛ ωm ′′ cos θ 1 + ⎜⎜ ⎝ 2ρ0c

⎞ ⎟ ⎟ ⎠

2⎡

2⎤

⎛ ⎞ ⎢1 − ⎜ ω sin θ ⎟ ⎥ ⎢ ⎜ ωc ⎟ ⎥ ⎠ ⎥⎦ ⎢⎣ ⎝ 2

2

,

(9-75)

in which ωc = 2πfc is the circular frequency corresponding to the so-called coincidence frequency fc , fc =

c2 2π

m ′′ . Dp

(9-76)

At that frequency, the bending wave speed is equal to c. Evidently, for the case of the lossless simple wall that are analyzing, complete transmission is obtained at a frequency corresponding to fc /sin2θ . At that frequency, the phenomenon of coincidence occurs, i.e., the bending wave speed is identical to the phase velocity (c/sinθ) of the sound wave moving along the plate in the y-direction. It is only for the special case of normal incidence (θ = 0°) that the simple wall does not exhibit a coincidence effect. For normal incidence, no bending waves are excited in the plate; instead, the entire plate moves in unison as a large piston, and only the mass of the plate is “felt” by the sound wave. With respect to frequency, we can distinguish three regimes that characterize the infinite simple wall. (i)

f < fc.

In this regime, the inertia terms dominate in the bending wave equation of the plate. If we ignore the bending stiffness term, then equation (9-75) provides a transmission factor of

τ (θ ) =

pˆ t

2

pˆ i

2

≈

{

}

⎛ ωm ′′ cos θ = antag ωm ′′ ρ 0 c » 1 ≈ ⎜⎜ 2 ⎝ 2ρ 0c ⎛ ωm ′′ cos θ ⎞ ⎟ 1 + ⎜⎜ ⎟ ⎝ 2ρ 0 c ⎠ 1

⎞ ⎟ ⎟ ⎠

−2

.

From that equation, we can obtain a sound reduction index, for normal incidence (θ = 0°), of

R⊥ = 20 ⋅ log

ωm′′ = { air} ≈ 20 ⋅ log f + 20 ⋅ log m′′ − 42, 2ρ0c

f < fc

(9-7

and for diffuse incidence (after integration), of Rd = R⊥ − 3 ,

(9-78)

for air at standard temperature and pressure. These equations, in which the only plate parameter accounted for is the mass, are variants of what is commonly referred to as the mass law for simple walls. A salient feature of that law is that the sound reduction index increases by 6 dB for each doubling of the frequency, or of the plate mass. In practice, we

322

Chapter 9: Energy Methods Applied to Room Acoustics always have finite walls that deviate somewhat from the result given above. As long as the plate is large, however, and the frequencies of interest are well below the coincidence frequency, then the mass law is a good approximation of reality; see figure 9-22. “Large”, in this context, refers to both of the wave types involved, i.e., the plate dimensions must be “large” in terms of both bending wavelengths and sound wavelengths in air. Reduction index Rd [dB] 40

30

20

10

0 100

500 Frequency [Hz]

1000

Figure9-22 Comparison of the calculated (equation (9-78)) – solid line – and measured (in thirdoctave bands) – stars – sound reduction Rd across a simple wall consisting of a 13 mm thick slab of gypsum (m” = 650 kg/m2, fc =2500 Hz).

(ii)

f ≈ fc .

At coincidence, the sound reduction index is controlled solely by the damping in the plate. Without damping, complete transmission, τ = 1, occurs. (iii)

f > fc .

In this frequency region, the bending stiffness term in equation (9-75) completely dominates. An expression for the sound reduction across an infinite wall, analogous to that for frequencies below coincidence, can be found. It turns out, however, that finite walls with small to moderate damping do not behave in accordance with the resulting expression. The reason for that is that the boundaries of the finite plate reflect the forced bending waves (direct field), bringing about a reverberant bending vibration field in the wall (compare section 9.2.2 Sound fields in rooms). The bending waves in that reverberant field (or standing wave field) consist of free bending waves that satisfy the homogeneous bending wave equation. As will be shown in section 8.6, sound radiation from free bending waves is negligible below the coincidence frequency fc, whereas very effective radiation

323

Chapter 9: Energy Methods Applied to Room Acoustics occurs above the coincidence frequency. As a result, the sound radiation above the coincidence frequency is dominated by the contribution from the reverberant field of the plate. The direct field and reverberant field radiate equally effectively to the surrounding fluid, but the latter is the dominating component of the plate’s vibration field when there is not a lot of damping. Below coincidence, on the other hand, the reverberant field is normally negligible, because free bending waves are very inefficient radiators at those frequencies. An expression for the sound reduction index of a finite plate, above the coincidence frequency, is provided without proof: R d = 20 ⋅ log

2η f ωm ′′ , + 10 ⋅ log + 10 ⋅ log π 2ρ 0c fc

f > fc

(9-79)

where η is the loss factor for the plate. Equation (9-79) accounts only for that part of the sound transmission due to the reverberant field in the plate. The fact that the sound radiation from the reverberant field is normally only significant above the coincidence frequency implies that added damping is only warranted if it is that frequency region in which improved sound insulation is sought. Finally, we briefly touch upon the sound insulation of a plate at low frequencies. With reference to our discussion in the introduction to the chapter, we consider the zeromodes and few-modes regions for a plate (see table 9-1). In the few-modes region, we have only a small number of resonant modes that determine the behavior of the plate. That implies that the sound reduction index varies strongly; at a resonance (Zin ≈ 0), practically no sound insulation at all is obtained, whereas at an anti-resonance (Zin ≈ ∞) very large sound reduction indices are obtained. Zero-mode Few-mode region region

R , [ dB ]

Multi-mode region

Mass law 6 dB / octave

log f , [

]

fc

Figure 9-23 Idealized insulation behavior of a simple wall. In the zero-mode region, the wall is stiffnesscontrolled; in the few-mode regions, the behavior is determined by a small number of resonances; and, in the multi-mode region (if, however, f « fc), the wall’s behavior is mass-controlled.

324

Chapter 9: Energy Methods Applied to Room Acoustics In the zero-mode region, the plate has no resonances and acts as a pure stiffness κ, the magnitude of which is determined by the plate’s material properties, geometry, and edge mounting (boundary conditions). In that region, the impedance takes the form Zin = κ / iω, which, at low frequencies, results in a very good sound reduction index. Summarizing, to serve as effective sound insulators, plates should be used in either the zero-mode region or the multi-mode region; in the latter, the mass law applies, provided that we are well below the coincidence frequency. In machinery-related applications, for which the insulators are typically metallic materials, the frequencies of interest are usually below coincidence (e.g., fc is 6 kHz for 2 mm steel plate). In building acoustics, on the other hand, the relevant frequencies are normally well above coincidence (e.g., fc is about 200 Hz for 100 mm thick concrete). 9.2.4.4 Double wall

A way to increase the sound insulation of a partition, without too drastic an increase in the mass, is to construct it of two or more layers (walls), with air pockets (possibly absorbentfilled) between the layers. When an acoustic wave passes through such a construction, the total transmission factor, ignoring all reflections (standing waves) between layers, is

τ tot = τ 1 τ 2 ...τ n ...τ N ,

(9-80)

in which we have assumed that there are N layers and the transmission factor of the n-th layer is given by τn. Equation (9-80) holds at sufficiently high frequencies, for large enough distances (in sound wavelengths) between each layer, or given sufficient damping, in the form of absorbent, between layers. At low frequencies, at which that doesn’t apply, standing wave effects (resonances) arise and modify the transmission. For practical reasons, two, or at most three, layers are used in constructions of this type, e.g., windows. The most common case is that of two layers (double walls), which therefore merit further discussion. If we assume that the mass law holds, then equation (977) gives the sound reduction index of a finite double wall

R⊥double = 20 ⋅ log

ω m1′′ ω m2′′ ω 2 m1′′m2′′ + 20 ⋅ log = 20 ⋅ log , 2ρ 0c 2ρ0c 4 ρ 02 c 2

f < fc ,

where m″1 and m″2 are the masses per unit area of each wall. Example 9-5 Express equation (9-81) for the special case of a double wall consisting of two identical elements. Solution double Set m1′′ = m 2′′ . That yields R⊥ = 40 ⋅ log

ωm′′ 2ρ0c

.

Hence, the sound reduction index increases by 12 dB for each octave band increase in frequency, or for each doubling of the mass. That can be compared to the earlier result of 6 dB for a simple wall.

325

(9-8

Chapter 9: Energy Methods Applied to Room Acoustics A double wall has a lower frequency bound corresponding to the lowest mechanical resonance of the system. That fundamental resonance corresponds to an oscillation in which the enclosed air acts a stiffness, and the two wall elements as masses. In order for the double wall to work as intended, it must be used at frequencies considerably higher than that resonance frequency. A typical rule of thumb is that the Helmholtz number, based on the thickness of the air pocket, must be greater than one. Finally, we derive an expression for the fundamental resonance frequency of an infinite double wall. By analogy to our analysis of a panel absorber, it follows that for normal plane wave incidence, the enclosed air volume has a stiffness per unit area of

κ=

ρ0c 2 h

,

in which h is the thickness of the air pocket. The system considered is mechanically equivalent to a system with two masses coupled by springs; see figure 9-24. As we know from basic mechanics, a system of that type has two degrees-of-freedom, but only one nonzero resonance frequency. That resonance frequency is given by f0 =

1 2π

ρ 0 c 2 (m1′′ + m 2′′ ) m1′′ m 2′′ h

.

(9-82)

The case of an air pocket filled with absorbent can be treated in the same way as is done in equation (9-53).

R , [dB

h

I

II

m''1

III

= m''1

f0

log f , [

k

m''2

m''2

]

Figure 9-24 Idealized frequency-dependence of the sound reduction across an infinite double wall obeying the mass law. See below for a description of regions I –-III. Note that the analogy between a double wall and a mechanical system requires kh « 1, i.e., low frequencies.

Region I corresponds to frequencies less than the fundamental resonance frequency f0. In that region, the enclosed air has a large stiffness, and the wall acts as a simple wall with a mass (m″1 +m″2). Region II corresponds to frequencies around, and including, the fundamental resonance frequency f0. In that region, there are specific frequencies at which

326

Chapter 9: Energy Methods Applied to Room Acoustics

the sound reduction index is poor, due to resonances of the enclosed airspace. Region III corresponds to frequencies at which the double wall acts as intended, and we obtain a sound reduction index in accordance with equation (9-81). As for coincidence and finite wall (boundary) effects, the discussion made in connection with simple walls also applies to the double wall, in principle. It is also worth noting that, in practice, double wall constructions are actually made with stiff connectors between the two wall elements; those may be either point or linear features. Such connecting links act as sound bridges between the wall elements, in which structure-borne sound short circuits the insulating air gap. Solid connections are therefore to be avoided, or at least realized in such a way that they have high mobilities compared to the wall elements (compare to vibration isolation, chapter 9).

13 mm gypsum board

Average reduction 37 dB

30 mm mineral wool

Average reduction 47 dB

30 mm mineral wool

Average reuction 55 db

Figure 9-25 A double wall in the form of two light, single wall elements, separated by an air gap, can insulate considerably better than a corresponding simple wall. Correctly dimensioned, a double wall can give the same sound reduction as a 5-10 times heavier single wall. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

9.2.4.5 Sound reduction across some common insulators Tables 9-4 and 9-5 provide sound insulation data for some commonly occurring insulating partitions.

327

Chapter 9: Energy Methods Applied to Room Acoustics Table 9-4 Sound reduction index R across simple wall constructions.

Material

Tile, 15 mm plaster Brick with 15 mm plaster

Concrete

Gypsum slab Plywood Particle Board Al-plate Steel plate Cover layer: 0.8 mm Alplate. Core: Polyurethane foam

Thickness [mm]

Sound Reduction Index R [dB] by octave band f [Hz]

145 270 95 105 145 270 40 70 120 150 190 7 10 10 15 25 4 19 0.5 1 3.5 7

Surface Density [kg/m2] 210 350 150 170 260 480 95 170 300 350 430 7 10 7 11 15 3 15 1.3 8 28 55

125 34 36 31 31 36 40 31 30 34 38 39 17 19 19 18 16 14 22 10 17 29 33

250 40 42 37 39 42 46 29 33 38 42 43 18 19 19 22 25 16 22 12 23 33 38

500 40 48 37 39 42 51 27 37 48 47 50 26 26 22 24 26 19 27 14 30 36 39

1000 46 55 34 37 48 54 36 44 53 54 55 28 31 25 27 24 21 28 19 32 39 40

2000 51 56 47 49 53 59 43 51 61 61 62 32 30 25 25 30 25 22 25 35 41 30

4000 56 59 52 53 58 62 48 59 63 64 66 27 34 19 32 36 28 24 28 38 31 42

50

6

10

16

18

20

24

32

80

8

12

17

19

16

32

30

328

Table 9-5 Sound reduction index R of double wall insulators. (Source: W Fasold, W Kraak, W Schirmer, Taschenbuch Akustik, 1982, VEB Verlag.)

Description

Concrete without damping in the air gap

Light concrete with 50 mm mineral wool layer in the air gap Gypsum with 30 mm mineral wool layer in the air gap Where d1 dh d2 m ′′

d1 [mm]

dh [mm]

d2 [mm]

m ′′ [kg/m2]

40 70 40 70 40 70 70 70 70 115 60 70 80

25 10 50 50 100 100 110 160 50 80 30 60 30

70 70 70 70 70 70 120 70 70 115 60 70 80

275 340 275 340 275 340 175 135 135 190 100 160 170

thickness of the 1st wall element, thickness of the air gap, thickness of the 2nd wall element, and total surface density of the construction.

Sound Reduction Index R [dB], by octave band f [Hz] 125 250 500 1000 2000 4000 33 38 43 50 57 55 43 44 50 54 55 60 35 42 45 53 58 60 44 42 48 54 59 58 44 42 47 55 58 62 43 41 48 54 59 65 42 44 46 48 53 60 38 41 42 44 52 60 37 43 41 44 55 63 45 42 46 59 56 64 39 40 40 48 55 64 35 40 41 46 56 63 36 41 39 43 52 67

Chapter 9: Energy Methods Applied to Room Acoustics

9.3

IMPORTANT RELATIONS

ENERGY METHODS IN GENERAL He = kl

Helmholtz number

(9-1)

Energy balance of single and coupled acoustic systems

Energy balance equation of a single system dE + ηωE = Win dt

(9-4)

Relation between wave theory and energy-based methods

ε c Id = d 4

Diffuse intensity

(9-24)

ROOM ACOUSTICS Sabine’s formula

Relation between reverberation time and loss factor T = (6 ⋅ ln 10)/ηω

(9-27)

Absorption factor for diffuse incidence π 2

αd =

∫ α (θ ) sin(2θ )dθ

(9-31)

0

Absorbed power from a diffuse sound field

ε c W diss = d α d ,m S m 4 m

∑

(9-32)

(24 ⋅ ln 10)V V = ( with c = 342 m/s) = 0.161 c αd S αd S

(9-35)

∑ α d ,m S m

(9-36)

Sabine’s formula for reverberation time

T=

Equivalent absorption area

A=

m

330

Chapter 9: Energy Methods Applied to Room Acoustics

Sound fields in rooms

Sound pressure level emitted by a source in a room ⎡ Γ 4⎤ dir Ltot + ⎥ p = LW + 10 ⋅ log ⎢ 2 ′⎦ A ⎣ 4πr ⎛ A′Γ ⎞ re = ⎜ ⎟ ⎝ 16π ⎠

Echo-radius

(9-42)

12

(9-43)

Acoustic absorbents Porous absorbents

Optimal flow resistance of absorbents mounted to a hard wall

φ h / ρ0c ≈ 2

(9-47)

In that case, α┴ ≥ 0.9 when kh is greater than about 1 Influence of a cover layer – lower bound on effective frequency fg =

ρ 0c 2π m′′

(9-48)

Fundamental resonance frequency of a panel absorber

f0 =

1 2π

κ m′′

=

1 2π

⎧ 60 ⎪ m′′h , adiabatic ρ0c 2 ⎪⎪ (9-53) = {for air} ≈ ⎨ m ′′h ⎪ 50 , isothermal ⎪ ⎪⎩ m′′h

Sound transmission through insulators

Transmission factor Sound reduction index

τ=

Wt Wi

R = 10 ⋅ log

(9-54) 1

(9-55)

τ

Sound transmission between two rooms L p,1 − L p,2 = 10 ⋅ log(1 +

331

A2 ) Sτ d

(9-64)

Chapter 9: Energy Methods Applied to Room Acoustics

If the factor 1 in parentheses in equation (9-64) is neglected, one obtains A L p ,1 = L p ,2 + Rd + 10 ⋅ log 2 S

(9-65)

Sound Reduction Index across composite walls R = 10 ⋅ log

S

∑ S n ⋅10 − R

n

10

(9-67)

n

where S = ∑ S n n

Simple Wall

Boundary frequency for coincidence fc =

Sound reduction index f < fc

R⊥ = 20 ⋅ log

c2 2π

m ′′ Dp

ωm′′ = {air} ≈ 20 ⋅ log f + 20 ⋅ log m′′ − 42 2ρ 0c Rd = R⊥ − 3

f > fc

R d = 20 ⋅ log

2η f ωm ′′ + 10 ⋅ log + 10 ⋅ log π fc 2ρ 0 c

(9-76)

(9-77) (9-78) (9-79)

Double wall

Sound Reduction Index f < fc ,

R⊥double = 20 ⋅ log

Fundamental Resonance

ωm1′′ ωm′2′ ω 2 m1′′m′2′ + 20 ⋅ log = 20 ⋅ log 2ρ 0c 2ρ 0c 4 ρ 02 c 2 f0 =

1 2π

ρ0c 2 (m1′′ + m2′′ ) m1′′m2′′ h

332

(9-81)

(9-82)

CHAPTER TEN SOUND RADIATION & GENERATION MECHANISMS

This chapter deals with the important question of how sound is really generated and radiated, i.e., what physical mechanisms bring about sound fields. Knowledge of that is crucial, because noise control measures taken at the sound source itself are normally the best and most effective strategies for elimination sound and vibration problems. An ever increasing interest in incorporating sound and vibration requirements into the design of new machines and vehicles also enhances the significance of that topic. In classical acoustics, which was largely developed during the 19-th century, the essential aspects of which were described in chapter 4, sound in a fluid can only be radiated by vibrating solid bodies. Driving the vibration of such bodies are dynamic forces of various kinds, e.g., inertial forces in connection with shocks and electromagnetic forces, as is the case with common loudspeakers. Thus, we will give due attention to sound emission from vibrating structures, particularly large plates. The most important contribution to the sound radiation from a plate is from bending waves. That type of wave primarily involves motions perpendicular to the plane of the plate, and can therefore excite sound waves in a surrounding fluid. It is, however, apparent to all who have listened to water boiling in a kettle, or the sound of airplane starting up, that there must be other mechanisms of sound radiation than vibrating solid bodies. 333

Chapter 10: Sound Radiation & Generation Mechanisms

The last-named examples suggest that flow-induced sound, associated with a time-varying flow field, is yet another important sound generation mechanism. The study of that type of mechanism got underway relatively late, and the first general theories were presented in the 1950’s. The noisy civil jet airplane that began to be used at that time was a driving force in the search for new knowledge in that area. Today, flow acoustics is an established part of the larger field of vibroacoustics, and an introduction to it is included here. With the theory of acoustic waves in fluids (see chapter 4) as a point of departure, this chapter begins by treating the various elementary radiating sources, or point sources. Because the equations of acoustics are linear, the superposition principle applies. It is therefore possible, as in electromagnetics, to first study various point sources and then take advantage of linearity to obtain fields from more general sources by regarding them as distributions of point sources, the fields of which are additive. 10.1

MONOPOLE

The monopole is the simplest type of point source. It can also be called a multipole of zeroth order. The sound field emitted by a monopole is spherically symmetric, as we studied in sections 4.2.5-6. For a harmonic time-variation, (4-112) provides the following general expression for a spherically symmetric field: p( r , t ) =

A + i (ω t − kr ) A − i (ω t + kr ) e + e . r r

With a monopole at the origin, in a free field without boundaries, only an outgoing wave exists. That implies that the second term in the equation given above, representing a wave that moves towards the origin as t increases, can be disregarded. In order to interpret a monopole physically, the amplitude A+ is related to what happens at the source. A clue to how that can be done might be obtained by considering that the physical dimension corresponding to that amplitude is [kg/s2], i.e., it represents the change in mass flow rate [kg/s] per unit time. Thus, in order to determine A+, it is necessary to calculate the acoustic mass flow rate generated by the monopole. For spherical symmetry, at a distance r, the volumetric flow rate is given by Q r = 4π r 2 u r ,

(10-1)

ˆ e iωt and where u is the particle velocity in the radial direction. The mass where Q r = Q r r flow rate is obtained from that expression by multiplying the volume flow rate by the density of the surrounding fluid which, in the context of a linearized theory, can be set to the fluid’s undisturbed density ρ0. According to equation (4-114), the particle velocity in the radial direction is given by u r (r , t ) =

A+

ρ 0 cr

(1 +

334

1 i (ωt − kr ) )e . ikr

Chapter 10: Sound Radiation & Generation Mechanisms

Putting that relation into equation (1), and taking the limit of a point source, yields ˆ = lim 4π r 2 A + (1 + 1 )e −ikr = 4πA + . Q 0 ρ 0 cr ikr iρ 0 ck r →0 By taking advantage of the relation ω = kc, the following relation between the mass flow rate and the amplitude of the radiated spherical sound wave is obtained:

A+ =

ˆ ρ 0 iω Q 0 . 4π

(10-2)

The spherical sound wave radiated from a harmonically oscillating monopole is, as such, given by pm =

ρ 0 iω Q 0 −ikr e , 4πr

(10-3)

where the index m indicates a monopole and Q0 is called the monopole’s source strength. Assume an arbitrary distribution of monopoles in a free field; the resulting sound field is then obtained through superposition, as pm = ∑ n

ρ 0 iω Q n −ikrn e , 4πrn

(10-4)

K K where rn is the position vector from monopole n to the field point r ; see figure 10-1. If the summation in equation (10-4) is interpreted as an integral, then even cases of a continuous source distribution can be treated.

K r

1

Q1

K r

K r

n

Figure 10-1 Sound field built up by superposition of monopoles.

Qn

The sound power radiated by a monopole is given by equation (4-123), and substituting in A+ from equation (10-2) gives the relation Wm =

ρ 0 ck 2 ~ 2 Q0 . 4π

335

(10-5)

Chapter 10: Sound Radiation & Generation Mechanisms

Finally, we take up the question of which specific physical mechanisms are capable of generating fluctuating mass or volume flow rates in a fluid, and, as such, constitute monopole-type sources. The physical realization of a monopole closest to the ideal is that of a small spherical shell, undergoing pure radial oscillations. The condition that the shell be small can be more specifically stated as a requirement that the wavelength be much larger than the radius of the shell. That is usually expressed by requiring that the Helmholtz number He = ka be much less than 1, where k is the wave number and a the radius of the shell (or, more generally, of the source). An example of a process that can be modeled as a small oscillating spherical shell is the sound radiation from cavitation bubbles. Such bubbles arise in a flowing liquid when the local pressure is so low that it approaches the liquid’s vapor pressure; see figure 10-2. A cavitation bubble is unstable, and normally implodes shortly after it is generated. That implosion is a strong source of sound, however, because it occurs in a very short span of time, and the sudden local change in volume yields a high value of the volume flow rate Q ~ d(ΔV)/dt , where ΔV is the volume of the bubble before collapse.

valve

collapsing gas bubbles

Very high pressure

low pressure

Figure 10-2 For sudden changes of the cross section in a pipe carrying flowing medium, the pressure drop can be so large that cavitation bubbles arise. The sound generated by the imploding cavitation bubbles can be described using the monopole model. Poorly designed valves are examples of that type of sound generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

A more widely familiar example of a monopole is that of a loudspeaker mounted in a box, as shown in figure 10-3, where only the front side of the loudspeaker can radiate sound into the surrounding air. At low frequencies, at which the He number, based on the radius of the box, is small, that sources acts as a monopole.

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Chapter 10: Sound Radiation & Generation Mechanisms

Figure 10-3 A loudspeaker element mounted in a box is an example of a sound source approximating a monopole in the low-frequency region. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

A relevant question that might be posed in is how deviations from spherical symmetry of the source region affect the validity of the monopole model. After all, for the case of the speaker in a box, just to take one example, the angular distribution of the flow rate in the source region deviates drastically from the ideal case of a radially oscillating sphere. A closer analysis, however, shows that the only factor of significance at low frequencies is the resultant mass or volume flow delivered by the entirety of the source. The deviations of the flow distribution from spherical symmetry contribute sound fields corresponding to higher order multipoles, such as dipoles and quadrupoles. It can be shown that, outside of the near field of the source, these multipole contributions are in a ratio, to the monopole contribution, of He raised to a power 1, 2, 3, ..., etc, equal to the order of the multipole. At low frequencies (small He numbers) the affect of asymmetry is therefore negligible. Other important examples of monopole sources are fluctuating heat sources, e.g., caused by combustion, as well as inlets and exhaust vents with pulsating flow, e.g., those of internal combustion engines and other reciprocating piston machines; see figure 10-4.

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Chapter 10: Sound Radiation & Generation Mechanisms

Figure10-4 The exhaust gas outlet of a fishing craft with a low-rpm ignition bulb engine radiates sound that can be described by a monopole model. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Example 10-1 Investigate whether the exhaust gas outlet from a car can be considered a monopole. Solution The exhaust gas outlet from the internal combustion engine of a passenger car has a typical radius of about 3 cm. The fundamental frequency f0 of the motor’s pulsations, i.e., the cylinder exhaust frequency, is given by: (( N / 2) × K ) / 60 , where N is the crankshaft rotational speed in revolutions per minute (rpm), K is the number of cylinders, and a 4stroke process is assumed. If, moreover, we assume that there are exactly four cylinders and that the idle speed is 1200 rpm, then f0 = 40 Hz. Because the engine does not generate a perfectly sinusoidal variation of the flow rate, a number of harmonics of the fundamental tone are also obtained: 80 Hz, 120 Hz, 160 Hz, etc. The condition that the exhaust outlet must fulfill to reasonably be considered a monopole is that the He number, based on the diameter of the exhaust pipe, be small. Putting in the values given above, as well as a sound speed of c = 340 m/s, yields He < 0.055 at 100 Hz. The sound radiation from the outlet of the exhaust pipe can therefore be modeled as a monopole, at least up to 400 500 Hz. (Note: it is the sound speed of the surrounding fluid, into which sound is radiated, rather than that of the hot exhaust gases, that is applicable to this computation. Similarly, the density ρ0 in the equations given above is always to be interpreted as that of the surrounding fluid medium.)

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Chapter 10: Sound Radiation & Generation Mechanisms

10.2

DIPOLE

From the monopole, which is the simplest type of point source (a multipole of zerothorder), new types of sources can be created by superposition. A systematic way to do so is to first superpose two monopoles with the same source strength, but opposite phases Q and -Q. If the He number based on the distance l between these two monopoles is much less than 1, a dipole is obtained; see figure 10-5. Such a source is also called a multipole of order one. The process can be repeated to obtain multipoles of arbitrary order. For example, a multipole of second order, or a quadrupole as it is usually called, is obtained by the superposition of two dipoles of equal strength, but opposite phase; see section 10.3. To calculate the resultant field from a dipole, and to interpret that source type physically, we now consider two harmonically oscillating monopoles Q and -Q as in figure 10-5. z

r1 Q

θ

l

r

r2

-Q Figure

10-5 Dipole obtained by superposition of two monopoles (kl « 1).

With notation as defined in figure 10-5, equation (10-3) leads to the result p tot =

ρ 0 iω Q ⎛⎜ e −ikr1 e −ikr2 − 4π ⎜⎝ r1 r2

⎞ ⎟ , ⎟ ⎠

(10-6)

G G G G G G G where r1 = r − 12 le z and r2 = r + 12 le z . Incorporating G (r ) = G ( x, y, z ) = e−ikr 4πr ,

equation (10-6) can then be written as p tot = ρ 0 iω Q(G ( x, y, z − 12 l ) − G ( x, y, z + 12 l )) ,

which, for small l, can be approximated by the differential p tot = ρ 0 iω Ql

(G ( x, y, z − 12 l ) − G ( x, y, z + 12 l )) l

≈ − ρ 0 i ω Ql

∂G . ∂z

(10-7)

ˆ l is Equation (10-7) represents the field from a dipole of finite extent. The product Q ˆ . To obtain an equality in equation usually called the dipole moment and indicated by D z

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Chapter 10: Sound Radiation & Generation Mechanisms

(10-7) (i.e., a point source!), we must let l approach zero. Carrying out that limit while constraining the dipole moment to be constant, one obtains p d = lim p tot = l →0

− ρ 0 iωD z ∂ ⎛⎜ e −ikr ∂z ⎜⎝ r 4π

⎞ ⎟ , ⎟ ⎠

(10-8)

which represents the field from a point source of the dipole type (index d) oriented in the zdirection. To interpret the dipole moment physically we use, as in the monopole case, a ˆ is [N], i.e., that expression can be dimensional perspective. The dimension of ρ 0 iωD z interpreted as a force. It can be shown that that force acts in the z-direction and corresponds to the force Fˆ z that the dipole exerts on the fluid. A dipole is therefore a process that does not provide a net volume flow to the fluid, but only a fluctuating force; that is evident by constructing a control volume around Q and -Q in figure 10-5. In summary, a harmonically oscillating point source F = Fˆ e iωt radiates a sound field of z

z

dipole type1 given by pd =

− F z ∂ ⎛⎜ e −ikr ⋅ 4π ∂z ⎜⎝ r

⎞ ⎟ . ⎟ ⎠

(10-9)

We will now study the appearance of the dipole field a little more closely, and therefore begin by calculating the derivative in equation (10-9). That gives pd =

− F z ∂ ⎛ e −ikr ⎞ − F z −ikr ∂ ⎛ 1 ⎞ 1 ∂ −ikr ⎟= ⎜ (e e )= ⎜ ⎟+ 4π ∂z ⎜⎝ r ⎟⎠ 4π ∂z ⎝ r ⎠ r ∂z =

−F z 1 ∂r ik ∂r −ikr (− 2 − )e . 4π r ∂z r ∂z

G The position vector r has the magnitude r = x 2 + y 2 + z 2 , from which ∂r = ∂z

z x2 + y2 + z 2

=

z = {see figure 10-5}= cosθ , r

(10-10)

and which then leads to the following expression for the sound field: pd =

ikFz cos θ 1 −ikr (1 + )e ikr 4π r

.

(10-11)

1 A dipole can, of course, be oriented in any direction, and not only in the z-direction relative to some predefined coordinate system.

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Chapter 10: Sound Radiation & Generation Mechanisms

The field consists of two parts, both the near field around the source corresponding to kr « 1 and the far field where kr » 1. In the near field, the sound pressure falls off as a rate 1/r2, and in the far field at a rate 1/r, with increasing distance r from the source.

z Figure 10-6 The variation of the sound pressure in the far field of a dipole; p d ∝ cosθ .

Note that for θ = 0, 180°, a maximum pressure is obtained, and for θ = 90°, p d = 0 holds. The vector Fz represents

θ Fz

the force the dipole exerts on the surrounding medium.

In order to find the radiated sound power, we can either use equation (4-121) for the sound intensity and integrate over all directions, or we can consider the far field. The latter is normally the simplest approach, because, for an arbitrary source in a free field, the far field can always be considered locally plane. Thus, the intensity can be calculated with the aid p−2 = 0 ). That relation, and equation (10of the plane wave relation, equation (4-83) (with ~ 11), gives the following expression for the intensity in the far field, ~ k 2 Fz2 cos 2 θ Ir = . (10-12) 16 ρ 0 cπ 2 r 2 The radiated sound power is obtained by summation of that intensity over a sphere Wd =

∫ I r dS = sfär

~ k 2 Fz2

π

∫

16 ρ 0 cπ 2 r 2 0

2π r 2 cos 2 θ sin θ dθ =

~ k 2 F z2 12πρ 0 c

(10-13)

where the surface element is expressed in spherical coordinates when integrating, i.e., dS = 2π r 2 sin θ dθ . The sound power can also be expressed as a dipole moment, using ˆ , the relation Fˆ z = ρ 0 iωD z ~

Wd =

ρ 0 ck 4 D z2 12π

.

(10-14)

For a dipole of finite size consisting of two closely spaced monopoles, as in figure 10-5, ˆ l applies; see equation (10-7). For small He numbers, the radiated sound power of ˆ =Q D z such a “finite” dipole is given very precisely by equation (10-14). We now compare the radiation from such a “finite” dipole with the sound power that a lone monopole of strength Q gives in the free field. Based on equations (10-5) and (10-14), Wd Wm = (kl ) 2 / 3 , kl