Ch01_ Introduction 2015

Mechanical Vibrations Singiresu S. Rao SI Edition Chapter 1 Fundamentals of Vibration

Course Outline 1. 2. 3. 4. 5. 6. 7.

Fundamentals of Vibration Free Vibration of Single DOF Systems Harmonically Excited Vibration Vibration under General Forcing Conditions Two DOF Systems Multidegree of Freedom Systems Determination of Natural Frequencies and Mode Shapes

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Course Outline 8.

Continuous Systems

9.

Vibration Control

10.

Vibration Measurement and Applications

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Chapter Outline 1.1 Preliminary Remarks 1.2 Brief History of Vibration 1.3 Importance of the Study of Vibration 1.4 Basic Concepts of Vibration 1.5 Classification of Vibration 1.6 Vibration Analysis Procedure 1.7 Spring Elements © 2005 Pearson Education South Asia Pte Ltd.

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Chapter Outline 1.8 Mass or Inertia Elements 1.9 Damping Elements 1.10 Harmonic Motion 1.11 Harmonic Analysis

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1.1 Preliminary Remarks • • • •

Examination of vibration’s important role Vibration analysis of an engineering system Definitions and concepts of vibration Concept of harmonic analysis for general periodic motions

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1.3 Importance of the Study of Vibration • Why study vibration?  Vibrations can lead to excessive deflections and failure on the machines and structures  To reduce vibration through proper design of machines and their mountings  To utilize profitably in several consumer and industrial applications  To improve the efficiency of certain machining, casting, forging & welding processes  To stimulate earthquakes for geological research and conduct studies in design of nuclear reactors © 2005 Pearson Education South Asia Pte Ltd.

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1.4 Basic Concepts of Vibration  Vibration = any motion that repeats itself after an interval of time  Vibratory System consists of: 1) spring or elasticity 2) mass or inertia 3) damper  Involves transfer of potential energy to kinetic energy and vice versa

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1.4 Basic Concepts of Vibration  Degree of Freedom (d.o.f.) = min. no. of independent coordinates required to determine completely the positions of all parts of a system at any instant of time  Examples of single degree-of-freedom systems:

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1.4 Basic Concepts of Vibration  Examples of single degree-of-freedom systems:

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1.4 Basic Concepts of Vibration  Examples of Two degree-of-freedom systems:

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1.4 Basic Concepts of Vibration  Examples of Three degree of freedom systems:

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1.4 Basic Concepts of Vibration  Example of Infinite-number-of-degrees-offreedom system:

 Infinite number of degrees of freedom system are termed continuous or distributed systems  Finite number of degrees of freedom are termed discrete or lumped parameter systems  More accurate results obtained by increasing number of degrees of freedom © 2005 Pearson Education South Asia Pte Ltd.

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1.5 Classification of Vibration  Free Vibration: A system is left to vibrate on its own after an initial disturbance and no external force acts on the system. E.g. simple pendulum  Forced Vibration: A system that is subjected to a repeating external force. E.g. oscillation arises from diesel engines Resonance occurs when the frequency of the external force coincides with one of the natural frequencies of the system © 2005 Pearson Education South Asia Pte Ltd.

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1.5 Classification of Vibration  Undamped Vibration: When no energy is lost or dissipated in friction or other resistance during oscillations  Damped Vibration: When any energy is lost or dissipated in friction or other resistance during oscillations  Linear Vibration: When all basic components of a vibratory system, i.e. the spring, the mass and the damper behave linearly © 2005 Pearson Education South Asia Pte Ltd.

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1.6 Vibration Analysis Procedure Step 1: Mathematical Modeling Step 2: Derivation of Governing Equations Step 3: Solution of the Governing Equations Step 4: Interpretation of the Results

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1.6 Vibration Analysis Procedure  Example of the modeling of a forging hammer:

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Example 1.1 Mathematical Model of a Motorcycle

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1.8

Mass or Inertia Elements

 Using mathematical model to represent the actual vibrating system E.g. In figure below, the mass and damping of the beam can be disregarded; the system can thus be modeled as a spring-mass system as shown.

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1.9

Damping Elements

 Viscous Damping: Damping force is proportional to the velocity of the vibrating body in a fluid medium such as air, water, gas, and oil.  Coulomb or Dry Friction Damping: Damping force is constant in magnitude but opposite in direction to that of the motion of the vibrating body between dry surfaces  Material or Solid or Hysteretic Damping: Energy is absorbed or dissipated by material during deformation due to friction between internal planes © 2005 Pearson Education South Asia Pte Ltd.

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Example 1.10 Equivalent Spring and Damping Constants of a Machine Tool Support

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Example 1.10 Solution

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Example 1.10 Solution F  k x;

i  1,2,3,4

F  c x ;

i  1,2,3,4

si

di

i

i

(E.1)

Let the total forces acting on all the springs and all the dampers be Fs and Fd, respectively (see Fig. 1.37d). The force equilibrium equations can thus be expressed as

F F F F F s

s1

s2

s3

s4

F F F F F d

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d1

d2

d3

d4

(E.2) 29

Example 1.10 Solution where Fs + Fd = W, with W denoting the total vertical force (including the inertia force) acting on the milling machine. From Fig. 1.37(d), we have

F k x s

eq

F  c x d

(E.3)

eq

Equation (E.2) along with Eqs. (E.1) and (E.3), yield

k  k  k  k  k  4k eq

1

2

3

4

c  c  c  c  c  4c eq

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2

3

4

(E.4) 30

1.7 Spring Elements  Linear spring is a type of mechanical link that is generally assumed to have negligible mass and damping  Spring force is given by: F  kx

1.1

F = spring force, k = spring stiffness or spring constant, and x = deformation (displacement of one end with respect to the other) © 2005 Pearson Education South Asia Pte Ltd.

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1.7 Spring Elements  Work done (U) in deforming a spring or the strain (potential) energy is given by: 1 2 U  kx 2

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1.2

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1.7 Spring Elements  Static deflection of a beam at the free end is given by: Wl 3  1 .6   st  3EI W = mg is the weight of the mass m, E = Young’s Modulus, and I = moment of inertia of cross-section of beam

 Spring Constant is given by: W 3EI k   l 3

 1 .7 

st

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1.7 Spring Elements  Combination of Springs: 1) Springs in parallel – if we have n spring constants k1, k2, …, kn in parallel, then the equivalent spring constant keq is:

keq  k1  k2  ...  kn

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 1.11

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1.7 Spring Elements  Combination of Springs: 2) Springs in series – if we have n spring constants k1, k2, …, kn in series, then the equivalent spring constant keq is:

1 1 1 1    ...  k k k k eq

1

2

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1.17 

n

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1.10 Harmonic Motion  Periodic Motion: motion repeated after equal intervals of time  Harmonic Motion: simplest type of periodic motion 1.30  x  A sin   A sin t  Displacement (x): (on horizontal axis)  Velocity:

dx   A cos t dt

 Acceleration: d 2x 2 2    A sin  t    x 2 dt © 2005 Pearson Education South Asia Pte Ltd.

1.31

1.32 36

1.10 Harmonic Motion  Complex number representation of harmonic motion:  1.35 X  a  ib where i = √(–1) and a and b denote the real and imaginary x and y components of X, respectively.

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1.10 Harmonic Motion  Definitions of Terminology: Amplitude (A) is the maximum displacement of a vibrating body from its equilibrium position Period of oscillation (T) is time taken to complete one cycle of motion 2 1.59 T  Frequency of oscillation (f) is the no. of 1  cycles per unit time 1.60 f   T 2 © 2005 Pearson Education South Asia Pte Ltd.

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1.10 Harmonic Motion  Definitions of Terminology: Natural frequency is the frequency which a system oscillates without external forces Phase angle () is the angular difference between two synchronous harmonic motions 1.61 x1  A1 sin t x2  A2 sin  t   

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1.62

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