VIBRATION ANALYSIS FOR ELECTRONIC EQUIPMENT

VIBRATION ANALYSIS FOR ELECTRONIC EQUIPMENT THIRD EDITION

Dave S. Steinberg Steinberg & Associates and University of California, Los Angeles

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC. New York

Chichester

Weinheim

Brisbane

Singapore

Toronto

This book is printed on acid-free paper. @ Copyright G2000 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected]. Library of Congress Cataloging in Publication Data:

Steinberg, Dave S. Vibration analysis for electronic equipment/Dave S. Steinberg.--3rd ed. p. cm “ A Wiley-Interscience publication.” ISBN 0-471-37685-X 1. Electronic apparatus and appliances-Vibration. I. Title. TK7870S8218 2000 621.381--dc21 Printed in the United States of America 109876

99-056617

To my wife, Annette, and to my two daughters, Con and Stacie

Electronic equipment continues its relentless expansion into virtually every area associated with commercial, industrial, and military applications throughout the entire world. Exotic technology has become commonplace in medicine, entertainment, communication, travel, transportation, manufacturing, education, and commerce. The internet has dramatically and forever changed the way many companies run their businesses, without the need for large offices and a large staff of people to perform clerical duties, largely due to the extensive use of the fast, small, powerful personal computers. This has resulted in improved profit margins in companies that embraced the new technologies, and disasters for companies that resisted the tidal wave of changes. Extensive changes were made in this third edition to reflect the changes that are taking place in the world of electronics. One of the big changes is the dramatic reduction in the costs of manufacturing the electronic equipment due to improved automation. This has resulted in the use of more electronics with improved performance over mechanical functions in a wide variety of products such as cameras, automobile controls for braking, ignition. air conditioners, transmission shifting, and combustion control, computers, computer printers, FAX machines, and televisions. One of the most dramatic changes has occurred in the military area, where high costs have forced the Department of Defense to scrap many of the military specifications in favor of best commercial practice electronic hardware for their new sophisticated equipment. Lower costs often lead to lower quality, which can result in a reduced reliability. This may only be an inconvenience when your television set fails or your automobile will not start. This can be a disaster in a military vehicle if the guns do not fire, or if the navigation system fails, or if the communications system fails. Therefore a new chapter was added that investigates and evaluates the effects of manufacturing methods and tolerances on the reliability of the electronic hardware. Another new chapter was added to show methods for improving the ruggedness of commercial hardware for improved reliability in harsh military environments. An additional chapter was added that relates to a better understanding of the bending distortion of the circuit board at its resonant condition, and how it effects the forces, stresses, and fatigue life in the component lead wires and solder joints. A new chapter was also added in the area of environmental stress screening, to provide a better understanding of how various combinations of xvii

XViii

PREFACE

thermal cycling and vibration can affect the accumulated damage and the amount of fatigue life used up by various screens. New equations are derived that show how to evaluate the effectiveness of a lead wire strain relief to reduce the dynamic forces and stresses in the lead wires and solder joints, and the resulting increase in the fatigue life. A simpler method for evaluating the fatigue life in random vibration is shown, which has the same accuracy as the three-band technique with much less work. A quick way of estimating the effectiveness of stiffening ribs on plug-in PCBs is shown, which is very simple to use and easy to apply. A new added section discusses a quick and easy method for evaluating the first three natural frequencies for flat plates and PCBs that are rectangular, octagonal, triangular, and round, with 16 different edge and point supports for each shape. A new and more accurate method for calculating the expected transmissibilty Q values for beams, circuit boards, and chassis is presented that relates to the natural frequency and the input acceleration G level. The data for this method are based on extensive vibration tests which showed that very low input acceleration G levels produced very low stresses and damping, which resulted in very high Q values. At the opposite end, high input G levels produced high stresses and damping, which resulted in low Q values. Many detailed sample problems are shown throughout this textbook, to demonstrate methods of analysis and evaluation that are based on 40 years of analysis and testing many different types of electronic systems. A section on vibration testing case histories was added to provide additional knowledge and insights for improving the testing results. Field failures in electronic equipment hardware compiled by the U.S. Air Force over a period of about 20 years show that about 40% of these failures are related to connectors, 30% to interconnects, and 20% to component parts. Failures in these areas can be due to handling, vibration, shock, and thermal cycling. Field failures related to operating environments show that about 55% of the failures are due to high temperatures and temperature cycling, 20% of the failures are related to vibration and shock, and 20% are due to humidity. Since high temperatures and temperature cycling events appear to be the major environmental causes of electronic failures, more information in these areas is very important. These subjects are not addressed in this book in great detail. More information is available on high temperatures and temperature cycling forces, stresses, and fatigue life effects [54]. Details on electronic components, electrical lead wires, solder joints, and plated through-holes for surface mounted and through-hole components, due to differences in the thermal coefficients of expansion (TCE), can be found in my book entitled Cooling Techniques for Electronic Equipment, Second Edition published by John Wiley & Sons. DAVES. STEINBERG Westlake Village, California September, 1999

DLIST OF SYMBOLS

A

Area (in.2), amplification g (dimensionless ratio) a length (in.) H ASIC Application-specific integrated circuit h Length (in.) Hz B Fatigue exponent, width (in.) I b Dynamic constant C C Length (in.) Lri C G Center of Gravity Distance from neutral axis J C K to outer fiber, length (in.), damping coefficient (lb . s/in.) Critical damping (lb .s/in.) cc D Dissipation energy KIN (lb .in./s) diameter (in.), plate bending stiffness Kt factor (lb . in.) Decibel dB KO D,, Plate torsional stiffness factor (Ib * in.) d L Diameter (in.), length (in.) DIP Dual inline package LCCC E Modulus of elasticity (lb/in.2) A4 e MT Bolted efficiency factor (%) ESS Environmental stress screen M , F Force (lb) Frequency (Hz) f f" Natural frequency (Hz) Rotational natural Mo fr MS frequency (Hz) FEM Finite element method Shear modulus (lb/in.2) G rn N with subscript: acceleration in gravity units (dimensionless)

Acceleration of gravity, 386 in./s2 Horizontal force Ob), drop heightr (in. or ft) Height (in.), thickness (in.) Frequency (cycles/s) Moment of inertia (area) (in.4) Mass moment of inertia (Ib in: s2) Torsional form factor (in.4> Linear spring rate (Ib in.), stiffness ratio (dimensionless), buckling form factor dimensionless) Kinetic energy Theoretical stressconcentration factor (dimensionless) Angular spring rate (Ib .in./rad) Length (in.) Leadless ceramic chip carrier Bending moment (lb . in.) Total moment (lb in.) Bending moment per unit length along the X axis (in: lb/inJ, bending moment at point X(lb. in.) Momentum (lb .s) Margin of safety (dimensionless) mass (lb s2/in.) Number of fatigue cycles to fail xix

XX

LIST OF SYMBOLS

Number of positive zero crossings, (Hz) n Actual number of fatigue cycles NS Number of sweeps through a resonance P Force (lb) P. PSD Power spectral density

N,+

wd W

X X X

x

z

Greek Symbols

(G*/HZ)

PCB d'

P

Q

4

R

Rn

RC

R, RMS RSS r sb

se

S SMD TCE T t

U V

W

Printed circuit board Dynamic force Unit load (lb/in.) Transmissibility (dimensionless ratio) Shear flow (lb/in.), dynamic pressure (lb/in.) Radius (in.), reaction (lb) Stress ratio (dimensionless), sweep rate (octave/min) Fatigue-cycle ratio (dimensionless) Damping ratio (dimensionless) Frequency ratio (dimensionless) Root mean square Root sum square radius (in.), relative position factor Bending stress (lb/in.2> Endurance limit stress (lb/in.*) Stress (lb/in.*) Surface mounted device Thermal coefficient of expansion (in./in./'C) Kinetic energy Ob-in.), torque (in/. lb) Time (min), temperature, Thickness (in.) Strain energy (lb in.), work (lb in.) Velocity (in./s>, force (lb) Weight (lb)

Dynamic load (lb) Unit load (lb/in.) Displacement along X axis, Xi, E;, Zcoordinate axes First derivative, velocity Second derivative, acceleration Displacement (in.)

CY

s 0 P

P

4 A

R

a"

Angle (degrees, radians), thermal coefficient of expansion (in./in./"C) Displacement (in.) Angular displacement (radians) Poisson's ratio (dimensionless) Density (lb/in.), mass per unit area (lb * in.^) Phase angle (degrees) Difference Angular velocity (rad/s) Natural frequency (rad/s) Subscripts

av b C

d e eq in m ax n 0

out st su t tu tY U

Y ST S T

Average Bending Critical Dynamic, desired Endurance Equivalent Input Maximum Natural Maximum Output or response Static Shear ultimate Tension Tensile ultimate Tensile yield Ultimate Yield Shear tearout Shear Torsion

-

CONTENTS

Preface

xvii

List of Symbols

xix

1. Introduction

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13

Vibration Sources Definitions Vibration Representation Degrees of Freedom Vibration Modes Vibration Nodes Coupled Modes Fasteners Electronic Equipment for Airplanes and Missiles Electronic Equipment for Ships and Submarines Electronic Equipment for Automobiles, Trucks, and Trains Electronics for Oil Drilling Equipment Electronics for Computers, Communication, and Entertainment

2. Vibrations of Simple Electronic Systems

2.1

2.2 2.3 2.4 2.5 2.6

Single Spring-Mass System Without Damping Sample Problem-Natural Frequency of a Cantilever Beam Single-Degree-of-Freedom Torsional Systems Sample Problem-Natural Frequency of a Torsion System Springs in Series and Parallel Sample Problem-Resonant Frequency of a Spring System Relation of Frequency and Acceleration to Displacement Sample Problem-Natural Frequency and Stress in a Beam Forced Vibrations with Viscous Damping Transmissibility as a Function of Frequency Sample Problem-Relating the Resonant Frequency to the Dynamic Displacement

1

1 2 3 3 5 5 6 7 10 13 15 16

16 17

17 19

21 22 23 25 26 27 30 34 34 vii

Viii

CONTENTS

2.7

Multiple Spring-Mass Systems Without Damping Sample Problem-Resonant Frequency of a System

3 Component Lead Wire and Solder Joint Vibration Fatigue Life

3.1 3.2

3.3 3.4 3.5

3.6

Introduction Vibration Problems with Components Mounted High Above the PCB Sample Problem-Vibration Fatigue Life in the Wires of a TO-5 Transistor Vibration Fatigue Life in Solder Joints of a TO-5 Transistor Recommendations to Fix the Wire Vibration Problem Dynamic Forces Developed in Transformer Wires During Vibration Sample Problem-Dynamic Forces and Fatigue Life in Transformer Lead Wires Relative Displacements Between PCB and Component Produce Lead Wire Strain Sample Problem-Effects of PCB Displacement on Hybrid Reliability

4. Beam Structures for Electronic Subassemblies

4.1 4.2

4.3

Natural Frequency of a Uniform Beam Sample Problem-Natural Frequencies of Beams Nonuniform Cross Sections Sample Problem-Natural Frequency of a Box with Nonuniform Sections Composite Beams

5. Component Lead Wires as Bents, Frames, and Arcs 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Electronic Components Mounted on Circuit Boards Bent with a Lateral Load-Hinged Ends Strain Energy-Bent with Hinged Ends Strain Energy-Bent with Fixed Ends Strain Energy-Circular Arc with Hinged Ends Strain Energy-Circular Arc with Fixed Ends Strain Energy-Circular Arcs for Lead Wire Strain Relief Sample Problem-Adding an Offset in a Wire to Increase the Fatigue Life

36 37 39 39 39 40

43 45 46 46 49 50 56

56 60 64 68 69

75 75 77 80 83 90 92 94 97

CONTENTS

6 Printed Circuit Boards and Flat Plates

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13

Various Types of Printed Circuit Boards Changes in Circuit Board Edge Conditions Estimating the Transmissibility of a Printed Circuit Board Natural Frequency Using a Trigonometric Series Natural Frequency Using a Polynomial Series Sample Problem-Resonant Frequency of a PCB Natural Frequency Equations Derived Using the Rayleigh Method Dynamic Stresses in the Circuit Board Sample Problem-Vibration Stresses in a PCB Ribs on Printed Circuit Boards Ribs Fastened to Circuit Boards with Screws Printed Circuit Boards With Ribs in Two Directions Proper Use of Ribs to Stiffen Plates and Circuit Boards Quick Way to Estimate the Required Rib Spacing for Circuit Boards Natural Frequencies for Different PCB Shapes with Different Supports Sample Problem-Natural Frequency of a Triangular PCB with Three Point Supports

7. Octave Rule, Snubbing, and Damping to Increase the PCB Fatigue Life

ix

103 103 106 108 111 116 120 122 127 131 132 137 141 141 142 144 149 150

7.1

Dynamic Coupling Between the PCBs and Their Support Structures 7.2 Effects of Loose Edge Guides on Plug-in Type PCBs 7.3 Description of Dynamic Computer Study for the Octave Rule 7.4 The Forward Octave Rule Always Works 7.5 The Reverse Octave Rule Must Have Lightweight PCBs Sample Problem-Vibration Problems with Relays Mounted on PCBs 7.6 Proposed Corrective Action for Relays 7.7 Using Snubbers to Reduce PCB Displacements and Stresses Sample Problem-Adding Snubbers to Improve PCB Reliability 7.8 Controlling the PCB Transmissibility with Damping 7.9 Properties of Material Damping 7.10 Constrained Layer Damping with Viscoelastic Materials 7.11 Why Stiffening Ribs on PCBs are Often Better than Damping 7.12 Problems with PCB Viscoelastic Dampers

150 154 154 155 155 156 157 159 161 162 162 163 164 164

X

CONTENTS

8. Preventing Sinusoidal Vibration Failures in Electronic Equipment 8.1 8.2

8.3 8.4

8.5 8.6

8.7

8.8

Introduction Estimating the Vibration Fatigue Life Sample Problem-Qualification Test for an Electronic System Electronic Component Lead Wire Strain Relief Designing PCBs for Sinusoidal Vibration Environments Sample Problem-Determining Desired PCB Resonant Frequency How Location and Orientation of Component on PCB Affect Life How Wedge Clamps Affect the PCB Resonant Frequency Sample Problem-Resonant Frequency of PCB with Side Wedge Clamps Effects of Loose PCB Side Edge Guides Sample Problem-Resonant Frequency of PCB with Loose Edge Guides Sine Sweep Through a Resonance Sample Problem-Fatigue Cycles Accumulated During a Sine Sweep

9. Designing Electronics for Random Vibration

9.1 9.2 9.3 9.4 9.5

Introduction Basic Failure Modes in Random Vibration Characteristics of Random Vibration Differences Between Sinusoidal and Random Vibrations Random Vibration Input Curves Sample Problem-Determining the Input RMS Acceleration Level 9.6 Random Vibration Units 9.7 Shaped Random Vibration Input Curves Sample Problem-Input RMS Accelerations for Sloped PSD Curves 9.8 Relation Between Decibels and Slope 9.9 Integration Method for Obtaining the Area Under a PSD Curve 9.10 Finding Points on the PSD Curve Sample Problem-Finding PSD Values 9.11 Using Basic Logarithms to Find Points on the PSD Curve 9.12 Probability Distribution Functions

166

166 167 168 169 171 174 175 177 179 182 185 185 187 188

188 188 189 190 192 193 193 194 195 197 198 200 200 20 1 202

CONTENTS

9.13 Gaussian or Normal Distribution Curve 9.14 Correlating Random Vibration Failures Using the Three-Band Technique 9.15 Rayleigh Distribution Function 9.16 Response of a Single-Degree-of-Freedom System to Random Vibration Sample Problem-Estimating the Random Vibration Fatigue Life 9.17 How PCBs Respond to Random Vibration 9.18 Designing PCBs for Random Vibration Environments Sample Problem-Finding the Desired PCB Resonant Frequency 9.19 Effects of Relative Motion on Component Fatigue Life Sample Problem-Component Fatigue Life 9.20 It’s the Input PSD that Counts, Not the Input RMS Acceleration 9.21 Connector Wear and Surface Fretting Corrosion Sample Problem-Determining Approximate Connector Fatigue Life 9.22 Multiple-Degree-of-Freedom Systems 9.23 Octave Rule for Random Vibration Sample Problem-Response of Chassis and PCB to Random Vibration Sample Problem-Dynamic Analysis of an Electronic Chassis 9.24 Determining the Number of Positive Zero Crossings Sample Problem-Determining the Number of Positive Zero Crossings

10 Acoustic Noise Effects on Electronics 10.1 Introduction Sample Problem-Determining the Sound Pressure Level 10.2 Microphonic Effects in Electronic Equipment 10.3 Methods for Generating Acoustic Noise Tests 10.4 One-Third Octave Bandwidth 10.5 Determining the Sound Pressure Spectral Density 10.6 Sound Pressure Response to Acoustic Noise Excitation Sample Problem-Fatigue Life of a Sheet-Metal Panel Exposed to Acoustic Noise 10.7 Determining the Sound Acceleration Spectral Density Sample Problem-Alternate Method of Acoustic Noise Analysis

xi

202 204 205 206 208 214 215 218 220 221 222 223 224 224 225 226 229 23 1 233

234 234 234 235 236 238 238 239 240 245 246

Xii

CONTENTS

11. Designing Electronics for Shock Environments 11.1 11.2 11.3 11.4

11.5 11.6 11.7 11.8

11.9

11.10 11.11

11.12 11.13 11.14 11.1s 11.16 11.17

11.18

11.19 11.20

11.21

Introduction Specifying the Shock Environment Pulse Shock Half-Sine Shock Pulse for Zero Rebound and Full Rebound Sample Problem-Half-Sine Shock-Pulse Drop Test Response of Electronic Structures to Shock Pulses Response of a Simple System to Various Shock Pulses How PCBs Respond to Shock Pulses Determining the Desired PCB Resonant Frequency for Shock Sample Problem-Response of a PCB to a Half-Sine Shock Pulse Response of PCB to Other Shock Pulses Sample Problem-Shock Response of a Transformer Mounting Bracket Equivalent Shock Pulse Sample Problem-Shipping Crate for an Electronic Box Low Values of the Frequency Ratio R Sample Problem-Shock Amplification for Low Frequency Ratio R Shock Isolators Sample Problem-Heat Developed in an Isolator Information Required for Shock Isolators Sample Problem-Selecting a Set of Shock Isolators Ringing Effects in Systems with Light Damping How Two-Degree-of-Freedom Systems Respond to Shock The Octave Rule for Shock Velocity Shock Sample Problem-Designing a Cabinet for Velocity Shock Nonlinear Velocity Shock Sample Problem-Cushioning Material for a Sensitive Electronic Box Shock Response Spectrum How Chassis and PCBs Respond to Shock Sample Problem-Shock Response Spectrum Analysis for Chassis and PCB How Pyrotechnic Shock Can Affect Electronic Components Sample Problem-Resonant Frequency of a Hybrid Die Bond Wire

248

248 249 25 1 252 253 257 258 260 260 262 264 265 269 269 274 274 275 276 277 278 28 1 282 284 285 285 286 288 288 29 1

292 296 298

CONTENTS

12. Design and Analysis of Electronic Boxes 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13

Introduction Different Types of Mounts Preliminary Dynamic Analysis Bolted Covers Coupled Modes Dynamic Loads in a Chassis Bending Stresses in the Chassis Buckling Stress Ratio for Bending Torsional Stresses in the Chassis Buckling Stress Ratio for Shear Margin of Safety for Buckling Center-of-Gravity Mount Simpler Method for Obtaining Dynamic Forces and Stresses on a Chassis

13. Effects of Manufacturing Methods on the Reliability of Electronics 13.1 13.2

13.3 13.4 13.5 13.6 13.7 13.8 13.9

Introduction Typical Tolerances in Electronic Components and Lead Wires Sample Problem-Effects of PCB Tolerances on Frequency and Fatigue Life Problems Associated with Tolerances on PCB Thickness Effects of Poor Bonding Methods on Structural Stiffness Soldering Small Axial Leaded Components on ThroughHole PCBs Areas Where Poor Manufacturing Methods Have Been Known to Cause Problems Avionic Integrity Program and Automotive Integrity Program (AVIP) The Basic Philosophy for Performing an AVIP Analysis Different Perspectives of Reliability

14. Vibration Fixtures and Vibration Testing

14.1 14.2 14.3 14.4 14.5 14.6

Vibration Simulation Equipment Mounting the Vibration Machine Vibration Test Fixtures Basic Fixture Design Considerations Effective Spring Rates for Bolts Bolt Preload Torque

Xiii

300 300 300 303 305 308 311 316 318 320 324 325 326 328

330 330 331 332 333 334 335 336 338 340 343

346 346 347 347 348 350 352

XiV

CONTENTS

Sample Problem-Determining Desired Bolt Torque Rocking Modes and Overturning Moments Oil-Film Slider Tables Vibration Fixture Counterweights A Summary for Good Fixture Design Suspension Systems Mechanical Fuses Distinguishing Bending Modes from Rocking Modes Push-Bar Couplings Slider Plate Longitudinal Resonance Acceleration Force Capability of Shaker Positioning the Servo-Control Accelerometer More Accurate Method for Estimating the Transmissibility Q in Structures Sample Problem-Transmissibility Expected for a Plug-in PCB

353 353 355 356 357 357 358 359 360 364 365 366

Vibration Testing Case Histories 14.19 Cross-Coupling Effects in Vibration Test Fixtures 14.20 Progressive Vibration Shear Failures in Bolted Structures 14.21 Vibration Push-Bar Couplers with Bolts Loaded in Shear 14.22 Bolting PCB Centers Together to Improve Their Vibration Fatigue Life 14.23 Vibration Failures Caused by Careless Manufacturing Methods 14.24 Alleged Vibration Failure that was Really Caused by Dropping a Large Chassis 14.25 Methods for Increasing the Vibration and Shock Capability on Existing Systems

369 369 370 371

14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 14.16 14.17 14.18

15. Environmental Stress Screening for Electronic Equipment (ESSEE) 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9

Introduction Environmental Stress Screening Philosophy Screening Environments Things an Acceptable Screen Are Expected to D o Things an Acceptable Screen Are Not Expected to D o To Screen or Not to Screen, That is the Problem Preparations Prior to the Start of a Screening Program Combined Thermal Cycling, Random Vibration, and Electrical Operation Separate Thermal Cycling, Random Vibration, and Electrical Operation

367 368

373 375 376 377

379 379 379 381 383 383 384 384 387 389

CONTENTS

15.10 Importance of the Screening Environment Sequence 15.11 How Damage Can Be Developed in a Thermal Cycling Screen 15.12 Estimating the Amount of Fatigue Life Used Up in a Random Vibration Screen Sample Problem-Fatigue Life Used Up in Vibration and Thermal Cycling Screen

XV

389

390 392 395

Bibliography

401

Index

405

-

CHAPTER 1

Introduction

1.I

VIBRATION SOURCES

Electronic equipment can be subjected to many different forms of vibration over wide frequency ranges and acceleration levels. It is probably safe to say that all electronic equipment will be subjected to some type of vibration at some time in its life. If the vibration is not due to an active association with some sort of a machine or a moving vehicle, then it may be due to transporting the equipment from the manufacturer to the customer. Vibrations encountered during transportation and handling can produce many different types of failures in electronic equipment unless the proper consideration are given to the mechanical design of the electronic structure and the shipping containers. Vibration is usually considered to be an undesirable condition, and in most cases it is. However, there are many applications where vibration is deliberately imposed to improve a function. Some sophisticated applications of vibrations are in the use of ultrasonics to clean medical instruments, measure wall thickness, and find flaws in castings. Vibration is also used in sorting rocks of different sizes by passing them over vibrating screens, which have groups of graduated holes. When the first jet airplanes were introduced, standard instruments from airplanes with piston engines and propellers were used. Theres instruments had a tendency to stick when they were used on the jet airplanes. The difference was due to the lower-frequency vibration levels developed in the piston engines. In order to make these same instruments work in the first jet airplanes, small vibrators had to be mounted on the instrument panels. Mechanical vibrations can have many different sources. In household products such as blenders and washing machines, the vibrations are due to the unbalance created by rotating and tumbling masses. In vehicles such as automobiles, trucks, and trains most of the vibration is due to rough surfaces over which these vehicles travel. In ships and submarines the vibration is due to the engines and to buffeting by the water. In airplanes, missiles, and rockets the vibration is due to jet and rocket engines and to aerodynamic buffeting. Most of the vibration in a missile, during subsonic flight, is due to 1

2

INTRODUCTION

+ Displacement ~

Period

-

Displacement

FIGURE 1.1. Simple harmonic motion.

the sound field developed by the rocket engines [1,6].* This is due to the extreme turbulence of the jet exhaust downstream from the rocket engine. 1.2

DEFINITIONS

Vibration, in a broad sense, is taken to mean an oscillating motion where some structure or body moves back and forth. If the motion repeats itself, with all of the individual characteristics, after a certain period of time, it is called periodic motion. This motion can be quite complex, but as long as it repeats itself, it is still periodic. If continuous motion never seems to repeat itself, it is called random motion. Simple harmonic motion is the simplest form of periodic motion, and it is usually represented by a continuous sine wave on a plot of displacement versus time, as shown in Fig. 1.1. The reciprocal of the period is known as the frequency of the vibration and is measured in cycles per second, or hertz (Hz), in honor of the German who first experimented with radiowaves. The maximum displacement is called the amplitude of the vibration. Crede [2,3] defines shock as a transient condition where the equilibrium of a system is disrupted by a sudden applied force or increment of force, or by a sudden change in the direction or magnitude of a velocity vector. Shock is not transmitted easily in the light airframe structures generally used in airplanes and missiles. Impact forces often result in a transient type of vibration, which is influenced by the natural frequencies of the airframe. Only steady-state linear vibrations are considered in this book. Linear vibrations occur with linear elastic media where the displacements are directly proportional to the applied force. If the force is doubled, the displacement is doubled. Only stresses up to the elastic limit of any given material are considered. Higher stresses generally result in permanent deformations, which extend into the plastic deformation region and are not considered here. *Numbers in brackets [ ] refer to bibliography entries in the back of the book.

DEGREES OF FREEDOM

3

FIGURE 1.2. Rotating vector simulating a single-degree-of-freedom system.

1.3 VIBRATION REPRESENTATION

A rotating vector can be used to describe the simple harmonic vibration of a single mass suspended on a coil spring (Fig. 1.2). The vector Yo rotates counterclockwise with a uniform angular velocity of R rad/s. The projection of the vector on the vertical axis represents the instantaneous displacement Y of the mass as it vibrates up and down. This can be written as

When the vector Yo rotates through one revolution, it rotates through an angle of 360°, which is 277- radians, for one complete cycle. The angular velocity is measured in radians per second, and the frequency f is measured in cycles per second. This leads to the relation

1.4

DEGREES OF FREEDOM

A vibrating system requires some coordinates to describe the positions of the elements in the system. If there is only one element in the system that is restricted to move along only one axis, and only one dimension is required to locate the position of the element at any instant of time with respect to some initial starting point, then it is a single-degree-of-freedom system. The same is true for a torsional system. If there is only one element that is restricted to rotate about only one axis so that only one dimension is required to locate the position of the element at any instant of time with respect to some initial starting point, it is a single-degree-of-freedom system. Some samples of systems with a single degree of freedom are shown in Fig. 1.3.

4

INTRODUCTION

Spring and mass

Rod and disk

Pendulum

(a)

(b)

(Ci

FIGURE 1.3. Single-degree-of-freedomsystems.

X+

F

Y

(a)

f6)

(di

(C)

FIGURE 1.4. Two-degree-of-freedomsystems.

A two-degree-of-freedom system requires two coordinates to describe the positions of the elements. Some samples of systems with two degrees of freedom are shown in Fig. 1.4. In rigid-body mechanics, a single body can have six degrees of freedom. These are translation along each of its three mutually perpendicular X , Y , and Z axes, and rotation about each of them, as shown in Fig. 1.5. A typical beam can have an infinite number of degrees of freedom since it can bend in an infinite number of shapes, or modes, as shown in Fig. 1.6.

FIGURE 1.5. A single mass with six degrees

of freedom.

+Z

-Y

VIBRATION NODES

5

_ _ - - -FIGURE 1.6. A simply supported beam showing several degrees of freedom.

k

....

Node point

,/--------

‘.._--

”

First harmonic mode

Second harmonic mode

(a)

(bi

FIGURE 1.7. First and second harmonic modes for a simply supported beam.

1.5 VIBRATION MODES

A standard manner in which a particular system can vibrate is known as a vibration mode. Each vibration mode is associated with a particular natural frequency and represents a degree of freedom. A single-degree-of-freedom system will have only one vibration mode and only one resonant frequency. The six-degree-of-freedom system shown in Fig. 1.5 can have six vibration modes and six resonant frequencies. The simply supported beam shown in Fig. 1.6 can have an infinite number of vibration modes; this is the same as saying this beam can have an infinite number of different shapes for each of its resonances, which are also infinite. The fundamental resonant mode of a vibrating system is usually called the natural frequency or the resonant frequency of the system. Sometimes it is called the first harmonic mode of the system. For example, a simply supported uniform beam vibrating at its fundamental resonant frequency has the . this beam is mode shape of a half sine wave as shown in Fig. 1 . 7 ~When vibrating at its second natural frequency, in its second harmonic mode, it has the mode shape of the full sine wave shown in Fig. 1.7b. The first harmonic mode of a system, with the lowest natural frequency, is the fundamental resonant mode; this often has the greatest displacement amplitudes and usually the greatest stresses. The second harmonic mode, or second resonance, usually has a smaller displacement than the first harmonic mode, so the stresses are usually smaller. The displacements continue to decrease for the higher resonant modes.

1.6 VIBRATION NODES

Vibration nodes are unsupported points on a vibrating body that have zero displacements. Nodes are generally associated with bending or torsion modes. At the first harmonic bending resonant mode in a beam there are no node

6

INTRODUCTION

First mode

Second mode

Third mode

Fourth mode

FIGURE 1.8. First four harmonic modes of a circular membrane.

First

mode

Second mode

Third

mode

FIGURE 1.9. First three harmonic modes of a square plate.

points. At the second harmonic mode there is one node point, the third harmonic mode has two node points, and so on. Figure 1 . 7 ~shows the bending mode of a vibrating beam with no node points, and Fig. 1.7b shows a beam with one node point. Vibrating plates can have straight-line bending nodes and circular nodes. The first four harmonic modes of a circular membrane are shown in Fig. 1.8. The plus ( + ) signs show positive displacements and the minus ( - > signs show negative displacements. The dashed lines represent nodal locations of zero displacement. The first three harmonic bending modes of a square plate with free edges are shown in Fig. 1.9.

1.7

COUPLED MODES

In a system with two or more degrees of freedom, the vibration mode of one degree often influences the vibration mode of the other degree. For example, in Fig. 1 . 4 if~ mass 2 is held rigidly while mass 1 is displaced in the vertical direction, mass 1 will oscillate up and down. Now if mass 2 is released, the motion of mass 1 will act upon mass 2 so mass 2 will begin to oscillate up and down. Because the motion of mass 1 has a direct effect on the motion of mass 2, these two vibration modes are defined as being coupled. In coupled modes, the vibration in one mode cannot occur independently of the vibration in the other mode. Coupled modes can occur in translation, rotation, and combinations of translation and rotation for systems with more than one degree of freedom. For coupled modes in translation and rotation, it is often possible to

FASTENERS

7

Y A

L

3

I I 1

I

> r

L+L+

FIGURE 1.10. A single mass with two springs.

determine whether the system is coupled or uncoupled by making a simple test. Apply a steady load to the body of the system at its center of gravity (CG), in a specific direction. If the body moves in the direction of the applied load without rotation, then the translational mode is not coupled with the rotational mode for motion along the direction of the applied load. Consider, for example, the mass with two springs as shown in Fig. 1.10. If a steady load is applied to the CG along the X axis, the mass will not only translate along the X axis, but it will also tend to rotate at the same time. This test shows that for vibration along the X axis, the translational mode is coupled with the rotational mode. If a steady load is applied to the C G along the Y axis, the mass will translate along the Y axis without rotation. Also, if a moment is applied to the mass about the CG, then rotation will take place without translation. These tests show that for vibration along the Y axis the translational mode is decoupled from the rotational mode.

1.8 FASTENERS

Many different types of fasteners are used in electronic equipment. These include screws, nuts, rivets, and clips. Fasteners are responsible for a very large percentage of field failures and, in most shock tests, they are the largest single source of failure. Although fasteners have been used in very large quantities in electronic equipment for many years, most of the applications are based on static installation considerations. The fastening techniques, which are generally chosen for ease of installation and low cost, are usually not satisfactory for severe shock and vibration environments. A contributing factor is that fasteners are such small items and their use so universal that their application tends to be semiautomatic without regard to their strength. This is particularly the case for machine screws. When large screws are used, it is often because someone has had some association with the automotive or aircraft industry, where very small screws are seldom used.

8

INTRODUCTION

Because fasteners play an important part in the overall reliability of the electronic equipment, extra consideration should be given in their selection as follows [ 5 ] : 1. Select the proper type of fastener (screws, rivets, etc.), considering such trade-offs as environment, strength, maintenance, and cost. 2. Select the correct fastener size and location based on dynamic loads and geometry of the structure. 3. Select the correct locking device for screws and nuts. 4. Select the right installation technique.

Most electronics manufacturers choose screws and rivets on a production basis. If the designer uses screws, the size and locking device are selected according to the tolerances as well as for ease in assembly. In installation, the designer depends on production personnel to use their own judgment in the proper installation of fasteners. The results may be that the wrong fasteners are used, the wrong sizes are used, the wrong lockwashers are used, and the wrong installation torques are used. It has been found, generally speaking, that cold-driven shank-expanding rivets are very satisfactory and should be used more frequently in electronic assemblies. It has been found, too, that when screws are used, they should be of larger sizes than those customarily considered and they should be made of better materials. Many locking devices commonly used are unsatisfactory. Some are often the source of many severe problems. It has also been found that a mechanic’s judgment in tightening a machine screw is usually faulty. Many failures in electronic equipment have resulted because bolts have come loose. Consider what can happen to a sensitive electronic chassis when a large transformer comes loose and rattles around during vibration. Although many investigations have been made with high-speed films and strain gages to study the loosening action of screws and nuts, the mechanism by which this occurs is still not well established. It appears, however, that the bolt stretches slightly under the action of a dynamic load, so that the interface friction forces in the area of the bolt are suddenly and sharply reduced. Since the threads in the screw or nut tend to return to their original shape during the small time increments, the small geometric changes become a driving force, which tends to loosen the screw and nut. Some specific recommendations can be made to improve the quality of the fasteners in electronic equipment: 1. Steel screws should be used in all screw fastenings. The steel should have the minimum properties of SAE 1010. 2. The screws should be tightened by a torque device, which can be preset to the required value.

FASTENERS

9

TABLE 1.1 ~~

Screw Size

Torque

Screw

Torque

(in: lb)

Size

(in: lb)

2-64 4-40 6-32 8-32 10-24 10-32

3-3.5 5-6.5 10-12 20-24 22-27 34-42

12-24 12-28 $20 -28 i-16 f -24

45-56 50-64 65-80

85-100 250-320 330-415

3. The tightening torque should be 6 0 4 0 % of the torque required to twist the head off the screw. These torque values for steel screws are given in Table 1.1 [lll. 4. The screw head should permit a positive nonslipping grip for the driving device and should also withstand the driving forces. A slottedhex-head machine screw seems to be, for general purposes, the most satisfactory. 5 . In all applications involving through-holes, locknuts should be used instead of lockwashers. Most of the standard steel locknuts are satisfactory. 6. Blind-tapped holes should be avoided, if possible. When they are necessary, locking devices such as lockwashers should be used under the screw head in order to prevent the screw from backing out during vibration. The tightening torque should be increased by an amount equal to the resisting frictional torque of the locking device. 7 . The fasteners for a unit should be distributed so that the failure of one fastener will not free the unit or cause it to malfunction. Even for very small components there should be a minimum of two fasteners. Lockwashers and screw-locking inserts should be used with care in electronic systems that will be used in the zero-G environment of outer space. These two devices create a binding friction in the screw by biting into the metal during installation. This action will very often shave metal particles from the screw. If these particles are not removed, they can float around in a zero-G environment and create electrical problems. An actual count was made of the metal particles after inserting 24 screws with external-star-type lockwashers, and a total of about 1000 metal particles were counted. The crossed-recess screwdriver slot (Phillips head) will also tend to be cut by the action of the screwdriver. When the screw is seated and the screwdriver is twisted, the screwdriver will very often twist out of the slot and shave small bits of metal out of the screw head.

10

INTRODUCTION

In order to avoid shaving small bits of metal from the screws with devices that bit into the metal, many electronic firms use liquids such as Loctite and Glyptal, which bind the screws. Some firms use nylon inserts in the screws. Nylon inserts are usually good for about a dozen insertion and removal cycles before the nylon cold flows and reduces the binding torque. NASA will not permit the use of volatile products in the area of displays and optical devices for their spacecraft. These products tend to outgas in a vacuum and then deposit themselves in a thin film; this can coat optical lenses and interfere with the operation of sensitive optical units such as cameras and telescopes. 1.9 ELECTRONIC EQUIPMENT FOR AIRPLANES AND MISSILES

Electronic boxes used in airplanes and missiles often have odd shapes that permit them to make maximum use of the volume available in tight spaces. Since volume and weight are generally quite critical, the electronic boxes usually have a high packaging density. This value normally ranges from about 0.03 to about 0.04 lb/in.3, depending on the severity of the environmental requirements. The weight of a typical electronic box will range from about 10 to about 80 lb. The vibration frequency spectrum for airplanes will vary from about 3 to about 1000 Hz, with acceleration levels that can range from about 1 G to about 5 G peak. The highest accelerations appear to occur in the vertical direction in the frequency range of about 100-400 Hz. The lowest accelerations appear to occur in the longitudinal direction, with maximum levels of about 1 G in the same frequency range. For helicopters, the frequency spectrum will vary from about 3 to about 500 Hz and acceleration will range from about 0.5 to about 4 G. The highest accelerations appear to occur in the vertical direction at frequencies near 500 Hz. The displacement at low frequency are very large, with values of about 0.20-in. double amplitude at about 10 Hz. Missiles have the highest frequency range in this group, with values that generally go up to 5000 Hz [7]. The lower frequency limit is about 3 Hz, and this appears to be due to bending modes in the airframe structure. Acceleration levels range from about 5 to about 30 G peak, with the maximum levels occurring during power-plant ignition at frequencies above 1000 Hz. The vibration environment in supersonic airplanes and missiles is actually more random in nature than it is periodic. However, sinusoidal vibration tests are still being used to evaluate and to qualify electronic equipment that will be used in these vehicles. Because the forcing frequencies in airplanes and missiles are so high, it is virtually impossible to design resonance-free electronic systems for these environments. Of course, it is always possible to completely encapsulate an entire electronic box with some expanding rigid type of foam, which could

ELECTRONIC EQUIPMENT FOR AIRPLANES AND MISSILES

11

drive the resonant frequency well above 1000 Hz (possibly to 2000 Hz) for a small box. This is generally considered to be impractical, however, because it becomes too expensive to maintain, troubleshoot, and repair such a system. The obvious conclusion is that the forcing frequencies present in airplanes and missiles will excite many resonant modes in every electronic box. What becomes equally obvious is that extra care must be taken in the design and analysis of an electronic system, or it can literally shake itself to pieces. The electronic support structure must be dynamically tuned with respect to the electronic components to prevent coincident resonances that can lead to rapid fatigue failures. The first thought that comes to the mind of an experienced mechanical design engineer, when confronted with a severe vibration specification, is to mount the electronic equipment on vibration isolators. There is no doubt that a set of isolators, properly designed, can control shock and vibration. There are four major factors that must be considered when isolation mounts are being discussed. 1. Sway space must be provided all around the electronic equipment to keep it from colliding with other objects. If volume is scarce, it might be more practical to pack more electronics into the same volume by using a larger electronic box with hard mounts. 2. Cold plates are being used more and more in electronic structures to remove the heat dissipated by the electronic equipment. If isolators are used, flexible couplings must be provided between the airframe structure and the electronic box to take care of the large displacements developed by the isolators. Reliable flexible couplings must be used because cooling effectiveness may be sharply reduced if a coupling fails. 3. Electrical wire cables and harnesses must be used to connect the typical electronic box to the main electrical system in the airplane or missile. If isolators are used, these cables and harnesses will be forced through large amplitudes because of the sway space required by the isolators. Special precautions must be taken to prevent fatigue failures in the cables and harnesses. 4. A good vibration isolator is often a poor shock isolator, and a good shock isolator is often a poor vibration isolator. The proper design must be incorporated into the isolator to satisfy both the vibration and the shock requirements.

Cold-plate designs for airborne electronic equipment have become more and more sophisticated with the use of air and liquids. Air heat exchangers make extensive use of multiple fins, wavy fins, split fins, and pin fins in order to improve the heat-transfer characteristics. The multiple-fin heat exchangers are usually dip-brazed aluminum with as many as 22 fins per inch. These fins may be only 0.006 in. thick, but the large number of fins in a typical heat

12

INTRODUCTION

exchanger makes it quite rigid for its weight. Airplanes make extensive use of electronic equipment where a cooling-air heat exchanger is built right into the electronic support structure. The heat exchanger is riveted, brazed, or cemented to the major structural members in the electronic box, so that the heat exchanger itself becomes a major load-carrying member of the system. The cooling air for the cold plate is usually taken from the compressor on the jet engine that powers the airplane. This air must be conditioned before it can be used for cooling, because the air temperature from the first stage is usually greater than 300°F. Liquid-cooled cold plates are usually used to cool electronic equipment on spacecraft or very-high-flying research airplanes. The cooling liquid can be a mixture of ethylene glycol and water, or some other liquids such as FC-75, a fluorocarbon or Coolanol 45, a silicone fluid. The liquid-cooled cold plates are usually made part of the spacecraft airframe structure instead of the electronic box structure. Then, when the electronic box must be removed from the spacecraft, it is not necessary to disconnect fluid lines, which can become quite messy. Since the cold plate stays with the airframe, the heat dissipated by the electronic box is usually transferred to the cold plate through a flat interface on the mounting surface of the electronic box that makes intimate contact with the cold plate. The trend in commercial and military electronics is toward the line replaceable unit (LRU), with which it is possible to replace a defective electronic box right on the flight line in a matter of minutes. This is accomplished by providing all of the required interface connections, both mechanical and electrical, at the back end of the electronic box. The box becomes similar to a printed circuit board (PCB) that can be plugged into its receptacle. If there are several large electrical connectors on the back end of the electronic box, it may be quite difficult to insert the box and engage the electrical connectors properly. Some connectors may require a force of 0.50 lb/pin for proper engagement. When there are 8 connectors, each with 100 pins, there is a total of 800 pins, which will require a 400-lb force to engage them. The plug-in electronic box is usually engaged and locked into position by some mechanism at the front of the box. Since the connectors are at the rear of the box, this means the force required to engage the connectors must pass through the box. When this type of electronic box is subjected to vibration, in many instances the vibration loads must be added to the installation loads to determine the total load acting on the structure. There has been an attempt to standardize electronic equipment used in military and commercial airplanes by establishing certain sizes for modular electronic units. These modular units are then mounted in a standard air transport rack (ATR), which provides rear-located dowel pins and connectors and a quick-release fastener at the front.

ELECTRONIC EQUIPMENT FOR SHIPS AND SUBMARINES

13

1. l o ELECTRONIC EQUIPMENT FOR SHIPS AND SUBMARINES

Ships and submarines will generally make use of console cabinets to support their electronic equipment, since there is usually more room available and weight is not very critical. The electronic components are usually mounted on panels and in sliding drawers. Panels are generally used to support dials, gages, manual controls, and test points. Only small masses are mounted on panels, because they are fastened to a frame or rack in the cabinet and they cannot withstand large dynamic loads. Drawers are often used to support the more massive electronic components such as those normally used in power supplies. The drawers are mounted on telescoping slides to provide access to the equipment. For safety, the drawers usually lock in the open and closed positions. For convenience, the drawers may also tilt to improve access in tight spaces. The vibration frequency spectrum for ships and submarines varies from about 1 to about 50 Hz, but the most common range is from about 12 to about 33 Hz. The maximum acceleration level in this range is about 1 G and appears to be due to vibrations set up by the engines and propellers. In military ships, shock is an important factor, due generally to various explosions, which can do extensive damage to electronic equipment, unless proper consideration is given to the design and installation. For example, it is not desirable to have a very rigid structure supporting the electronics, because a very rigid structure may not deform enough to absorb much stain energy. Theoretically, any structure that does not deform when it is subjected to an impact load will receive an infinite acceleration. A large displacement is desirable, since it can substantially reduce acceleration levels. Either this displacement must be confined to the structure or shock mounts must be used. In either case, provisions must be made in the design and installation to make sure parts will not collide and equipment will not break loose [36]. If shock isolators are used, they should be designed to deflect enough to absorb the shock energy without transmitting excessive loads to the electronic equipment. The shock mounts should have a minimum resonant frequency of about 25 Hz [2,51. Ideally, the resonant frequency of the electronic components should be at least twice that of the shock mounts, but never below 60 Hz. If the electronic components have resonances substantially below 60 Hz, it might bring them into the range of the most common vibration forcing frequencies, which, as previously mentioned, are as high as 33 Hz. If this should happen, the electronic components would be driven continuously near their resonance and could suffer fatigue failures. When the forcing frequency of the ship’s structure is near its higher frequency limit (around 25 Hz), the resonant frequency of the electronic equipment cabinet will be excited, since the shock isolators also have their resonance at 25 Hz. This condition should not impose high stresses on the electronic components mounted in the cabinet if the component resonance is

14

INTRODUCTION

twice that of the isolators. The cabinet support structure will, however, have to withstand the dynamic vibration loads. These loads will be determined by the amplification characteristics of the shock isolators during vibration. Shock isolators are available that provide a vibration amplification of about 3 for the conditions described above. Since the general vibration acceleration input level in these frequency ranges is normally quite low, an amplification factor of 3 for the isolators does not result in high stresses in the equipment cabinet. It is generally not desirable to increase the resonant frequency of the electronic equipment as high as possible. If the equipment is very stiff, it may be good for the vibration condition but poor for the shock condition. A very high spring rate may result in very high shock stresses because of the high acceleration loads. Lee [5] recommends a maximum resonant frequency of about 100 Hz and a maximum acceleration of 200 G on the electronic components. On tall narrow cabinets the load-carrying isolators should be at the base and stabilizing isolators should be at the top. A rigid structure must be used to support the stabilizing isolators at the top. If there is excessive deflection in the top support structure, it can change the characteristics of the entire system because of severe rocking modes. If shock isolators are not used, the shock energy must be absorbed by deflections in the electronic equipment cabinet and in the structure of the ship supporting the cabinet. In this case the natural frequency of the assembly, which consists of the cabinet and the ship’s structure, should be about 60 Hz. The natural frequency of the electronic components mounted on the equipment cabinet should be twice that of the assembly, or about 120 Hz. The ship’s structure must provide a good part of the deflection required to attenuate the shock force, or the dynamic stresses in the equipment cabinet may be high enough to cause structural failures. When the vibration forcing frequency in the ship’s structure is near its normal maximum limit of 33 Hz, the dynamic loads in the equipment cabinet will not be amplified to any great extent since the resonant frequency of the cabinet, at 60 Hz, is almost twice the forcing frequency. Furthermore, the electronic components mounted in the equipment cabinet have a still higher resonant frequency (120 Hz), so their dynamic vibration loads should be relatively small. The materials that are best suited for shock are ductile materials with a high yield point, a high ultimate strength, and a high percentage elongation. In general, metals that are mechanically formed are more desirable than cast metals, which have a relatively low percentage elongation. Since acceleration forces become progressively smaller as they propagate into the interior of the equipment cabinet, the electronic components that can withstand the highest G forces should be mounted near the exterior of the cabinet. Electronic components that have their own rigid structures

ELECTRONIC EQUIPMENT FOR AUTOMOBILES, TRUCKS, AND TRAINS

15

should be used to lend additional strength to the outer structural elements in the cabinet. Electronic components that cannot withstand high G forces should be mounted at the maximum elastic distance from the application points of the shock load. This will usually be at the center of the cabinet. The load path for each mass element in the system should be examined closely to determine the path the load will take as it passes through the structure. For example, a large transformer should be mounted close to a major structural support in order to reduce the length of the load path. This will result in smaller deflections and stresses.

1.1 1 ELECTRONIC EQUIPMENT FOR AUTOMOBILES, TRUCKS, AND TRAINS

Electronic equipment for automobiles and trucks has grown very rapidly in the past few years. Electronics are being used in their antilocking brake systems, fuel-air mixture control, radios, ignition systems, air conditioners, heaters, automatic transmission shifting points, rear view mirrors, door locks, instruments, global positioning systems (GPSs), windows, sun roofs, cruise control, air bags, television, telephones, and many other features, with more being added every year. Anticollision systems are being developed that will automatically engage brakes when vehicles are dangerously close to each other at high speeds. There are plans for computerized car trains that will link several automobiles together going to similar areas. Antitheft systems such as Lojack are available that can be activated by radio to send out a silent alarm to police when an automobile is stolen. Automobile and truck electronics must operate in harsh environments that include wide temperature swings, high under-hood operating temperatures of 140"C, high humidity with condensation, high-velocity splashing water, and subfreezing temperatures with only moderately high vibration and shock levels for the automobiles. Trucks are often required to travel over unpaved country roads for deliveries in outlying areas. Test data on a 2.5-ton truck instrumented with several accelerometers showed acceleration levels between about 15 and 19 G peak at speeds of 10-15 mph, at frequencies between about 15 and 40 Hz. At truck speeds above about 35 mph over rough roads the frequency appears to be random in nature. Diesel electric trains have been making use of electronics on their drive wheels to sense when the wheels are just reaching the slipping point on the tracks. A large engine may have six pairs of driving wheels. The electronics can sense when any one pair of drive wheels will slip. Sand is automatically dispensed to increase friction and the electric motor driving the one set of drive wheels is automatically slowed slightly to prevent slopping on the

16

INTRODUCTION

tracks. A single diesel electric engine with this new feature can go up steeper grades and pull three times as many cars as previous diesel electric trains. 1.12 ELECTRONICS FOR OIL DRILLING EQUIPMENT

Oil drilling equipment probably has the most severe thermal and vibration requirements for electronic systems. Deep drilling at 30,000 feet can encounter temperatures as high as 200°C. The electronics are usually located in a 6-foot long tube section just above the cutters. The coolant is the 200°C mud washing over the electronics tube section. Sometimes the cutters get stuck in a rock section while drilling at 30,000 feet. The turntable at the surface keeps turning while the cutters are stationary. The long drilling pipe winds up and when it breaks loose it generates random vibration acceleration levels as high as 30 G RMS. This means that the electronics section in an oil drilling rig must be very rugged to withstand the high vibration levels and the high temperatures and still provide a high reliability. When an electronic failure occurs, it is a very expensive and time-consuming task to remove 30,000 feet of pipe, replace the electronics section, and drop 30,000 feet of pipe down the hole again. 1. I 3 ELECTRONICS FOR COMPUTERS, COMMUNICATION, AND ENTERTAINMENT

According to many people in the three fields of computers, communication, and entertainment, they are rapidly converging into one giant industry with the internet. Sometimes this is called bandwidth, because fiber optic cables are being added, which can carry a very wide range of frequencies that are required to provide services for television, radio, data processing, stock trading, banking, internet, telephone, security, and more energy efficient homes and offices.

-

CHAPTER 2

Vibrations of Simple Electronic Systems

2.1

SINGLE SPRING-MASS SYSTEM WITHOUT DAMPING

The natural frequency of many single-degree-of-freedom systems can be determined by evaluating the characteristics of the strain energy and the kinetic energy of each system. Considering a single spring-mass system, for example, if there is no energy lost, then the maximum kinetic energy of the mass must be equal to the maximum strain energy in the spring, if the spring mass is negligible. All real systems have some damping. If a mass is suspended on a real coil spring and the spring is stretched and released, the mass will vibrate up and down. This free vibration may continue for a long time, but eventually all free vibrations die out and the mass stops vibrating. This is because damping in the spring dissipates a little energy with each cycle and eventually the mass stops moving. If there is no damping, the mass will theoretically keep on vibrating up and down with the same amplitude and frequency forever. In many systems, the damping is so small that it has very little effect on the natural frequency. Under these circumstances the natural frequency of the damped system can conveniently be approximated by the natural frequency of the undamped system. The maximum kinetic energy of a vibrating spring-mass system with no damping is at the point of maximum velocity. This occurs as the vibrating mass passes through the zero-displacement point, as shown in Fig. 2.1. From elementary physics the maximum kinetic energy T of the vibrating mass is

To = i m V 2 (2.1) The instantaneous tangential velocity I/ can be expressed in terms of rotational velocity as shown in Fig. 1.3:

v=

Yon (2.2) Substituting Eq. 2.2 into Eq. 2.1, the kinetic energy of the system becomes To = ;my,Zn*

(2.3) 17

18

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

T

,

+Displacement

t

< -<

rJ-1

;- - _ - - - - - --

Maximum strain energy

J Time

'Maximum velocity -Displacement

FIGURE 2.1. A vibrating spring-mass system.

The maximum strain energy can be determined from the work done on the spring by the mass as it stretches and compresses the spring during vibration. Since the spring is linear, the deflection is directly proportional to the force as shown in Fig. 2.2. The area under the curve represents the work done on the spring. Since the work is equal to the strain energy, the strain energy becomes

U" = +P,Y"

(2.4)

The spring rate K can be defined in terms of the maximum load P, and the maximum deflection Yo as K = P,/Y,, so

Po = KY,,

(2.5)

Substituting Eq. 2.5 into Eq. 2.4, the maximum strain energy becomes

uo= ;KY; If damping is assumed to be zero, then at resonance the kinetic energy will be equal to the strain energy: i,y2flz 2

=' K y 2

(I

2

0

[E) 1/ 2

Q" =

SINGLE SPRING-MASS

SYSTEM WITHOUT DAMPING

I9

This is the natural frequency in radians per second. Using Eq. 1.2, the natural frequency in cycles per second (Hz) is

The natural frequency equation can be written in a slightly different form by considering that the static deflection a,, of the spring is due to the action of the weight W acting on the spring. The spring rate can then be written as

Expressing the mass in terms of weight W and acceleration of gravity g , we have

Substituting Eqs. 2.8 and 2.9 into Eq. 2.7 gives another relation for the natural frequency: (2.10)

Frequency of a Cantilever Beam

Sample Problem-Natural

Determine the natural frequency of the cantilevered beam with an end mass, as shown in Fig. 2.3, in the vertical direction. Solution. When the weight of the aluminum beam is small compared to the weight of the end mass, it can be ignored without too much error. The cantilever beam can then be analyzed as a concentrated load on a massless beam, which retains its elastic properties. The static deflection at the end of

1

A

y

q

g

O 50

-1rL3 B-r Sect A A

1.4 -100

-

in

---+

FIGURE 2.3. Cantilever beam with an end mass.

20

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

this beam can be determined with the use of a structural handbook: (2.11) where W = 10.0 Ib (end weight) L = 10.0 in. (length) E = 10.5 x l o 6 lb/in.’ (modulus of elasticity, aluminum) I = bh3/12 = (0.50)(3.0)3/12 = 1.125 in.4 (moment of inertia) (10.0)(10.0)~ =

3(10.5

X

106)(1.125)

=

0.282

in.

X

(2.12)

Substitute Eq. 2.12 into Eq. 2.10 and note that the acceleration of gravity, g , is 386 in./s2. The natural frequency becomes 1 =

187 HZ

(2.13)

The natural frequency of the cantilever beam can also be determined from the spring rate, K , of the beam using Eq. 2.11: (2.14) Using the values previously calculated, the spring rate becomes

K=

3( 10.5 x l o 6 )(1.125)

( 10.0)3

= 3.54 X

l o 4 lb/in.

Also,

w 10 m=-=-= g 386

0.0259 Ib . s’/in.

Substituting into Eq. 2.7, the natural frequency becomes

”’-[-)

1

K

27r m

f,

=

1/2

1

3.54 x l o 4 0.0259

=z(

187 HZ

The results are exactly the same as Eq. 2.13.

1

1/ ?

(2.15)

SINGLE-DEGREE-OF-FREEDOM TORSIONAL SYSTEMS

2.2

21

SINGLE-DEGREE-OF-FREEDOM TORSIONAL SYSTEMS

The energy method is convenient for determining the natural frequency of a torsional system with one degree of freedom, as shown in Fig. 2.4. The torsional system is similar to the spring-mass system shown in Fig. 2.1, except that the spring action is due to the twisting of the rod and the inertia is due to the moment of inertia of the disk about an axis perpendicular to the plane of the disk. Assume the rod mass is small. The maximum kinetic energy of the oscillating disk is

KIN = ;zme2 where I

(2.16)

= mass

moment of inertia of disk 6 = rotational angular velocity

The maximum angular velocity for the oscillating system moving through the angle Bo is

io= eon

(2.17)

Substituting Eq. 2.17 into Eq. 2.16, the maximum kinetic energy becomes (assuming a small rod mass)

(2.18) The maximum strain energy can be determined from the work done on the twisting rod by the disk as it oscillates. Since the spring rate of the rod is linear, the angular deflection will be directly proportional to the torque T applied, as shown in Fig. 2.5. The area under the curve represents the work

22

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

done on the rod. Since the work is equal to the strain energy, the strain energy becomes

u, = +T,O,

(2.19)

The torsional spring rate K , can be defined in terms of the torque To and the angular displacement On as follows:

K 0 = -TO

(2.20a)

To = K , Bo

(2.20b)

On

so

Substituting Eq. 2.20 into Eq. 2.19, the maximum strain energy becomes

U, = i K , O i

(2.21)

If damping is assumed to be zero, then at resonance the kinetic energy (Eq. 2.18) must equal the strain energy (Eq. 2.21):

(2.22) The natural frequency (in Hz) becomes

j""('= f

Sample Problem-Natural

1/2

2 T I,,

(2.23)

Frequency of a Torsion System

Determine the natural frequency of the torsional system shown in Fig. 2.6.

Aluminum

{-X7" 'i FIGURE 2.6. A single-degree-of-freedom torsional system.

o&

L-

f

R =' 6 0 1" = 0 50 in

23

SPRINGS IN SERIES AND PARALLEL

Solution. The torsional spring rate K , of the rod can be determined using its angular displacement 8 under the action of an external torque To: TOL e= GJ

TO K O =-

and

e

so

where L = 10.0 in. (length of rod) G = 4.0 X l o 6 1b/ine2 (shear modulus of aluminum) J = r d 4 / 3 2 = ~ ( 1 . 0 ) ~ / 3=20.0981 in.4 (polar moment of inertia)

(4.0 X 106)(0.0981) K, =

10.0

= 3.92 X

IO4 in:lb/rad

The mass moment of inertia of the aluminum disk must be taken about the axis perpendicular to the plane of the disk:

I,

’

WR

=-

2g

where W = ~(6.0)~(0.50)/(0.10l b / i r ~ . ~=)5.65 lb (disk weight) R = 6.0-in. disk radius g = 386 in./s2 (gravity)

I,

(5.65) (6.0)2 =

2( 386)

= 0.264

1b.in:s’

The torsional resonant frequency can now be determined from Eq. 2.23:

’=

I

%[

3.92 x 104 0.264

1

1,/2

=

61.5 HZ

(2.24)

2.3 SPRINGS IN SERIES AND PARALLEL If a mass is suspended on two different springs in such a manner that the load path is directly from the mass through one spring, and then through the second spring before it reaches the support, the springs are said to be in series. The series spring, therefore, implies a series load path where the load must first pass through one of the springs before it can pass through the

24

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

Tension a n d compression

-

Bending

Torsion

FIGURE 2.7. Springs in series.

other spring. Cutting either spring would completely destroy the system. Some samples of springs in series are shown in Fig. 2.7. Springs in series can be combined into one equivalent spring using the relation

1

1 -

Ke,

K,

1

1

K?

K3

+-+-+

(2.25)

If a mass is suspended on two different springs in such a manner that the load path from the mass to the support is split between the springs, then the springs are said to be in parallel. The parallel path then permits the load to pass through one spring without passing through the other. Cutting one spring would still permit the other spring to carry the load. Some samples of springs in parallel are shown in Fig. 2.8. Springs in parallel can be combined into one equivalent spring using the relation

K,,

=K ,

+ K , + K , + ...

(2.26)

Kz = 10001b/in.

-

SPRINGSIN SERIES AND PARALLEL

25

< < K3 = 1000 Ib/in. 6

1 <

5

3

> K4 = 800 Ib/in. >

FIGURE 2.9. A combination of series and parallel springs.

Frequency of a Spring System

Sample Problem-Resonant

Determine the natural frequency of a mass supported by several springs in series and parallel combinations as shown in Fig. 2.9, by obtaining the equivalent spring rate for the system. Solution. Springs K, and K, are in parallel, so they can be combined using Eq. 2.26:

K,

= K,

+ K, = 1000 + 1000

Now there are three springs-K,, combined using Eq. 2.25: 1

--

K,,

_-

1

K,

=

2000 lb/in. series, so they can be

K,, and K,-in

1

1

1

1

K,

K,

4000

2000

+-+-=-+-

+- 1

800

The system now consists of one spring and one mass (Fig. 2.10). The natural frequency can be determined from Eqs. 2.7 and 2.9 as follows:

f,,= 49.5 HZ

3; K,,= 500 >

1

Ib/in.

W = 2 Ib

FIGURE 2.10. A single spring-mass system.

(2.27)

26

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

2.4 RELATION OF FREQUENCY AND ACCELERATION TO DISPLACEMENT

Many important approximations can be related to the dynamic displacements developed during resonant conditions. For example, if the geometry of a specific structure can be defined, then dynamic bending moments and bending stresses can be calculated from the dynamic displacements. Since displacements are very seldom measured during vibration tests, because optical measurements are often difficult to make, accelerometers are generally used. This permits test data to be taken in terms of frequency and acceleration G forces. The relation of frequency and acceleration to the displacement must therefore be determined to use the test data. Consider a rotating vector that is used to describe the simple harmonic motion of a single mass suspended by a spring. The vertical displacement Y of the mass can be determined by the projection of the vector Yo on the vertical axis as shown in Fig. 2.11. The vertical displacement can be represented by the equation

Y = Yosin nt

(2.28)

The velocity is the first derivative, I/= Y =

dY -=

dt

Szy, cos nr

The acceleration is the second derivative.

A

..

=

Y=

~

d2Y = dt’

-a2yosin f i t

The negative sign indicates that acceleration acts in the direction opposite to displacement.

27

RELATION OF FREQUENCY AND ACCELERATION TO DISPLACEMENT

The maximum acceleration will occur when sin Rt is 1: A,,,

= R2Y0

(2.29)

The acceleration in gravity units (G) can be determined by dividing the maximum acceleration by the acceleration of gravity: g = 32.2 ft/s2

= 386 in./s2

Also, changing radiants to cycles per second, we have fl=2.rrf Substituting into Eq. 2.29,

G = -A,,, g

- 4X2f2YJ 386 (2.30)

Note that the displacement Yo is the single-amplitude displacement. Sample Problem-Natural

Frequency and Stress in a Beam

Determine the resonant frequency, dynamic displacement, and maximum dynamic stress in the bracket with a transformer, as shown in Fig. 2.12, for a 5.0-G peak sinusoidal vibration input. Solution. Consider the transformer as a concentrated mass at the center of a massless beam. Since small screws are used to support the beam, the ends are considered to be simply supported. The natural frequency of a simply supported beam with a concentrated center mass can be determined from the static displacement as shown in Fig. 2.13, using Eq. 2.10. The static displacement can be obtained from a handbook: 6,,

W Transformer Screw

WL3 48 EI

(2.31)

=-

Bracket

L ---

0503 Sect A A FIGURE 2.12. A transformer mounted o n a bracket.

28

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

W=201b

-- --

FIGURE 2.13. A simply supported beam with a concen-

trated load.

__

~

-8O---.--.

where W = 2.0 Ib (transformer weight) L = 8.0 in. (length) E = 10.5 x l o 6 (aluminum-beam modulus of elasticity) I = bh3/12 = (1.2)(0.50)'/12 = 0.0125 i n 4 (moment of inertia) g = 386 in./s2 (2.0) ( 8.0)3 =

48(10.5

X

106)(0.0125)

=

1.62 X

in.

Substitute into Eq. 2.10 for the natural frequency:

f,

= 246

Hz (simply supported ends)

(2.32)

Equation 2.30 can be used to determine the dynamic deflection at the center of the transformer bracket, using the 246-Hz resonance: 9.8G 9,8Gi,Q yo=-- -

f2

(2.33)

f 2

where G,, = 5.0 G (input acceleration) Q = 30 (transmissibility at resonance) f , = 246 Hz (simply supported beam)

Yo=

9.8(5 .O) (30) =

(246)'

0.0243 in.

(2.34)

The approximate transmissibility for a beam-type structure can be obtained from the resonant frequency. For an electronic subassembly, experience has shown that a value equal to two times the square root of the resonant frequency, or about 30, gives reasonable results.

RELATION OF FREQUENCY AND ACCELERATION TO DISPLACEMENT

29

The dynamic load acting on the bracket can be approximated by letting Eq. 2.34 equal Eq. 2.31 and solving for the dynamic load wd:

w, =

w,=

48 EW, ~

L3 48(10.5 X 106)(0.0125)(0.0243)

(8.0)3

W, = 300 lb

(dynamic load)

(2.35)

If the acceleration loads were assumed to act directly on the transformer, the dynamic load would be ?t$ = WG,,Q

(2.36)

where W = 2.0 lb (transformer weight) Gin= 5.0 G (peak input) Q = 30 (transmissibility at resonance) Thus W, = (2.0) ( 5 .O) (30)

= 300 Ib

(2.37)

This result is exactly the same as that shown in Eq. 2.35, which was obtained from the dynamic deflection of a simply supported beam with a concentrated load. The dynamic bending stress in the simply supported transformer bracket can be determined from the geometry of the structure, using the dynamic loading as shown in Fig. 2.14. The maximum dynamic bending moment

Lr7

Wd = 300 Ib

4.0

Jlransfcyy

bracket

8.0

R = 150

R = 150

_L50_t.-150-~-

Shear diagram

FIGURE 2.14. Shear and bending-moment diagram for a beam

30

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

occurs at the center of the bracket: =150(4) =600in:lb

(2.38)

The maximum dynamic bending stress can be determined from the standard bending-stress equation:

s, =

MC r

(2.39)

where M = 600 in: lb (bending moment) c = h/2 = 0.50/2 = 0.25 in. (ref. Sect. A.A. Fig. 2.12) I = bh3/12 = (1.2)(0.50)'/12 = 0.0125 ine4 Hence

s, =

(600)(0.25) 0.0125

=

12,000 lb/in.2

(2.40)

A stress concentration factor must be used when several million stress reversals are expected. When only a few thousand stress cycles are expected in a ductile material, then a stress concentration factor is not critical. 2.5

FORCED VIBRATIONS WITH VISCOUS DAMPING

When a harmonic shaking force acts on a spring-mass system with damping, the resulting forced vibration will also be harmonic. The final frequency of the mass will be the same as that of the shaking force, because the initial transient vibrations will eventually be dissipated by the damping. Consider the harmonic shaking force Po cos R t acting on a damped spring-mass system. A free-body force diagram is shown in Fig. 2.15. The differential equation of motion with an external force exciting the system is mY+

C Y + Ky=

P" cos at

(2.41)

The solution of this equation consists of a complementary function plus a particular function. The complementary solution is the free vibrations. These will die out because of the damping. The particular solution can be taken in the form Y=Y,cos(Rt-

e)

(2.42)

FORCED VIBRATIONS WITH VISCOUS DAMPING

t

1

Po cos Ltt

31

Po cos nt

FIGURE 2.15. Forced vibration forces acting on a simple system.

The maximum displacement, Yo,can be expressed in terms of the maximum impressed force, Po, as follows:

Yo=

PO

[ ( K - mQ2)' + c 2 R 2 ]'''

(2.43)

Dividing the numerator and denominator of the above equation by K and substituting Eq. 2.6 leads to the following:

Po -

--

K

(2.44:)

Let Y,,= P o / K , where Y,, is the deflection of the system due to the maximum dynamic input load acting as a static load. For additional simplification, let

R R,=-

a"

C

and

R , = -cc

(2.45)

This leads to the general amplification (not transmissibility) equation A = -YO =

1

(2.46)

A plot of the dynamic amplification ratio A against the frequency ratio R , is shown in Fig. 2.16. This is not a force-transmissibility curve. A force-transmissibility curve is shown in Fig. 2.17.

32

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

10 8

6 4

2

10 08

06 04

02 01

0

10

0.5

15

R"

=

25

20

Forcing frequency

30

f

--

f,

Natural frequency

FIGURE 2.16. A dynamic amplification curve for a simple system.

10

a 6 4

02 01

0

05

10

R"

15

25

20

30

Forcing frequency - f Natural frequency f,

_-

=

FIGURE 2.17. A transmissibility curve for a simple system.

33

FORCED VIBRATIONS WITH VISCOUS DAMPING

The instantaneous magnitude of the force experienced by the support is the vector sum of the spring force and the damper force. These two forces have a 90” phase angle between them, so the force becomes

Fo = Yo(K 2

+~~fl’)’~’

Substituting Eq. 2.43 into the above expression,

Fo =

+C2fl2)li2 [ ( K - mn’)2 + C W ] 1/ 2 Po( K 2

The transmissibility Q, which represents the ratio of the maximum output force F,, to the maximum input force Po, can be determined by dividing the numerator and denominator by K and substituting Eq. 2.6 into the above equation:

By substituting Eq. 2.45 into Eq. 2.47, the transmissibility Q becomes

1 + (2R,Rc)2

(2.48)

(1 - R $ ) ’ + ( 2 R , R c ) 2

A plot of the force transmissibility Q against the frequency ratio R , is shown in Fig. 2.17. This curve is very similar to the amplification curve shown in Fig. 2.16. In the resonant area, where R , = 1.0, the two curves are almost identical for systems with a low damping ratio. For lightly damped systems, where R: is small enough to be assumed zero, the transmissibility expression reduces to

1 e=--1-Rk

(2.49)

34

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

A very convenient relation can be obtained from Eq. 2.48 by considering the transmissibility at resonance, where R , = 1.0: 1

+ (2RC)' W C l 2

For lightly damped systems, the ratio R: is very small compared to the value of 1 in the numerator; thus the transmissibility equation can be reduced to Q=-

1

(2.50)

2Rc 2.6

TRANSMISSIBILITY AS A FUNCTION OF FREQUENCY

The transmissibility of a system can be related to the resonant frequency and damping as follows: c=

/E Q2- 1

(2.51)

For a lightly damped system, where Q is greater than about 10, the relation can be solved approximately for the transmissibility:

Q=

(Km)"l

(2.52)

When the spring rate K is increased, the resonant frequency is increased. This decreases the dynamic displacement and the stresses, so the damping c is decreased, which increases the transmissibility Q. When the mass m is increased, the resonant frequency is decreased. This increases the dynamic displacement and the stresses, so the damping is increased, which decreases the transmissibility. Sample Problem-Relating the Resonant Frequency to the Dynamic Displacement

The natural frequency of a printed circuit board (PCB) is 100 Hz, with a transmissibility of 10 at the center of the circuit board. It is desirable to keep the maximum dynamic deflection at the center of the circuit board to 0.020 in. (single amplitude), in order to keep from overstressing the electrical lead wires on the resistors, capacitors, and diodes mounted on the circuit board. What should the natural frequency of the circuit board be for a 5-G peak sinusoidal vibration input?

TRANSMISSIBILITY AS A FUNCTION OF FREQUENCY

35

Solution. The PCB can be approximated as a single-degree-of-freedom system when it is vibrating at its fundamental resonant mode. Therefore Eq. 2.33 can be used to estimate the single-amplitude displacement for the 100-Hz resonant frequency:

Yo=

9.8Gi, Q

f2

where Gin= 5.0-G input (peak) Q = 10 (transmissibility at resonance) f, = 100 Hz (resonant frequency)

Yo=

9.8( 5 .O) (10) (

= 0.0490

in.

(single amplitude) (2.53)

This displacement is much too high. If the resonant frequency is increased, the displacement will decrease very rapidly as shown by Eq. 2.33. However, Eq. 2.52 shows that the transmissibility will also increase as the resonant frequency increases. Extensive test data on PCBs, with various edge restraints, indicate that many epoxy fiberglass circuit boards, with closely spaced electronic component parts, have a transmissibility that can be approximated by the relation

Q=fl

(2.54)

Substitute into Eq. 2.33 and solve for the resonant frequency:

( 2 3)

To obtain a single-amplitude displacement of 0.020 in. with a 5-G peak vibration input, the circuit board resonant frequency must be increased to the following: (2.56)

36 2.7

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

MULTIPLE SPRING-MASS

SYSTEMS WITHOUT DAMPING

The fundamental resonant frequency of a multiple spring-mass system, without damping, can be determined by considering the strain energy and the kinetic energy of the system. If there is no energy lost, then the total kinetic energy of the masses must be equal to the total strain energy of the springs. Consider the multiple spring-mass system as shown in Fig. 2.18. The total kinetic energy of the system is the sum of the individual kinetic energies of each mass as shown by Eq. 2.3:

T,,,,,

=

$m,Y:02 + i2 m 2 Y’O’ 2

+ $m,Y;’02

This can be written as

(2.57)

The total strain energy of the system is the sum of the individual strain energies of each spring, as shown by Eq. 2.4:

This can be written as

( 2S 8 )

FIGURE 2.18. A multiple spring-mass system without damping.

rF1 yj

MULTIPLE SPRING-MASS

SYSTEMS WITHOUT DAMPING

37

Since the total kinetic energy must equal the total strain energy, set Eq. 2.57 equal to Eq. 2.58 and solve for the natural frequency:

(2.59)

Note that the deflections in the above equation are the total deflections. For example, in Fig. 2.18 the total deflection of mass 3 must include the deflections of masses 1 and 2, and the total deflection of mass 2 must include the deflection of mass 1. Also, the effective weight of mass 1 on spring 1 must include all three weights. Sample Problem-Resonant Frequency of a System

Calculate the fundamental resonant frequency of the two spring-mass system shown in Fig. 2.19. Solution. The static deflection of weight 1 will be influenced by weight 2, since weight 2 is attached to weight 1; thus both weights will act on spring 3 . We have

Yl =

wl+ w,=--2 + 4 Kl

1000

- 0.006 in

(2.60)

The static deflection of weight 2 must include the static deflection of weight 1, since weight 2 is attached to weight 1:

Yz

1

1

w2

Yl + - = 0.006 K,

4

+ -- 0.00733 in. 3000

(2.61)

2.0 Ib

. 5

3 K2 = 3000 Ib/in. 4.0 Ib

FIGURE 2.19. A two-degree-of-freedom system without damping.

38

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

FIGURE 2.20. A free-body force diagram for a two-mass system.

Substitute into Eq. 2.59:

[(2)(0.006)

1

f f,

=-[

+ (4)(0.00733)](386)

(2)(0.006)2 + (4)(0.00733)*

2~

= 37.3

(2.62)

HZ

The static deflections of weights 1 and 2 in Fig. 2.19 could also have been determined by the simultaneous solution of the deflection equations for each mass. A free-body force diagram of the system is shown in Fig. 2.20. Consider the forces in the vertical direction: For weight 1,

- K , Y , + W , + K 2 ( Y 2 - Y Y =i )O -(

K , + K 2 ) Y ,+ K 2 Y 2+ W , = O

(2.63)

For weight 2,

- K , ( Y 2 - Y,)+ w,= 0 -KzYz

+ K,Y, + Wz = 0

(2.64)

Substituting values for K , , K 2 , W,, and W, into the above equations and solving,

+2 = 0

+

- 4000Y1 3000Y2 3000Y, - 3000Y,+ - IOOOY,

4= 0

+6=0

-

h

Y , = -= 0.006 in 1000

Yz =

3000(0.006) 3000

+4

(2.65) = 0.00733

in.

(2.66)

These results are exactly the same as those shown by Eqs. 2.60 and 2.61.

Component Lead Wire and Solder Joint Vibration Fatigue Life

3.1

INTRODUCTION

Electronic components are available in a large variety of types, sizes, and materials for leaded and leadless mounting and for through-hole or surface mounted applications. Most of these components will end up on printed circuit boards (PCBs) for use in everything electronic from cameras and washing machines to telephones and space shuttles. Some types of electronic equipment will be used in homes and offices that are air conditioned with very quiet environments. Other types of equipment will be used in military programs with high vibration and shock levels. It does not matter what the electronic equipment is doing, or what functions it performs, or how the equipment is used-in all cases the buyers or the consumers want a low cost, reliable, easy to use piece of equipment. Sometimes they get it and sometimes they don’t. Luck often seems to play a large part in the quality of electronic products. With good luck they last for years. With bad luck they last for weeks. A satisfied customer is a repeat customer. A quality product will usually attract repeat customers, even when the price of the product is slightly higher than the price of the competition. Most electronic failures are mechanical in nature. Many of these mechanical failures occur in the component lead wires and solder joints. Extensive military testing experience over a period of many years has shown that about 80% of the electromechanical failures are due to some type of thermal condition and about 20% of the failures are due to some form of vibration and shock. This chapter is devoted to the examination and evaluation of sinusoidal vibration, and its fatigue effects on the electrical lead wires and solder joints on different types of components and PCBs [25]. VIBRATION PROBLEMS WITH COMPONENTS MOUNTED HIGH ABOVE THE PCB

3.2

Electronic components are usually attached to PCBs with the use of solder. Different attachment methods can be used including dip soldering, wave 39

40

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

soldering, and infrared, vapor phase, oven, and even hand soldering. Eutectic solder is a mixture of 63% tin and 37% lead. It has a melting temperature of about 183°C. A wave soldering machine for through-hole devices uses a solder temperature of about 230°C to get the proper wicking action up the plated through-holes with the lead wires. Most semiconductor electronic components are rated below a temperature of about 200°C. Some types of capacitors are only rated at 85°C ambient. When these components are wave soldered to a PCB, the heat from the solder can be conducted up the metal lead wires directly into the silicon semiconductor chips, causing them to malfunction. The cure for this problem is often to raise the components high above the PCB. Longer lead wires increase the length of the heat flow path to the temperature-sensitive chips. This increases the thermal resistance so the temperature rise to the chip is reduced. The thermal problem is solved but it creates a vibration problem as shown below. Sample Problem-Vibration TO-5 Transistor

Fatigue Life in the Wires of a

A TO-5 transistor is mounted on the top of a 0.25-in. high plastic tube, which is sitting on the top surface of a PCB. The purpose of the high mounting is to reduce the excessive heat that is conducted up the three wires during the wave soldering operation, causing electrical malfunctions in the transistor. The longer wires reduce the temperature rise, which solves the thermal problem. Will the tall transistor wires survive a qualification dwell test requirement of 30 min with a 4.0-G peak sinusoidal vibration input level up to a frequency of 2000 Hz?

Solution Find the Response of the Transistor and the Wire Fatigue Life. The approximate natural frequency of the tall transistor can be obtained by considering it to be similar to a cantilevered beam with a concentrated end mass, using the static displacement natural frequency relationship:

2%-

(see Eq. 2.10)

where W = 1.0 g = 0.0022 lb (transistor weight) L = 0.25-in. (length of three lead wires) E = 20 x lo6 lb/in.* (kovar modulus of elasticity)

I

=

%-dJ/64= %-(0.016)'/64 = 3.22 X (one wire moment of inertia)

in.4 (3.1)

VIBRATION PROBLEMS WITH COMPONENTS MOUNTED HIGH ABOVE THE PCB

g

= 386

in./s2 (acceleration of gravity)

WL3

Sst =

S,,

41

(0.0022) (0.25)

E + (3)(20 X 106)(3 X 3.22 X l o p 9 )

= 5.93 X

f,,= % .2

lo-’ in.

(see Eq. 2.11)

(displacement for three wires)

/-

5 . 9 3 x lo-’

= 406

Hz (natural frequency)

(3.2) (3.3)

The dynamic force Pd acting on the transistor can be obtained from the acceleration and the mass of the transistor. The acceleration depends on the transmissibility expected at the natural frequency of the transistor. A good approximation for a beam-type structure is about two times the square root of the natural frequency. However, since the transistor is sitting on (and not cemented to) the top of a plastic tube, the interface slapping and impacting of the transistor on the tube during vibration will tend to reduce the expected transmissibility by about 50%. A transmissibility equal to the square root of the natural frequency will therefore be used here:

Pd = WGinQ (see Eq. 2.36) where W = 0.0022 lb (transistor weight) Gin= 4.0 peak (acceleration input level) = 20 (approximate transmissibility expected) Q=

Pd= (0.0022) (4.0) (20)

= 0.176

Ib

(dynamic force on transistor) (3.4)

The dynamic bending stress in the three lead wires can be obtained from the standard bending stress equation shown below. Stress concentrations due to cuts, scrapes, and scratches in the wires must be considered when many thousand stress reversal cycles are expected. The stress concentration can be used in the bending stress equation below, or it can be used to define the slope of the S-N fatigue curve for the kovar lead wire material. The stress concentration factor is normally used only once in either location. In this problem the stress concentration factor will not be used in the bending stress equation below. The stress concentration will be used in the kovar S-N fatigue curve, which is shown in Fig. 3.1. S,

where M

MC = -

I

(see Eq. 2.39)

= PdL = (0.176)(0.25) = 0.044

lb *in.(dynamic bending moment (3.5) c = d / 2 = 0.016/2 = 0.008 in. (wire radius)

42

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

Kovar wire

I

1

N , cycles to fail

FIGURE 3.1. S-N log-log fatigue curve for kovar wire, with a stress concentration factor of 2.

I

=

(3X3.22 x = 9.66 x wires) (see Eq. 3.1)

s, =

in.4 (moment of inertia, three

(0.044) (0.008)

9.66 x 10-9

= 36,439

lb/in.'

(3.6)

The fatigue life expected for the transistor kovar lead wires can be obtained from the S-N fatigue curve shown in Fig. 3.1. This is a log-log plot of the stress S and the number of cycles N to fail. Fatigue tests are run to failure on many similar closely machined samples. The results show there is a lot of scatter in the failure data. The best average straight line is drawn through the scattered test points to represent the typical fatigue properties of the material. The slope of the fatigue curve is adjusted to include a stress concentration factor of 2.0. This results in a fatigue exponent slope of b = 6.4. Every point on the straight sloped line represents a failure. The equation for every point on the sloped line is shown below: N,SP

=

N2Sf

(3.7)

Then

where N2 = 1000 cycles to fail at 84,000 lb/in.' (kovar wire reference point) S, = 84,000 lb/in.* (ultimate tensile strength of kovar lead wire)

VIBRATION FATIGUE LIFE IN SOLDER JOINTS OF A TO-5 TRANSISTOR

43

S , = 36,439 lb/in.’ (wire bending stress) (see Eq. 3.6) b = 6.4 (slope of kovar fatigue curve with stress concentration of 2)

N,

=

Wire life

=

84,000 6 . 4 (1000) 36,439

]

(

= 2.096 X

2.096 x l o 5 cycles to fail (406 cycles/s) (60 s/min)

l o 5 wire cycles to fail (3.8)

= 8.6

minutes to fail

(3.9)

3.3 VIBRATION FATIGUE LIFE IN SOLDER JOINTS OF A TO-5 TRANSISTOR Vibration forces generate overturning moments in the three lead wires. These moments result in shear tearout stresses in the solder joints, which can be obtained from the equation shown below. Microscopic examinations of failed solder joints have been made. This shows the solder appears to fail someplace between the outer diameter of the wire and the inner diameter of the plated through-hole. Therefore, an average of the wire diameter and the plated through-hole diameter is used to find the average of the solder joint [121.

S,,

=

A4 (solder shear tearout stress) hA

(3.10)

where A4 = 0.044 lb in. (see Eq. 3.5) h = 0.062 in. (solder height assumed equal to PCB thickness) A = (3>.rrd2/4 = ( 3 ) ~ ( 0 . 0 2 2 ) ~ /= 4 0.00114 i n 2 (average area of three solder joints)

”‘

=

0.044 (0.062) (0.00114)

=

622.5 Ib/in.*

(3.10a)

The vibration fatigue life for solder can be obtained from the log-log fatigue curve shown in Fig. 3.2, and with the use of Eq. 3.7. Very rapid fatigue cycles are considered here since the vibration frequency is so fast that the solder will not have a chance to creep and relax the stresses. Solder creep is common in thermal cycling environments, where the stress cycles are often very slow. This allows the solder to creep at high temperatures and relax a large percentage of any high stresses that are developed. Note that if forced

44

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

63/37 Tinilead solder

I

h’,cycles to f a i l FIGURE 3.2. S-N log-log vibration and thermal cycling fatigue curve for 63/37 tin/lead solder.

cooling is used immediately after the soldering operation it will increase the solder fatigue life.

1000 cycles to fail at 6500 Ib/in.2 (solder vibration life reference) S , = 6500 lb/in.l (solder ultimate shear reference) S , = 622.5 Ib/in.* (see Eq. (3.10a) b = 4.0 (slope of solder fatigue curve for vibration rapid cycles)

where N ,

=

N,

=

Solder life

=

6500 (1000) 622.5

[

‘

=

1.19 X lo7 solder cycles to fail (3.11)

1.19 x IO’ cycles to fail (406 cycles/s) (3600 s/h)

=

8.14 hours to fail

(3.12)

Comparing the solder fatigue life in Eq. 3.12 with the wire fatigue life of 8.6 minutes in Eq. 3.9 shows the wires are far more critical than the solder joints in vibration. Sinusoidal vibration tests were run using the same 4.0-G peak input level to establish the approximate fatigue life of the lead wires. The test results

RECOMMENDATIONS TO FIX THE WIRE VIBRATION PROBLEM

45

showed the wires failed after several minutes of vibration. The solder joints showed no visible cracks using a 35 power microscope. The vibration analysis and the vibration tests showed the 0.25-in. high transistor wires are critical. They will not be able to survive a 30-minute sinusoidal vibration qualification test using an input acceleration level of 4.0 G peak. Several changes were recommended to improve the fatigue life of the lead wires.

3.4 RECOMMENDATIONS TO FIX THE WIRE VIBRATION PROBLEM 1. Remove the 0.25-in. tall tube spacer and bring the transistor down to a height of about 0.060 in. above the top surface of the PCB. Attach heat sink clips to each of the transistor lead wires just before the wave soldering operation. This will thermally short-circuit the heat flow into the clips and reduce the high temperatures produced in the transistor. Remove the heat sink clips after the wave soldering operation. Experience with this approach has been very good in reducing hot spot temperatures. The reduced lead wire length will drive the transistor natural frequency above the 2000-Hz range, so there will be no vibration problems. This fix is labor intensive because it requires hand labor to add and to remove the heat sink clips. It is only cost effective for a production run of several hundred units. This fix would not be acceptable for production runs that involve several million units, as in the communication and automobile business. 2. Cement the 0.25-in. tall tube spacer to the PCB and to the transistor body to increase the stiffness of the assembly. This should drive the transistor natural frequency above the highest forcing frequency range of 2000 Hz, so there will be no vibration problems. This fix can be automated for high production runs. It may still not be acceptable for the super high communication and automobile production runs because their costs are calculated down to 0.001 penny. 3. Automate an operation that dimples the wires to automatically position the transistor 0.025 in. above the PCB for the wave solder operation. After soldering, automate another operation that adds a measured amount of viscous material, such as RTV, that encloses the three wires and fills most of the 0.025-in. space under the transistor. It is not necessary to fill the entire area under the transistor to substantially increase the damping and easily reduce the transmissibility more than 60%. This will reduce the wire bending stresses more than 60% and increase the wire fatigue life to about 50 hours. A fatigue life of only 30 minutes is required, plus some additional time to account for the usual false starts in most qualification test programs. This fix will solve the vibration problem with a slight increase in the production costs.

46

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

3.5 DYNAMIC FORCES DEVELOPED IN TRANSFORMER WIRES DURING VIBRATION

Transformers are made with copper and iron so they are often quite heavy. When these components are mounted on PCBs that are exposed to vibration, high dynamic forces and stresses can be produced in the lead wires and solder joints. Heavy components must be carefully mounted, properly restrained, and adequately supported to keep the lead wires and solder joints from failing in severe vibration and shock environments. Transformers are often in the shape of cylinders or cubes. Surface mounted transformers have the lead wires extending from the sides so they can be bent down 90" to meet the PCB. For through-hole mounting, straight wires are typically used, where they extend from the bottom surface of the component directly into the PCB. When transformers are mounted on through-hole PCBs that are exposed to vibration, high acceleration forces can be generated in the lead wires and solder joints when the natural frequency of the PCB is excited. The arching of the PCB under these conditions produces axial forces and bending forces in the lead wires. When the lead wires are not spaced very far apart, there will be very little bending in the wires, so almost all of the inertia force produced in the transformer wires will be carried in direct tension. Copper wires are typically used for the transformer core winding and for the external leads. The fatigue properties of the copper wire must be known to find the approximate fatigue life expected in vibration conditions.

Sample Problem-Dynamic Transformer Lead Wires

Forces and Fatigue Life in

A 0.80-lb transformer, with four 0.028-in. diameter copper wires, is mounted near the center of a 0.062-in. thick plug-in PCB that has a natural frequency of 225 Hz. The system is required to operate in a 3.0-G peak sinusoidal vibration environment. Find the following: (a) Dynamic force in the wires and solder joints. (b) Dynamic stresses in the lead wires and solder joints. (c) Approximate fatigue life expected in the lead wires and solder joints. Solution Use the Square Root of the Natural Frequency to Find the PCB Transmissibility. (a>The dynamic force Pd can be obtained from the following equation: Pd = WG,,Q

(see Eq. 2.36)

47

DYNAMIC FORCES DEVELOPED IN TRANSFORMER WIRES DURING VIBRATION

where W = 0.80 lb (transformer weight) Gin= 3.0 (peak sine input) Q= = \/225 = 15 (estimated PCB transmissibility)

a

Pd

=

(0.80)(3.0)(15)

=

36.0 lb

( 3.13)

(b) The dynamic tensile stresses in the wire can be obtained from a handbook, as shown below:

s,=

Pd

-

where Pd = 36.0 lb (shown above) d = 0.028 in. (copper wire diameter) A , = n d 2 / 4 = [ ~ ( 0 . 0 2 8 > ~ /X4 4] wires copper wires) 36.0

sw=---= 0.00246

(3.14)

A,

= 0.00246

i n 2 (area of four

14,634 lb/in.* (wire tensile stress) (3.15)

The dynamic shear stress in the solder joint can be obtained from Eq. 3.14 with a slight change as shown below. Test data show that solder joints appear to fail at some point between the wire and the plated through-hole. Therefore, the average solder joint diameter is obtained by using the average of the lead wire diameter plus the plated through-hole diameter. The thickness of the PCB,without any solder fillets, is normally used for the solder shear area. This is a conservative estimate since visual inspections show solder fillets of about 0.010 in. on the top and bottom surfaces of the PCB for through-hole mounted components. Pd

ss= A,

(3.16)

where Pd= 36.0 lb (dynamic load) (see Eq. 3.13) d = (0.028 + 0.040)/2 = 0.034 in. (average solder joint diameter) A s = (4)(n)(0.034)(0.062) = 0.0265 ine2(area of four solder joints) 36.0

s,=-0.0265

- 1358 lb/in.2

(solder shear stress) (3.17)

(c) The approximate fatigue life of the lead wire can be obtained from the log-log S-N fatigue curve for the copper lead wire shown in Fig. 3.3. This curve includes a stress concentration factor of 2.0, to account for cuts,

48

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

Copper wire

108

103 iV, cycles to f a i l

FIGURE 3.3. S-N log-log fatigue curve for copper wire, with a stress concentration factor of 2.

scrapes, and scratches normally found in component lead wires. The fatigue damage relation is used to find the number of stress cycles required to produce a fatigue failure, as well as the expected fatigue life.

N,S,b = N,S,b (see Eq. 3.7) where N , = 1000 stress cycles to fail at 45,000 lb/in., (copper reference point) S , = 45,000 lb/in., (stress reference point) b = 6.4 (reference slope of fatigue curve for copper lead wire with k = 2) S , = 14,634 lb/in.’ (wire stress) (see Eq. 3.15) Nl

=

Wire life

=

(1000)

[

~

45,000 14,634

1

6.4

=

1.33 x l o 6 wire cycles to fail (3.18)

1.33 x lo6 cycles to fail

(225 cycles/s) (3600 s/h)

=

1.64 hours

(3.19)

The approximate fatigue life of the solder can be obtained from the log-log fatigue curve for solder shown in Fig. 3.2. Solder creep is not considered here because the stress cycles are very fast, so the solder does not have a chance to relax any internal stresses. The fatigue damage equation (Eq. 3.7) is used to find the number of stress cycles to fail, as well as the expected fatigue life.

DISPLACEMENTS BETWEEN PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

49

where N2 = 1000 stress cycles to fail at 6500 Ib/in2 (solder reference point) S , = 6500 lb/in.* (stress reference point) SI = 1358 Ib/in.* (solder shear) (see Eq. 3.17) b = 4.0 (reference slope of fatigue curve for solder vibration)

[

6500

1

N,= (1000) Solder life =

1358

= 5.25 X

5.25 x 10' cycles to fail (225 cycles/s) (3600 s/h)

IO5 solder cycles to fail (3.20) = 0.65

hour

(3.21)

Comparing Eq. 3.19 (for the wire life) and Eq. 3.21 (for the solder life) shows the solder is more critical than the lead wire in this case. 3.6 RELATIVE DISPLACEMENTS BETWEEN PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

Plug-in PCBs can produce large dynamic displacements in high vibration and shock environments, when their natural frequencies are excited. This often results in large relative displacements between the PCB and any large electronic components mounted on the PCB. The body of a large component may be thicker and stiffer than the PCB that supports the component. When the component is attached to the PCB with many lead wires in a through-hole or surface mounted arrangement, the relative displacement between the PCB and the component can generate a significant amount of strain in the lead wires, as shown in Fig. 3.4. The lead wires that are soldered to the PCB will

50

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

stay perpendicular to the PCB when it bends away from the component body. The lead wires will also stay perpendicular to the component body. This motion will generate an axial force and a bending moment in the lead wires. When the length of the component body is small compared to the length of the PCB, the axial force in the lead wire will be much greater than the bending moment in the lead wire. The bending moment in the wire can then be ignored without introducing a large error. Finite element methods (FEMs) of analysis show that a long component body will bend much more than the lead wires will stretch. Also, when the component body is stiff and when it has many relatively stiff wires, such as a dual inline package (DIP), the PCB will be stiffened in the local area of the component. This will tend to reduce the relative displacement between the component and the PCB. In the following analysis only the component body bending and the lead wire stretching will be considered. The change in the PCB curvature is too complex to consider here for hand calculations. Approximations mentioned above will significantly reduce the amount of calculation time-and still give reasonable results that are conservative-and can be used to estimate the fatigue life. A trigonometric expression can be used to describe the PCB displacement at any point. The relative displacement ( 6 ) between the component and the PCB will be the sum of the component body bending deflection and the lead wires stretching deflection. The component considered in the analysis is a ceramic hybrid with kovar lead wires spaced on 0.050-in. centers. It becomes too complex to examine the forces in every wire so only four end wires are assumed to carry the dynamic loads produced by the relative displacements. Chapter 8 has a section that relates to finding the minimum desired PCB natural frequency to achieve a 10 million cycle fatigue life with sinusoidal vibration, based on the component type, the PCB geometry, and the expected environment, This information was based on testing many different types of PCBs with different types of components until they failed. The data were used with FEMs to cross check the results analytically. The following sample problem starts with the minimum desired PCB natural frequency equation to show how the approximate fatigue life in the lead wires and solder joints can be established using approximate hand calculation methods. Sample Problem-Effects

of PCB Displacement on Hybrid Reliability

A ceramic hybrid 1.5 in. long, with 0.025-in. diameter kovar lead wires, is through-hole mounted 0.050 in. above the top surface and at the center of a 0.062-in. thick epoxy fiberglass plug-in PCB, parallel to the 9.0-in. edge, as shown in Fig. 3.4. The PCB is required to operate in a 5.0-G peak sinusoidal vibration environment. Simplify the analysis by assuming only four wires at each end of the component carry the dynamic load. Find the following:

(a> Minimum desired PCB natural frequency for a 10 million cycle fatigue life.

DISPLACEMENTS BETWEEN PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

51

(b) Relative dynamic displacement expected between the PCB and the component. (c) Dynamic forces and stresses in the lead wires and solder joints. (d) Approximate fatigue life expected in the lead wires and solder joints. (e) Bending stress in the ceramic component body.

Solution. (a) The minimum desired PCB natural frequency equation is shown below. -

[

9.8GinChr&

)2’3

0.00022B

(see Eq. 8.13)

where Gin= 5.0 (peak sine vibration) C = 1.26 (component type for hybrid with pins extending from bottom surface) h = 0.062 in. (thickness of PCB) r = 1.0 (relative position factor for component at center of PCB) L = 1.5 in. (length of component) B = 9.0 in. (length of PCB parallel to length of component)

fd =

[

(9.8) (5 .O) ( 1.26) (0.062) ( 1.O) (0.00022)(9.0)

=

177 Hz (3.22)

(b) The relative dynamic displacement difference ( 6 ) between the component and the PCB can be obtained from the dynamic displacement curve, which is trigonometric, as shown below. 7TX

2,= Znsin-

and

U

6= Z ,

(3.23)

- 2,

Then 6 = Z , 1 -sini

)

(3.24)

7aT x

where Gin= 5.0 (peak sine vibration) f , = 177 Hz (to achieve 10 million cycle fatigue life) = 13.3 (approximate PCB transmissibility expected) Q= =

fi

2,= 9.8Gi, Q/(fnI2 = (9.8)(5.0)(13.3)/(177)2

= 0.02080 in.

(3.25)

x = 3.75 in. (to edge of component) u

=

9.0 in. (length of PCB)

1

.rr(3*75) 9.0

= 0.00071

in.

(3.26)

52

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

This represents the deflection between the PCB and the edge of the component. This also represents the bending deflection of the component and the stretching deflection of the four lead wires assumed to carry the dynamic force. (c) There is not enough information at this point to find the forces and stresses in the lead wires and solder joints. Additional information must be developed. There are two unknowns-the deflection of the component and the deflection of the lead wires-so two equations must be generated to solve the problem. The first equation says that the force in the component must be the same as the force in the lead wires. The second equation says that the sum of both deflections must equal 0.00071 in. Start with the deflection of the component body (Z,) as a cantilevered beam with an end load generated by the four lead wires. PC L3, z,= 3% 1,

where E , I, L,

so

Pc

=

3ECICZC L3,

(3.27)

= 40 X

lo6 1b/in2 (ceramic component modulus of elasticity) = bh3/12 = (0.562)(0.170)'/12 = 2.301 X l o p 4 = 0.75 in. (half the full length of the component) (3)(40 x 106)(2.301X 10-4)Zc P- =

'L

( 0.75)

= h 545 .- X

10JZc (3.28)

Find the axial deflection of the four lead wires (Z,) assuming pure tension only. 2,

where d, A, E, L,.

=

PWL, ( 4 wires) AWE,

so

P,

4A,EwZ, =

(3.29)

L,

in. (wire diameter) = 7r(d,)2/4 = ~(0.025)'/4 = 0.000491 in.2 (area of one wire) = 20 x 10 Ib/ine2 (kovar wire modulus of elasticity) = 0.050 2(0.025) 2(0.025) = 0.150 in. (includes two wire diameters into the component and two wire diameters into PCB for equivalent length) = 0.025

+

P,

+

(4)( 0.000491)( 20 x 1 0 6 ) Z , =

0.150

= 2.618

x 105ZW (3.30)

For equilibrium conditions the bending force in the component must equal the axial force in the four wires so Eq. 3.28 must equal Eq. 3.30. 6.545 x 104Z,

= 2.618

x 105Zz, then

Z,

=

4.002,

(3.31)

DISPLACEMENTS B E W E E N PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

53

The relative displacement (6) between the component and the PCB will be equal to the bending displacement of the component and the axial displacement of the four wires. 6=Z,+Z, where S Z,

= 0.00071 = 4.002,

(3.32)

in. (see Eq. 3.26) (see above)

Then 0.00071 = 4.002,

+Z,

= 5.00Z,

so Z,

=

1.420 X

in.

(expected wire displacement)

(3.33)

The resulting dynamic force in one lead wire can be obtained from Eq. 3.29 considering one wire. This will be the same force acting on the solder joint of the lead wire.

P,

A,E,Z, =

L

-

,

(0.000491)(20 X 106)(1.420X 0.15

= 9.296

lb (3.34)

(d) The dynamic tensile stress expected in the lead wire can be obtained as follows:

P,

sW -- - A=,

9.296 0.000491

=

18,933 lb/in.,

(3.35)

The approximate fatigue life expected for the kovar lead wire can be obtained from Eq. 3.7 and the kovar fatigue curve shown in Fig. 3.1, using the information shown below.

where

N,= 1000 stress cycles (reference point at 84,000 lb/in.*) = 84,000 lb/in.2 (reference point) S , = 18,933 lb/in.* (tensile stress shown above) b = 6.4 (slope of fatigue curve with a stress concentration of 2.0 to account for cuts, scrapes, and scratches in the lead wires)

S,

Nl

=

84,000 6.4 (1000) 18,933)

[

=

13.83 X 106 wire cycles to fail (3.36)

54

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

This is close to the 10 million cycle fatigue life typically expected for components in a sinusoidal vibration environment when the PCB natural frequency is obtained using Eq. 8.13. The approximate wire fatigue life for a sinusoidal vibration resonant dwell test condition can be obtained as shown below. Life

13.83 X l o 6 cycles to fail =

(177 cycles/s) (3600 s/h)

= 21.7

hours to fail

(3.37)

The dynamic shear stress expected in the solder joint can be obtained using the following information.

& = - ps -

ps --

rrdh

where P, d h A,

A,

= 9.296

lb (same as Eq. 3.34) = (0.025 + 0.040)/2 = 0.0325 in. (average solder joint diameter) = 0.062 + 2(0.015) = 0.092 in. (solder joint height including fillets) = n-dh = n-(0.0325)(0.092) = 0.00939 in.* (solder joint shear area)

Ps 9.296 S S -- - = A, 0.00939 ~

=

990 lb/in.2

(3.38)

The approximate fatigue life for the solder joint can be obtained using Eq. 3.7 and the vibration portion of the solder fatigue curve shown in Fig. 3.2, with the following data:

where N , = 1000 cycles to fail at 6500 1b/im2 (reference point) S , = 6500 1 b / h 2 (stress reference point) S , = 990 1 b / h 2 (shear stress in solder) b = 4.0 (slope of solder fatigue curve for vibration) N,

=

[ 7;: il

( 1000) -

=

1.86 x l o 6 solder cycle to fail (3.39)

The approximate solder joint time to fail in a sinusoidal vibration condition will be as follows: Life

1.86 x l o 6 cycles to fail =

(177 cycles/s) (3600 s/h)

= 2.92

hours to fail

(3.40)

DISPLACEMENTS BETWEEN PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

55

The solder joint with this geometry and application is more critical than the lead wire. (e) The bending stress in the ceramic hybrid body can be obtained from the standard bending stress equation with the following information:

where P = 9.296 lb (in one wire, but four wires carry the load) (see Eq. 3.34) L = 0.75 in. (half of the component length) M = PL = (4)(9.296)(0.75) = 27.88 lb . in. c = 0.17/2 = 0.085 in. (half the component thickness) in.4 (moment of inertia) (see Eq. 3.28) I = 2.301 X s b =

(27.88)(0.085) 2.301 X

=

10,300 lb/in.’

(3.41)

Ceramics have a compressive strength of about 250,000 1b/ins2 but the tensile strength is only about 25,000-30,000 lb/in.2, so the ceramic component body appears to have a good safety factor.

-

CHAPTER 4

Beam Structures for Electronic Subassemblies

4.1 NATURAL FREQUENCY OF A UNIFORM BEAM The natural frequency of a uniform beam can usually be determined by considering the strain energy and the kinetic energy of the vibrating beam. There are several different ways of expressing these quantities. Some methods are approximate, while others are exact. The exact methods usually require more work, and the approximate methods are often satisfactory for engineering solutions, so the latter will be given more consideration. Boundary conditions are very important in exact methods as well as in approximate methods. For approximate methods, if the geometric boundary conditions are satisfied, the resulting natural frequency equations will usually be quite accurate for the fundamental resonant frequency. The geometric boundary conditions consist only of the slope and the deflection. However, these boundary conditions must be met by the type of deflection curve that is used to describe the vibrating beam. The technique of assuming a deflection curve for a vibrating system, from which the natural frequency equation is determined, is called the Rayleigh method, after Lord Rayleigh [28], who first proposed it. If the deflection curve happens to be the exact curve for the boundary conditions, then the frequency equation will also be exact. A trigonometric function can be used to describe the deflection curve of the simply supported beam shown in Fig. 4.1. Consider the deflection equation

TX

Y = Yosin L This curve satisfies the geometric boundary conditions of slope and deflection as follows: Deflections:

x = 0, x =L , X=L/2,

56

Y=O Y=O Y=Y,

NATURAL FREQUENCY OF A UNIFORM BEAM

57

I yo

FIGURE 4.1. Deflection curve for a simply supported beam.

Slopes:

If then

X=O,

8=-Y0 L

X=L,

8=--Y LO

x = -2 ’

8=0

77

The strain energy can be obtained from the general bending moment equation d2Y E I y = M dX

as

From Eq. 4.1,

dY 7~ “iX _- -Yo cosdX L L d2Y “i2 - - -Yo dX L2

--

($1

2

T4 =

“iX sin L

“iX

F Y i sin’-- L

(4.4)

58

BEAM STRUCTURES FOR ELECTRONIC SUBASSEMBLIES

Substituting Eq. 4.4 into Eq. 4.3 for the strain energy of the beam, we find

U=

(4.5)

The kinetic energy of the vibrating beam will be as follows:

At resonance, the kinetic energy is equal to the strain energy, so Eq. 4.6 is equal to Eq. 4.5:

EIli Y i 4 L3

-

w O2Y;L

4g li 'EIg

=-

="i"i wL4

f

1/ 2

(4.7)

21i w L 4

This results in L e exact frequency equation because L e exact deflection curve was used in Eq. 4.1. A polynomial can also be used to describe the deflection curve for a simply supported beam, shown in Fig. 4.2. Consider the deflection equation

w (lb/in )

[La-* I

E -x

FIGURE 4.2. A simply supported beam with the Y axis at the center.

NATURAL FREQUENCY OF A UNIFORM BEAM

59

This curve satisfies the geometric boundary conditions of slope and deflection as follows: Deflections:

x =0 ,

Y=Yo

X=a, X=-a,

Y=O Y=O

e = -d=Y- -

2Y0X

Slopes:

dX

so

x =0 ,

a*

o=o

From Eq. 4.8,

--

dY

2Y0X

dX

a*

d2Y dX

2Y0 a*

--

(4.9) For the strain energy, substituting Eq. 4.9 into Eq. 4.3,

2 EIY: u= 2 EIY,, ( a ) = ___ ~

a'

a3

(4.10)

The kinetic energy can be obtained by using Eq. 4.8 as follows:

(4.11)

60

BEAM STRUCTURES FOR ELECTRONIC SUBASSEMBLIES

At resonance, the kinetic energy must equal the strain energy, so Eq. 4.11 must equal Eq. 4.10:

2EIY;

--

U3

Since u

-

4wR2Y,'u 15g

=L/2,

(4.12) Comparing these results with the exact frequency equation (Eq. 4.7) shows the polynomial results in a frequency that is about 11.1% too high. In fact, the Rayleigh method always results in a frequency value that is greater than the true value. unless the exact deflection curve is used. Sample Problem-Natural

Frequencies of Beams

Determine the approximate resonant frequency of the beam in Fig. 4.3 when it is made of aluminum and when it is made of steel, and vibration is in the vertical direction. Solution. If the beam is made of aluminum, = 10.5 x 106 1 b / h 2 (modulus of elasticity) b = 0.50 in. (beam width) h = 1.0 in. (height of beam) I = b h 3 / 1 2 = (0..50)(1.0)'/12 = 0.0417 in.4 (moment of inertia) g = 386 in./s2 (gravity) L = 10.0 in. (length) p = 0.10 1 b / h 3 (density of aluminum) W = b h L p = (0.50)(1.0)(10.0)(0.10)= 0.50 lb (weight of beam) w = W / L = 0.50/10.0 = 0.05 lb/in.

E

Section A A

FIGURE 4.3. A uniform beam with simply supported ends.

NATURAL FREQUENCY OF A UNIFORM BEAM

61

Substituting into Eq. 4.12,

fn= fn =

i

10.96 (10.5 X 106)(0.0417)(386)

(0.05) ( 10.0)'

1013 Hz

(4.13)

Now assume the beam is made of steel: 29 x lo6 lb/in.* (modulus of elasticity) p = 0.283 lb/in.3 (density) W = (0.50)(1.0)(10.0>(0.283)= 1.415 lb (weight) w = W / L = 1.415/10.0 = 0.1415 lb/in.

E

=

Substituting into Eq. 4.12,

fn= fn =

i

10.96 (29 x 106)(0.0417)(386) (0.141) ( 10.0)'

1005 HZ

(4.14)

Comparing Eqs. 4.13 and 4.14 shows the natural frequency of the steel beam is almost the same as that of the aluminum beam, even though the steel beam is almost three times heavier. A closer examination of the natural frequency equation shows that it depends on the ratio of the modulus of elasticity to the density. These two factors can be examined for the beam shown in Fig. 4.3, considering several different materials:

E

6.5 x lo6

100 x l o 6 in.

Magnesium:

-P

Aluminum:

-P

E

10.5 X l o 6 = 105 x l o 6 in. 0.10

Steel :

-=

E

29.0

Beryllium :

0.065

=

P

lo6 0.283

E

42.0 x lo6

P

0.068

_ --

X

=

102 x IO6 in.

= 619

x l o 6 in.

(4.15)

The only two items that will change in the natural frequency equation are the modulus of elasticity and the density. An examination of the ratios shown above indicates that the natural frequency of the beam will be about the

62

BEAM STRUCTURES FOR ELECTRONIC SUBASSEMBLIES

same for magnesium, aluminum, and steel. A beryllium beam, however, would have a much higher natural frequency. The Rayleigh method can be used to determine the fundamental resonant frequency for beams with various end conditions. This method can be used with a trigonometric or a polynomial expression as long as the geometric boundary conditions of slope and deflection are satisfied. Some typical deflections for different boundary conditions are shown, along with the resulting natural frequency equations, in Fig. 4.4. The resonant frequencies of beams with different edge conditions are shown in Fig. 4.5. CASE

1

Cantilever Beam- Uniform Load

Trigonometric

Polynomial

Y

Y - L -

-x

(

Y = Y,, l-cos-

CASE

2

2L

-

-X

Y

=

Y,,

(;I2

Fixed-Fixed Beam- Uniform Load

Trigonometric

Y

Polynomial

Y

FIGURE 4.4. Displacement curves and natural frequency equations.

fn =

C

J@$

Where C = Modal constant

BEAM TYPE

Cantilever

t-

$JL

k C = 56

Simply supported ends or hinged-hinged

MODE 3

MODE 2

MODE 1

,644

C = 351 500

r====7 4-7

&

C = 982 ___ ___~

Fixed ends

=

C = 6.28

m C

F ree-free

1.57

=

224

+ I-+C = 1 4 1

I

19.2

=

500

~

C

=

_

252

_

Fixed-hinged

=

C

=

c =192

9.82

132

277

723

868

I , , i

356

C

=

b (' =

C = 394

F & C = 166

245

('=

795

446

1X

Fixed

--

& i Q-x

_-

c

FIGURE 6.9. Axes at the center of a plate with four fixed edges.

z

This polynomial will result in the following natural frequency:

The natural frequency of a rectangular plate with three supported edges and one free edge can be derived using a trigonometric function or a polynomial function. A sketch of the coordinate axes for each plate, along with the deflection expression and the resulting natural frequency equations, for a trigonometric function and for a polynomial function, are shown in Fig. 6.10. A comparison can be made of the deflection equation and the resulting natural frequency equation using a trigonometric function or a polynomial function for plates with combinations of free edges, supported edges, and fixed edges, as shown in Figs. 6.11 and 6.12. Trigonometric functions can also be combined with polynomial functions in the same deflection equation to describe the deflection of a uniform rectangular plate. For example, considering a plate that is simply supported on two opposite edges and fixed on two opposite edges, the coordinate axes and the deflection equation that meets the geometric boundary conditions will be as shown in Fig. 6.13:

z=z, ( l-cos-

2"x)/I c1

-

$1

NATURAL FREQUENCY EQUATIONS USING THE RAYLEIGH METHOD Trigonometric

125

Polynomial

Y

Y

a m a

a m a

X

U

Supported

X

(61

fa)

i?xsin irY z = Z,,sin 2a h

z = z , , x1-d'

y2)

FIGURE 6.10. T h r e e sides supported and o n e side free: ( a ) trigonometric and ( b ) polynomial.

Trigonometric

a a m

Polynomial

n Free

a

a

m

X

i?X

Z = Z , ,sin (I

Free

( 1 -):7

z = z,,

FIGURE 6.11. Two opposite edges free and two opposite edges supported: ( a ) trigonometric and ( b ) polynomial.

126

PRINTED CIRCUIT BOARDS AND FLAT PLATES Trigonometric

Y

I Supported

Supported

I 1 ----

I

X

(

z=z,1 - - :*2)2(

-

p)

FIGURE 6.12. Two opposite edges fixed and two opposite edges supported: ( a > trigonometric and ( b ) polynomial.

Y

Supported

FIGURE 6.13. Combining a trigonometric function and a polynomial function.

1

-x

Supported

There has been an extensive amount of work on the derivation of natural frequency equations for uniform, flat, rectangular plates with various edge conditions. Little [30], Warburton [31], and Laura and Saffel [33] made extensive use of trigonometric and polynomial series to determine the various resonant modes of different plates. If only the fundamental resonant mode of a uniform, flat, rectangular plate is desired, then the simplified types of trigonometric and polynomial expressions, just outlined, can be combined to determine the fundamentalresonant-frequency equations for many different types of plates. Many of these equations for plates with various edge conditions are shown in Figs. 6.14-6.16.

DYNAMIC STRESSES IN THE CIRCUIT BOARD

127

xxxxxxxxxxxxxxxxx

Free edge

Supported edge

Fixed edge

Equation

Plate a

X X

b I

XL

xxxxxxxxxxxxxxxxxxxxxx

j b " !

xxxxxxxxxxxxxxxxxxxxxx

X

xxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxx

ZTbji X

X

xxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxx

\ " I

xxxxxxxxxxxxxxxxxxxxxx

FIGURE 6.14. Natural frequency equations for uniform plates.

6.7

DYNAMIC STRESSES IN THE CIRCUIT BOARD

Dynamic bending stresses in a uniformly loaded PCB can be determined for the resonant condition by considering a sinusoidal load variation for the dynamic load over the board surface when the edges of the board are simply supported. This follows from the relative dynamic deflections, which will be maximum at the center of the board and zero at the edges (Fig. 6.17).

128

PRINTED CIRCUIT BOARDS AND FLAT PLATES xxxxxxxxxxxxxxxxx

Free edge

Supported edge

Fixed edge Plate

Equation l

0

I

j, =

+)3 55 (i-

D p

h

I "

$ " l

xxxxxxxxxxxxxxxxxxxxxx

a

031'

b

FIGURE 6.15. Natural frequency equations for uniform plates.

The load intensity at any point can be expressed in a trigonometric form that satisfies the boundary conditions: q

?Tx

= q,, sin-sin-

a

7TY

b

The differential equation for the deflection of the circuit board then becomes [291

a4z

a4z

a4z qo

7Tx

7TY

- -sin -sin axj+2aX2aY'+iiYc-D a b

(6.43)

DYNAMIC STRESSES IN THE CIRCUITBOARD

Plate

Equation

f n

D 2 [p

= -x

D 2.45

Nib'

2.68 2 45 f n = -2 - -+-(,4 c r 2 b b "h + J x

Lzizi3 xxxxxxxxxxxxxxxxxxxxx

-+-+F)] 0.127 0.707 2.44

( u4

[, (

129

il I"'

xxxxxxxxxxxxxxxxxxxxxxx X

a

I '' \

xxxxxxxxxxxxxxxxxxxxxx

:v

xxxxxxxxxxxxxxxxxxxxxxx

:m

xxxxxxx

x x

x xx

FIGURE 6.16. Natural frequency equations for uniform plates.

FIGURE 6.17. Dynamic deflection mode of a circuit board.

130

PRINTED CIRCUIT BOARDS

A N D FLAT PLATES

The deflection of the circuit board at any point can also be represented by a trigonometric expression similar to the one shown in Eq. 6.1:

%-x

%-Y

Z = A sin ---sina

(6.44)

b

Performing the operations on Eq. 6.44 as required by Eq. 6.43,

a4z ax4 a4z --

-- -

a4z

%-Y b

a

a4

%-x

ITTT1

-

dY4

%-x

IT4

-Asin-sin-

%-Y

,Asin-sinb a

b

%-x

7i4

A sin -sin ax2ay2 a2b2 a =-

TY b

-

(6.45) (6.46) (6.47)

The deflection form factor A can be determined by substituting Eqs. 6.45-6.47 into Eq. 6.43: A=

40 1 2 n4D T + F [a a2b-

1

(6.48)

+g)

Substitute Eq. 6.48 into Eq. 6.44, and the deflection at any point on the circuit board is

%-x

q0 sin -sin Z=

ITY -

b

a

2

i l

The maximum deflection, Z,, will occur at the center of the board where X = a / 2 and Y = b / 2 . Substituting into the above equation, 40

z0= r4D

1

2

1

?+- a2b2' 2 )

(6.49)

[ a

Assuming that the circuit board acts like a single-degree-of-freedom system at its fundamental resonant mode, the maximum dynamic displacement can be approximated by Eq. 2.30 as follows:

DYNAMIC STRESSES IN THE CIRCUIT BOARD

131

Substitute this expression into Eq. 6.49 and solve for the maximum dynamic pressure intensity qo as follows: 2 4n

=

(6.50)

f,'

The plate stiffness factor D can be determined from the natural frequency equation for a uniform, simply supported rectangular plate, using Eq. 6.21: 4f,2 P 2

D=

1

(6.51)

Substitute Eq. 6.51 into Eq. 6.50, and the maximum dynamic load intensity at the center of the plate simplifies to the following expression: (6.52) The maximum dynamic bending moment in the rectangular circuit board will occur in the center. The bending moment must be greater for the section along the shorter plate dimension, b, where it has the value M y because the bending is along the Y axis. The maximum bending moment then becomes [291

My=

Sample Problem-Vibration

(6.53)

Stresses in a PCB

Determine the dynamic bending stresses in a 1.0-lb epoxy fiberglass PCB 7.0 X 8.0 X 0.062 in. thick when it is exposed to a 2-G peak sinusoidal vibration input and a transmissibility Q of 8.0 (see Fig. 6.7). Solution. Substitute into Eq. 6.53 for the dynamic bending moment, where

a = 8.0 in. (board length) b = 7.0 in. (board width) Gout= G,,Q = (2)(8.0) = 16 at center W = 1.0 lb (board weight) p = 0.12 (Poisson's ratio for G-10 epoxy fiberglass) 40 = WG,,,/ab = (1.0)(16)/(8.0)(7.0) = 0.286 Ib/in.2

132

PRINTED CIRCUIT BOARDS AND FLAT PLATES

Substituting into Eq. 6.53 for the bending moment, we have

M y=

= 0.496

in.lb/in.

(6.54)

The dynamic bending stress at the center of the plate can then be determined from the standard plate equation. Since PCBs normally have holes in them for the component lead wires, a stress-concentration factor should be used: (6.55) where M y = 0.496 in: lb/in. (see Eq. 6.54) h = 0.062 in. (circuit board thickness) K , = 3.0 (theoretical stress-concentration factor for a small hole in the circuit board) Substituting into the above equation,

s, =

6(3.0)(0.496) =

(0.062)

2320 lb/in.’

(6.56)

An examination of the fatigue S-N curve for G-10 epoxy fiberglass shows that the circuit board will never fail with a stress so low. 6.8

RIBS ON PRINTED CIRCUIT BOARDS

If the stiffness of a PCB is increased without a significant weight increase, the natural frequency will also increase and the deflection at the center of the board will decrease rapidly. A decrease in the circuit board deflection means a decrease in the electrical lead wire stresses and the circuit board stresses during vibration. One simple method for increasing the stiffness of a circuit board is to add ribs. If the ribs are made of thin steel, copper, or brass, they can be cemented or soldered to the copper cladding on the circuit board. This forms a very stiff section because these materials have a high modulus of elasticity. These ribs can be undercut between supports to allow the printed circuits to run under the ribs without causing electrical shorts. Consider the PCB shown in Fig. 6.7. If two steel ribs are soldered to the board in a symmetrical pattern along the length, the circuit board will appear as shown in Fig. 6.18.

133

RIBS ON PRINTED CIRCUIT BOARDS

Board supported on 4 sides

Y h=O062 epoxy

/

"if

n

a

and cemented to board

b=70

-

t

= 0040 rib (steel)

1

b/2 = : 5 0 = d -L

U, -

b/4 = 175

1

X

'

Connector

FIGURE 6.18. Stiffening ribs soldered to a circuit board.

If a 2-G peak sinusoidal vibration input is used to excite the circuit board, the four edges of the board will act as though they were simply supported. If a substantially higher G force is used, such as 10 G, the connector edge may act more like a free edge. Considering four simply supported edges, the natural frequency for the circuit board with ribs can be determined by making a slight modification in Eq. 6.21 to take account of the different stiffnesses along the X and Y axes. The natural frequency equation then takes the following form [291:

+3 ] l ' *

(6.57)

Here D, is the bending stiffness of the composite board along the X axis, and D , is the bending stiffness along the Y axis. Since the circuit board ribs are parallel to the X axis, they will increase the bending stiffness along the X axis but not along the Y axis. The torsional stiffness of the plate-and-rib combination is represented by DxY, If the ribs are in a symmetrical pattern on one side of the board, then the moment of inertia of one T section can be used to compute the bending stiffness along the X axis. A table (see Tables 6.2 and 6.3) is used to make it easier to compute the moment of inertia of the composite T section shown in Fig. 6.19.

z f

0062J

L-----35

-_I

Epoxy board

@

FIGURE 6.19. Dimensions for a rib section soldered to a circuit board.

134

PRINTED CIRCUIT BOARDS AND FLAT PLATES

TABLE 6.2

Item

Area

E X 10'

Z

A E x 10'

A E Z X 10'

I,, x 10-3 3.5(0.062)'

1

0.2170

2.0

0.031

0.434

0.0134

2

0.0152

29.0

0.252

0.441

0.1110

0.875

0.1244

= 0.070

12 0.040(0.380)3

Total

= 0.183

12

TABLE 6.3

2

EI,, x 103

C

2

0.140 5.310

0.111 0.110

Total

5.450

Item

AEC'

x 10'

~~~~~~~~~

1

0.0123

5.34 5.34

0.0121

10.68

The centroid of the T section is at

z=XCAAEEZ

0.1244 -

X

30'

0.875 X l o 6

= 0.142

in.

The bending stiffness of the T section is

C E I = EI, +AEc'

= 5.450

x 10'

+ 10.68 x l o 3 (6.58)

C E I = 16.13 x l o 3 lb.in.2

The bending stiffness of the circuit board along the X axis, with the ribs, is

D,=T where EI d

EI

(6 3 9 )

16.13 x l o 3 1 b . i n 2 (see Eq. 6.58) = b/2 = 3.5 in. (see Fig. 6.18) =

Substituting into the above equation, D,

16.13 X 10' =

3.5

= 4610

1b.h.

(6.60)

RIBS ON PRINTED CIRCUIT BOARDS

135

The bending stiffness along the Y axis will be approximately the same as the epoxy board:

Eh

D,= where E h

(6.61)

12(1 - p2)

=2X

l o 6 1b/in2 (G-10 epoxy fiberglass) = 0.062 in. (circuit board thickness) p = 0.12 (Poisson’s ratio)

Substituting into the above equation, (2 X 106)(0.062)3

D,=

12[ 1 - (0.12)?]

= 40.2

1b.in.

(6.62)

The torsional stiffness can be determined by considering a unit board width along with one rib (Fig. 6.19). The subscripts e and r refer to the epoxy board and to the rib, respectively:

D,,

= G, J ,

Gr +2d Jr

(6.63)

where G, = 0.90 X l o 6 1b/ine2 (shear modulus, epoxy fiberglass) J, = $h3 (unit torsional stiffness, epoxy fiberglass) in3 J, = 3(0.062>3= 79.3 X GI = 12 X l o 6 lb/in.2 (shear modulus, steel rib) J , = i L t 3 (torsional stiffness, steel rib) JI = $(0.38>(0.040>3= 8.10 x l o p 6 in.4 d = b/2 = 3.5 in. (rib spacing; see Fig. 6.18) Substituting into the above equation,

D,,=

(0.90 X 106)(79.3 X

D,,=

85.3 1b.in.

+ (12

X

106)(8.10 X 2( 3 S ) (6.64)

With the addition of the two steel ribs, the circuit board weight will increase to about 1.06 lb. The mass per unit area will then be mass P = --

area

p = 0.490 X

W

-

1.06 (386)(8.0)(7.0)

lb

in.^

gab

(6.65)

136

PRINTED CIRCUIT BOARDS AND FLAT PLATES

The natural frequency of the circuit board with the two ribs can be determined by substituting Eqs. 6.60, 6.62, 6.64, and 6.65 into Eq. 6.57: 4610

1

f,

=

1/2

4( 85.3)

251 HZ

(6.66)

It would appear from Eq. 6.66 that the natural frequency of the circuit board with two ribs will be about 251 Hz. However, since the center section of the circuit board, between the ribs, is only 0.062 in. thick, as shown in Fig. 6.18, it may be possible for this section to develop a resonance below 251 Hz. This can be checked by considering the center section of the circuit board, between the ribs, as a simply supported rectangular plate. Equation 6.21 can then be used to determine the natural frequency as follows:

-[-I 7T

f

"

=

2

D P

1

1

(2+ 2 )

(seeEq.6.21)

where a = 8.0 in. (board length) b = 3.5 in. (board width) D = 40.2 lb .in. p = 0.463 x lb . s 2 / i d Substituting into the above equation,

'= f,

7T

=

40.1 0.463 X l o p 4

142 HZ

1

1

)1'2[m m] +

(6.67)

Since the natural frequency of the unstiffened center section is only 142 Hz, a third rib may have to be added to raise the circuit board fundamental resonant frequency to 251 Hz. When the fundamental resonant frequency of the circuit board is increased to 251 Hz, the transmissibility of the circuit board will also increase. Considering the geometry and the mounting of the circuit board, if a low-input vibration-acceleration force with a 2-G peak is used, the transmissibility at resonance will probably be slightly greater than the square root of the natural frequency, as explained in Section 6.3. Vibration test data on many circuit boards indicate that the transmissibility is related to the natural

RIBS FASTENED TO CIRCUIT BOARDS WITH SCREWS

137

frequency as follows: Q = 1.2(fn)”’ Q = 1.2(251)”’

=

19.0

The dynamic deflection at the center of the circuit board with the ribs can be determined from Eq. 2.30 by approximating the circuit board as a single-degree-of-freedom system in its fundamental resonant mode:

6,=

9.8GinQ

f,’

where Gin= 2.0 (peak input acceleration) Q = 19.0 (transmissibility at resonance) f, = 251 Hz (natural frequency of board with ribs) Thus 6, =

(9.8) (2.0) (19.0) (251)*

= 0.00591

in.

(6.68)

Electronic component parts mounted on the circuit board will experience lower stresses in their electrical lead wires when the circuit board deflections are reduced. These stresses are directly proportional to the circuit board deflections at the fundamental resonant mode. This means that the deflections can be used in a direct ratio to determine the new stresses developed in the circuit board components when the ribs are added. 6.9 RIBS FASTENED TO CIRCUIT BOARDS WITH SCREWS

Extensive vibration tests on bolted assemblies show that bolted joints will experience a substantial amount of relative motion during major structural resonances. This relative motion may add damping to the system, which reduces the transmissibility at resonance. It also reduces the stiffness of the structure, and this tends to reduce the natural frequency. The ability of two bolted interfaces to remain rigid is a function of the stiffness of the joint, the number of screws, the screw size, the screw spacing, the screw torque, the interface conditions, the vibration force, and the vibration frequency. In order to correlate test data with a theoretical analysis, a bolted efficiency factor is very convenient. This efficiency factor will range from 0%, for a joint that has no physical connection, to loo%, for a joint that is welded. Evaluating the types of bolted joints normally found in electronic

138

PRINTED CIRCUIT BOARDS AND FLAT PLATES

Z 7 k 0040 original thickness

' 2

-~+0010

equivalent thickness

Bolted steel'

__

._._ ~.

I

b

y

r Bolted ~ interface

Y

4

L = 0 38 -T

h=0062

subassemblies such as brackets, covers, ribs, and stiffeners, which must transfer an alternating load, the typical efficiency of a bolted joint used for airborne electronic structures appears to be about 25%. It may be as high as 50% for small sheet-metal panels with large, closely spaced screws, and as low as 10% for large sheet-metal panels held down with quarter-turn quickdisconnect fasteners. The natural frequency of a PCB, with stiffening ribs bolted to the board, can be determined by using a bolted efficiency factor. For example, if the ribs on the circuit board shown in Fig. 6.19 were bolted to the board, instead of soldered, the resulting T section through one rib would be similar to the section shown in Fig. 6.19, except for the efficiency of the bolted interface. The 25% efficiency factor for this bolted joint will be analyzed by reducing the effective thickness of the rib to 25% of its true thickness (Fig. 6.20). The moment of inertia of the composite T section, with the bolted rib, will be determined with the use of Tables 6.4 and 6.5. Note that the equivalent thickness of the rib has been reduced from 0.040 to 0.010 in. to account for the 25% bolted efficiency factor. The centroid of the T section is at '=

CAEZ 0.O411X1O6 CAE 0.544 X 106

=

0.0756 in

TABLE 6.4 ~~

Item

Area

E X lo6

Z

1

0.2170

2.0

0.031

0.434

0.0134

2

0.0038

29.0

0.252

0.110

0.0277

0.544

0.0411

Total

A E X l o 6 A E Z X lo6

I" x lo-' 3.5(0.062)3 12

(0.010)(0.380)3 12

= 0.070

= 0.046

RIBS FASTENED TO CIRCUIT BOARDS WITH SCREWS

139

TABLE 6.5

EI, x 103

Item

C

C2

A E C x~ io3

0.0446 0.1764

0.00199 0.0311

0.864 3.420

~~

1 2

0.140 1.335

Total

1.475

4.284

The bending stiffness of the T section is

LEI

= El,

XEI

= 5.76

+ A E c 2 = 1.475 X l o 3

+ 4.28 X l o 3

x lo3 lb.in.2

(6.69)

The bending stiffness of the circuit board along the X axis, with the bolted rib, is

where EI d

= 5.76 X

lo3 1 b . h 2 (see Eq. 6.69) in. (see Fig. 6.18)

= b/2 = 3.5

Substituting into the above equation,

D,

5.76 x 103 =

3.5

=

1645 1b.in.

(6.70)

The bending stiffness along the Y axis will be approximately the same as that of the epoxy board, which is shown by Eq. 6.62 to be DY=40.2lb.in.

(6.71)

The torsional stiffness of the bolted T section can be determined by considering a unit board width, along with one rib, as shown in Fig. 6.20. The subscripts e and r refer to the epoxy board and to the rib, respectively: Gr Jr D,yy=G,J,+ - (see Eq. 6.63) 2d where G, = 0.90 X l o 6 lb/in.’ (shear modulus, epoxy fiberglass) J, = +A3 = +(0.06213= 79.3 X in3 G, = 12 X lo6 1b/in2 (shear modulus, steel rib) Jr = iLt3 3 = f(0.38)(0.010)3 = 0.126 X in.4 d = b/2 = 3.5 in. (rib spacing; Fig. 6.18)

140

PRINTED CIRCUIT BOARDS AND FLAT PLATES

Substituting into the above equation,

D,,

=

+

(0.90 X 106)(79.3 X

(12 X 106)(0.126 X

2( 3 S)

DX,=71.6lb.in.

(6.72)

If the circuit board weight does not change, the mass per unit area will be the same as that shown in Eq. 6.65. The natural frequency of the circuit board with the bolted ribs can then be determined by substituting Eqs. 6.70-6.72 and 6.65 into Eq. 6.57:

f=Z[ ”

f,

2 0.490 =

[-+1645

1 X

lo-‘

(8.0)4

“il

1/2

4( 71.6) (8.0)2(7.0)2

160 HZ

+

(7.0)‘

(6.73)

Since the center section of the circuit board is not stiffened between the ribs, as shown in Fig. 6.18, it may be possible for this section to develop a resonance below 160 Hz. The natural frequency of this section was previously checked in the approximation of a simply supported rectangular plate. These results are shown in Eq. 6.67, which shows a natural frequency of 142 Hz. If a natural frequency of 160 Hz is required, then a third rib may have to be bolted to the circuit board. The deflection at the center of the circuit board, with the bolted ribs, can again be determined from Eq. 2.30 by approximating the board as a singledegree-of-freedom system at its resonance. The transmissibility developed with the bolted ribs should be slightly less than with the soldered ribs because of the additional damping at the rib interfaces and the bolt interfaces. Vibration test data on several groups of circuit boards using a 2-G peak sinusoidal vibration input indicate the transmissibility was approximately related to the natural frequency as follows:

Q

=

( 160)1’2 = 12.7

Substituting into Eq. 2.30. the single-amplitude displacement with the bolted ribs becomes 6, = 6,

9.8G,,Q - 9.8(2.0)(12.7) f ;1 (160)‘

= 0.00972

in.

(6.74)

Circuit board displacements are important because they can be used to estimate the fatigue life, or the change in the fatigue life, of the electronic components mounted on the circuit board.

PROPER USE OF RIBS TO STIFFEN PLATES AND CIRCUIT BOARDS

141

Y

t

I

Circuit

board

h-rrmrmrrrml FIGURE 6.21. A circuit board with ribs in two directions.

6.10 PRINTED CIRCUIT BOARDS WITH RIBS IN TWO DIRECTIONS

If two sets of ribs are fastened to one side of a circuit board in a symmetrical pattern, a T section will have to be analyzed to determine the bending stiffness along the Y axis as well as the X axis (Fig. 6.21). Again, be sure to check each plate section between the ribs to make sure each section has a natural frequency greater than the full-plate natural frequency including the ribs.

6.11 PROPER USE OF RIBS TO STIFFEN PLATES AND CIRCUIT BOARDS

Ribs will not increase the stiffness of a plate or a circuit board if they are not properly used. In order for a rib to be effective, it must carry a load. Since all loads must eventually be transferred to the supports, the rib is most effective when it carries a load directly to the supports. For example, consider a flat square plate simply supported on two opposite edges and free on two opposite edges. If ribs are added to the plate so that they join the two free edges, the ribs will not carry the load directly to the supports and these ribs will not be very effective in stiffening the plate. If the ribs are added to the plate so that they join the two supported edges (Fig. 6.22), then these ribs will carry the load directly into the supports and they will be very effective in stiffening the plate. If ribs cannot be carried directly into the supports, then a secondary member should be provided to carry the load to the supports to make the

142

PRINTED CIRCUIT BOARDS AND FLAT PLATES Poor

Good Free

r-----l

Free

Free

FIGURE 6.22. Adding ribs to stiffen a flat plate effectively.

Poor

Good

Free

Free

FIGURE 6.23. Adding an additional rib to stiffen a flat plate.

ribs more effective. For example, consider a flat square plate simply supported on three sides and free on a fourth side. If ribs must be added to the plate so they end at the free edge, then an additional rib should be placed across the free edge to act as a secondary member, which will carry the load to the supports (Fig. 6.23).

6.12 QUICK WAY TO ESTIMATE THE REQUIRED RIB SPACING FOR CIRCUIT BOARDS

A quick method for estimating the required spacing for stiffening ribs on flat plates was shown by Bruhn [13]. H e uses stiffening rib spacing criteria based on the thickness of the plate. The stiffness (or modulus of elasticity) of the rib material should be at least equal to or greater than the stiffness of the plate material. The height and thickness of the ribs and the attachment to the plate must add enough local stiffness to the assembly to substantially reduce the displacements when the plate is subjected to a transverse load. The effective plate width B that the rib can reinforce is shown in Fig. 6.24 as 30t, where t is the plate thickness. The required rib height for a metal rib will vary from about 0.25 in. for a small PCB about 4 in. square, to a metal rib height of about 0.50 in. for a large PCB about 10 in. square.

143

QUICK WAY TO ESTIMATE THE REQUIRED RIB SPACING FOR CIRCUIT BOARDS

Rib,

\n

,Bonded

I

I

if

I

I-

B----l

B = 30t

t

FIGURE 6.24. Effective width of a flat uniform plate with a stiffening rib.

An examination of the sample problem rib pattern in Figs. 6.18 and 6.19 shows the rib is centered on a PCB section that is 3.5 in. wide. When the Bruhn criteria are used for the spacing, the allowable PCB plate width should be limited to the value shown below:

B

= 30t =

(30)(0.062) = 1.86 in.

(6.75)

The obvious conclusion from this quick approximation is that the rib spacing in Figs. 6.18 and 6.19 is inadequate. More stiffening ribs will be required to reach the 251-Hz natural frequency shown in Eq. 6.66. The same conclusion was reached after a great deal of work in the paragraph following Eq. 6.67, and in the paragraph following Eq. 6.73. If a third rib is added to the PCB as shown in Fig. 6.25, the effective plate width associated with each rib will be 2.33 in. This is still greater than the maximum allowable plate width of 1.86 in. shown in Eq. 6.75. This means that even with three ribs on the PCB, the natural frequency will still be

T 7.0 I

1 -

FIGURE 6.25. Suggested stiffening rib placement to achieve effective circuit board frequency.

144

PRINTED CIRCUIT BOARDS AND FLAT PLATES

slightly under the desired 251-Hz level. If the thickness of the PCB is increased to about 0.0776 in., the effective allowable plate width will be as shown below:

B

= 30t =

( 3 0 ) ( 0 . 0 7 7 6 ) = 2.33 in.

(6.76)

The desired plate width of 2.33 in. now matches the plate width shown in Fig. 6.25 for three ribs.

6.13 NATURAL FREQUENCIES FOR DIFFERENT PCB SHAPES WITH DIFFERENT SUPPORTS

Any time the natural frequency of a PCB is excited, problems can develop. Electrical lead wires, solder joints, and connector pins can crack, closely spaced PCBs can impact against each other and cause short circuits, and screws can become loose. Most of the PCBs made today are rectangular in shape. Some of these PCBs are plug-in types and other are bolted to some type of support. Natural frequencies for uniform rectangular plates and PCBs with various supports are readily found in technical literature. However, there has been a large increase in the use of electronics in unusual new applications, where volume and size are limited, so new shapes with new support methods are often required. There is very little data available for finding natural frequencies for odd-shaped plates and PCBs with different edge and point supports. Computers are available with various FEM software programs that can calculate the natural frequencies and mode shapes for plates with different shapes and supports. When access to a computer is available, when the operator is familiar with the FEM software, and when there is time available, odd-shaped plates and PCBs are easy to evaluate. When time is short, or when there are no computers available, the analysis methods shown here can be very convenient for obtaining the natural frequencies of some odd-shaped PCBs. The analysis methods shown here can be used for a sanity check of FEM analyses of odd-shaped plates. When the computer analyst is tired, especially right after lunch or late in the afternoon, it is very easy to make data entry errors. One bad data entry error on a computer can ruin weeks and months of hard work. In addition, it is often very difficult to locate and correct the bad data entry point. Natural frequencies for a variety of PCB shapes with different supports were calculated using a computer. The results for the first three natural frequencies are shown in Figs. 6.26-6.29 for different shapes. The first number is for the fundamental or the lowest natural frequency. The next two numbers are for the higher harmonics. Most of the vibration and shock damage will usually occur at the lowest natural frequency where the dynamic displacements are the greatest. The higher harmonics usually have smaller

NATURAL FREQUENCIES FOR DIFFERENT PCB SHAPES

145

Rectangles

[XI

Case 1

(1) 450.60 (2) 647.31 (3) 887.04

Hz

Case 5

Case 2

Case 3

243.10 Hz 452.94 731.52

72.21 Hz 123.16 208.36

1

Case 4

1

111.27 Hz 136.32 252.29

I

1

Case 6

iU (1) 72.12 (2) 200.80 (3) 223.21

71.78 174.72 210.68

Case 9

Case 10

174.72 210.68 243.48

217.88 270.59 278.93 I

a3 (1) 270.59 (2) 278.93 (3) 281.00

42.8 1 164.27 275.81

Case 13

Case 14

, , 107.72 225.38 343.32

Caai5

(1) 217.30 (2) 255.83 (3) 423.79

142.20 212.87 305.91

160.49 166.69 176.11

164.74 217.30 342.75

I

[ a i 7

160.49 176.11 178.21

FIGURE 6.26. Natural frequencies for flat uniform rectangular plates with different types of supports.

displacements so they produce less damage. These frequencies were tabulated for easy reference. The frequencies are for 0.10-in. thick epoxy fiberglass PCBs with a 24-in.’ area and a density of 0.208 Ib/ine3. This resulted in slightly different dimensions for the different shapes. The density was based on the total weight of a fully assembled PCB. This included all of the electronic component parts as well as the weight of the bare epoxy circuit board. The density was based on the total weight divided by the volume of

146

PRINTED CIRCUIT

BOARDS AND FLAT PLATES Hexagons

Case 1

( 1 ) 379.62 Hz ( 2 ) 718.78 (3) 8 0 6 . 3 1

Case 5

00 G Case 2

Case 3

Case 4

212.42 Hz 513.24 534.05

72.52 H z 89.17 235.97

117.56 Hz 134.47 272.42

Case 6

Case 7

Case 8

00 (1) 61.36 (2) 166.94 ( 3 ) 184.45

97.55 142.27 233.46

142.27 169.85 36 1.43

147.86 184.45 216.28

Case 9

Case 10

Case 11

Case 12

170.27 210.54 350.92

135.45 340.55 36 1.43

@ ( 1 ) 184.45 (2) 216.45 (3) 260.60

0 55.11

208.86 229.82

Case 1 3

( 1 ) 340.55 (2)361.43 ( 3 )3 9 3 . 3 1

189.24 265.18 265.98

@ 184.45 194.95 214.54

184.45 214.54 233.23

FIGURE 6.27. Natural frequencies for flat uniform hexagonal plates with different types of supports.

the epoxy board by itself. The PCB shapes are rectangles, hexagons, triangles, and circles with 16 different support arrangements for each shape shown in Figs. 6.26-6.29. The natural frequencies of a PCB that matches the dimensions, material, and support method of one of the tabulated PCBs can be obtained directly from the tabulated natural frequencies shown for each geometry. The approximate natural frequencies for PCBs with similar shapes and similar supports, but with different sizes and different materials, can be obtained with some simple conversion calculations using the equation shown below.

NATURAL FREQUENCIES FOR DIFFERENT PCB SHAPES

147

Triangles

I

Case 3

I

Case 1

Case 2

Case 4

(1) 443.23 Hz (2) 858.95 (3) 971.22

270.02 Hz 657.43 759.54

44.01 Hz 87.93 99.66

75.25 Hz 139.12 150.74

Case 5

Case 6

Case 7

Case 8

(1) 88.10 (2) 89.97 (3) 132.27

88.70 118.79 241.04

88.74 120.78 259.15

88.13 89.97 134.44

10

Case 11

Case 12

( 1 ) 39.31 ( 2 ) 98.49 (3) 126.02

77.21 290.66 335.03

263.63 306.07 456.64

226.94 237.77 261.01

Case 13

Case 14

Case 1 5

Case 16

(1) 95.48 (2)127.65 13) 227.91

227.54 245.94 326.47

AAAA Case 9

Case

A AIA1A I

I

115.57 122.67 199.19

122.50 141.72 210.29

--1 1

1

~~

Free edge

Supported edge

Fixed edge

Point support ~

FIGURE 6.28. Natural frequencies for flat uniform triangular plates with different types of supports.

148

PRINTED CIRCUIT BOARDS AND FLAT PLATES

Circles Case 1

Case 2

@ (1) 384.96Hz (2)775.09 (3)775.40 Case

5

(1) 59.55

(2)183.34 ( 3 ) 196.05 Case

9

4

Case

@ 177.02Hz 505.87 506.06 Case

6

119.73 166.89 249.89 Case

10

81.15Hz 94.16 258.04 Case

124.97Hz 148.03 306.04

7

166.89 182.12 441.24 Case

8

Case

156.94 196.05 255.55

11

Case

12

~

(1) 196.05 (2) 255.56 (3)255.91 Case

67.04 219.59 234.61

13

172.81 441.23 442.49

162.43 234.61 342.68 Case

15

Case

@olf '

(1) 441.23 (2)442.49 (3)458.35

Free edge

210.37 290.44 335.79

16

!$-

196.05 216.08 252.96

196.05 252.96 253.17

Fixed edge

Point s u p p o r t

............//////////// -I

Supported edge

FIGURE 6.29. Natural frequencies for flat uniform circular plates with different types of supports.

NATURAL FREQUENCIES FOR DIFFERENT PCB SHAPES

149

The subscript 1 refers to the tabulated values and geometry shown for the four different PCB shapes. The subscript 2 refers to the properties of a new specimen plate or PCB.

(6.77)

where f = natural frequency (Hz) a = width (in.) b = height (in.) E = modulus of elasticity (lb/in.,> h = thickness (in.) y = density (lb/in.3) p = Poisson’s ratio (dimensionless) Sample Problem-Natural Three Point Supports

Frequency of a Triangular PCB with

Find the natural frequency of a polyimide glass triangular PCB that is 8.5 in. wide, 7.5 in. high, and 0.090 in. thick, with a total weight of 0.70 Ib. It is fastened to a support with three small screws located at the center of the three sides, near the edges, similar to Fig. 6.28 Case 5. The following information is required to obtain a solution: a, = 7.44 in. wide

a, = 8.5 in.

b , = 6.45 in. high E , = 2.0 X l o 6 lb/in.’ modulus h , = 0.10 in. thick p1 = 0.12 dimensionless

in. E , = 3.0 X l o 6 lb/in.’ h , = 0.090 in. p2= 0.12 dimensionless

y , = 0.208 1b/ins3

f,= 88.10 Hz

b,

= 7.5

0.70 lb

yz

(8.5)(7.5)(0.090)(0.50) in.

0.244 1 b / h 3

f, = ? (3.0 X 106)(0.090)2(0.208) (2.0 X 106)(0.10)2(0.244)

= 67.5

Hz (6.78)

I CHAPTER 7

Octave Rule, Snubbing, and Damping to Increase the PCB Fatigue Life

7.1 DYNAMIC COUPLING BETWEEN THE PCBs AND THEIR SUPPORT STRUCTURES

Vibration and shock failures in PCBs can often be avoided by understanding how energy is transferred directly to the PCBs from their support structures. The support structures are used to provide easy access to the PCBs for troubleshooting, repairs, or replacement. Since the PCBs are attached to the support structure, the dynamic response of the support structure becomes the input to the PCBs. In any type of dynamic vibration or shock environment, the natural frequencies of the support structure and the PCBs can become excited. If the natural frequencies of the PCBs are close to the natural frequency of the support structure, their transmissibility Q values can couple, which means that their Q values will multiply. This will force the PCB response acceleration G levels to be magnified substantially, resulting in rapid PCB failures. For example, when the chassis has an input acceleration level of 10 G and a transmissibility Q of 10, the chassis will experience 10 X 10 or 100 G. If the PCB also has a transmissibility Q of 10 at the same frequency, the PCB will respond at a level of 100 X 10 or 1000 G. Acceleration levels this high can result in very rapid fatigue failures. The natural frequencies of the chassis and the various PCBs must be well separated to prevent severe coupling. An octave apart represents a good separation. Octave means to double. The natural frequencies of the chassis and the PCBs should be separated by a ratio of more than 2 : 1. The properties, locations, and arrangements of the various PCBs and the properties of the support structure will affect the responses and the fatigue life of the PCBs. There are also feedback effects, where the dynamic feedback from the PCBs to the support structure can produce acceleration G levels high enough to damage the support structure, in poorly designed electronic systems. Electronic systems with various PCBs enclosed within some type of chassis, form a complex dynamic structure. This usually requires the use of a 150

DYNAMIC COUPLING BETWEEN PCBs AND SUPPORT STRUCTURES

-,

151

PCB

Chassis mass 1

FIGURE 7.1. Chassis and PCB modeled as a two-degree-of-freedom lumped mass dynamic

system.

high-speed computer with the proper software to perform a dynamic frequency response analysis and to obtain the coupling properties between the chassis and the PCBs. This can be a very expensive and time-consuming analysis. This study has already been done using a computer to perform a parametric analysis of many two-degree-of-freedom systems with springs, masses, and dampers. The chassis is represented as mass 1, since it receives the energy first, so it is the first degree of freedom. The PCB is represented as mass 2, since it receives its energy from the chassis, so it is the second degree of freedom. This is shown in Fig. 7.1. The spring rates were changed to change the natural frequencies, the masses were changed to change the weight ratios between the chassis and the PCBs, and the dampers were changed to change transmissibility Q values of the chassis and the PCBs. The energy transferred from the chassis to the PCB for different dynamic combinations was used to find the acceleration G levels acting on the PCB. The information was arranged in the form of curves with several different PCB and chassis weight ratios, shown in Fig. 7.2-7.5, to make it easier to find the dynamic acceleration G response of different types of PCBs mounted in different types of chassis. Vibration test data show that most of the damage in electronic assemblies occurs at the fundamental natural frequency of the PCBs. These data also show that the response of the chassis is important since this represents the input to the PCB. The manner in which the chassis influences the PCB response determines the coupled acceleration G level acting on the PCB. The coupled response of the PCB is shown as Q,. The uncoupled response of the chassis acting by itself is shown as ql. The uncoupled response of the PCB acting by itself is shown as q2. The ratio R is used to show the Q2/q1 values that were obtained from the computer parametric studies. The fundamental natural frequency for the chassis is shown as f l and the fundamental natural frequency of the PCB is shown as f i . An examination of Fig. 7.2 shows that when the natural frequency of the PCB is about the same as the natural frequency of the chassis, the frequency ratio f2/fi is about 1.0. The

152

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE

-

Dangerous area

-

T

E=4.0

/k

\\

\

7 %=0.05

l.-==u.3u

i

1

0 0

0.5

2

1

3

Ratio = f2/fl

FIGURE 7.2. Dynamic coupling effects between the chassis and the P C B for various ratios of transmissibility when the weight of the chassis is 20 times the P C B weight.

0

1

2

3

Ratio = f2/fi

FIGURE 7.3. Dynamic coupling effects between the chassis and the P C B for various ratios of transmissibility when the weight of the chassis is four times the P C B weight.

DYNAMIC COUPLING BETWEEN PCBs AND SUPPORT STRUCTURES

153

FIGURE 7.4. Dynamic coupling effects between the chassis and the PCB for various ratios of transmissibility when the weight of the chassis is two times the PCB weight.

FIGURE 7.5. Dynamic coupling effects between the chassis and the PCB for various ratios of transmissibility when the weight of the PCB is two times the chassis weight.

154

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE

weight ratio of the PCB to the chassis is shown as W,/W,. When the weight of one of the PCBs is less than one-tenth the weight of the chassis, the weight ratio will be less than 0.10. Under these conditions the dynamic coupling between the PCB and the chassis, shown as R = Q 2 / q 1 ,can reach a value that is close to about 5.0. This means that the acceleration G levels acting on the PCB will be about 5 times greater than the expected acceleration G levels acting on the chassis. These conditions can produce high forces and stresses on the PCB, which can result in very rapid PCB fatigue failures. This condition can be avoided by following the octave rule. 7.2

EFFECTS OF LOOSE EDGE GUIDES ON PLUG-IN TYPE PCBS

The computer parametric studies for the octave rule were based on plug-in types PCBs that have three or four restrained edges that are simply supported (hinged) or clamped (fixed). Vibration test data show that a typical plug-in type of electrical connector on one edge of the PCB will act like a support. If there are any two unsupported PCB edges that are free to translate instead of being restrained in some way, the increased displacement amplitude will increase the period of vibration. This condition can occur, for example, when loose PCB edge guides are used. This tends to decrease the effective natural frequency of the PCB. This condition often brings the PCB natural frequency closer to the chassis natural frequency, which increases the dynamic coupling between the chassis and the PCB. This is a dangerous condition that can sharply increase the acceleration G levels produced in the PCB, resulting in more rapid PCB fatigue failures. The purpose of the octave rule is to reduce the dynamic coupling effects between the chassis and the PCBs, to improve the fatigue life of the PCBs in severe vibration and shock environments. Loose PCB edge guides will often result in increased coupling between the PCB and the chassis, which is not desirable. Loose PCB edge guides should be avoided to improve the reliability of the PCBs in dynamic conditions. DESCRIPTION OF DYNAMIC COMPUTER STUDY FOR THE OCTAVE RULE 7.3

Dynamic response data generated by the computer for the chassis and the PCB were translated into a graphic form with curves, to cover a wide range of parameters in order to simplify analysis methods that are normally very complex. The purpose of this technique was to permit the use of hand calculations for finding quick. but approximate, dynamic acceleration G levels acting on the PCB. These analysis methods have also been used as a sanity check for finite element modeling (FEM) calculations performed on the computer. When there are large differences in the results of the computer analysis and the hand calculations, something is usually wrong, so

THE REVERSE OCTAVE RULE MUST HAVE LIGHTWEIGHT PCBs

155

further investigations should be made. It is very easy to make data entry errors on a large computer program. Data input errors are very easy to make, but very hard to find in a large program. One bad data entry can ruin months of hard work. This is called GIGO, for garbage in garbage out, by computer users. 7.4

THE FORWARD OCTAVE RULE ALWAYS WORKS

Natural frequencies for the chassis and the PCBs should be planned in advance to take advantage of the octave rule. Once the design has been completed and orders have been placed, it is usually too late to try to utilize the octave rule. In the analysis phase the natural frequency for the chassis, with the weights of the PCBs included, must be obtained. Then the natural frequency of each PCB, assumed to be acting by itself outside the chassis, must be obtained. The resulting natural frequencies of the chassis and the various PCBs should be separated by more than one octave to avoid severe dynamic coupling. For example, when the chassis considered as mass 1 has a natural frequency of 100 Hz, then each PCB considered as mass 2 should have a natural frequency greater than 2 X 100 or greater than 200 Hz. This is called the forward octave rule. The forward term is used to describe the direction of the load path. The chassis, mass 1, gets the dynamic load first and transfers the load in the forward direction to the PCBs, mass 2. An examination of Figs. 7.2-7.5 shows that the dynamic coupling ratio R between the PCBs and the chassis is always sharply reduced when the forward octave rule is used, where the frequency ratio f2/fi is greater than 2.0. This means that the acceleration G levels transferred from the chassis to the PCBs are also sharply reduced. This reduction increases the fatigue life of the PCBs in severe dynamic environments.

7.5 THE REVERSE OCTAVE RULE MUST HAVE LIGHTWEIGHT PCBs

The reverse octave rule is used to describe the load path in the opposite direction from the PCB (mass 2) back to the chassis (mass 1). For the reverse octave rule to work the chassis should have a natural frequency that is more than two times greater than the natural frequency of the PCB. When the chassis has a natural frequency of 200 Hz, then the PCB should have a natural frequency less than 100 Hz. An examination of Fig. 7.2 reveals that a frequency ratio f2/fl of 0.5 or less represents the reverse octave rule. This shows the ratio R is approximately 1.0. This means that the acceleration level acting on the PCB will be about the same as the acceleration level acting on the chassis. This is good for vibration and shock, because the acceleration G levels transferred to the PCB from the chassis are low. Remember, these conditions are for a very small PCB weight, where the weight ratio W,/W, is

156

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE

typically less than about 0.10. This covers the vast majority of the electronic equipment produced. Care must be used when the PCB weight ratio is increased to a value of 0.25, as shown in Fig. 7.3. Then the reverse octave rule, with a frequency ratio of f 2 / f l of 0.5, shows an R value of about 4.5. This means that the acceleration level acting on the PCB will be 4.5 times greater than the acceleration level acting on the chassis. The reverse octave rule in this case is a very dangerous condition because it can lead to high PCB acceleration G levels with a reduced PCB fatigue life. Some companies in the automobile industry tend to use a single PCB clamped to a sheet metal tray, where the PCB weight may be half the weight of the tray, This results in a weight ratio W,/W, of 0.50. An examination of Fig. 7.4 shows that if the reverse octave rule is used with the frequency ratio of 0.50, the coupling ratio R is about 4.0. The PCB acceleration G level under these conditions can be about 4 times greater than the G levels developed in the supporting tray. This condition may result in a reduced PCB fatigue life. An examination of Fig. 7.4 for the forward octave rule, with a frequency ratio of 2 or more, shows a coupling R value of about 1.2. This means that acceleration G levels acting on the PCB will be about 1.2 times greater than the acceleration G levels acting on the chassis. This shows that the forward octave rule will always work well, even for conditions that do not work well for the reverse octave rule. Sample Problem-Vibration

Problems with Relays Mounted on PCBs

Sinusoidal vibration tests were run on a small electronic chassis that contained several relays, with an acceleration rating of 75 G peak, mounted at the center of a plug-in PCB. During a 5 G peak qualification test, using a sinusoidal vibration sweep from 20 to 2000 Hz, the relays chattered, resulting in a test failure. An examination of the PCB by itself showed it had a weight of 2.5 lb, a natural frequency of 81 Hz, and a transmissibility Q of 9.0. An examination of the chassis by itself showed it had a weight of 10.0 lb, a natural frequency of 120 Hz, and a transmissibility Q of 3.0. The questions here are why did the relays chatter, and what can be done to the system so it will pass the test? Solution The Curves in Fig. 7.3 Can Be Used to Solve This Problem. Several different relations must be obtained to solve this problem as shown below.

fi/fl W,/W,

=

=

81/120 = 0.675 (uncoupled natural frequencies of PCB to chassis)

(7.1)

2.5/10

(7.2)

=

0.25 (weight ratio of PCB to chassis)

q 2 / q 1= 9/3 = 3.0 (uncoupled transmissibility ratio of PCB to chassis)

(7.3)

PROPOSED CORRECTIVE ACTION FOR RELAYS

G,

= Ginql

G 2 = G,Q, R

= Q2/q1

157

(uncoupled acceleration response of chassis)

(7.4)

(coupled acceleration response of PCB)

(7.5)

(ratio of the coupled transmissibility of PCB Q2 to uncopuled transmissibility of chassis q11

(7.6)

(coupled transmissiblility of the PCB) (rewriting Eq. 7.6) (7.7)

Q2 = Rq,

Substitute Eqs. 7.4 and 7.7 into Eq. 7.5 to obtain the coupled response of the PCB: G,

=

(Gi,q,)( Rq,)

= RGin(q , ) ,

(coupled response of the PCB) (7.8)

An examination of Fig. 7.3 shows there is no curve with a ratio q2/ql = 3, so a curve with this ratio must be generated by following the profiles between uncoupled transmissibility ratios 2 and 4. When this new curve is added with a dashed line, using the frequency ratio of 0.675 from Eq. 7.1, it results in the coupled transmissibility ratio R from the chassis to the PCB as shown below:

Substitute Eq. 7.9 into Eq. 7.8 along with an input Gin level of 5.0 and a chassis uncoupled transmissibility q , of 3.0 to obtain the coupled acceleration G, response of the PCB as shown below:

G,

=

(3.6)(5.0)(3.0),

=

162-G peak response of PCB

(7.10)

The 162-G peak response of the PCB far exceeds the 75-G peak rating for the relays, so the relays are not compatible with the PCB design and the environment. Some type of corrective action must be taken to allow the system to pass the qualification test. 7.6 PROPOSED CORRECTIVE ACTION FOR RELAYS

1. Mount the relays away from the center of the PCB, where acceleration levels are the highest. Mount the relays at the edges of the PCB where the acceleration levels are the lowest. This will require new PCB layouts with new PCB fabrication and assembly costs and delayed delivery schedules. 2. Make a search of the various relay manufacturers to see if they may have another relay off the shelf, with the same form, fit, and function, that can meet the requirements. 3. Examine the PCB layout to see if it is possible to add stiffening ribs to increase the natural frequency to a value that is two times the chassis natural

158

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE

frequency. This would result in a new PCB natural frequency of 2 X 120 = 240 Hz, which would then follow the forward octave rule. Increasing the PCB natural frequency will also increase the PCB Q value. 4. The reverse octave rule with a PCB natural frequency of 120/2 = 60 Hz will not work here because the coupling ratio R for the PCB would be about 3.3. An examination of Eq. 7.10 shows the PCB acceleration response would be about 148 G peak, which is still well above the 75-G peak relay rating. 5. Adding snubbers to the PCB will reduce the PCB dynamic displacement and stress, but that is not the problem here. The problem is high acceleration levels. Snubbers can cause impacting shocks when the snubbers strike each other, which can produce high shock acceleration pulses. This can make the relays chatter and fail. 6. Damping is often recommended. Increased damping will decrease the transmissibility Q value during vibration. The best damping is constrained layer damping. This consists of alternate layers of thin high-energy dissipating viscoelastic strips separated by thin strips of aluminum foil. Damping works well on structures with low natural frequencies, below about 50 Hz. The dynamic displacements are then large enough to produce large relative shear displacements that convert kinetic energy into heat. When the natural frequencies are above about 100 Hz, the relative displacements are not large enough to provide significant damping. The amount of room required by the constrained layer dampers can usually be better utilized by employing stiffening ribs to increase the PCB natural frequency. The best option is corrective action item 3 above, adding stiffening ribs to the PCB to increase the natural frequency to 240 Hz. This may be difficult to do on a PCB that is fully populated so it may require a new design. This change will follow the forward octave rule. The new information required is shown below. =

f2/f,

g,

ql/q,

R

=

=

=

240/120

=

2.0 (frequency ratio of PCB/chassis)

(7.11)

= 15.5 (approximate uncoupled transmissibility of PCB)

(7.12)

15.5/3 = 5.16 (uncoupled transmissibility ratio of PCB/chassis)

(7.13)

1.4 (value of Q2/g1 estimated from Fig. 7.3)

(7.14)

Substitute Eq. 7.14 into Eq. 7.8 to find the coupled acceleration response of the PCB: G,

=

( 1.4) (5 .O) ( 3 ) 2 = 63 .O-G peak PCB response

(7.15)

USING SNUBBERS TO REDUCE PCB DISPLACEMENTS AND STRESSES

159

The peak response acceleration level of 63.0 G for the modified PCB is now well below the relay rating of 75 G peak, so the new PCB design should be acceptable. 7.7 USING SNUBBERS TO REDUCE PCB DISPLACEMENTS AND STRESSES

Snubbers can often be used to make PCBs work smarter in severe vibration and shock environments. Sometimes electronic equipment designers pay no attention to the octave rule so their PCBs have a high failure rate when they are exposed to vibration. Once the design is in full production and equipment is failing in the field, it can be very difficult to find a cost-effective fix. This is where snubbers can often save the day [431. High procurement costs for military electronic equipment have forced governments to go to the best available commercial electronic equipment manufacturers for many of the new military programs. Sometimes the commercial equipment lacks the ruggedness necessary for military programs, so the ruggedness must be increased, but at a reasonable cost. Again snubbers can often save the day. Snubbers are very effective for reducing problems in PCBs associated with large dynamic displacements. This condition can cause high lead wire and solder joint stresses, cracking of components, short circuits due to impacting between adjacent PCBs, and broken pins on electrical connectors. Snubbers, sometimes called bumpers, are small devices that can be attached to adjacent PCBs so they strike each other in dynamic conditions. This reduces the bending displacements in severe vibration and shock conditions. Reducing the deflection reduces the dynamic forces and stresses. This increases the fatigue life of the assembly. The best locations for snubbers and bumpers are close to the centers of the various PCBs, where the maximum displacements are expected. Soft rubber snubbers work best on PCBs with natural frequencies below about 50 Hz. Half-sphere shaped snubbers with diameters of about 0.20 in. can be used on small PCBs. Large PCBs work well using similar half-sphere shapes with diameters up to about 0.50 in. Soft rubber snubbers should be installed so they interfere slightly with other snubbers on adjacent PCBs for improved performance, as shown in Fig. 7.6. PCBs with natural frequencies above about 100 Hz must use more rigid materials for snubbers to reduce the PCB relative motion. Higher natural frequencies result in smaller dynamic displacements, so soft rubbery materials will not work. High-level vibration and shock qualification test programs using plug-in PCBs with epoxy fiberglass snubbers have been very successful. The snubbers in these programs were short posts with a cross-section diameter of 0.25 in., epoxy bonded to the center of each PCB. Hard snubber materials require a small clearance between adjacent snubbers to allow the PCBs to be inserted and removed with no interference. Smaller clearances

160

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE

FIGURE 7.6. Soft rubber snubbers recommended for low natural frequency circuit boards.

work better. The best snubbers are ones that just touch each other with no space between them. This acts like a center support, which substantially increases the natural frequency. Small gaps are very desirable for reducing displacements, but they can be expensive because they require very close manufacturing tolerance control. The clearance between adjacent snubbers should be less than half of the dynamic PCB displacement expected in some particular dynamic environment, as shown in Fig. 7.7. Most snubber applications will be related to modifying existing hardware. The best location is at the center of the PCB. It will be very difficult to find any space at the center of any existing hardware. Therefore, a search must be made of the various PCBs to find free spaces for attaching the snubbers. One method that works well is to make transparent copies of all the PCB layouts. These are then arranged in their proper assembled order. Looking through the transparent assembled layouts will often reveal desirable places for locating adjacent snubbers.

Small gap

--I&

Rigid

FIGURE 7.7. Rigid epoxy glass rod snubbers with a small clearance gap, recommended for high natural frequency circuit boards.

USING SNUBBERS TO REDUCE PCB DISPLACEMENTS AND STRESSES

Sample Problem-Adding

161

Snubbers to Improve PCB Reliability

Several different military and commercial electronic systems have been experiencing component failures on plug-in type PCBs during sinusoidal vibration tests. The PCBs that have been experiencing problems are listed below. 1. PCBs with a natural frequency of 125 Hz during a 5-G peak sine vibration test. 2. PCBs with a natural frequency of 160 Hz during a 5-G peak sine vibration test. 3. PCBs with a natural frequency of 260 Hz during a 6-G peak sine vibration test.

An examination of the problem PCBs shows there is room available near the center for mounting small snubbers. The proposed plan is to use 0.25-in. diameter epoxy glass dowel rods as snubbers, bonded to the PCBs, leaving a nominal clearance gap of 0.012 in. between adjacent snubbers. Is this an acceptable course of action?

Solution Find the Expected Dynamic Displacements and Compare with 0.012-in. Gap. PCB dynamic displacements can be obtained from the general deflection equation: 2, =

9.8GinQ

(see Eq. 2.33)

(fn)'

Table 7.1 lists the results when values are inserted in above equation. The recommended gap between the snubbers is 0.012 in. for all three problem PCBs. The first PCB needs a gap of 0.017 in. or less so a smaller gap of 0.012 in. is good. The second PCB needs a gap of 0.012 in. or less so it just meets the requirements. The third PCB needs a gap of 0.007 in. or less, so a gap of 0.012 in. does not meet the requirements. Such a small gap will be very difficult and very expensive to achieve due to the large number of tolerances involved in the materials, the manufacturing processes, and the

TABLE 7.1 PCB

Gin

fn

1 2 3

5.0

125 Hz 160 Hz 260 Hz

5.0 6.0

Q

dm = 11.2 dm 12.6 =

= 16.1

ZO

Gap = Z0/2

Acceptable

0.035 in. 0.024 in. 0.014 in.

0.017 in. 0.012 in. 0.007 in.

Yes Yes no

162

OCTAVE RULE T O INCREASE T H E PCB FATIGUE LIFE

assembly. However, it can be done with the use of matched assemblies. This involves matching each individual PCB with its own set of snubbers in its own chassis. There can be no switching of similar PCBs to another chassis because the accumulation of tolerances might result in a much large gap that can reduce the PCB fatigue life.

7.8 CONTROLLING THE PCB TRANSMISSIBILITY WITH DAMPING

Damping is usually defined as the conversion of kinetic energy into heat. Material damping relates to the energy lost due to internal friction or hysteresis in the molecular structure of the material. Structural damping relates to the energy lost due to friction in rubbing, scraping, slapping, and impacting at various interfaces and joints. The total system damping is the sum of the material damping and the structural damping. All real systems produce some damping when they are vibrated. When a system with light damping is disturbed, it will continue to oscillate for a long time after the disturbing force has been removed. A system with heavy damping may oscillate once or twice after the disturbing force has been removed, similar to a good automobile shock absorber. The damping in all systems will always bring the system back to a state of rest some time after the disturbing force has been removed. Damping removes some of the energy within the system so there is less energy available to distort and damage the system. This means there is less energy available to deform structures such as PCBs in vibration and shock environments. A reduction in the kinetic energy reduces the forces and stresses in PCBs, which increases the fatigue life, so damping increases the PCB reliability. Increasing the damping in a PCB increases the amount of energy that is lost, which decreases the transmissibility, and further increases the fatigue life and the reliability of the PCB. 7.9

PROPERTIES OF MATERIAL DAMPING

Internal energy is lost, or converted into heat, any time a structure is deformed. Applying a tensile load to a bar does not instantly deform the bar. There is a small time lag as the bar elongates to its new stabilized length. This entire process is reversed when the applied load is removed. However, the bar never returns to its exact original length since some of the strain energy has been converted into heat, so less energy is available to restore the original bar length. Applying an equal compressive load on the bar reverses the direction of the events just described. The positive and negative strains can be plotted on a graph to show the typical hysteresis loop formed by materials that are deformed. The area in the enclosed hysteresis loop becomes a measure of the energy lost with each stress cycle. A larger area

CONSTRAINED LAYER DAMPING WITH VISCOELASTIC MATERIALS

163

enclosed within the loop shows there is more damping. Again, more damping reduces the PCB transmissibility, which increases the fatigue life and the reliability. 7.10 CONSTRAINED LAYER DAMPING WITH VISCOELASTIC MATERIALS

Most of the damage produced in a vibrating PCB occurs at the natural frequency, where the transmissibility is the greatest, so the displacements, forces, and stresses are also the greatest. When the damping is increased, the PCB reliability is increased. More damping in a PCB is desirable, if it does not result in an increase in the size, weight, and cost. One good method for increasing the damping is with the use of viscoelastic materials with constrained layers to increase the damping. Viscoelastic materials are very similar to rubber materials. They are quite elastic so they can deform, compress, and stretch through large displacements without failing. These materials can also dissipate large amounts of energy when they are deformed. Even more energy can be dissipated with the use of materials such as asbestos mixed in with the viscoelastic materials. Pure natural rubber has similar problems-it does not have much strength or much damping. The addition of carbon black improves these properties. Low temperatures and very high frequencies make viscoelastic materials act like hard materials. Their ability to perform well under these conditions is reduced. High temperatures are also a problem, because these materials lose much of their damping properties under these conditions. Their modulus of elasticity is difficult to measure because it changes with the speed of the applied load. The modulus for a rapidly applied load can be three times greater than for a slowly applied load. Viscoelastic materials can be effective in reducing the dynamic displacements, forces, and stresses in vibrating beam and plate structures that are not required to operate over a wide range of high to low temperatures. These materials are available in various forms of paints and tapes that can be applied in single and multiple layers. Multiple layers of damping materials are much more effective than single-layer applications. More damping layers are more effective but they also add more weight and may make repairs more difficult. One of the most effective methods for increased damping is constrained layer damping. This is the application of many layers of damping material that are applied in the form of a high narrow strip. Alternate layers of viscoelastic materials are separated by layers of aluminum foil about 0.005 in. thick. The typical height is about 0.50 in. This can often be added across the center of a plug-in type PCB, in the form of a strip similar to a stiffening rib, as shown in Fig. 7.8. The center of the PCB is the best place for a damping strip because the displacements are the greatest at that position. This results in large relative shear deflections in the alternate damping layers, which improves the damping.

164

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE Constrained layer viscoelastic

Adhesive bonding

PCB

Alumi

fo ‘Connector

Section BB enlarged

FIGURE 7.8. Constrained layer viscoelastic damper for possible use on low-frequency

boards.

7.1 1 WHY STIFFENING RIBS ON PCBs ARE OFTEN BETTER THAN DAMPING

Vibration testing experience with damping techniques has shown that damping can be quite effective for PCBs with low natural frequencies, below about 50 Hz. However, the tests show that stiffening ribs usually work even better. When the natural frequencies are well above about 100 Hz damping is not a very effective means for reducing the transmissibility. High operating temperatures also tend to reduce the damping properties and effectiveness of viscoelastic materials. Constrained layer damping strips take away valuable space that could be used for mounting additional electronics. When this space is already lost, experience has shown that it is often much better to use the space for bonding stiffening ribs to the PCB, instead of constrained layer damping strips. Stiffening ribs will increase the natural frequency, which will more rapidly reduce the dynamic displacements, because displacements are inversely related to the square of the natural frequency.

7.12

PROBLEMS WITH PCB VISCOELASTIC DAMPERS

A very large supplier of sophisticated electronic systems received a large contract for a military electronic system that had to operate with a high reliability in severe thermal, vibration, and shock environments. There was not enough sway space available for an isolation system, so the mechanical design engineers selected to use viscoelastic damping techniques to reduce the plug-in PCB response accelerations and displacements. The design ap-

PROBLEMSWITH PCB VISCOELASTIC DAMPERS

165

proach used two, one-sided surface mounted PCBs assembled back to back, with a constrained layer viscoelastic material bonding both halves of the PCBs together into one double PCB plug-in module. Vibration and shock tests were run on prototype models using extensive instrumentation. The test data looked very good, so the design was released to the production group. The first few production models were shipped to the customer for a preliminary inspection and evaluation. Production unit number 8 was selected for the qualification test program. Many military qualification test programs require running electrically operating vibration tests at low temperature, room temperature, and high temperature. The first two vibration tests were satisfactory. However, the high-temperature vibration test resulted in many PCB failures. Component lead wires were breaking, solder joints were cracking, and components were flying off the PCBs. The high-temperature vibration tests were a complete disaster. The mechanical engineers were stunned. They could not understand what went wrong. The previous vibration tests on prototype models had been very successful. An investigating failure analysis team was formed and additional vibration tests were run on the viscoelastic damped PCBs. The tests were instrumented and run at room temperature. The resulting test data looked good. The vibration tests were then run again at high temperatures. The test results were dramatically different, with substantially lower PCB natural frequencies producing sharp increases in the dynamic displacements, with components flying off the PCBs. This was caused by the dramatic reduction in the stiffness and the damping properties of the viscoelastic materials at the high temperatures. The system had to be completely redesigned, using stiffening ribs on the PCBs, to obtain a high natural frequency to keep the dynamic displacements and stresses low to achieve the required fatigue life. The production group had to redesign their tools for the new production program. Needless to say that a lot of money was lost on this program, and product deliveries were substantially delayed, because the mechanical engineering group overlooked the adverse effects of high temperatures on the damping properties of the viscoelastic materials.

DCHAPTER 8

Preventing Sinusoidal Vibration Failures in Electronic Equipment

8.1

INTRODUCTION

Sinusoidal vibrations are often found in machines and devices that rotate or oscillate, such as electric motors. wheels, engines, turbines, gears, washing machines, blenders, oil drilling equipment, springs, and even bridges. Sinusoidal vibrations are also extremely useful as a diagnostic tool for evaluating the dynamic characteristics of various types of structures, and for examining resonant conditions with the use of a strobe light. Vibration failures are often difficult to trace, since there are so many different factors to be considered. Not only must the environment be considered, but also the mechanical and electrical design, manufacturing tolerances, resonant frequency of the outer housing structure, resonant frequency of internal elements, possible dynamic coupling effects, sensitivity of electronic components, cables and harnesses, effects of conformal coatings or lack of coatings, humidity effects, and dozens of other factors. Sometimes there does not appear to be any connection between parts that are failing and exposure to vibration, as shown in the following case history. A large auto manufacturer was having problems with dead batteries on a particular automobile model, when the auto was shipped across the country by rail. The batteries were charged when they left the factory and were discharged when they arrived at their destination. The most obvious sources for the problem were the batteries themselves and the electrical connections related to the battery. When these suspected sources were cleared of blame, the investigation was expanded to other areas, but always within the engine compartment where the battery was located. It appeared logical that the battery failures must be related to the battery location. After many hours of investigation and many meetings, one clever individual observed that the horn ring on this particular car model had a very soft touch. The horn would sound when a small force was applied to the horn ring. H e had a hard time convincing the company officials to send an observer along with the next rail shipment of cars across the country. His theory was correct. Along one long stretch of track through a desolate area, the speed of the train and the 166

ESTIMATING THE VIBRATION FATIGUE LIFE

167

condition of the tracks set up a disturbing force that excited the resonant frequency of the horn ring on its springs. Over an extended period of many hours, the constant sounding of the horn drained the battery. The springs on the car horn ring were made stiffer, which raised the resonant frequency of the horn ring above the disturbing force, and the problem was solved. ESTIMATING THE VIBRATION FATIGUE LIFE

8.2

The approximate fatigue life of a vibrating system can often be estimated from the fatigue characteristics of the various members that carry the major structural loads. These fatigue properties are usually obtained from controlled stress cycle tests performed on many parts that have been manufactured from the same material, with close dimensional tolerances. These parts are stress-tested to failure; then the data points are plotted on log-log paper, with stress on the vertical axis and the number of cycles to failure on the horizontal axis. A straight line that represents the best average fatigue properties is drawn through the scattered data points as shown in Fig. 8.1. The equation for the sloped portion of the curve can be expressed as follows [l]:

N,S,b = N,S,b where N = number of stress cycles to produce a fatigue failure S = stress level at which these failures occurred b = fatigue exponent related to the slope of the line

(8.1)

The exponent b is related to the fatigue life of the structure, and it is useful in predicting the expected fatigue life of other structures fabricated of the same material and exposed to similar environments. In order to reflect the real structural properties of these materials, a stress-concentration factor K must be included in the evaluation of the exponent b. A typical stress-concentration factor for an electronic box structure or for a PCB within the box is about 2.0. Since the stress concentration has little importance with a small number of fatigue cycles, and more importance with more fatigue cycles, the slope of the S-N curve increases with the stress concentration as shown by the dashed line in Fig. 8.1.

I Cycles to failure.

I

FIGURE 8.1. Typical plot of the data obtained

A’

from fatigue tests of metal structures.

168

PREVENTING SINUSOIDAL VIBRATION FAILURES

The value of the exponent b can be determined for aluminum by using typical physical properties for aluminum alloys, where the endurance stress is typically one-third of the ultimate tensile stress. With a stress concentration of 2, the endurance is one-sixth of the ultimate tensile stress. Rewrite Eq. 8.1 as follows:

where Nl = lo8 cycles N2 = l o 3 cycles S , = S, = +St, (one-sixth the ultimate tensile stress) S, = S,, = ultimate tensile stress Substitute into Eq. 8.2 and simplify:

Take the logarithm of both sides and solve for the exponent b:

b=

log,, 105

--

log,, 6

5

0.778

=

6.4

This fatigue exponent b can be used to determine the approximate fatigue life for a number of different conditions. For linear systems, the number of fatigue cycles will be directly related to the time, so the time T can be used to replace the number of cycles ( N ) .Also, the stress S will be directly related to the acceleration G and to the displacement Z . The dynamic load can also be included if it is desired. These relations can be written as follows:

T,G:

=

N,ZF

= N2Z!

(8.4b)

N,G!

=N , G ~

(8.4~)

Sample Problem-Qualification

T,G~

(8.4a)

Test for an Electronic System

An electronic control system is to be used on subway trains that are expected to operate 6 hours per day, 4 days per week, for 52 weeks per year, with a desired life of 20 years. Field test data show that the subway track system produces harmonic vibrations over a frequency range from 10 to 300 Hz, with typical accelerations of 1.1 G peak. A sinusoidal vibration qualification test is

ELECTRONIC COMPONENT LEAD WIRE STRAIN RELlEF

169

desired to prove that the electronic system will be capable of operating in this environment for the 20-year period. The proposed qualification test plan specifies a sinusoidal test where the forcing frequency sweeps back and forth over the observed bandwidth using a 4.5-G peak acceleration input level. Determine the length of time that should be used for these sweep tests to qualify the electronic system. Solution. Equation 8.4a can be used to determine the period of time for the qualification test. The following information is required:

T2 = (6 h)(4 day)(52 wk)(20 yr) = 24,960 h (life desired) G, = 1.1 G (peak field test data) GI = 4.5 G (peak proposed qualification test level) b = 6.4 (vibration fatigue exponent for electronic structures) Substitute into Eq. 8.4a and solve for the desired vibration qualification test time period:

T , = (24,960)

(

:::r4

-

= 3.03

hours

(8.5)

The 3.03 hours represents the testing time with no safety factor. Since there are many variables involved in the determination of the field test data and in the design, manufacture, and assembly of this electronic equipment, some safety factor must be included. Therefore, recommendations should be made to vibrate the electronic equipment for 3.03 hours along each of three mutually perpendicular axes for a total of 9.09 hours of vibration for the qualification test. The above recommendation is equivalent to a safety factor of 3.0, which will require a slightly heavier structural assembly to meet these levels. The weight penalty is not very large here, because the fatigue life changes very rapidly with a slight reduction of the stress level. For example, when the stress level changes by a factor of 2, the fatigue life changes by a factor of 2 raised to the 6.4 power, which is 84.4.

8.3 ELECTRONIC COMPONENT LEAD WIRE STRAIN RELIEF Electronic systems that are used in any vibration or shock environments will expose the electronic components to some level of vibration and shock. These levels may be amplified or attenuated, depending on the characteristics of the dynamic system. When PCBs are used in these electronic systems, the resonant frequencies of the PCBs are often excited, which forces the PCBs to bend back and forth. This motion forces the electrical lead wires on the components to also bend back and forth, as shown in Fig. 8.2.

170

PREVENTING SINUSOIDAL VIBRATION FAILURES

Strained lead wires

+L

\

Relative

PCB Bending’

1

B -_____i_/

FIGURE 8.2. Relative motion produced between component and circuit board when the board resonance is excited.

Different lead wire geometries will result in different stress levels in the lead wires and solder joints of these components. Even when the dynamic displacements of a group of PCBs are the same, the stresses developed in the lead wires and solder joints of similar components can still be sharply different, depending on the geometry of the wire. When the dynamic displacement of a particular PCB has been determined for a particular environment and a particular PCB resonant frequency, the displacement amplitude can be treated as a fixed condition. Under these circumstances, it can be shown that the best way that the forces and stresses can be reduced in the lead wires is to reduce the spring rate of the wires. This appears to be contrary to the normal approach generally taken when a structure fails in a vibration or shock environment. Stronger is normally considered to be better, so most of the time the structure will be made stiffer (stronger) when a failure occurs. However, in the case of lead wires, a stiffer wire will fail faster. This can be demonstrated by examining the force P developed in a structure as a function of the spring rate K and the displacement Y as follows:

P=KY

(8.6)

When the displacement is held constant, the best way that the force can be reduced is to reduce the spring rate. This is the same condition that exists in the component lead wires for a specific dynamic displacement generated by the resonant frequency and acceleration level. In the typical application, where the electrical lead wires are forced to bend back and forth during a resonant condition, the lead wire stiffness is related to the following parameters:

K wire .

EI = -

L3

(wire flexing stiffness parameters)

The above relation shows that increasing the length L of the wire will reduce the stiffness K very rapidly, since it is a cubic function. For example, when the wire length is doubled, the stiffness, the force, and the stress in the wire are reduced by a factor of two cubed, or eight. The above relation also shows

DESIGNING PCBs FOR SINUSOIDAL VIBRATION ENVIRONMENTS

171

that decreasing the moment of inertia I of the electrical lead wire will rapidly reduce the wire stresses, since the moment of inertia varies as the cube of the wire height. When the wire height, or thickness, is reduced by a factor of two, the spring rate, stiffness, force, and stress in the wire are decreased by a factor of two cubed, or eight. This can be accomplished by coining the lead wire, that is, squeezing it into a flat thin metal strip. The cross-sectional area of the wire stays the same, but its thickness is reduced, so the spring rate, stiffness, force, and stress in the wire are also reduced. The above relation also applies to surface mounted electronic components as well as the poke-through type of mounting, where electrical lead wires are forced to bend during resonant conditions. It also applies to leadless ceramic chip carrier (LCCC) devices, which are surface mounted with or without electrical lead wires. In applications that do not use lead wires, the above relations then apply to the shear stiffness of the solder joints that bond the component to the PCB. Here again the stiffness of the solder joints must be reduced to reduce the forces and stresses in them. An examination of the shear stiffness will show which parameters will affect the solder stresses: AG Kshear =

(solder shear stiffness) L

(8.8)

where A = cross-sectional area of the solder joints (he2) L = height of solder joint (in.) G = shear modulus of the solder (lb/in.’> The shear stiffness can be decreased by reducing the cross-sectional area of the solder, reducing the shear modulus of the solder, or increasing the height of the solder joint. Some experiments are being carried out to reduce the solder shear modulus, but so far this has been unsuccessful. Some success has been achieved by reducing the solder cross-sectional area. The greatest success in improving the fatigue life of the solder has been in increasing the height of the solder joint. The typical solder joint height for LCCC devices is about 0.003-0.005 in. This has successfully been raised to 0.010 in. with the use of small solder mask pads, 0.010 in. thick, under the LCCC devices. 8.4 DESIGNING PCBs FOR SINUSOIDAL VIBRATION ENVlRONMENTS

In typical electronic systems, the electronic components are mounted on plug-in PCBs. This makes it easy to repair an electronic system if an electrical failure occurs and the failure can be traced to a faulty PCB. The faulty PCB is simply replaced with a good one.

172

PREVENTING SINUSOIDAL VIBRATION FAILURES

f

PC pcB

2nd

degree

Chassis

I

1st degree Rigid fixture

3

1 Vibrat

3n

lnpu!

I

1

V i brat ion

input

FIGURE 8.3. A simple electronic chassis assembly with PCBs, simulated as a

lumped-mass system. When the plug-in PCBs are mounted within a chassis housing, and the system is required to operate in a sinusoidal vibration environment, then the chassis will act as the first degree of freedom, because the vibration energy will excite the chassis structure first. Since the PCBs are normally attached to the chassis structure (side walls or base of the chassis), the PCBs receive their dynamic excitation from the chassis. This makes the PCBs act like the second degree of freedom. A typical system is shown in Fig. 8.3. Resonant conditions can occur in the chassis and also in the PCBs during excitation by sinusoidal vibration. Since the chassis and the PCBs are coupled together, any resonant condition in any one of the masses will force some response in the other masses. When the resonant frequency of the chassis is close to the resonant frequency of any of the PCBs, and when the chassis and the PCBs have high transmissibilities Q, then the chassis response can augment the PCB response. This will produce very high acceleration levels in the PCBs, which can result in rapid fatigue failures. In order to prevent this from occurring, the chassis resonance and the PCB resonances must be well separated. This can be accomplished by using the octave rule. An octave means a factor of two in frequency. When the resonances of the chassis and the PCBs are separated by one octave or more, then the coupling effects between those members are sharply reduced. It makes very little difference whether the chassis or the PCBs have the higher resonant frequency, as long as their resonant frequencies are well separated. Vibration testing experience and associated finite element studies have shown that the fatigue life of many different types of electronic components can be related to the dynamic displacements developed by the PCBs during vibration [38]. These studies have shown that these components can achieve a fatigue life of about 10 million stress reversals in a sinusoidal vibration environment, when the peak single-amplitude displacement of the PCB is limited to the value shown below and in Fig. 8.4 [371:

Z=

0.00022 B Chra

(maximum desired PCB displacement)

(8.9)

DESIGNING PCBs FOR SINUSOIDAL VIBRATION ENVIRONMENTS Bending lead wires

I

FIGURE 8.4. Axial-leaded component wires will bend when the PCB resonance is excited.

-2 displacement where B

173

of PCB edge parallel to component (in.) L = length of electronic component (in.) h = height or thickness of PCB (in.) C = constant for different types of components (see Eq. 9.50) r = relative position factor for components (see Eq. 9.50) = length

The peak single-amplitude displacement expected at the center of a PCB can be estimated by assuming the PCB acts like a single-degree-of-freedom system at its resonant condition, as shown in Eq. 2.30: 9.8G 9.8Gi,Q z=-

f'

f,'

The displacement value obtained from the above relation will be a little less than the correct value obtained from the actual dimensions of the PCB plate geometry. When the PCB is approximated as a single degree of freedom, and a single mass is used to represent the PCB, the single mass will show the displacement of the centroid of the PCB and not the actual displacement of the PCB itself. Vibration test data have shown that the transmissibility Q of a PCB can typically be related to its resonant frequency. Although the test data show that high input acceleration levels (above 8 G) will reduce the Q values, and low input acceleration levels (below 2 G ) will increase the Q values, a good approximation of the Q value can still be obtained with the following relation (also see Eq. 14.21):

Q=&-

(8.10)

The desired PCB resonant frequency that will provide a fatigue life of about 10 million stress cycles for the component lead wires and their solder joints can be determined by combining Eqs. 2.30, 8.9, and 8.10:

( fd =

9.8Gi,Chra 0.00022 B

)2'3

(minimum desired PCB frequency) (8.11)

174

PREVENTING SINUSOIDAL VIBRATION FAILURES

Sample Problem-Determining

Desired PCB Resonant Frequency

A 40-pin DIP with standard lead wires will be mounted at the center of a 6.0 x 8.0 x 0.10-in. thick plug-in PCB. The DIP will be mounted parallel to the 8-in. edge. The system will be subjected to a sinusoidal vibration qualification test where the maximum acceleration input level is expected to be 7.0 G peak. Determine the minimum desired PCB resonant frequency, and the approximate fatigue life considering a resonant dwell condition.

Solution. The octave rule must be used to make sure the resonant frequency of the PCB is well separated from the resonant frequency of the chassis, so coupling effects between the chassis and the PCB are reduced. Under this condition it is convenient to assume that the acceleration input to the PCB is the same as to the chassis. This is not really true, but it is convenient for a quick approximate answer. Then

B = 8.0 in. (length of PCB parallel to component) h = 0.10 in. (PCB height or thickness) L = 2.0 in. (length of 40-pin DIP) C = 1.0 (constant for standard DIP geometry) P = 1.0(relative-position factor for component at center) Substitute into Eq. 8.11 for the desired PCB resonant frequency:

fd =

[

0.666

i

9.8( 7.0) ( l . O ) ( O . l O ) ( 1. O ) m 0.00022(8.0)

= 310

HZ

(8.12)

The fatigue life of the DIP can be estimated from the expected 10million-cycle fatigue life:

Life

10 x l o 6 cycles to fail =

(310 cycles/s) (3600 s/h)

= 8.96

hours to fail

(8.13)

A resonant frequency of 310 Hz will be difficult to achieve; a stiffening rib will probably be required. A stiffening rib will use up some of the valuable surface area of the PCB, which may reduce the number of electronic components that can be mounted on a tightly packaged PCB. When several PCBs require stiffening ribs, the amount of surface area may be reduced, so an additional PCB may have to be added to the system. This may result in a larger, heavier, and more expensive assembly, which will make many company executives very unhappy. These factors must be considered with respect to a possible qualification test failure, or to failures in the field.

HOW LOCATION AND ORIENTATION OF COMPONENT ON PCB AFFECT LIFE

175

The probable fatigue life of 8.96 hours to fail, as shown by Eq. 8.13, is not really a long period of time. An examination of MIL-STD-810D shows it requires sinusoidal sweep tests of 1 hour per axis for three axes, plus resonant dwell periods of 30 minutes at each of four major resonances, in each of three axes, for a total resonant dwell period of 6 hours, and a total testing time of 9 hours for one qualification test. Past experience in qualification test programs has shown that many different types of problems and failures can occur, which will require the test to be repeated. Sometimes these tests may have to be repeated several times before the whole qualification test can be completed. A safety factor is therefore necessary to ensure the successful completion of the tests. A recommended practice is to design the electronic system with enough fatigue life to permit the system to pass five qualification tests. This does not represent a safety factor of 5, as typically computed using stress levels as criteria. Instead it represents a safety factor of 1.286, since the fatigue life is related to the stress level raised to the 6.4 power as shown by Eq. 8.3: Fatigue-life change

=

( ~ 2 8 6 )=~5.0 '~

(8.14)

8.5 HOW LOCATION AND ORIENTATION OF COMPONENT ON PCB AFFECT LIFE

Electronic components can be mounted in any location and in any orientation on a PCB. Rectangular components are usually mounted with their sides parallel to the sides of rectangular PCBs. Smaller components, less than about 1 in., very seldom cause any vibration problems unless the component happens to be a very tall and heavy transformer. This type of component is not considered here. Larger components, greater than about 1 in. are where vibration problems typically start, depending on the type of PCB and the vibration-level requirements. Vibration problems become very severe when the component size reaches 2 in. or more. When plug-in PCBs are being evaluated, the most critical location for large components will be at the center of the PCB. This is where the most rapid change of curvature occurs during the fundamental resonant frequency of a PCB with simply supported edges. As the component is moved away from the center, the curvature is reduced, so the relative motion between the PCB and the component body is reduced, as shown in Fig. 8.5. This reduces the forces and stresses in the lead wires and solder joints, so the fatigue life is increased. The most dangerous orientation for a long component is parallel to the shorter side on a rectangular plug-in PCB. This is due to the more rapid change of curvature of the shorter side than of the longer side of the PCB. This results in more relative motion between the PCB and the component, which increases the loads and stresses in the lead wires and solder joints as

176

PREVENTING SINUSOIDAL VIBRATION FAILURES Large relatlve rnat10n

__

,

Small relative no!icl J

1'-

FIGURE 8.5. Relative motion between the component and the P C B is reduced when the component is mounted near the P C B edge.

it--L---L-

Sma I rela!lve notlo'-

Large relative rnotlon

S a n e d i s p l a c e m e i t a t center

FIGURE 8.6. Relative motion between the component and the P C B is reduced when the component is mounted parallel to the long edge of the PCB.

shown in Fig. 8.6. Increased stresses are undesirable because they reduce the fatigue life of the system. The forces and stresses in the lead wires and solder joints of electronic components mounted on PCBs can be related to the relative curvature of the PCB during its resonant condition. The relative curvature can be related to the displacement of the PCB, and the displacement can be related to the location of the component on the PCB. When the edges of the PCB are simply supported (or hinged), then the displacement at any PCB location can be determined from the following relation:

Z

=Z,

7Tx

sin-sina

irY b

(see Eq. 6.1)

HOW WEDGE CLAMPS AFFECT THE PCB RESONANT FREQUENCY

177

When the component is located at the center of the PCB, X is a / 2 and Y is b/2. The PCB displacement at the center simply becomes 7 T 7 T

Z

=Z,

sin-sin2 2

=Z,

(8.15)

When the component is located off center, at the position where X is a / 2 and Y is b/4, then the PCB displacement is 7

Z

T

X

= 2, sin -sin - = 0.707Z,

2

4

(8.16)

When the component is located at the quarter mounting points, X is a / 4 and Y is b/4. The PCB displacement at this point is

Z

=Z,

X

T

sin -sin - = o.50ZO 4 4

(8.17)

These relative displacements at different positions on the PCB will determine the forces and stresses in the component lead wires and solder joints, which will have a direct effect on the fatigue life of these components. These relative displacements can therefore be used to find the approximate fatigue life of different types of components, in different environments, based on the component location on the PCB. 8.6 HOW WEDGE CLAMPS AFFECT THE PCB RESONANT FREQUENCY

Wedge-clamp-type edge guides are often used to grip the side edges of the PCB to reduce the thermal resistance for lower temperatures and to increase the PCB resonant frequency. A typical wedge clamp is shown in Fig. 8.7. Vibration tests with wedge-clamp edge guides have shown that their effectiveness is related to the stiffness of the PCB, and therefore to the resonant frequency of the PCB. A PCB with a low resonant frequency, less than 100 Hz, is not very stiff. The wedge clamp is then capable of holding the PCB edge guide rigidly to prevent any rotation during the resonant condition, when the input acceleration level is less than about 8 G peak.

FIGURE 8.7. Wedge-clamp edge guides grip the PCB edges firmly, which increases the PCB stiffness Wedge clamD

and resonant frequency.

178

PREVENTING SINUSOIDAL VIBRATION FAILURES

? -

z

P,

L

FIGURE 8.8. Curve for estimating the per-

centage fixity for a wedge-clamp edge guide.

;r

i -i*-

I?C 366

693

1,

ireaJercy

-2

A PCB with a high resonant frequency, greater than about 600 Hz, is very stiff. The wedge clamp is not capable of holding the PCB edge guide rigidly to prevent rotation, so some rotation will occur at this interface. The higher the frequency, the greater the amount of relative motion. Thus the frequency can be used to estimate the percentage of fixity for the stiffness characteristics of a wedge clamp, as shown in Fig. 8.8. When wedge clamps are used as side edge guides for plug-in PCBs, the accuracy of the resonant-frequency calculation can be improved by using the following equation:

f n =fs

+Pf(ff - f s >

(8.18)

where f , , = expected natural frequency of PCB f , = natural frequency of PCB with simply supported sides f t = natural frequency of PCB with fixed (clamped) sides Pf = percentage fixity of PCB side edges from Fig. 8.8 The fixity cannot be determined until the resonant frequency is known, and the resonant frequency is not known yet. However, a closed-form solution can be obtained by using similar triangles in Fig. 8.8 as follows:

f,,- 100 600 - 100 (fn

-

-

1.00 - Pf 1.00 - 0.50

100)(0.50) =(500)(1.00-Pf) Pt = 1.1 - O.0Olfn

(8.18a)

Substitute Eq. 8.18a into Eq. 8.18 and solve for f,:

f n = f s + (1.1 -0.00lfn>(ff-fs) f s + l W f *-f,> fn =

1

+ 0.001( ff - f , )

(8.19)

The use of this equation can be demonstrated with a sample problem.

HOW WEDGE CLAMPS AFFECT THE PCB RESONANT FREQUENCY

Sample Problem-Resonant Clamps

179

Frequency of PCB with Side Wedge

Determine the resonant frequency of the 0.5-lb plug-in PCB with side wedge clamps shown in Fig. 8.9. Rubber strips under the top cover of the chassis are expected to grip the top edge of the PCB, so the top of the PCB will act as a simply supported (hinged) edge. The electrical connector across the bottom edge of the PCB is also expected to support this edge, so it acts like a hinge. For a preliminary analysis, assume a uniformly distributed weight across the face of the PCB. Solution. When the resonant frequency of the PCB is less than about 100 Hz, the side wedge clamps will prevent rotation of the side edges. The PCB resonant frequency can then be determined by assuming the side edges are clamped. When the PCB resonant frequency is greater than about 600 Hz, the side wedge clamps are not capable of preventing rotation, so the side edges cannot be evaluated for a clamped condition. Under these circumstances, the sides will act somewhere between a fixed edge and a hinged edge. A good approximation of the resonant frequency for the PCB with side wedge clamps can be obtained from Eq. 8.19. In order to use this expression, it is necessary to determine the resonant frequency of the PCB with two different side edge conditions. The first condition is with hinged side edges, and the second condition is with fixed side edges. First Condition: PCB with Hinged Side Edges For the first condition, all four edges of the PCB will act as hinges. The resonant frequency can be determined from Eq. 6.21: 1 \ m i 1 p ( a- 2+ - b 2 )

f "= -21:'

Top rubber st' ps

6 0-

FIGURE 8.9. Plug-in PCB with supported (hinged) top and bottom edges, and wedge clamps at the sides (dimensions in inches).

180

PREVENTING SINUSOIDAL VIBRATION FAILURES

where E h

= 3.0

x lo6 I b / h 2 (modulus of epoxy glass with copper planes)

= 0.10

in. (thickness of epoxy fiberglass PCB) (Poisson's ratio for epoxy glass with copper planes) W = 0.50 Ib (weight of PCB) g = 386 in./s2 (acceleration of gravity) a = 6.0 in. (length of PCB) b = 4.0 in. (width of PCB) D = Eh3/12(l - p ' ) = (3 X 106)(0.100)'/(12)(1 - 0.0324) = 258.4 Ib . in. p = W/gab = 0.50/(386)(6.0)(4.0) = 0.000054 Ib * s 2 / h 3 p = 0.18

Then f n = ~ 0.000054 / ~ ( i +36 i / 16 = 3 ~ o H z

(8.20a)

Second Condition: PCB with Fixed Side Edges The resonant frequency of the PCB with opposite edges clamped and opposite edges supported can be determined from Fig. 6.12. Note that clamped edges as defined here mean that the side edges of the PCB do not undergo any rotation during the resonant condition. We have 7~

where D

8

10116

31

258.4 Ib in. p = 0.000054 Ib s'/inS3 =

n = 6.0

b

=

in.

4.0 in.

Thus ir fn =

4

3.46 45:::.0

(&

8

++ 576

-1

3 256

=387Hz

(8.20b)

P CB with Side Wedge Clamps, Supported on Top and Bottom Edges The resonant frequency of the PCB with side wedge clamps can be determined using two different methods. In the first method a trial-and-error solution is used with Fig. 8.8 to find the percentage fixity. In the second method a direct solution for the PCB resonant frequency is obtained in closed form, using Eq. 8.19.

HOW WEDGE CLAMPS AFFECT THE PCB RESONANT FREQUENCY

181

Trial-and-Error Method .for Resonant Frequency. Assume a resonant frequency of 366 Hz. From Fig. 8.8 find the percentage fixity using similar triangles:

X

-

100 - 50

366 - 100 600 - 100

or X = 2 6 . 6 %

This value is measured from the top down, so that one has 100 - 26.6 = 73.4% fixed edges at the wedge clamps. Substitute into Eq. 8.18 to obtain the calculated PCB resonant frequency,

f, = 310 + (0.734)(387 - 310)

= 366.5

HZ

(8.21a)

Since the assumed PCB resonant frequency of 366 Hz is approximately the same as the calculated value of 366.5 Hz, the correct PCB resonant frequency with the side wedge clamps is about 366 Hz. Closed-Fom Solution for Resonant Frequency. The resonant frequency of the PCB with side wedge clamps can also be determined directly with the use of Eq. 8.19, where

f,= 310 Hz for simply supported (hinged) sides (see Eq. 8.20a), and

ff = 387 Hz

for clamped sides

(see Eq. 8.20b). Thus 310 fn =

+ (1.10)(387 - 310)

1 + (0.001)(387 - 310)

= 366.4

HZ

(8.21b)

Wedge-clamp side edge guides will increase the resonant frequency of the

PCB,which will reduce the dynamic displacements and improve the fatigue life of large components mounted on the PCB.The fatigue-life improvement is not quite as good as might be expected from the reduction of the dynamic displacement alone. This is due t o the differences in the mode shapes for the simply supported side edge guides and the clamped side edge guides as shown in Fig. 8.10.

Supported sides

ClamFed s des

Small -3 -Large

P C 6 Gendtng

"

FIGURE 8.10. Differences in relative motion between the component and the PCB for supported sides and clamped sides.

182

PREVENTING SINUSOIDAL VIBRATION FAILURES

An examination of the relative displacements between the component body and the PCB for large components shows that the clamped-edge condition will result in a slightly larger relative displacement. This means that the fatigue life for a large component mounted at the center of a PCB with a clamped-edge condition will not be quite as good as the fatigue life of the same component mounted on the same PCB with a simply supported edge condition, when the dynamic displacements of both PCBs are the same. 8.7

EFFECTS OF LOOSE PCB SIDE EDGE GUIDES

Plug-in PCBs must have some type of guide at the sides to control the position and maintain alignment accuracy for the blind-mating electrical connector at the bottom edge of the PCB. Many different types of edge guides are available, from spring clips to wedge clamps. Sometimes slots or grooves are cast or machined in the chassis side walls to act as edge guides. These slots or grooves are usually made extra wide to allow for manufacturing and assembly tolerances on the chassis and the PCBs. The end result is a loose side edge guide, which will not adequately support the PCB in a vibration or shock environment. The idea behind this looseness is to reduce manufacturing costs and still provide a rugged electronic chassis assembly. At first glance it might appear that a loose PCB side edge guide might even act like an isolation system, which would reduce the acceleration levels transferred to the PCBs by the chassis. However, vibration test data show that just the opposite is true. A loose PCB side edge guide can actually increase the acceleration transferred to the PCBs in a vibration and shock environment, with light damping [17]. Investment castings are often used for electronic chassis housings, and it is tempting to cast slots in the chassis side walls to be used as PCB edge guides. These slots are almost free, since you only pay for the slots in the initial tooling cost; so machining costs are reduced. However, these cast-in slots should not be used, because they can sharply increase the acceleration levels and stresses in the PCBs. This can reduce the fatigue life of the system. Vibration test data on chassis assemblies with loose PCB side edge guides showed that the PCB resonant frequency was always coupled to the chassis resonant frequency, even when the chassis resonant frequency was increased to try to separate the two resonances. When the PCB has loose side edge guides, the period of the vibration depends on its amplitude. The clearance of the loose PCB edge guide as well as the spring rate of the PCB and the acceleration level will determine the resonant frequency of the PCB. For a given set of conditions, a higher input acceleration level will result in a higher PCB resonant frequency. A smaller clearance will also result in a higher PCB resonant frequency. The effects of a loose PCB side edge guide can be evaluated by considering the PCB to be a mass vibrating between two springs when there is a clearance distance a between the mass and the spring for some initial condition, as shown in Fig. 8.11.

EFFECTS OF LOOSE PCB SIDE EDGE GUIDES

4 a L-

y\

183

4Fa

Clearance

Plug-in connector acts as a hinge

FIGURE 8.11. A loose edge guide permits the PCB to slap back and forth through the clearance, resulting in a reduced PCB resonant frequency.

The resonant frequency of a PCB with loose side edge guides can be determined by computing the PCB period of vibration in three steps:

Step 1: First compute the resonant period of the PCB with a zero side edge clearance. The period is simply the reciprocal of the resonant frequency of the PCB with simply supported (or hinged) edges. Step 2: Next compute the period from the velocity of the PCB as it swings through the loose side edge clearance. The velocity will be proportional to the input acceleration and the transmissibility. Step 3: Simply add the periods from steps 1 and 2, and invert, to obtain the true PCB resonant frequency with the clearance. The velocity of the PCB can be determined from the edge clearance a, considering harmonic motion and starting with the displacement: Y = a sin R t

(8.22)

The first derivative of displacement is the velocity:

dY

I/= - = Ru cos Rt

(8.23)

dt

The maximum velocity will occur when cos R t

=

1.0:

v= f l u Let R

=

2 r f , so

v= 2 T f U

(8.24)

184

PREVENTING SINUSOIDAL VIBRATION FAILURES

The single-amplitude displacement due to vibration was shown in Eq. 2.30. Substitute this value into Eq. 8.24 for the velocity: V=

19.6rrGinQ

fn

(8.25)

The total period of the PCB can be obtained from the motion through one complete cycle. When the PCB is in contact with the springs, it is hard against the stops with no more side edge clearance. For step 1, the period is just the reciprocal of the PCB resonant frequency:

P,= 2 s g

(period of PCB)

(8.26)

For step 2, the PCB period depends on the clearance a and the velocity of the PCB as it passes through the clearance. Since the total clearance during one full cycle is 4a, and velocity x time is distance, the period of the PCB passing through the clearance becomes 4a

7

(8.27)

4afn 19.6rrGinQ

(8.28)

P2

=

Substitute Eq. 8.25 into Eq. 8.27,

P,

=

The total period of the PCB is the sum of the two periods:

P = P , + P2 = 27T +

4afn 19.6rrGinQ

(8.29)

The resonant frequency of the PCB is the reciprocal of the period P : f

1

= -

"

P

(8.30)

If a periodic disturbing force has a frequency lower than the PCB resonant frequency (with no side edge clearance), it will always be possible to set up a condition where the PCB resonant frequency (with side edge clearance) will become equal to the disturbing-force frequency and produce severe coupling effects. In other words, when loose PCB edge guides are used in a chassis and the chassis has a lower resonant frequency than the PCB (with no side edge clearance), it will always be possible to set up a condition where the PCB resonant frequency (with side edge clearance) will become equal to the chassis resonant frequency and produce severe coupling effects on the PCB. This increases the stress levels in the PCB, which reduces the fatigue life [171.

SINE SWEEP THROUGH A RESONANCE

Sample Problem-Resonant Loose Edge Guides

185

Frequency of PCB with

A plug-in PCB has a resonant frequency of 187 Hz when the PCB is simply supported on all four sides. The PCB is to be used in a chassis that has clearance slots in the side walls instead of side edge guides. The single-amplitude clearance expected between the PCB and the chassis side wall slots will be 0.008 in. minimum and 0.012 in. maximum. Determine the expected PCB resonant frequency range for a sinusoidal vibration input level of 6.0 G peak. Solution 0.008-in. Clearance Between PCB and Chassis Side Wall Slot Test data show that loose side edge guides result in higher PCB transmissibility Q values, because of more severe dynamic coupling between the chassis and the PCB. Therefore a slightly higher than normal Q must be used in this solution. The following information is required:

f,, = 187 Hz (PCB resonant frequency, no side edge clearance) PI = 1/187 = 0.005347 s (PCB period, no side clearance) a = 0.008 in. (minimum side edge clearance) Gin= 6.0 G (peak input acceleration level) Q = 1 . 2 m = 16.4 (expected PCB transmissibility) Substitute into Eq. 8.29 for the resonant frequency of the PCB with side edge clearance:

P = 0.005347

(4)(0.008)( 187) + (:19.6)( T )(6)( 16.4)

= 0.00633

second

1L

fn=

o.00633 = 158 HZ

(for PCB)

(8.31)

0.072-in. Clearance Between PCB and Chassis Side Wall Slot

P = 0.005347

(4) (0.012) (187) + ( 19.6) ( (6) (16.4)

=

0.00683 second

T )

1 fn=--

8.8

0.00683

-

146 Hz

(for PCB)

(8.32)

SINE SWEEP THROUGH A RESONANCE

Several military specifications (MIL-E-5400, MIL-T-5422, and MIL-STD-810) recommend using sinusoidal sweeps through resonant points as part of a qualification test on many different types of electronic systems. Such a sine

186

PREVENTING SINUSOIDAL VIBRATION FAILURES

I

--J 1' - /n (c'

--

I

-L_*

rreaiie"

I

I

F-

FIGURE 8.12. Half-power points for a transmissibility curve.

sweep test is also convenient for evaluating electronic subassemblies, such as PCBs, to determine their resonant frequencies and transmissibilities. The test will usually start at a low frequency, around 10 Hz, with a sweep up to 2000 Hz, then back to 10 Hz once again. A logarithmic sweep is usually used, so that the time to sweep from 10 to 20 Hz is the same as the time to sweep from 1000 to 2000 Hz. The sweep rate is usually specified in terms of octaves per minute. an octave being a factor of 2 in frequency. This rate is thus related to the time it will take to sweep from 10 to 20 Hz, from 20 to 40 Hz, from 40 to 80 Hz, and so on. Most of the damage accumulated during a sweep through the resonant points of an electronic structural assembly will occur near the peak response points. A convenient reference is the half-power points, used extensively by electrical engineers to characterize resonant peaks in electronic circuits. These are the points where the power that can be absorbed by damping is proportional to the square of the amplitude at a given frequency. For a lightly damped system, where the transmissibility Q is greater than about 10, the curve in the region of the resonance is approximately symmetrical. The half-power points are often taken to define the bandwidth of the system as shown in Fig. 8.12. The time it takes to sweep through the half-power points can be determined from the following expression:

t=

R log, 2

where t = time (minutes) R = sweep rate (octaves/min) Q = transmissibility at resonance (dimensionless)

(8.33)

SINE

Sample Problem-Fatigue

SWEEP THROUGH A RESONANCE

187

Cycles Accumulated During a Sine Sweep

A qualification test program requires several sine vibration sweeps through the 320-Hz resonant frequency of a PCB with a transmissibility Q of 20. Previous vibration tests were run on a similar PCB using the same input acceleration level with a resonant dwell condition, which demonstrated a component fatigue life of approximately 10 min. (a) How many fatigue cycles are accumulated during one sweep through the resonance using a sweep rate of 0.5 octaves/min? (b) How many single sweeps through the resonant point are required to produce a failure? Solution. (a) The time it takes to sweep through the PCB resonant point can be determined from Eq. 8.33, where t = time (minutes) R = 0.5 octaves/min (sweep rate) Q = 20 (transmissibility of PCB)

The number of fatigue cycles accumulated during one sweep through the resonant frequency of 320 Hz will be

n = (320 cycles/s) (8.66 s)

= 2771

cycles per sweep

(8.35)

(b) The number of single sweeps through the resonant point to produce a fatigue failure can be determined as follows:

10 min to fail NS =

0.144 min/sweep

= 69.4

sweeps to fail

(8.36)

-

CHAPTER 9

Designing Electronics for Random Vi bration

9.1

INTRODUCTION

Random vibration is being specified for acceptance tests, screening tests, and qualification tests by commercial, industrial, and military manufacturers of electronic equipment, because it has been shown that random vibration more closely represents the true environments in which the electronic equipment must operate. This includes airplanes, missiles, automobiles, trucks, trains, and tanks as well as chemical processing plants, steel rolling mills, foundries, petroleum drilling machines, and numerically controlled milling machines. Random vibration has also proved to be a very powerful tool for improving the manufacturing integrity of electronic equipment by screening out defective components and defective assembly methods, which results in a sharp improvement in the overall reliability of the system. Electronic packaging designers and engineers must understand the fundamental nature of random vibration and fatigue, in order to design, develop, and produce cost-effective and lightweight structures that are capable of operating in the desired environments with a high degree of reliability. They must examine the path that the dynamic load takes as it passes through the structure, to make sure there are no weak links that can cause catastrophic failures. They must also examine the load-carrying capability of the structural elements, to make sure they will not buckle under the expected dynamic loads. They must constantly be on the alert for major structural resonances that can magnify dynamic loads and stresses. If two major structural resonances occur close to one another, they may produce severe dynamic coupling effects that can produce rapid fatigue failures.

9.2

BASIC FAILURE MODES IN RANDOM VIBRATION

There are four basic failure modes that must be considered and controlled in order to produce a reliable electronic system. These failures are the results of 188

CHARACTERISTICS OF RANDOM VIBRATION

189

the following conditions: 1. 2. 3. 4.

High acceleration levels. High stress levels. Large displacement amplitudes. Electrical signals out of tolerance.

Considering high acceleration levels, there are a number of electronic components, such as relays and crystal oscillators, that will malfunction electrically when their internal resonances are excited. These components will not suffer catastrophic failures during the vibration; they will just not operate properly. For example, if the relays chatter, the electrical signal may be impaired and an electrical failure can occur. When the resonance is excited in a crystal oscillator, the electronic signal may be distorted enough to produce an electrical failure. Therefore, it is important to use electronic components that will be capable of proper operation in the expected environment. It is also important to locate these sensitive electronic components in areas that will not respond to severe resonant conditions. For example, mount the components at the edges of a plug-in PCB and not at the center. Considering high stress levels, if they occur in major structural elements, then catastrophic failures can occur. This can usually be avoided by increasing the stiffness of the structure to raise the resonant frequency. This reduces the dynamic displacements and stresses, so the fatigue life is improved. Considering large displacement amplitudes, this often results in collision between adjacent PCBs when they are too close together, or when insufficient attention has been given to the placement tolerances. Collision between components on adjacent PCBs can result in broken and cracked components, circuit traces, and solder joints. PCB resonant frequencies must be high enough to limit the dynamic displacements, and positioning tolerances must be controlled to properly locate the PCBs. Considering electrical signals out of tolerance, this can be caused by relative motion in cables and harnesses, high temperatures, relative motion within some capacitors, resonances within crystal oscillators, a slipping potentiometer, relative motion in a transformer core, and excessive motion in a tube filament.

9.3 CHARACTERISTICS OF RANDOM VIBRATION

The most obvious characteristic of random vibration is that it is nonperiodic. A knowledge of the past history of random motion is adequate to predict the probability of occurrence of various acceleration and displacement magnitudes, but it is not sufficient to predict the precise magnitude at a specific instant.

190

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

1

D sp'ace-nent-trace

FIGURE 9.1. Typical plot of acceleration o r displacement with respect to time.

FIGURE 9.2. R a n d o m motion is made up of many different overlapping sinusoidal curves.

A typical time curve for random vibration would appear as shown in Fig. 9.1. Analysis of Fig. 9.1 reveals that the random motion can be broken up into a series of overlapping sinusoidal curves, with each curve cycling at its own frequency and amplitude, as shown in Fig. 9.2.

9.4 DIFFERENCES BETWEEN SINUSOIDAL AND RANDOM VIBRATIONS

Random vibration is unique in that all of the frequencies within a given bandwidth are present all of the time, and at any instant of time. This means that when an electronic system is subjected to a random vibration environment over a frequency bandwidth from 20 to 2000 Hz, all of the structural resonances of the electronic system within the same bandwidth will be excited at the same time. In other words, the fundamental resonant frequency of the chassis will be excited along with many of the higher harmonics. The same thing is true for all of the circuit boards in the system.

DIFFERENCES BETWEEN SINUSOIDAL AND RANDOM VIBRATIONS ,--Mass

191

2

V bration

direction

I

b Oil film

/

FIGURE 9.3. Two spring-mass systems with two different resonant frequencies on an

oil-film slider plate.

When the same electronic system is subjected to a sinusoidal sweep vibration input from 20 to 2000 Hz, each structural resonance in the given frequency band for the chassis and the various circuit boards will be excited individually. Since electronic systems respond differently in random vibration than in sinusoidal vibration, this means that fatigue failures due to random vibration can be different than fatigue failures due to sinusoidal vibration. This can be demonstrated by considering two cantilevered structures with two different resonant frequencies as shown in Fig. 9.3. Mass 1 is shown as having a low resonant frequency, and mass 2 a high resonant frequency. During a sinusoidal sweep, mass 1 with a lower resonant frequency will develop its resonant peak first. Mass 2, with a higher resonant frequency, will develop its resonant peak later, as shown in Fig. 9.4. While the resonant frequency of mass 1 is being excited, mass 2 is quiet, so mass 1 will not strike mass 2 and there will be no damage to the system. When the resonant frequency of mass 2 is excited, mass 1 will be isolated and so will be quiet. Therefore, mass 1 will not strike mass 2, and there will be no damage to the system.

Resonance of mass 1

81

fl

fz

fl

fz

FIGURE 9.4. Two masses with different resonant frequencies.

192

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

In the random vibration environment all of the exciting frequencies within the given bandwidth are present at the same time. Therefore, the forcing frequencies that correspond to the resonant frequencies of masses 1 and 2 are present at the same time, so masses 1 and 2 will be excited at their individual resonant frequencies at the same time. Both masses can then produce large displacement amplitudes at the same time, so they can strike each other and produce impact failures. This shows that random vibration environments can produce failures that cannot be duplicated in a sinusoidal vibration environment. 9.5

RANDOM VIBRATION INPUT CURVES

There are many different types of curves that can be used to show the random vibration input requirements. The most common curve, which is also the simplest, is the white-noise curve shown in Fig. 9.5. Random vibration input and response curves are typically plotted on log-log paper, with the power spectral density, expressed in squared acceleration units per hertz (G2/Hz), plotted along the vertical axis, and the frequency (Hz) plotted along the horizontal axis. The power spectral density P is often referred to as the mean squared acceleration density, and it is defined by G2 P = lim Af-0 Af

In the above equation, G is the root mean square (RMS) of the acceleration expressed in gravity units, and Af is the bandwidth of the frequency range expressed in hertz. Root mean square acceleration levels are related to the area under the random vibration curve. The input RMS acceleration levels can be obtained by integrating under the input random vibration curve, and the output (or response) RMS acceleration levels can be obtained by integrating under the

FIGURE 9.5. Typical white-noise curve with a constant input power spectral density (PSD).

fi

/1

iog (frequency)

.Y

RANDOM VIBRATION UNITS

10

193

2000 Frequency, Hz

FIGURE 9.6. White-noise input curve with a constant input PSD of 0.20 G2/Hz.

output (or response) random vibration curve. The square root of the area then determines the RMS acceleration level, with units

Sample Problem-Determining the Input RMS Acceleration Level

Determine the input RMS acceleration level resulting from the white-noise (flat-top) random vibration input specification shown in Fig. 9.6.

Solution. GRMs

=

GRMS =

9.6

JET= d0.20(2000 - 10) 19.95 (RMS input acceleration level)

(9.3)

RANDOM VIBRATION UNITS

Random vibration environments in the electronics industry normally deal in terms of the power spectral density P (or mean squared acceleration density), which is measured in gravity units so that it is dimensionless. That is, the acceleration is divided by the acceleration of gravity: a

acceleration

g

gravity

G=-=

(dimensionless)

An acceleration level of 10 G means that the acceleration has a magnitude that is 10 times greater than the acceleration of gravity. The most common method used for evaluating random vibration is in terms of the power spectral density. However, random vibration can also be expressed in terms of velocity spectral density, and in terms of displacement

194

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

\

Negative

(-

1 slope

spectral density as shown below: Quantity

Dimensions

Unit

(acceleration)'

Power SD

Velocity SD

(in./s'):

frequency

Hz

(velocity)' frequency

(in/s)' Hz

(displacement)'

Displacement SD

in.' -

Hz

frequency

Random vibration input curves are usually plotted on log-log paper, using straight sloping lines. A straight line sloping up to the right is considered to have a positive ( + ) slope, and a straight line sloping down to the right is considered to have a negative ( - ) slope, as shown in Fig. 9.7. Accelerations, velocities, and displacements for random vibration are typically expressed in terms of root mean square (RMS) values. These should not be confused with root sum square (RSS) values. The difference between the RMS and the RSS values for a set of numbers can be shown by selecting the set 2,6,10,14, and determining the RMS and the RSS values: RMS =

4

2'-+ 6'

+

RSS = \/22 + 6* 10' 9.7

+ 14'

=

9.165

+ 142 = 18.330

SHAPED RANDOM VIBRATION INPUT CURVES

Input random vibration curves can come in a wide variety of shapes, depending on the type of environment or condition the curve is trying to simulate. Two of the more common types of curves are shown in Fig. 9.8. The square root of the area under the curve still represents the input RMS acceleration level. Since these curves are plotted on log-log paper, special equations must be used to determine the areas under the sloped sections of the curves, Only straight vertical lines must be used to break up

SHAPED RANDOM VIBRATION INPUT CURVES

Frequency

195

Frequency

FIGURE 9.8. Two typical random vibration input PSD curves.

the area segments under the sloped sections of the curves. Horizontal lines must not be used. The subscript 1 below refers to the left side of the area segment being analyzed, and the subscript 2 refers to the right side. Areas under the positive- or negative-slope sections can be determined with the following equations, when the slope is not -3:

or (9.4a) When the sloped section of the PSD curve has a value of -3, then the following equations can be used to find the area under the curve:

A

= -f 2

fl

P, log, f 2

(9.5)

or A

f 2

= flP, loge-

(9 S a )

fl

The area under the flat-top section, where the slope is zero, can be determined from the following relation:

Sample Problem-Input

RMS Accelerations for Sloped PSD Curves

Determine the input RMS acceleration level for the random vibration curve when it is drawn on log-log paper, as shown in Fig. 9.9.

196

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

~;:l~-y Slope

Slooe

~

6 dB octave

FIGURE 9.9. Breaking u p an input PSD curve using straight vertical lines to produce three area sections.

5

75

200 2000

Frequency i i z

Solution. Vertical lines are used to break up the area into three sections. The area of each section is determined individually; then they are added together to obtain the total area under the curve. Area 7

Using Eq. 9.4, where fl

=5Hz

f2 =

75 HZ

P? = 0.20 G ~ / H ~

we have S

=3

Substitute into Eq. 9.4:

A, =

Area 2

3(0.20) ~

3+3

dB/octave

1 [ 7t)i3 75 - -

( 5 ) =7.47G2

Using Eq. 9.6, A , = (0.20)(200 - 75)

Area 3

(slope)

=

25.0 G 2

Using Eq. 9.4, where p 2 = 0.002 G ~ / H Z

f,= 200 Hz fi = 2000

Hz

(9.7)

RELATION BETWEEN DECIBELS AND SLOPE

S = - 6 dB/octave

3(0.002)

A,

=

3-6

[

(slope)

[ ,'"o"o

2000-

)-6'3

(200)

-

]

= 36.0

G2

197

(9.9)

The total area under the curve leads to the RMS acceleration: A,=A, +A, +A,

7.47

=

GRMS= 468.47 = 8.27 G

+ 25.0 + 36.0 = 68.47 G2

(RMS input acceleration)

(9.10)

The areas under the PSD curves in the previous sample problem can also be determined by using integration methods. The slope in dB per octave must first be changed to a pure slope number. 9.8 RELATION BETWEEN DECIBELS AND SLOPE

The term decibel, as it is used in random vibration, is used to measure PSD ratios P,/P, as follows: Number of decibels (dB)

=

p2 10 log,, -

(9.11)

PI

When a PSD plotted on log-log paper gives a straight line, its slope can conveniently be expressed in dB/octave. For example, when the PSD at point 2 is two times greater than the PSD at point 1, then in terms of decibels this ratio is expressed as follows:

1010g1,(2)

=

lO(0.301)

= 3.01 dB

(9.12)

Another way of saying the same thing is that to increase by 3 dB means (nearly) to double; to decrease by 3 dB means (nearly) to cut in half. Figure 9.10 shows several frequently used positive and negative slopes expressed in dB/octave, using straight lines on a log-log plot. Note that a slope of 3 dB (which is the same as 3 dB/octave) is represented by a line at a 45" angle, which represents a pure-number slope value of one. A slope of 6 dB/octave then represents a pure-number slope value of two, which is represented by an angle of 63.4'. Another relation that is often convenient for plotting straight lines on log-log paper is as follows: 3 dB/octave

=

10 dB/decade

(9.13)

198

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

30 dB

1003

- 9 dB’octave

9 dB octave N

I

100

>. 5 d B ortave L

-6 dB’octave a ‘A a

10

3 dB octave

- 3 dB octave 1

‘1

2

10

100

1000

Frequency, Hz

FIGURE 9.10. Representing the sloped lines on a PSD plot in terms of decibels.

9.9 INTEGRATION METHOD FOR OBTAINING THE AREA UNDER A PSD CURVE

Areas under PSD curves can also be determined from the basic area relation using logarithm equations, Consider the following point-slope equation from analytical geometry:

Y=SX+b

(9.14)

where Y = log P (where P = PSD in G’/HZ> S = slope (pure number) X = log f (frequency) b = log b intercept on the Y (or PSD) axis Then log P

= S log f

+ log b

(9.15)

Rewriting the above expression in exponent form, P=bfs

(9.16)

The area under the curve can be determined from the standard integration equations as follows: A

= /’YdX 1

= /’Pdf 1

= 12bf 1

df

(9.17)

INTEGRATION METHOD FOR OBTAINING THE AREA UNDER A PSD CURVE

199

Starting with area 1 in Fig. 9.9, the b intercept on the P axis must be determined where the frequency f equals 1 Hz. This can be obtained from point 1 or point 2, since they are both on the same line with the same slope:

P,

= 0.0133

G2/Hz

f,=5Hz or

P,

= 0.20

G*/HZ

f 2 = 75 HZ

The slope S = 1.0 (equivalent to 3 dB/octave). Substitute into Eq. 9.16 for the intercept b: b=-

0.20 or -- 0.00266 G2/Hz (5)’ (75)’

0.0133

(9.18)

Substitute into Eq. 9.17 and integrate between the limits of 5 and 75 Hz for area 1 as shown in Fig. 9.9:

[;J

A , = ~ ~ 0 . 0 0 2 6 6 dff ’= 0.00266 -

0.00266 A, = 7 [ ( 7 5 ) , - ( 5 ) 2 ]= 7.46 G 2

(9.19)

i-

Comparing the results with Eq. 9.7 shows that the two different methods of analysis agree very well. The integration method for finding area 2 will be exactly the same as Eq. 9.8. The integration method for finding area 3 will be the same as the method shown for area 1, except that a new intercept b must be obtained using Eq. 9.16. The 200-Hz point or the 2000-Hz point can be used in Fig. 9.9, since they are both on the same line:

P,

= 0.2

f l = 200

G2/Hz Hz

or

p2 = 0.002 G ~ / H ~

f 2 = 2000 Hz

200

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

The slope S = - 2 (equivalent to - 6 dB/octave). Substitute into Eq. 9.16 for the intercept b: b=

0.20 (200) - 2

0.002 Or

(2000) -*

= 8000

G2/Hz

(9.20)

Substitute into Eq. 9.17 and integrate between the limits of 200 and 2000 Hz for area 3, as shown in Fig. 9.9: 2000

A,

=~

A,

=

~ ~ 0 8 0 0 0df f -=28000

8000 (200)-’] ~ [ ( 2 0 0 0 )- ~

= 36.0

G2/Hz

(9.21)

Comparing the results with Eq. 9.9 shows the two areas and the two different methods of analysis agree very well. 9.10 FINDING POINTS ON THE PSD CURVE

Random vibration input PSD curves are often specified in terms of the frequency break points and the slope in dB, with only one G2/Hz point defined as shown in Fig. 9.11. When it is necessary to find the PSD level at these break points, then it is convenient to use the relation

(9.22)

Sample Problem-Finding

PSD Values

Determine the PSD values at break points 1 and 2 as shown in Fig. 9.11.

Slope

FIGURE 9.11. Locating break points on a PSD curve.

Sloae

Frequency Hz

USING BASIC LOGARITHMS TO FIND POINTS ON THE PSD CURVE

201

Solution. At break point 1, f1=5Hz 75 HZ

f,

=

P,

= 0.20

S

=3

G~/HZ

dB/octave

Substitute into Eq. 9.22: 5

P, = 0.20(

3/3

E)

= 0.0133

G2/Hz

(9.23)

Use Eq. 9.22 to find the PSD level at break point 2:

Pz =

0.20 (200/2000)

= 0.002

G~/HZ

(9.24)

-6’3

USING B SIC LOGARITHMS TO FIND POINTS ON THE PSD CURVE

9.1

The two-point slope equation from analytical geometry can also be used to obtain PSD values at break points 1 and 2 in Fig. 9.11:

y , -Y,

= S ( x 2 -1 .>

(9.25)

Since the PSD relation is plotted on log-log paper, Fig. 9.25 has the following meaning: log Y2- log Y,= S(l0g x,- log X , )

(9.26)

At point 1 substitute the known values from Fig. 9.11 into Eq. 9.26 and note that a slope of 3 dB/octave is the same as a pure-number slope of 1.0 at point 1: log 0.2 - log Yl = ( 1.O) (log 75 - log 5) log Yl = - 1.875

Yl = 0.0133 G2/Hz

(9.27)

At point 2 substitute the known values from Fig. 9.11 into Eq. 9.26 and note

202

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

that a slope of - 6 dB/octave is the same as a pure-number slope of -2: log Y2- log 0.2 = ( - 2) (log 2000 - log 200) log Y, = - 2.7

y2= 0.002 G?/HZ

(9.28)

Comparing Eq. 9.23 with 9.27 and Eq. 9.24 with 9.28 shows they agree very well. 9.12

PROBABILITY DISTRIBUTION FUNCTIONS

In order to predict the probable acceleration levels the electronic equipment will see in a random vibration environment, it is necessary to understand probability distribution functions. The distribution most often encountered, and the one that lends itself most readily to analysis, is the Gaussian (or normal) distribution, which is defined by

e -x Y=

/2u

(9.29)

a+%

The right side of the above equation represents the probability density function, or the probability, per unit of X , for the ratio of the instantaneous acceleration ( X I to the RMS acceleration ( a1. 9.1 3

GAUSSIAN OR NORMAL DISTRIBUTION CURVE

The Gaussian distribution curve shown in Fig. 9.12 represents the probability for the value of the instantaneous acceleration levels at any time. The abscissa is the ratio of the instantaneous acceleration to the RMS acceleration, and the ordinate is the probability density, sometimes called the

-4

-3

-2

-1

0

1

2

3

4

-

'1

FIGURE 9.12. Gaussian distribution curve. T h e probability of a range of X is given by the area under the curve in that range.

GAUSSIAN OR NORMAL DISTRIBUTION CURVE

203

c x

0.1

46 (95.4%) 20

n W

-

001

R

fl

0

0027 (99 73%,) 3n

0001

Q

0 0001

0

1

2

3

n/n

FIGURE 9.13. Another method for showing the Gaussian distribution.

probability of occurrences. The total area under the curve is unity. The area under the curve between any two points then directly represents the probability that the accelerations will be between these two points. For example, the shaded area under the curve in Fig. 9.12 shows that the instantaneous accelerations will be between l a and - l a about 68.3% of the time. Figure 9.12 can be presented in another way, as shown in Fig. 9.13. This plot shows the probability that a given acceleration level will be exceeded. Figures 9.12 and 9.13 show how the Gaussian distribution relates to the magnitude of the acceleration levels expected for random vibration. The instantaneous acceleration will be between the + 1a and the - l a values 68.3% of the time. It will be between the +2a and the - 2 a values 95.4% of the time. It will be between the + 3 a and the - 3 a values 99.73% of the time. Another way of expressing the Gaussian distribution is shown in Fig. 9.13 as follows. The instantaneous acceleration will exceed the l a value, which is the RMS value, 31.7% of the time. It will exceed the 2a value, which is two times the RMS value, 4.6% of the time. It will exceed the 3 a value, which is three times the RMS value, 0.27% of the time. It is important to remember that in a random vibration environment, all of the frequencies in the bandwidth are present instantaneously and simultaneously. Likewise, the l a (or RMS), the 2a, and the 3 a acceleration levels are all present at the same time in the proportions shown above. Remember also that the square root of the area under the PSD-versus-frequency curve represents the RMS accelerations in gravity units (GI. The square root of the area under the input PSD curve represents the input RMS acceleration level, and the square root of the area under the response (or output) PSD curve represents the response RMS acceleration level. The maximum acceleration levels considered for random vibrations are the 30 levels, because the instantaneous accelerations are between the + 3 a and the - 3 a levels 99.73% of the time, which is very close to 100% of the time. Higher acceleration levels of 4a and 5a can occur in the real world, but they are usually ignored because virtually all of the test equipment for random vibration have 3 a clippers built into the electronic control systems.

+

204

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

FIGURE 9.14. A time history of a typical random vibration acceleration trace.

These clippers limit the input acceleration levels to values that are 3 times greater than the RMS input levels. A real plot of the random vibration acceleration or displacement with respect to time will appear as shown in Fig. 9.14. This same practice is typically applied to the response (or output) acceleration levels. For the purposes of analysis, the maximum acceleration levels considered are the 3a levels. This means that when a 10-G RMS random vibration environment is being evaluated (it does not matter if it is an input or a response), then 2a accelerations of 2 X 10 or 20 G can be expected some of the time, and 3a accelerations of 3 X 10 or 30 G can be expected some of the time. Most of the damage will be generated by the 3a levels, since they are the maximum levels expected in this environment. Displacements, forces, and stresses will occur in exactly the same proportions as the accelerations described above, for linear systems. In other words, the maximum displacements, forces, and stresses expected in a random vibration environment will be 3 times greater than the RMS displacements, forces, and stresses. Random vibration qualification tests performed in a laboratory will typically be run using much higher acceleration levels than the values found in the actual environments. The laboratory may use input test levels of 8.0 G RMS for a period of 2 hours, to try to accumulate the same amount of damage that is expected in a 1.5-G RMS environment over a 15-year period. Random vibration qualification tests are often considered to be accelerated life tests.

9.14 CORRELATING RANDOM VIBRATION FAILURES USING THE THREE-BAND TECHNIQUE

Random vibration is a complicated phenomenon that often produces rapid and unexpected fatigue failures. Many large companies have extensive computer databases that contain a vast amount of stress and fatigue data from testing programs on specific types of structures. The computers are used to sort the fatigue data to evaluate the approximate fatigue life of other similar structural members in similar environments. Many small companies do not have the time or money to run large testing projects with sophisticated

RAYLEIGH DISTRIBUTION FUNCTION

205

programs on high-speed digital computers. A simplified approach to the evaluation of random vibration fatigue, without the use of a large computer, would therefore be a valuable tool. The three-band technique offers such a tool. This technique is based on a large volume of test data that was arranged and rearranged to see if a simplified method could be developed to analyze random vibration fatigue failures with reasonable accuracy. The basis for the three-band technique is the Gaussian distribution. The l a and - l a are assumed to act at instantaneous accelerations between the l a level 68.3% of the time. The instantaneous accelerations between the + 2a and the - 2 a are assumed to act at the 2a level 95.4 - 68.3, or 27.170, of the time. The instantaneous accelerations between +3a and -3a are assumed to act at the 3a level 99.73 - 95.4, or 4.3370,of the time. These values are shown below for the three bands:

+

l a values occur 68.3% of the time 2a values occur 27.1% of the time 30. values occur 4.33% of the time

(9.30)

RAYLEIGH DISTRIBUTION FUNCTION

9.15

The Rayleigh distribution function shows the probability for the distribution of the peak accelerations in a random vibration environment. The curve is shown in Fig. 9.15. The total area under the curve is equal to unity. The area under the curve between any two points then represents the probability that a peak amplitude will be between these two points. Another way in which the Rayleigh distribution can be presented is shown in Fig. 9.16. This plot shows what the probability is for exceeding some peak acceleration level. The Rayleigh distribution shows the following relations: Peak accelerations will exceed the l a level 60.7% of the time. Peak accelerations will exceed the 2a level 13.5% of the time.

0

1

2 no

3

4

FIGURE 9.15. The curve.

Rayleigh

probability

distribution

206

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

0 0!2

0 01

(98 B o o )

0 001 0 0003 (999 7 % ) 0 0031

0

1

2

3

4

0,'"

FIGURE 9.16. Another method for showing the Rayleigh distribution.

Peak accelerations will exceed the 3a level 1.2% of the time. Peak accelerations will exceed the 4a level 0.03% of the time. Extensive random vibration testing and analysis have shown that fatigue failures can be correlated with much greater accuracy using the Gaussian distribution rather than the Rayleigh distribution. Therefore, only the Gaussian distribution will be used for fatigue failure analysis with random vibration in this book.

9.16 RESPONSE OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM TO RANDOM VIBRATION

When a single spring-mass system is excited by random vibration, the system responds by vibrating at its resonant frequency, with a varying displacement amplitude, as shown in Fig. 9.17. When the base of the single-degree-of-freedom system is excited by a continuous random vibration whose spectrum is nearly flat in the area of the resonance, the mean square acceleration response of the mass can be obtained by considering the area under the PSD-versus-frequency curve as defined in Eq. 9.2. The area can be obtained by integrating across the

Randon

!

Input

FIGURE 9.17. A displacement history for the response of a spring-mass system to a

random vibration input.

207

RESPONSE OF A SINGLE-DEGREE-OF-FREEDOMSYSTEM

frequency band from f i and

f2

Area

using the output PSD (Pout)as follows: = G& = /"Pout df

(9.31)

fl

The response (or output) Poutis needed to obtain the response of the mass to the random vibration input. Since only the input PSD P is usually known, it is desirable to use P in the above equation. This can be accomplished by using the following expression [l]:

Pout= Q2P

(9.32)

The transmissibility Q for the single-degree-of-freedom system is obtained from Eq. 2.48. For a lightly damped system, where the Q is greater than about 10, the transmissibility will be as shown in Eq. 2.46. Substitute Eqs. 2.46 and 9.32 into Eq. 9.31:

The above equation can be integrated to obtain the mean square acceleration response of the mass to the random vibration input: (9.34)

In a lightly damped system the damping ratio can be related to the transmissibility as follows: (9.35) Substitute into Eq. 9.34 to obtain the RMS response of the mass to the broadband random vibration input: (9.36) where P = input PSD in units of G2/Hz at the resonant frequency f, = resonant frequency (Hz) Q = transmissibility at the resonant frequency

208

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

--Random

--_,Response

v bration input curve

for a simple system

Freauency. Hz

FIGURE 9.18. The random vibration input PSD curve should be flat in the area of

the resonance.

The above equation is valid when the random vibration PSD input is flat in the area of the resonance, as shown in Fig. 9.18. The amount of error is small even when the slope of the random vibration input curve is 6 dB/octave in the area of the resonant frequency. Therefore, Eq. 9.36 can be used to obtain good values under most conditions. Sample Problem-Estimating

the Random Vibration Fatigue Life

A 6061-T6 cantilevered aluminum beam 6.0 in. long by 0.50 in. wide by 0.30 in. high has a transformer mounted at the end of the beam as shown in Fig. 9.19. The total weight of the assembly is 0.50 Ib, and it is restricted to move only in the vertical direction. The assembly must be capable of operating in a white-noise random vibration environment with an input PSD level of 0.30 G*/Hz, from 20 to 2000 Hz, for a period of 4.0 hours. Determine the approximate dynamic stress and the expected fatigue life of the assembly. Solution No. 7 Use the Three-Band Technique and Miner's Cumulative Damage Ratio. The natural frequency is the heart of the system. The analysis can be simplified by finding the static displacement of the beam with an end mass and consider it to be a single-degree-of-freedom system. This will save time and the resulting error is small.

l+jJ

-

6.0 Sect. AA

FIGURE 9.19. Cantilever beam with an end mass.

RESPONSE OF A SINGLE-DEGREE-OF-FREEDOMSYSTEM

209

A. Static Displacement of the Cantilever Beam with an End Mass

y

WL3 =-

3EI

St

(see Eq. 2.11)

where W = 0.50 Ib (total weight) L = 6.0 in. (beam length) E = 10.5 X lo6 Ib/im2 (aluminum modulus of elasticity) I = (0.50)(0.30)3/ 12 = 0.00112 in.4 (moment of inertia)

(0.50) ( 6.0)3

E’=

(3)(10.5 X 106)(0.00112)

st

= 0.00306

in. (static displacement)

The natural frequency can be obtained using Eq. 2.10 when gravity is 386 in./s2:

6. Natural Frequencj Based on the Static Displacement and Gravit]

C. Response of Beam to Random Vibration The response of the beam to random vibration can be obtained from Eq. 9.36 using the following information:

P = 0.30 G2/Hz (PSD input) f,, = 56.5 Hz (beam natural frequency) Q =2fi =2 m = 15 (approximate beam transmissibility) G (RMS)

=

/:(0.30)(56.5)(15)

= 20.0

G

(RMS acceleration response)

(9.38)

D. Dynamic la RMS Bending Stress in the Beam Acting 68.3% of the Time Alternating stresses must always include stress concentrations when thousands of stress cycles are expected in a structure. The stress concentration K can be used in the stress equation or in defining the slope b of the S-N fatigue curve for alternating stresses. The stress concentration should be used only once in either place. For this sample problem a stress-concentration factor K = 2 will be used in the S-N fatigue curve as shown in Fig. 9.22, where slope b = 6.4. MC

S, = - (see Eq. 2.39) I

210

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

Frequency,

Hz

FIGURE 9.20. Transmissibility Q response plot of the cantilever beam with a n e n d mass.

-67.5

56.5 Frequency, Hz

FIGURE 9.21. PSD response plot of the cantilever beam with a n e n d mass.

WG (RMS) L = (0.50)(20.0)(6.0) = 60.0 lb .in. (RMS bending moment) c = 0.30/2 = 0.15 in. (distance to neutral axis) I = 0.00112 in.4 (moment of inertia)

where M

=

A plot of the transmissibility curve versus frequency will appear as shown in Fig. 9.20. A plot of the PSD response curve can be obtained with the use

RESPONSE OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM

I

21 1

6061-T6 Aluminum

I

103

108

N,cycles to fail FIGURE 9.22. S-N fatigue curve for 6061-T6 aluminum beam with a stress concen-

tration of 2. of Eq. 9.32. The resulting curve is shown in Fig. 9.21. la&=

(60.0)( 0.15) 0.00112

= 8036 lb/in.2

(RMS bending stress) (9.39)

E. Number of Stress Cycles Needed to Produce a Fatigue Failure Using the Three-Band Method The approximate number of stress cycles Nl required to produce a fatigue failure in the beam for the l a , 2 a , and 3a stresses can be obtained from Fig. 9.22 and the following equation: b

(seeEq.3.7) where

N,= 1000 cycles to fail at 45,000 lb/in.* (reference point) S,

= 45,000

lb/in.* (stress to fail reference point) S, = 8036 1 b / h 2 ( l a RMS stress) b = 6.4 (slope of fatigue line with stress concentration K careful to use the stress concentration K only once)

RMS l a N ,

(

45,000

]

be

6.4 = 6.14 X

lo7 cycles to fail

2 0 N2= (1000)

= 7.27 X

l o 5 cycles to fail

3a N3 = (1000)

= 5.43

(1000) 8036

= 2;

x i o 4 cycles to fail

I

(9.40)

21 2

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

F. Actual Number of Stress Cycles Accumulated During a 4-Hour Vibration Test The actual number of fatigue cycles n accumulated during 4 hours of vibration testing can be obtained from the percent of time exposure for the l a , 2 a , and 3 u values in the three-band method of analysis shown in Eq. 9.30. 1a n , = (56.5 cycles/s)(3600 s/h)(4 h)(0.683)

= 5.56 X

10’ cycles

2a n 2 = (56.5 cycles/s)(3600 s/h)(4 h)(0.271)

=

2.20

10‘ cycles

3 u n3 = (56.5 cycles/s)(3600 s/h)(4 h)(0.0433)

X

= 3.52 X

104 cycles

I

(9.41) G. Use Miner’s Cumulative Fatigue Damage Ratio to Estimate Fatigue Life Miner’s cumulative fatigue damage ratio is based on the idea that every stress cycle uses up part of the fatigue life in a structure. It does not matter if the stress cycle is due to sinusoidal vibration, random vibration, thermal cycling, shock, or acoustic noise. A fatigue cycle ratio of n / N is used to show the percentage of life used up. The actual number of stress cycles generated in a specific environment is shown as M . The number of cycles required to produce a fatigue failure in a specific environment is shown as N . When the ratios are all added together, a sum of 1.0 or greater means that all of the life has been used up so the structure should fail. The cumulative fatigue damage ratio equation is shown below:

(9.42) Substitute Eqs. 9.40 and 9.41 into Eq. 9.42 to obtain Miner’s fatigue damage cycle ratio: 2.20 X l o 5 3.52 X l o 4 7.27 x 10’ 5.43 x l o 4 R,, = 0.009 +- 0.303 + 0.648 = 0.960 5.56

X

loi

R1’= 6.14 x 10’

+

+

(9.43)

An examination of the above fatigue cycle ratio shows that the 1a RMS level does very little damage even though it acts about 68.3% of the time. Most of the damage is generated by the 3a level even though it acts only about 4.33% of the time. The 3a level generates more than two times as much damage as the 2 a level, which acts about 27.1% of the time. The above fatigue cycle ratio shows that about 96% of the life is used up by the 4-hour vibration test. This means that 4% of the life is left. The expected life of the structure can be obtained as shown below. Expected life

= 4.0

h

+ (4.0)(1.0 - 0.960) = 4.0 + 0.16 = 4.16 hours

(9.44)

RESPONSE OF A SINGLE-DEGREE-OF-FREEDOMSYSTEM

21 3

Since fatigue has a large amount of scatter, the proposed beam does not have a sufficient safety factor to ensure its fatigue life for the environment. The beam design should be changed to provide a fatigue life of about 8 hours. This is equivalent to a safety factor of 2 on the fatigue life. The beam S-N fatigue curve already included a safety factor of 2 based on the structural stress level. This resulted in a fatigue exponent slope b of 6.4. Therefore, the real fatigue life is expected to be greater than about 4.16 hours. Because fatigue is very difficult to predict accurately, it is better to play it safe by adding another safety factor of 2 on the life. This is only equivalent to a structural safety factor of 1.11, as shown below. Structural safety factor

=

(2)1'6.4 = 1.11

Upper management executives may grumble a little bit when there is a slight increase in the size, weight, and cost, as long as the product passes the qualification tests and shows a high reliability with low maintenance costs in the field. However, if the product fails its required qualification tests, or if it has a very poor reliability record in the field, the people associated with the poor product will probably be looking for new jobs. Solution NO. 2 Quick Method for Finding Approximate Random Fatigue Life. The three-band technique for estimating the random vibration fatigue life was based on test data failures accumulated over a period of several years, to permit the use of hand calculations before computers were available. The previous sample problem required a substantial amount of work to reach a final conclusion. A simplified method for finding the random vibration fatigue life was developed, based on the total damage accumulated in the three-band method of analysis. This new quick method of analysis is based on the percentage of time exposure at the l a , 2 a , and 3 a levels as shown below. Damage D

=

C N S " = N,S,b + N2S,b+ N,S,b

Substitute the Gaussian probability distribution function shown in Eq. 9.30 for the percent of time spent at each level for the three bands, using the fatigue exponent b of 6.4:

+

+

D = (0.683)(1.0)6'4 (0.271)(2.0)6.4 (0.0433)(3.0)6'4= 72.55 (9.45) Now find one random v acceleration level that will generate the same total damage as the three-band technique for the full Gaussian distribution shown above, assuming that it acts 100% of the time.

G6,4= 72.55

SO

G = 1.95

(9.46)

214

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

This means that an acceleration response level of 1 . 9 5 ~can ~ be assumed to act 100% of the time to quickly estimate the dynamic stress level and the number of fatigue cycles required to produce a fatigue failure in a structure. The approximate fatigue life can then be obtained from the natural frequency of the structure. The previous sample problem can now be reexamined using this quick method for calculating the approximate fatigue life of the cantilevered beam structure. The previous l o RMS stress level of 8036 lb/in.2 was shown in Eq. 9.39. The number of fatigue cycles needed to produce a failure can be obtained by using the 1 . 9 5 ~level in Eq. 9.40 as shown below: 1 . 9 5N ~

=

[

(1000) ( 1 . ~ ~ ~ )6'~4 = ~ 8.55 ~ 3 X610') cycles to fail (9.47)

The approximate fatigue life of the beam can be obtained from the natural frequency of 56.5 Hz shown in Eq. 9.37: Life

=

8.55 x 10' cycles to fail (56.5 cycles/s)(3600 s/h)

= 4.20

hours to fail

(9.48)

Comparing the fatigue life of 4.16 hours using the three-band technique in Eq. 9.44 with the fatigue life of 4.20 hours using the quick analysis method shows good correlation. One note of caution is advised here. The quick method of analysis shown above only applies to the determination of the approximate fatigue life. This method of quick analysis must not be used for finding the maximum e.xpected displacements or maximum acceleration levels. The real maximum displacements expected will still be three times the RMS displacement levels and the maximum expected acceleration levels will still be three times the RMS acceleration levels. Some people like to round off the 1.95 factor to an even 2.0 factor because it is easier to remember. There is nothing wrong with this idea since it provides a slightly greater safety factor. This can be demonstrated by using Eq. 9.40, which shows a 2 o fatigue life of about 7.27 X l o 5 cycles. The approximate fatigue life can be obtained as shown below: Life 9.17

7.27 x 10' cycles to fail =

(56.5 cycles/s) (3600 s/h)

= 3.57

hours to fail

(9.49)

HOW PCBs RESPOND TO RANDOM VIBRATION

Extensive testing and analysis of PCBs show that they can be evaluated as single-degree-of-freedom systems if they are supported or restrained around the perimeter. In a random vibration environment, Eq. 9.36 can therefore be used to predict the response characteristics of typical PCBs (circular, rectangular. etc.). This applies to plug-in PCBs and to PCBs that are bolted around the perimeter. As long as the fundamental resonant frequency of the PCB

DESIGNING PCBs FOR RANDOM VIBRATION ENVIRONMENTS

/Plug-in

215

Low-level higher harmonics

PC.B

osci~ioscope

FIGURE 9.23. Oscilloscope pattern shows a PCB acts like a single-degree-of-freedom system when it is exposed to random vibration.

can be determined, the approximate response can be obtained for a random vibration environment. When an accelerometer is mounted at the center of a plug-in PCB that is excited by random vibration, and when the accelerometer response signal is picked up and displayed on an oscilloscope, it shows that the fundamental resonant frequency comes through with a large amplitude, as shown in Fig. 9.23. Since the input and the response are both random, the amplitude and the frequency can be observed to be constantly changing with small variations. The higher harmonic resonant frequencies come through as very low, constantly changing amplitudes. 9.18

DESIGNING PCBs FOR RANDOM VIBRATION ENVIRONMENTS

Electronic equipment must be designed with an understanding of the operating environment, the capability of the electronic components, and the characteristics of the support structure in order to achieve a high degree of reliability for systems exposed to random vibration. In most electronic assemblies the PCBs are fastened to a chassis, and the chassis is mounted to a structure that is excited by random vibration. Since the chassis receives the dynamic energy first, the chassis becomes the first degree of freedom. Since the PCBs must receive their dynamic energy from the chassis, the PCBs become the second degree of freedom, as shown in Fig. 9.24. Chassis

PCB /

Leaf sorine \

2nd degree of freedon PCB mass

FIGURE 9.24. A typical electronic chassis with PCBs, modeled as a spring-mass system.

216

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

FIGURE 9.25. PCB resonances can produce stresses in the electrical lead wires due to relative motion between the component body and the PCB.

Care must be used in the design of the chassis and the PCBs to avoid severe coupling of the chassis resonances and the PCB resonances. This is usually accomplished by following the octave rule: the resonant frequencies of the chassis and the PCBs are separated by an octave (a factor of 2). When the resonant frequency of the chassis is at least two times the resonant frequency of the PCB, then the PCBs will not be severely amplified by the chassis resonance. The same is also true when the resonant frequency of the PCB is at least two times the resonant frequency of the chassis. The PCBs are usually considered to be the heart of the electronic system, because almost all of the electronic components are mounted on PCBs. This makes it convenient and easy to service and repair electronic equipment by simply removing and replacing defective PCBs. The typical PCB consists of several flat, thin plastic plates, with printed electrical traces, all laminated together. This type of construction results in an oil-canning of the PCB in any vibration or shock environment. Since the electronic components are mounted on these flexible assemblies, and since some of the electronic components are large and heavy, a significant amount of relative motion can occur between the PCB and the electronic component, as shown in Fig. 9.25. Extensive finite element analysis and vibration testing of electronic systems show that the fatigue life of many different types of electronic components can be related to the dynamic displacements experienced by the PCBs that support these components. When the PCB resonant frequency is excited, the plate structure is forced to bend back and forth. When the displacement amplitudes are high, the relative motion between the components and the PCB can be high, which often results in cracked solder joints and broken electrical lead wires. The fatigue life of these components can often be increased by reducing the dynamic displacements of the PCB. Since the displacements can be controlled through the resonant frequency, it follows that the fatigue life of many different types of components can also be controlled by controlling the PCB resonant frequency. When the dynamic single-amplitude displacement at the center of a perimeter-supported PCB is limited to the value shown below, the component can be expected to achieve a fatigue life of about 20 million stress

217

DESIGNING PCBs FOR RANDOM VIBRATION ENVIRONMENTS

reversals in a random vibration environment, based on the 3a PCB displacements [38]: Z=

where B L h

0.00022 B

Chrc

(maximum desired PCB displacement)

(9.50)

length of PCB edge parallel to component (in.) = length of electronic component (in.) = height or thickness of PCB (in.) C = constant for different types of electronic components =

'1.0 1.26 1.26

1.o -

2.25 1.o 1.75 0.75

t r

=

for a standard dual inline package (DIP) for a DIP with side-brazed lead wires for a pin grid array (PGA) with two parallel rows of wires extending from the bottom surface of the PGA for a PGA with wires around the perimeter extending from the bottom surface of the PGA for a leadless ceramic chip carrier (LCCC) for leaded chip carriers where the lead length is about the same as for a standard DIP for a ball grid array (BGA) far axial-leaded component resistors, capacitors, and fine pitch semiconductors

relative position factor for component on PCB (see Eqs. 6.1 and 8.15 and Fig. 6.5) 1.0 0.707

when component is at center of PCB (i point X and Y ) when component is at point X and

f point Y on a PCB supported on four sides 0.50

when component is at

point X and

f point Y on a PCB supported on four sides The actual dynamic single-amplitude displacement expected for the PCB is shown by Eq. 2.30. This relation can also be used for random vibration when the RMS displacements are utilized in the evaluation. Since this analysis is based on a single degree of freedom, the true resonant frequency

21 8

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

can be used in place of the number of positive zero crossings associated with random vibration. For a single-degree-of-freedom system, the number of positive zero crossings is the same as the resonant frequency. Thus (9.51) The approximate transmissibility Q of the PCB can be related to the resonant frequency as follows (also see Eq. 14.21): (9.52)

Q=&

The desired PCB resonant frequency that will provide a fatigue life of approximately 20 million stress reversals can now be determined by combining Eqs. 9.50, 9.51, 9.52, and 9.36. Equation 9.36 must be modified slightly by multiplying by 3, because we are using 3 a acceleration levels. The desired PCB resonant frequency is then 2 9 . 4 C h r d m 0.00022 B Sample Problem-Finding

(minimum desired)

(9.53)

the Desired PCB Resonant Frequency

A 40-pin dual inline package (DIP) with side-brazed lead wires is to be flow-soldered to a typical 7.0 X 9.0 X 0.090-in. thick plug-in PCB. The DIP is to be mounted parallel to the 7.0-in. edge. The input PSD in the area of the PCB resonant frequency is expected to be flat at 0.080 G2/Hz. Determine the minimum desired PCB resonant frequency and the approximate fatigue life for the DIP wires on the simply supported PCB for the following mounting positions:

(i

(a) Component mounted at the center of the PCB point X and Y > . (b) Component mounted at the quarter points ( f point X and point Y ) on the PCB.

Solution

When the octave rule is followed, the PCB resonant frequency will be well separated from the chassis resonant frequency, so coupling between the chassis and the PCB will be reduced. Under these conditions it is convenient to assume that the PSD input to the PCB is the same as the input to the chassis. This is not really true, but in most cases the error is small, and the method provides quick approximate answers. This method is conservative (so the PCB will have a resonant frequency that is a little too high) when the PCB resonant frequency is two or more

Part (a)

DESIGNING PCBs FOR RANDOM VIBRATION ENVIRONMENTS

21 9

times greater than the chassis resonant frequency, for the forward octave rule. This method is slightly unconservative (so the PCB will have a resonant frequency that is slightly too low) when the chassis resonant frequency is two or more times greater than the PCB resonant frequency, for the reverse octave rule. When these conditions are followed, the fatigue life of the electronic component lead wires and solder joints will be approximately 20 million stress cycles. We have

B

= 7.0

in. (length of PCB parallel to DIP) h = 0.090 in. (PCB thickness) L = 2.0 in. (length of 40-pin DIP) C = 1.26 (constant for side-brazed DIP lead wires) P = 0.080 G2/Hz (input PSD to PCB) r = 1.0 (for component at center of PCB) Substitute into Eq. 9.53 to obtain the desired PCB frequency:

f d =

[

(29.4)( 1.26)(0.090)( 1.0)4( v/2)(0.08)(2.0)

f d = 268

(0.00022) (7 .O)

Hz

(9.54)

(minimum desired)

The fatigue life of the component can be estimated as follows: Life

20 x l o 6 cycles to fail =

(268 cycles/s) (3600 s/h)

= 20.73

(9.55)

hours to fall

A resonant frequency of 268 Hz is difficult to achieve with the PCB described above. Some type of stiffening will be required to raise the resonant frequency to this level. This may require removing some electronic components from the PCB to make room for the ribs, which will make many electrical engineers and program managers very unhappy. Reducing the number of electronic components that can be mounted on each PCB means that more PCBs will be required in the system, which may increase the size, weight, and cost. These factors must be carefully considered against the possibility of qualification test failures, or failures in the field. Part (b) When the component is mounted at the quarter points (+ point X and point Y),the relative position factor Y is reduced to 0.5 because the

relative motion between the component and the PCB is less. Substitute a

220

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

value of r = 0.5 into Eq. 9.53 and solve: fd =

154 Hz

(minumum desired)

(9.56)

Substitute into Eq. 9.55 to obtain the approximate fatigue life when the PCB has a resonant frequency of 154 Hz: Life

= 36.1

hours to fail

(9.57)

9.19 EFFECTS OF RELATIVE MOTION ON COMPONENT FATIGUE LIFE

The component fatigue life was based on the relative motion that occurs between the component body and the PCB, because this relative motion produces stresses in the lead wires and solder joints. When the stress levels are high, the fatigue life is low, and vice versa. Therefore, the stress levels in the lead wires must be reduced. This can be done in several different ways: 1. Increase the damping, which will reduce the displacements. This works well for PCBs that have a low resonant frequency, below about 50 Hz, because then dynamic displacements are large, so there is a good transfer of kinetic energy into heat. However, when PCB resonant frequencies are high, above about 200 Hz, the displacements are small, so damping is not very effective. 2. Increase the PCB resonant frequency. This is a very effective method for improving the fatigue life, because the PCB displacements are reduced very quickly. However, it is not always easy to increase the stiffness. Ribs may have to be added, which reduces the surface area available for electronic components. If the thickness is increased to increase the stiffness. electrical lead wires from components such as DIPs may not extend through the PCB, so flow-soldering reliability is reduced. 3. Reduce the spring rate of the electrical lead wires. This can be done by increasing the length of the wires, perhaps by looping them. The spring rate can also be reduced by coining the wires. In this operation the wire cross section is squeezed to make it thin and flat, to reduce the moment of inertia of the cross section. These operations will, of course, increase the manufacturing costs. 4. Add local stiffeners under the problem electronic components. Stiff metal shims can be epoxy-cemented to the PCB, under components such as large DIPs, to reduce the relative motion between the PCB and the DIP body. This will, of course, increase the cost of the assembly. 5. Reduce the acceleration input level to the system. Sometimes it is possible to obtain a change in the environmental requirements by just asking

EFFECTS OF RELATIVE MOTION ON COMPONENT FATIGUE LIFE

221

the customer for a reduction. This is always easier than modifying the design. Many companies increase the acceleration requirements to provide an additional safety factor. Sometimes it is possible to show them that their high safety factor will increase their weight and cost, so a less severe requirement is desirable. 6. Mount the large components away from the center of the PCB. Since the maximum curvature change on the PCB usually occurs at the center, it is desirable to mount large components near the sides. Of course, it is not always possible to mount electronic components where they are best suited for mechanical reasons. After all, the purpose of an electronic system is to work electrically. However, if the mechanical requirements are ignored, then it is possible for many fatigue failures to occur. Extensive failures may not only be very expensive to repair, they may ruin the reputation and the business of a company.

Sample Problem-Component

Fatigue Life

Determine the fatigue life of the PCB component lead wires in the previous sample problem, when the PSD input level is increased from 0.080 to 0.120 G2/Hz, and the PCB resonant frequency remains at 268 Hz, as shown in part (a).

Solution. Use the fatigue-damage method as shown in Eq. 8.1, and rewrite the fatigue expression to utilize the time ( T ) to fail and the acceleration ( G ) values as follows:

T,Gf

=

T,G,b

(see Eq. 8.4a)

where b = 6.4 (fatigue exponent) P , = 0.120 G2/Hz (PSD level at 268 Hz) P, = 0.080 G2/Hz (PSD level at 268 Hz) T2= 20.73 hours to fail (see Eq. 9.55) = 16.37 Q , = Q, = Substitute into Eq. 9.36 to obtain the RMS G level: G,

=

4(1.57)(0.12)(268)(16.37)

= 28.75

G RMS

G , = ~(1.57)(0.08)(268)(16.37)= 23.47 G RMS

222

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

Then for the expected life we have

1

23.47 6 , 4 T , = 20.73 - = 5.66 hours to fail 28.75

(

(9.58)

The approximate fatigue life can also be determined by considering a ratio of the displacements or the stresses. Since this is a linear system, these quantities are all directly proportional to each other. IT’S THE INPUT PSD THAT COUNTS, NOT THE INPUT RMS ACCELERATION

9.20

References are often made to the magnitude of the input RMS acceleration levels to establish the severity of some random vibration test or environment. Although the magnitude of the acceleration level can be of some importance, the magnitudes of the input PSD levels in the regions of the structure’s resonant frequencies are far more important. The input (and also the output or response) acceleration level was shown by Eq. 9.2 to be directly related to the square root of the area under the PSD curve. However, there can be an infinite number of input PSD curves that have the same area, and therefore the same input RMS acceleration level. Several typical curves with the same area, but with different shapes and different PSD levels, are shown in Fig. 9.26. When the input PSD and the resonant frequency are known, then Eq. 9.36 shows that the response acceleration is related to the value of the input PSD level at the resonant frequency. A low input PSD level will result in a low RMS acceleration response value. A high input PSD level will result in a high response value. Therefore, the RMS response levels can be different for different-shaped PSD curves, even when the RMS acceleration input levels are exactly the same. A large area under the high-frequency segment of the input curve may not produce a high acceleration response level, unless the structure has its resonant frequency and a high Q i n the region of the curve where the PSD is also high. Similarly, a low input acceleration level does not mean that the system must have a low acceleration response level.

FIGURE 9.26. Various random vibration input curves with the same areas, but with different PSD values at the same frequencies.

CONNECTOR WEAR AND SURFACE FRETTING CORROSION

9.21

223

CONNECTOR WEAR AND SURFACE FRETTING CORROSION

Extensive testing experience with electronic equipment in random vibration environments has shown that fretting corrosion can occur at electrical contacts when high interface pressures and high relative velocities are present for long periods of time. What appears to happen is that wear occurs through the outer layer of gold (approximately 20-50 millionths of an inch thick) at the contact area first. With continuing random vibration, wear then occurs through the nickel undercoating (approximately 100-200 millionths of an inch thick) so that now the electrical contact interface is between a brass pin and a brass socket. The brass-on-brass interface with a high relative velocity produces a high local temperature, which produces an oxide of brass that is an electrical insulator. Although there is mechanical contact between the pin and the socket at the electrical interface, there is no electrical continuity for approximately one microsecond of time. In an analog electrical system, this is no problem. However, in a high-speed digital system where multiplexing is being used, the loss of information for that time period can result in electrical failures. These failures are very difficult to find, because the problem appears to be very similar to a broken wire or a cracked solder joint. Also, when the vibration is stopped the problem always disappears. A great deal of time is then lost in looking for broken cables, wires, solder joints, and components [521. Inspecting electrical contacts with a microscope for fretting corrosion can be very difficult. Polished brass looks very much like gold on the contacts, so the wearthrough cannot be observed easily and is almost always overlooked. However, when a 5% solution of sodium sulfide in water is brushed on the connector surface in the contact area, a small spot will turn black in a couple of seconds when wear through the gold and nickel has occurred and there is any copper base alloy present. This is a nondestructive test; the solution can be washed off with water with no harm to the contacts. This problem with contact fretting corrosion and wear appears to occur only in severe random vibration environments and is most severe on electrical connectors that have only two points of contact. Contacts with a flat blade pin and a tuning-fork socket (and only two points of electrical contact) have shown very high failure rates in severe random vibration tests for long time periods. This problem has never been observed to occur in any sinusoidal vibration environment. or in random vibration tests on connectors with more than three points of electrical contact. This does not mean that fretting does not occur. This means that all of the electrical points of contact do not open (lose continuity) at the same time under these conditions [52]. Test data show that this condition may first occur after extensive highacceleration random vibration exposure, and then not reappear even after many hours of additional vibration testing. The failure mode appears to follow the previously defined exponential fatigue laws (Eq. S.l), except that the fatigue exponent b appears to have a value of about 4.0. When high-level

224

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

random vibration tests are run on individual PCBs in a rigid vibration fixture, and when blade-and-tuning-fork electrical contacts are used, the first fretting corrosion failures have been observed to occur after about 15 min [521 using a white-noise input level of 40 G RMS, with a PSD of 0.80 G2/Hz. Sample Problem-Determining

Approximate Connector Fatigue Life

A blade-and-tuning-fork electrical connector, with two contact points, is being proposed for plug-in PCBs that must be capable of passing a random vibration qualification test program with an input acceleration level of 10 G RMS for 1 hour along each of three axes, for a total test time of 3 hours. Will the proposed contacts be capable of reliable operation in this environment?

Solution. In any type of qualification test program, there is such a high probability of malfunctions occurring due to improper instrumentation, calibration, and data accumulation and to recording-device failure, accidents, and human errors, that these tests often have to be repeated several times in order to complete only one qualification test. Therefore, electronic systems must be designed with some margin of safety. One good approximate rule is to design the system so it is capable of withstanding five qualification tests. This safety factor adds very little weight, and it provides that extra safety margin that is often necessary to complete a difficult qualification test schedule. Therefore, let

b = 4.0 (fatigue exponent for electrical contact failure) T2 = 15 min (baseline time to fail, from test data [52]) G2 = 40 G (RMS acceleration, baseline for failure) Substitute into Eq. 8.4a for the approximate time to fail:

[

T , = ( 15 min) 40 RMS 10 G RMS

14”

= 3840

minutes

= 64.0

hours

(9.59)

Since the maximum qualification testing time expected is 3 X 5 = 15 hours, and the system is capable of withstanding about 64.0 hours of random vibration testing, the proposed design will be satisfactory. The fatigue life of these electrical contacts will be affected by the manufacturing tolerances, spring rate of the tuning-fork socket, surface finish, gold thickness, nickel thickness, and any lubrication used. 9.22

MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS

When a multiple-degree-of-freedom system is subjected to a random vibration input, the motion of each mass in the system will influence the motion of every other mass in the system.

OCTAVE RULE FOR RANDOM VIBRATION

225

-.

I .._ 3. L

d

A '

a

r"?, 1

1

1

I

/

FIGURE 9.27. Integrating under the transmissibility Q response and the input PSD levels to obtain the RMS acceleration re-

sponse.

The response of the system can be determined by using an integration technique. The l a RMS acceleration response can be obtained for any or all masses in the system by integrating under the input PSD curve and under the frequency response curve for any of the masses at the same time, as shown in Fig. 9.27 and in Eq. 9.60:

Go,, =

P,A f , Qf

(9.60)

This is just another form of Eq. 9.36, as can be demonstrated by substituting the expression for the bandwidth Af of the resonance into the above relation:

f" Q

Af= -

9.23 OCTAVE RULE FOR RANDOM VIBRATION In Chapter 7 it was shown that severe coupling can occur between the chassis and the PCBs within it when the octave rule is not followed in a sinusoidal vibration environment. Similar problems can occur in random vibration environments when the octave rule is not followed. When the resonant frequencies of the chassis and the PCBs are not separated by at least one octave, then the chassis resonance can magnify the PCB resonance. This will increase the PCB acceleration and displacement levels, which will reduce the PCB fatigue life. When the chassis and PCB resonances are separated by one octave or more, then the acceleration and displacement levels are reduced, so the fatigue life of the PCB is increased. It makes no difference whether the forward octave rule or the reverse octave rule is followed, as long as the chassis and the PCB resonances are well separated, and the relative PCB mass is small. The approximate responses of the chassis and PCBs in a random-vibration environment can be determined by assuming that these elements will act like lumped spring-mass systems. When the resonances are well separated, the chassis mass and each PCB mass can be treated as a single-degree-offreedom system.

226

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

FIGURE 9.28. Chassis and PCB modeled as a spring-mass system.

Vlb

A

c

10

Irlplt

2x0

: 400 Hz

~

FIGURE 9.29. Response of the chassis, mass 1, to sinusoidal vibration.

Sample Problem-Response

1L1

i '

10,

I

I 1 102

1

103

104

Frequency H r

of Chassis and PCB to Random Vibration

An electronic chassis with a total weight of 30 lb and a resonant frequency of 400 Hz holds several plug-in PCBs, each with a weight of 1 Ib and an approximate resonant frequency of 200 Hz. Figure 9.28 shows the chassis as mass 1 and one of the PCBs as mass 2. The system must be capable of operating in a white-noise random vibration environment with a PSD input of 0.100 G2/Hz. Determine the RMS acceleration response levels and the RMS displacement levels for the chassis and PCB, masses 1 and 2. Solution Part 1-Chassis

RMS Response The RMS response of the chassis can be determined with the use of Eq. 9.36. Since the chassis and PCB resonances are well separated, and since the PCB mass is small compared to the chassis mass, there will be very little effect of the PCB resonance on the chassis resonance. The chassis will therefore have the typical single-degree-of-freedom response as shown in Fig. 9.29, where the chassis resonance is strong at 400 Hz and the 200-Hz PCB resonance shows up as a small spike on the chassis response curve. Then

P = 0.10 G'/Hz (PSD random vibration input to chassis) Q = 0 . 5 m = 10 (estimated chassis transmissibility) f , = 400 Hz (chassis resonant frequency)

OCTAVE RULE FOR RANDOM VIBRATION

227

Substitute into Eq. 9.36 to obtain the RMS acceleration level:

GRMS = (5(0.10)(400)(10)

= 25.06

(9.61)

The dynamic single-amplitude displacement expected for the chassis can be obtained from Eq. 2.30. This represents the relative motion of the chassis, mass 1, with respect to the support: G R b f S = 25.06

f,

= 400

(RMS acceleration response) (chassis resonant frequency)

Hz

(9.8) (25.06) ‘RMS

=

(400)’

= 0.00153

in.

(9.62)

Part 2-PCB RMS Response The PCB has its resonant frequency at 200 Hz, and the chassis has its resonant frequency at 400 Hz. When the PCB is excited at its resonant frequency, the uncou )ed transmissibility normally expected for the PCB would be about 200 = 14.14. The PCB can also receive additional energy from the chassis due to the coupling with the chassis at 200 Hz. The uncoupled transmissibility expected for the chassis at a frequency of 200 Hz can be obtained with Eq. 2.49 and the following information:

9

ff = 200 Hz (forcing frequency) f , = 400 Hz (chassis resonant frequency) R = ff/f, = 200/400 = 0.5 (frequency ratio) Thus 1 e=-= 1- R 2

1 1 - (0.5)’

=

1.33 (chassis at 200 Hz)

This shows that the PCB resonance will couple with the chassis resonance, and that the PCB transmissibility will be 1.33 times greater than previously calculated at a frequency of 200 Hz. The coupled transmissibility for the PCB at 200 Hz then becomes Q p = (1.33)(14.14)

=

18.8 (PCB at 200 Hz)

(9 -63)

A second resonant peak can be expected for the PCB at a frequency of 400 Hz, because the chassis has its resonance at 400 Hz. The chassis uncoupled transmissibility at 400 Hz was shown to be about 10 in part 1 above. The PCB uncoupled transmissibility at 400 Hz can be obtained from

228

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

Eq. 2.49 with the following information: ff = 400 Hz (forcing frequency)

f,,

= 200

R

= 400/200 = 2.0

Hz (PCB resonant frequency) (frequency ratio)

Thus

Q=

1 1 - (2)2

= - 0.333

(PCB at 400 Hz)

The negative sign means that the 200-Hz response for the PCB is out of phase with the 400-Hz response. This is not important here, so the sign can be ignored. The coupled transmissibility for the PCB at 400 Hz becomes Q p = (10)(0.333)

=

3.33

(PCB at 400 Hz)

(9.64)

The frequency response curve for the PCB with its two resonant peaks will now appear as shown in Fig. 9.30. The relative RMS acceleration response of the PCB with respect to the chassis can be determined with the use of Eq. 9.36 or Eq. 9.60. When Eq. 9.36 is used, the two PCB resonant peaks at 200 and 400 Hz can be treated as two single-degree-of-freedom peaks, to obtain the RMS response at the center of the PCB using Eqs. 9.63 and 9.64: 7T

GRMs=

- (0.10) (200) (18.8) 2

G R M S = 28.3

7T

+ -2 (0.10) (400) (3.33)

(response at center of PCB)

(9.65)

Equation 9.36 is very accurate when the transmissibility of each resonant peak is greater than 10. Some accuracy is lost when the transmissibility is

I

200 Hz

I 10

FIGURE 9.30. Response of the PCB, mass 2, to sinusoidal vibration.

'1 10'

1 1

102

" loa

Frequency Hz

104

OCTAVE RULE FOR RANDOM VIBRATION

229

significantly less than 10. When this condition exists, better accuracy will be obtained with Eq. 9.60. The relative dynamic single-amplitude displacement of the PCB, mass 2, with respect to the chassis, mass 1, can be determined with the use of Eq. 2.30 and the following information:

f, = 200 Hz

(PCB resonant frequency)

Thus (9.8) (28.3) =

'RMS

(200) *

Sample Problem-Dynamic

= 0.00693

in. (PCB)

(9.66)

Analysis of an Electronic Chassis

A 30-lb chassis with four mounting flanges contains some 1.0-lb plug-in PCBs with relays as shown in Fig. 9.31. The relays have a maximum rating of 40 G peak up to a frequency of 2000 Hz. A lumped spring-mass system was used to model the assembly with two degrees of freedom as shown in Fig. 9.28. The transmissibility curves expected for the PCB and the chassis were shown in Figs. 9.30 and 9.29, respectively. The operating environment will have a PSD input level of 0.10 G'/Hz. Determine the following:

(a) Will the relays be capable of operating in the environment without chattering? (b) Will the chassis mounting lugs be capable of operating in the environment without failing?

Solution. (a> The RMS response of the PCB to the random vibration was shown by Eq. 9.65 to be 28.3 G. Since the relays are rated at 40 G peak, it

-

. . ~

! I

Relays on PCB

/

4 '

Vibration

fi. i ~ p u t

FIGURE 9.31. PCBs with relays, mounted within an electronic chassis (dimensions in

inches).

230

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

would appear at first glance that the relays are acceptable for the random vibration environment. However, the 3 a peaks for the random vibration must be considered even though they only occur about 4.33% of the time. Therefore, the maximum levels the relays are expected to see would be 3 x 28.3 or 84.9 G . This is well above the 40-G peak rating for the relay, so these relays are unacceptable for the application when they are mounted at the center of the PCB. If the relays are mounted at the edges of the PCB, they will see the 25.06-G RMS levels developed by the chassis resonance at 400 Hz, as shown by Eq. 9.61. The 317 acceleration levels will be 75 G. so again the 40-G peak relay rating will be exceeded. Therefore, these relays are unacceptable for this application. (b) The dynamic stresses in the mounting feet can be determined from the spring rate of the chassis using Eq. 2.7 with the following information: f, = 400 Hz (chassis resonant frequency) W = 30 lb (chassis total weight) g = 386 in./s2 (acceleration of gravity)

Thus

K=

47r (400) '( 30) 386

=

490,940 lb/in.

(chassis spring rate)

The dynamic load acting on the chassis can be determined from the chassis dynamic displacement shown in Eq. 9.62:

Pd= KZ

=

(490,940)(0.00153)

= 751.1

lb RMS on four lugs

The dynamic bending moment acting on the lugs can be obtained from the mounting-lug geometry shown in Fig. 9.31:

M=

(751.1) ( 1.4) 4 lugs

=

263 1b.in. RMS

Substituting into Eq. 2.39 for the dynamic bending stresses the quantities

I = bh'/12 = (0.75)(0.50)'/12 C = 0.50/2 = 0.25 in. K = 1.5 (stress concentration)

= 0.00781

in.'

we obtain

s, =

( 1.5) (263) (0.25) 0.00781

=

12.628 lb/in.' RMS (bending)

(9.67)

DETERMINING THE NUMBER OF POSITIVE ZERO CROSSINGS

231

The 3 a bending stress in the mounting lug is S,(3a)

=

(3)(12,628)

= 37,884

Ib/in.’

(bending)

(9.68)

Since this exceeds the 12,000-lb/in.2 fatigue bending endurance limit for the aluminum-alloy mounting lug, the design is unacceptable. In order to improve the structural integrity of the design, gussets can be added to the mounting lug as shown in Fig. 9.32. Gussets will increase the stiffness and strength of the mounting feet, which will increase the resonant frequency of the chassis. The previous analysis must now be revised to account for the new chassis stiffness. In the preliminary design phase of a new electronic chassis design, this process is repeated until a satisfactory design has been achieved for the environment.

9.24 DETERMINING THE NUMBER OF POSITIVE ZERO CROSSINGS The number of fatigue cycles accumulated during a resonant dwell condition in a sinusoidal vibration test is easy to determine. It is simply the product of the resonant frequency (cycles/second) and the time (seconds). The number of fatigue cycles accumulated during a random vibration test is also related to the duration of the test. However, the actual number of stress reversals is more difficult to evaluate. Since all of the exciting frequencies within the bandwidth are present all of the time, the number of fatigue cycles must relate to the frequencies that will do the most damage. These are the frequencies with the highest transmissibilities, where the damping is small. These frequencies will influence the number of times, on the average, where the displacement trace crosses the zero axis with a positive slope. This is called the number of positive zero crossings and is denoted by the symbol

No+. The number of crossings, with both positive and negative slopes, was determined by Rice [41] to be (see Fig. 9.1)

(9.69)

232

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

In order to obtain position, to show zero crossings, it is necessary to divide by

R4 and simplify as follows:

1

N’

= -

2T

O

For lightly damped systems with several resonant peaks, the above equation can be simplified by considering each peak to act like a single degree of freedom. Then using Eq. 9.36 and simplifying,

There are a number of other random vibration relations that are convenient for determining displacements, velocities, and accelerations in random vibration environments for single-degree-of-freedom systems. For example, when the input PSD is known and the RMS response displacement is desired, the following relation is convenient:

z,,, = 1

2 2 F

(9.72)

where Q = transmissibility at resonance (dimensionless) P = PSD at resonance f, (G*/Hz) f, = resonant frequency of system (Hz) The displacement, velocity, or acceleration response of a single-degreeof-freedom system can be determined form Eq. 9.36, or from the same equation written in a slightly different form, as follows: Output

where R , f,

=

[0.7”i( 1 +R4R: , ](f,,)(input)]

0.5

1/2Q = damping ratio (dimensionless) = resonant frequency (Hz) = C/C, =

(9.73)

DETERMINING THE NUMBER OF POSITIVE ZERO CROSSINGS

233

The units are as follows: Quantity

Input

Displacement Velocity Acceleration

Output (RMS)

in.’/Hz (in./s>’/Hz G~/HZ

in. in./s G

Sample Problem-Determining the Number of Positive Zero Crossings

Sinusoidal vibration tests were run on an electronic system that had an accelerometer mounted at the center of a plug-in type PCB. The fundamental natural frequency and the next two higher harmonics with the following transmissibility Q values were obtained: Frequency (Hz)

Transmissibility Q

345 604 1010

18.7 8.4 4.8

Determine the number of positive zero crossings, considering a white-noise random vibration excitation with a 0.10-G2/Hz PSD input level.

1 No - 2

[ 2 7 (345)]’

i-

[2 a (604)12

+

+ - _

( a/2) (0.1)(345)( 18.7)

2.155X lo-‘ 4.587x l o - ”

+ 5.531 X

+

( a/2) (0.1)(604) (8.4)

+

+ 1.889 X lo-’

+ 3.840 X lo-’* + 4.692 X

=

[ 2 a (lolo)]* ( a/2) (0.1) ( 1010) (43)

383 Hz

(9.74)

-

CHAPTER 10

Acoustic Noise Effects on Electronics

10.1

INTRODUCTION

Acoustic noise (pressure waves) can be generated by almost any source, from a loud radio or a rocket engine to a siren or a ship’s propeller. This noise can be radiated through the air or conducted through liquids and solids. High acoustic energy levels can lead to rapid fatigue failures in large, thin panels and can also affect the operation of certain sensitive electronic components. Acoustic noise is measured in decibels (dB), implying a ratio of the mean square sound pressure to a reference mean square pressure. The reference selected is the threshold of hearing. Thus

Number of decibels (dB)

P

= 20 log,, -

(10.1)

Pref

where P

= sound

pressure level (lb/in.’)

Pref= sound pressure reference = 2.90

x

lb/in.*

= 0.0002

dyn/cm*

The above expression should not be confused with Eq. 9.11, where the coefficient is 10 rather than 20 because a power ratio rather than an amplitude ratio is used. The sound pressure P is of interest because it represents the level of the pressure wave that can cause acoustic damage.

Sample Problem-Determining the Sound Pressure Level

Acoustic noise tests yielded sound pressure-level measurements of 150 dB. Determine the equivalent sound pressure level in terms of a pressure in pounds per square inch RMS. 234

MICROPHONIC EFFECTS IN ELECTRONIC EQUIPMENT

235

Solution. Substitute into Eq. 10.1: P

150 = 20 log,,

2.9 x 10-9

150

P - log,, 2.9 x 10-9 20 lOg,,P=7.5+(-8.537) = -1.037 - - - log,,

P = 0.092 lb/in.2 RMS

(10.2)

The threshold of acoustic damage is normally considered to be at about 150 dB. Values below this level generally do not result in any serious structural damage in electronic equipment. Microphonic effects can, however, occur at lower values. 10.2

MICROPHONIC EFFECTS IN ELECTRONIC EQUIPMENT

When an electronic component is sensitive to acoustic noise, it is called microphonic. Sound pressure waves can physically alter the shape of the component slightly, and therefore its electrical characteristics will fluctuate. Certain types of ceramic and paper capacitors are affected by acoustic noise, as well as certain types of crystals, which are often used in oscillators and piezoelectric accelerometers. However, most of the so-called microphonic effects are not really caused by microphonic components. Instead, the signal distortions, modulations, and noise are most often due to loose wires, loose ground screws, poor wire routing, improper use of shields, poor solder joints, and many other electromechanical defects. Acoustic tests on some types of electronic components have shown that they can produce malfunctions in the electronic equipment. The equipment items most susceptible to malfunctions are components whose function involves motion of flexible parts (e.g., relays and pressure switches) and components that contain relatively flexible structural parts but are otherwise rigid (e.g., vacuum tubes and piezoelectric devices). Capacitors, resistors, and inductors also have a history of malfunctions in severe acoustic environments. Film capacitors from some manufacturers have shown problems at acoustic levels as low as 132 dB. Similar components from other sources showed no problems at 166 dB. Ceramic capacitors showed problems at levels of 159 dB. Electrolytic capacitors showed problems at acoustic levels of 125 dB [451. In general, structural members that have large, flat, thin, lightweight sections are most sensitive to high-level acoustic environments, so members with these characteristics should be stiffened or damped to reduce the acceleration response levels. When large flat thin panels are subjected to acoustic noise, the acoustic fluctuating pressures force the panels to vibrate. The panels will have a

236

ACOUSTIC NOISE EFFECTS ON ELECTRONICS

dynamic response related to the geometry, method of support, and construction. The response influences the stress patterns and the stress levels, which determine the fatigue life.

10.3 METHODS FOR GENERATING ACOUSTIC NOISE TESTS Acoustic noise tests are usually generated by air jets, sirens, and loudspeakers. Air jets and loudspeakers generate a broader band of frequencies; sirens generate several narrow bands. A siren can generate discrete frequencies close to the fundamental resonant frequency of the panel, which will increase the response of the panel. Air jets and loudspeakers generate a broader frequency spectrum, so the panel has a lower response. Therefore, the stress levels in a panel excited by a siren will be greater than the stress levels developed by an air jet or loudspeaker, for the same sound pressure level. The time to failure for a 0.032-in. thick flat 2024-T3 aluminum panel subjected to acoustic noise from a siren and from an air jet is shown in Fig. 10.1.These tests show that the siren produces more rapid fatigue failures at the same sound pressure level. This is probably due to the greater amount of energy developed by the siren at the fundamental resonant frequency of the panel [461. The fatigue life of the panel can be increased by adding a bonding in addition to the rivets holding the panel to the supports. A further increase in the fatigue life can be achieved by forming a radius in the panel to increase the panel stiffness. This increases the resonant frequency, which reduces the deflections and stresses. These structural modifications are shown in Fig. 10.2 [46].

3

Time to f a i l u r e hours

FIGURE 10.1. Test data showing failures of 0.032-in. thick 2024-T3 flat aluminum panels subjected to air jet and siren acoustic noise (dimensions in inches).

METHODS FOR GENERATING ACOUSTIC NOISE TESTS Fixture opening

237

0 032 in

Radius = 8 ft

Differential

pressure = 6 l b

in

1

* 0'---r%-++0 Relative time to failure

FIGURE 10.2. Effects of different mounting methods and curvature on the panel fatigue life (panel gage and size constant). 5

1 2 000,

0 01

-

01

Air

jet

Sireq

10

Time to failure, hours

FIGURE 10.3. Effects on the RMS stress levels and failures induced in 0.032-in. thick 2024-T3 aluminum panels subjected to air jet and siren acoustic noise.

Strain gages on the flat panel show that, for a given RMS stress level, the air jet produces more rapid fatigue failures than the siren. This may be because the peak stress responses for the air jet are several times greater for a given RMS value than they are for the constant-level siren tests. The test results are shown in Fig. 10.3. No significant differences were noted in the nature of the failures experienced for the two types of loading. Similar acoustic noise tests were run on a similar aluminum panel 0.064 in. thick for a comparison. It was found that doubling the panel thickness increased the panel fatigue life by a factor of about 20. These results were confirmed for both the air jet and the siren test. Air jet and loudspeaker acoustic noise tests are very similar to random vibration tests, since they cover a broad frequency spectrum. Random vibration analysis methods can therefore be used to determine the fatigue life of

238

ACOUSTIC NOISE EFFECTS ON ELECTRONICS

thin flat panels exposed to these acoustic noise environments. The method of analysis is to determine the pressure spectral density in units of pressure squared per hertz [(lb/in.2)2/Hz] in place of the power spectral density in units of acceleration squared per hertz (G2/Hz) used for random vibration, as shown in Eq. 9.36.

10.4

ONE-THIRD OCTAVE BANDWIDTH

Sound pressure levels are usually measured with filters set up for one-third octave bands. Each octave is then divided into three bands. The ratio of the upper band frequency to the lower band frequency for the one-third band increments can be shown to be +log,,

f2

=

+log, 2 = f(0.693)

= 0.231

Take the antilogarithm of both sides: f 2

-=

1.2598 or

f2 =

1.2598f,

(10.3)

f l

The center frequency within the band is obtained from

f, =

rn

(10.4)

The bandwidth at the center frequency of the one-third octave band becomes Af

=f

, -f l

= 0.231f,

(10.5)

upper frequency of band (Hz) f l = lower frequency of band (Hz) f, = center frequency of band (Hz)

where

f2 =

10.5

DETERMINING THE SOUND PRESSURE SPECTRAL DENSITY

The sound pressure spectral density P, is defined as the sound pressure ( P ) squared and divided by the one-third octave bandwidth:

P' p, = Af

[ (lb/in.2)2/Hz]

(10.6)

SOUND PRESSURE RESPONSE TO ACOUSTIC NOISE EXCITATION

239

The sound pressure P can be determined with the use of Eq. 10.1 in a slightly different form as follows:

or iog,, P

=

dB -log,, 20

io + log,,

2.9 x 10-9

Taking the antilogarithm of both sides results in the sound pressure shown in a slightly different form:

P = (2.9 x 10-9) x I O ~ B / * O

(10.7)

Substitute Eqs. 10.7 and 10.5 into Eq. 10.6:

P, =

(2.9 x 10-9)* x 0.231f ,

iod~/10

[ (lb/in.')*/Hz]

(10.8)

10.6 SOUND PRESSURE RESPONSE TO ACOUSTIC NOISE EXCITATION

In a lightly damped structural system excited by acoustic noise, the RMS sound pressure response, or output (P,,,), can be obtained using the same method as for random vibration, because the two methods of excitation are similar. From Eq. 9.36 PRMS =

{Z

(10.9)

where P, = sound pressure spectral density (SPSD) in (lb/in.*>*/Hz at the resonant frequency f, = resonant frequency (Hz) Q = transmissibility at the resonant frequency (dimensionless) The RMS sound pressure response can be used to obtain the stresses in the structure, and the fatigue life can be obtained by following the Gaussian probability distribution for a single-degree-of-freedom system using Miner's cumulative damage index as shown in Eq. 9.42.

240

ACOUSTIC NOISE EFFECTS ON ELECTRONICS

Octave band center frequency. Hz

FIGURE 10.4. Acoustic noise qualification test spectrum.

Panel

Bulkheads

I

I

I

I

FIGURE 10.5. An electronic box with a thin flat aluminum side panel.

Sample Problem-Fatigue to Acoustic Noise

Life of a Sheet-Metal Panel Exposed

An electronic box must be capable of withstanding an acoustic noise qualification test spectrum as shown in Fig. 10.4 for a period of 30 minutes. Weight is a problem, so 2024-T3 aluminum side panels with a thickness of 0.040 in. are proposed, as shown in Fig. 10.5. Will the thin side panels survive the acoustic environment? Solution. The resonant frequency of the panel must be determined first to establish the dynamic characteristics, and to find the acoustic noise level acting on the side panels. The panel is considered to be a flat uniform square plate clamped on all four edges. Using the relations shown in Chapter 6 for plates.

(10.10) 10 x l o 6 1b/im2 (modulus of elasticity for aluminum) in. (thickness of panel) p = 0.30 (Poisson’s ratio, dimensionless)

where E h

=

= 0.040

SOUND PRESSURE RESPONSE TO ACOUSTIC NOISE EXCITATION

g

241

= 386

in./s2 (acceleration of gravity) y = 0.10 1 b / h 3 (density of aluminum) a = 10.0 in. (length of square side panel) Eh3 (10 x 106)(0.040)3 D= = 58.6 lb in. 12(1- F’) (12) 1 - (0.3)’]

[

yh

(0.10)(0.040)

P = -=

386

g

= 0.00001036

lb . s2/in3

Substitute into Eq. 10.10: (10.11) An examination of Fig. 10.4 shows an acoustic noise level of 150 dB at a frequency of 141 Hz. Deflections and stresses in the side panels will be obtained for this noise level, using the fundamental resonant frequency of the panel as the center frequency of a one-third octave band. Substitute into Eq. 10.8 for the SPSD level: P, =

(lb/in .2 )’ (2.9 x 1 0 - 9 ) ’ ( 1 0 ) ~ ~ ~ ~ = 0.000258 (0.231) (141) Hz

Substituting into Eq. 10.9 for the sound pressure response, where

f,,= 141 Hz (resonant frequency of panel) Q=

fi=

=

11.9 (approximate transmissibility)

we have 77

PRMS =

-(0.000258)(141)(11.9) 2

= 0.824

lb/in.*

(10.12)

The dynamic displacement of the aluminum panel must be determined to see if the structural system is linear or nonlinear. When large displacements occur in the flat panel, the panel will carry some of the load in direct bending and some of the load as a membrane. When the panel is loaded as a membrane, it is possible to carry large loads with a relatively small displacement. When the dynamic displacements in the panel are less than one-third of the panel thickness, linear plate-bending theory can be used to calculate the plate-bending stresses. When the panel deflections are greater than one-third the panel thickness, large-deflection theory must be used to obtain an accurate estimate of the true stress condition. This theory goes into

242

ACOUSTIC NOISE EFFECTS ON ELECTRONICS

membrane stresses and shows the proportion of the load carried by direct bending and the proportion of the load carried as a membrane, where the load is carried in pure tension. The total RMS acoustic pressure response acting on the panel is determined from Eq. 10.12. Part of this pressure will be carried as a membrane acoustic pressure, and part will be carried as a bending acoustic pressure when the plate deflection exceeds one-third of the panel thickness: qm = membrane acoustic pressure (Ib/in.* RMS) q h = bending acoustic pressure (Ib/in.* RMS)

Deflections resulting from membrane loading can be determined from Timoshenko's formula for a square plate [29]:

q,c 1'3 Y = ~ . N Q c ( ~ ) or

Y 'Eh q,= 0.516C4 (Ib/in

.2)

(10.13)

Deflections resulting from the plate bending due to a uniform load on a square plate with clamped edges can be determined as follows:

Y = 0.224-

4bC4 Eh3

Or

"=

YEh 0.224C4

(lb/in

.2

)

(10.14)

The total acoustic pressure acting on the panel is the sum of the membrane pressure and the bending pressure, which is the acoustic response pressure shown by Eq. 10.12. (Note: C = 10/2 = 5.0 in.).

P,,,

=

( 1 . 9 4+ ~ 4.46 y*

i

(10.15)

In the above equation, everything is known except the deflection Y , so Y can be determined. Substitute Eq. 10.12 into Eq. 10.15: 0.824 = 4.567Y+ 1241.6Y3 Y = 0.0735 in.

(RMS displacement)

(10.16)

Since this displacement is much greater than the panel thickness of 0.040 in., the acoustic stresses will be nonlinear. Membrane and direct bending stresses in the panel will have to be calculated.

SOUND PRESSURE RESPONSE TO ACOUSTIC NOISE EXCITATION

243

The portion of the load that is carried as a membrane can be obtained from Eq. 10.13, where

Y = 0.0735 in. (RMS displacement) E = 10 x 106 lb/in.* (modulus of elasticity of aluminum) h = 0.040 in. (thickness) C = 10/2 = 5.0 in. (half of panel dimension) Thus (0.0735)3(10 x 106)(0.040) 9,

=

(0.516)(5.0)4

= 0.492

1b/ina2 RMS

(10.17)

The membrane stress in the panel due to the acoustic noise can be determined from the following expression [29]:

S,

= 0.396(

S,

=

0.396

1

qiEC2 7

i

(10.18)

(0.492)2(10 x 106)(5.0)2 =

( 0.040)2

1318 lb/in.2 RMS (10.19)

The portion of the load that is carried in direct bending can be determined from Eq. 10.14: (0.0735)(10 x 106)(0.040)3 qb

=

(0.224) (5 .0)4

= 0.336

1b/ina2 RMS

(10.20)

The maximum bending moment developed in the square panel with fixed edges can be determined from the following relation [29]:

M

= 0.0513qbn2

M = (0.0513)(0.336)(10.0)* = 1.724 in:lb/in.

RMS

(10.21)

The bending stress is determined from the relation 6M

sb -- - =h2

(6) (1.724)

(0.040) *

= 6465

lb/in.2 RMS

( 10.22)

The maximum total stress at the edge of the panel will be the sum of the membrane stress and the bending stress. Since reversed stresses are involved,

244

ACOUSTIC NOISE EFFECTS ON ELECTRONICS

fatigue must be considered, so stress concentrations ( K ) must be included in the fatigue life evaluation. Typical stress risers found in electronic structures are about K = 2.0 at holes and corners. Thus

+ S,) (total stress) S, = 2( I318 + 6465) = 15,566 Ib/in.2 S, = K ( S,

(10.23) RMS

(10.24)

Miner's cumulative damage can be calculated to obtain an accurate estimate of the fatigue life expected for the panel. A quicker way to obtain the fatigue life, but with less accuracy, is to consider only the 3 a stresses, which do most of the damage, but occur only 4.33% of the time. The 3 a stresses are as follows: S,

(3)(15,566)

=

= 46,698

lb/in.2

(10.25)

The approximate fatigue life of the 2024-T3 aluminum panel can be determined from the S-N fatigue curve shown in Fig. 10.6. An examination of this curve shows that an approximate fatigue life of about 1.8 X IO4 fatigue cycles can be expected. The time it will take for the panel to fail can be obtained from the relations established for random vibration, as follows:

T=

T= =

N

(10.26)

0.0433f , 1.8 x l o 4 cycles to fail (0.0433) ( 141 cycles/s) (60 s/min)

(10.27)

49.1 minutes to fail

The acoustic noise qualification test only lasts for 30 minutes. At first glance it would appear that the panel could meet the requirements. However, when manufacturing tolerances are considered, and when there is a strong

' ~,

.

:a

x 104

57000

0 -

18C22~-

-

v

I

Q

c

s

I

.L___

i.p--_l13-

106

105 \

106

cycles t o f a

1:-

.c8

I

FIGURE 10.6. Fatigue curve for 2024-T3 aluminum sheet.

245

DETERMINING THE SOUND ACCELERATION SPECTRAL DENSITY

possibility that these tests must often be repeated several times due to calibration and electrical problems, it must be concluded that the panel design is deficient. The panel design must be changed. The performance can be improved by making the panel slightly thicker, or by increasing the damping in the panel to reduce the transmissibility Q to reduce the stresses. These changes will, of course, increase the weight slightly. 10.7 DETERMINING THE SOUND ACCELERATION SPECTRAL DENSITY

An alternate method of solution can be used to solve acoustic noise problems associated with sound pressure acting on flat panels. This is the sound acceleration method, where the acoustic sound pressure is converted into a sound acceleration acting on the panel. The acceleration response of the panel is then determined, which leads to the dynamic load. The dynamic load is then converted back to a dynamic pressure response acting on the panel, The which is the same as the sound pressure response of the panel (PRMS). stresses in the panel are then determined as before. The sound acceleration spectral density A , is defined as the square of the acceleration level expressed in gravity units, divided by the frequency bandwidth across the one-third octave points. This is the same idea that was used before, except that sound acceleration terms are now being used instead of sound pressure terms: (10.28) The acceleration level G, due to the sound pressure P can be determined from the specific weight w of the thin flat panel as follows: (10.29) where P = sound pressure (lb/in.*> from the dB acoustic level (RMS) w = specific weight of the panel (lb/in.*> y = density of the panel ( l b / i ~ ~ . ~ > h = thickness of uniform panel (in.) The RMS acceleration response to the broadband acoustic noise can be determined using the same method previously shown for random vibration with Eq. 9.36 since the methods of excitation are the same: (10.30)

246

ACOUSTIC NOISE EFFECTS ON ELECTRONICS

Sample Problem-Alternate Method of Acoustic Noise Analysis

Use the sound acceleration spectral density method to solve the previous sample problem.

Solution. Determine the acceleration level due to the sound pressure of 150 dB using Eq. 10.29, where lb/in.* (sound pressure due to 150 dB; see Eq. 10.2) y = 0.10 I b / i r ~ .(density ~ of aluminum panel) 12 = 0.040 in. (thickness of panel) G, = 0.092/(0.10)(0.040) = 23.0 (acceleration due to sound) (RMS)

P

= 0.092

Determine the acoustic acceleration spectral density using Eq. 10.28, where

G,

f,

= 23.0 =

(RMS)

141 HZ

We obtain

A, =

(23 .O)’ (0.231)( 141)

=

16.24 G2/Hz

(10.31)

Substitute into Eq. 10.30 to determine the acceleration response to the sound pressure excitation, where A , = 16.24 G2/H2 (sound acceleration spectral density) f, = 141 Hz = 11.87 (approximate transmissibility) Q= We obtain

G,,,

-/

= 206.6

=

(10.32)

This acceleration response must now be converted back to a sound ). This is obtained from the acceleration force pressure response (PRhIs divided by the area of the panel, using one square inch of panel area:

PRMS =

wp

RMS

(lb/in.* RMS)

(10.33)

A,

. ~0.004 ) lb (panel weight) where Wp = (1 in.)(l in.)(0.04 in.)(O.lO l b / i ~ ~ = A , = 1.0 i n 2 (panel surface area) G,,, = 206.6 (acceleration response of panel)

DETERMINING THE SOUND ACCELERATION SPECTRAL DENSITY

247

We obtain

(0.004) (206.6) PRMS =

1.o

= 0.826

lb/in.*

(10.34)

Comparing the acceleration response result with the sound pressure response result in Eq. 10.12 shows they are the same. The panel stresses due to the acoustic noise can now be obtained for the nonlinear system, starting with the membrane loading condition defined by Eq. 10.13 through Eq. 10.27.

Designing Electronics for Shock Environments

11.1

INTRODUCTION

Shock is often defined as a rapid transfer of energy to a mechanical system, which results in a significant increase in the stress, velocity, acceleration, or displacement within the system. The time in which the energy transfer takes place is usually related to the resonant frequency, or to the natural period of the system. Shock will often excite many of the natural frequencies in a complex structure, which can produce four basic types of failures in electronic systems. These failures are due to (1) high stresses, which can cause fractures or permanent deformations in the structure: (2) high acceleration levels, which can cause relays to chatter, potentiometers to slip, and bolts to loosen; (3) high displacements, which can cause impact between adjacent circuit boards-cracking components and solder joints, breaking cables and harnesses, and fracturing castings; and (4) electrical malfunctions that occur during the shock but disappear when the shock energy dissipates. This last effect can occur in crystal oscillators, capacitors, and hybrids. A large thin hybrid cover can displace and cause a temporary short circuit with the internal die bond wires. The hybrid typically appears normal after the shock exposure, which makes it difficult to find and correct the failure. Fatigue is usually not an important consideration in shock, unless a million or more stress cycles are involved. When less than a few thousand stress cycles are expected, fatigue stress concentrations are ignored because they do not have a great influence on how or when the structure will fail. Stress-concentration factors must be included in any shock evaluation where fatigue effects are expected to accumulate. The impact sensitivity, notch sensitivity, and brittleness of a structure are important, especially when high-strength steels and castings are used to carry high shock loads. When the ductility of any structural load-carrying material is less than about 370, problems can develop, unless the dynamic stresses are carefully evaluated with respect to the anticipated environment. However, in most structural elements the ductility is normally greater than about 5%, so 248

SPECIFYING THE SHOCK ENVIRONMENT Onedegree of freedom

v + : < , ;

249

Twodegrees of freedo-n

s i s s a h [ , , , i ; Ori , : : !

2 K

2 KI

FIGURE 11.1. Lumped mass-spring systems with one and two degrees of freedom.

the limiting factor in the design is usually based on the yield strength of the material. Isolation systems are often used to protect sensitive electronic equipment in severe shock environments. Care must be exercised to allow sufficient sway space around the equipment to prevent impact against other surrounding structures. Shock analysis techniques can become quite complex, unless some simplifying approximations are made. For this reason, sophisticated electronic systems are often simulated using simple masses, springs, and dampers to estimate the dynamic characteristics of the system. Figure 11.1 shows typical one- and two-degree-of-freedom systems used to approximate electronic systems. The single degree can be used to represent a chassis or a PCB. The two degrees can be used to represent a PCB mounted within the chassis. The chassis represents the first degree of freedom, and the PCB the second.

11.2 SPECIFYING THE SHOCK ENVIRONMENT Many different methods have been used to specify shock motion or its effects. The three most popular methods are (1) pulse shock, ( 2 ) velocity shock, and (3) shock response spectrum. Pulse shock deals with accelerations or displacements in the form of well-known shapes such as the square wave, half sine wave, and various types of triangular waves (vertical rise, vertical decay, and symmetrical). Pulse shocks are easy to work with because the mathematics are simple and convenient. However, pulse shocks do not represent the real world. The true shock environment is seldom a simple pulse. Nevertheless, simple pulses are often effective in revealing weak areas in many different types of structures. Some typical pulse shocks are shown in Fig. 11.2. Velocity shock is concerned with systems that experience a sudden velocity change, such as a falling package whose velocity abruptly goes to zero when

250

DESIGNING ELECTRONICS FOR SHOCK ENVIRONMENTS

I(',

Versed slne

101

FIGURE 11.2. Various types of shock pulses.

Vertical "se (bast pulse)

(1

I

Symmetrica trlangular

I f )

Vertical decay triangular

the package strikes the ground. This is a common test that is called the drop shock. Sometimes an inclined plane is used, where a package gains velocity as it slides down the plane and hits a rigid wall. Another type of velocity shock makes use of a heavy hammer that slams into a fixture that supports a test specimen. The hammer imparts a sudden velocity to the fixture and the test specimen. This type of velocity shock test is used extensively by the Navy to

FIGURE 11.3. Lightweight Navy shock machine. with a 400-lb hammer, for shocktesting electronic assemblies that weight less than 250 lb.

PULSE SHOCK

Shock response acceleration

251

iiG I

1

1

fi

f2

f3

fa

f5

f6

Transient shock

input

FIGURE 11.4. Development of a shock response spectrum curve, showing how a

large number of single-degree-of-freedom systems, with different resonant frequencies, are used to describe the dynamic characteristics of a transient wave.

simulate the effects of explosions on ship and submarine equipment. Figure 11.3 shows a typical lightweight shock machine used by the U.S. Navy to test equipment weighing less than 250 Ib. The shock response spectrum deals with the way in which a structure responds to the shock motions, rather than trying to describe the shock motion itself. The spectrum is a plot of the peak acceleration response of an infinite number of single-degree-of-freedom systems to a complex transient wave form. The individual single-degree-of-freedom masses are usually specified as having a transmissibility Q of 10 when excited at their resonant frequency with a sinusoidal vibration input. This method of analysis is more representative of the real world, but the mathematics is far more complex than the mathematics of the simple shock pulse. Figure 11.4 shows how the peak shock response acceleration levels are used to develop the shock response spectrum.

11.3 PULSE SHOCK

Pulse shocks are often specified for testing electronic equipment, and many military specifications such as MIL-E-5400, MIL-STD-810, and MIL-T-5422

252

DESIGNING ELECTRONICS FOR SHOCK ENVIRONMENTS

define different types of shock pulses and detailed methods for testing with these pulses. The half-sine shock pulse is the most common shock pulse used for testing almost any kind of commercial, industrial, or military product. This is because the half-sine shock pulse is the easiest to generate, and because it is easy to analyze and to evaluate. This does not mean that the half-sine shock pulse accurately represents the actual shock expected in the real environment. Impact machines are generally used to generate the shock pulses shown in Fig. 11.2. Some of these machines have rigid tables that drop on shaped lead pellets, some have compressed air cylinders to sharply increase the velocity for a short drop height, and some use a flexible beam or plate to produce a strong rebound that can almost double the velocity change through a change in the direction of motion. Pulse shocks are conveniently measured with an accelerometer, so it is convenient to define pulse shocks in terms of acceleration and time. With this method the area under the shock pulse represents the change in velocity. and the area under the velocity curve represents the displacement. When a drop test is used to generate a shock pulse, the rebound condition must be evaluated. The rebound coefficient can vary from a value of 1.0 for zero rebound, to a value of 2.0 for a full rebound. The velocity change for a drop test can then be related to the coefficient of rebound:

v = c&E

(11.1)

where I/ = velocity change (ft/s) C = coefficient of rebound (1.0-2.0) g = acceleration of gravity (usually 32.2 ft/s’) H = drop height (ft)

11.4 HALF-SINE SHOCK PULSE FOR ZERO REBOUND AND FULL REBOUND The half-sine shock pulse for a drop shock can be generated using any condition between zero and full rebound. The shock pulse will be the same for zero and for full rebound when the acceleration and time are the same. The velocity change will also be the same, since the area under the acceleration curve represents velocity change. However, the velocity curves will look different because there is a change in the direction for full rebound. The displacements produced by zero rebound and full rebound will be different for the same drop heights. Also, the drop heights required to produce the same acceleration levels will be different for the two conditions. These results can be demonstrated with a sample problem.

HALF-SINE SHOCK PULSE FOR ZERO REBOUND AND FULL REBOUND

253

20 5 Full rebound -Time

c 0 0Y 3 .l - c

>

-273 -41 0

I

c

Time

c

E -za z -

n

FIGURE 11.5. Relation between acceleration, velocity, and displacement for a 100-G, 0.020-second half-sine shock pulse for zero rebound, 50% rebound, and full rebound.

Sample Problem-Half-Sine

Shock-Pulse Drop Test

An electronic box must be shock-tested using a 100-G half-sine shock pulse with a 0.020-second time base for a qualification test. Determine the expected velocity changes, the dynamic displacements, and the drop heights required for a zero-rebound and a full-rebound test. Figure 11.5 shows how the acceleration, velocity, and displacements will vary with respect to time for zero rebound, 50% rebound, and full rebound 1441. Solution Zero Rebound The zero-rebound condition will be generated by a drop machine that smashes a lead pellet as shown in Fig. 11.6. When the electronic box is rigidly fastened to the drop table, the shock pulse will be transmitted from the rigid table to the box. The area under the acceleration curve than becomes the velocity change. The area can be obtained using integration methods, as follows:

Y=A,sinArea

ITX

t

= /x=tYdr x=o

(shock pulse)

(11.2) (11.3)

254

DESIGNING ELECTRONICS FOR SHOCK ENVIRONMENTS

'

Iatior i (I

)

r ,+:/ ~

/""

Rc'o'cn f

bJ

.,

-

.

.-

e \,

v,-3. b>

Bend 16

(0

FIGURE 14.16. T h r e e different vibration modes on a test fixture.

There are three different vibration modes that can have these characteristics: pure translation, rotation, and bending (Fig. 14.16). Unless more accelerometers are used or more points are monitored, it is very difficult to tell what is happening without the use of a strobe light.

14.14 PUSH-BAR COUPLINGS Oil-film slider plates are usually bolted directly to the shaker head. This is generally adequate if the test frequencies are below about 500 Hz and if the test specimen is light enough. When the test frequencies go up to 2000 Hz and the test specimen becomes very heavy, the few bolts holding the slider plate to the shaker head may not have a spring rate high enough to produce the required force at the high frequencies. What often happens is that the shaker head has an output of 30 G but the slider plate receives only 3 G because of a soft bolt-spring system. In order to increase the stiffness of the bolt system, a push-bar coupling is often used. The coupling permits the use of many more bolts at the junction. although another bolted interface is added, the increase in the number of bolts will increase the overall stiffness of the system.

PUSH-BAR COUPLINGS

(Slider plate

361

I

u

FIGURE 14.17. View of a slider plate bolted to a shaker head

tap in

a \ ; : J

shaker head

$

/

,

\

,

16.0 dia.

FIGURE 14.18. A typical bolt pattern in a shaker head.

Consider the vibration system shown in Fig. 14.10. If the slider plate is bolted directly to the shaker head, the interface would probably appear as shown in Fig. 14.17. Considering the bolt-hole pattern shown in Fig. 14.18 only five b o l t s could be used in t h e slider p l a t e . The bolted interface system shown in Fig. 14.17 is nonlinear, since a pulling force on the slip plates places the bolts, which have a small cross-sectional area, in tension. A pushing force, however, acts directly on the slider plate and places that member, which as a large cross section, in compression. A load-deflection diagram for this type of system is similar to that shown in Fig. 14.19. Nonlinear systems do not have the same response characteristics as linear systems, and the natural frequency equations are not the same. However, if the nonlinear system is approximated as a linear system, then linear-system equations can be used to estimate the probable performance characteristics, as shown in the following section.

362

VIBRATION FIXTURES AND VIBRATION TESTING

1+

/i

Load

Sdder plate in .o-?pre;sion

- Loaa

FIGURE 14.19. A load-deflection curve for a nonlinear spring.

300 ib Bolts

FIGURE 14.20. Simulating the slider plate as a two-mass system.

H-

1

100 Ib

-

I\/Wii~ I 1

6

-

1

The natural frequency of the vibration system shown in Fig. 14.10 can be approximated by lumping the masses into a two-mass system with one spring where the bolts act as the spring (Fig. 14.20). Weight W , is as follows: Slider plate Vibration fixture Electronic box

150 lb 90 60 W , = 300 lb

(14.9)

Weight W , is the shaker head. Assuming an MB vibration machine Model C-126, which has a rating of 9000 force pounds, the shaker head weighs Wz = 100 lb

(14.10)

The spring rate of the bolts can be determined from Eq. 14.3:

where E = 30 x lo6 lb/in.’ (steel bolts) L = 0.625 1 bolt diameter (see Fig. 14.17) L = 0.625 + 0.375 = 1.00 in. (length) A , = 0.0876 in., (stress area for a in.-diameter fine thread on one bolt)

+

PUSH-BAR COUPLINGS

363

For five bolts, the spring rate is

K=

5(0.0876)(30 X l o 6 )

13.10 X 106 Ib/in.

=

1.oo

(14.11)

Substituting Eqs. 14.9-14.11 into Eq. 14.7, (13.1 X 106)(3.86 X 10')

1/?

3.00 x 10' (14.12)

f, = 1310 HZ

Since the system is really nonlinear, the natural frequency would be somewhat higher than this. However, this system would not be adequate to perform vibration tests as high as 2000 Hz if high G forces are required. Since the spring system is relatively soft, a high G force in the shaker head produces a relatively small G force, at frequencies well above the resonance of the bolted system, in the slider plate. Even if the spring rate of the bolts indicates that a resonance might occur in the vibration oil-film-slider system, a high preload in the bolts may not permit a resonance to occur. In order for a resonance to occur, the dynamic loading on the oil-film slider plate must exceed the preload in the bolts. If steel bolts with a yield strength S, of 35,000 Ib/in.? are used to fasten the slider plate directly to the shaker head, then the maximum allowable preload in each bolt is P = A S S ,= (0.0876)(35,000)

=

3060 Ib

(14.13)

The installation torque required on each bolt can be determined from Eq. 14.4:

T = 0.2DP = 0.2(0.375)(3060) = 230 in:lb

(14.14)

The total preload on five bolts is P, = 5(3060)

=

15,300 Ib

(14.15)

The dynamic load acting on the slider plate can be approximated from the input G force, the weight, and the transmissibility at resonance: Pd

Q

= G,"Wi

(14.16)

Assuming a 10-G peak sinusoidal vibration input with a transmissibility of about 6, the dynamic load is Pd = 10(300)(6) = 18,000 lb

(14.17)

364

VIBRATION FIXTURES AND VIBRATION TESTING /Push-bar

‘I

coupler using 11 bolts,

St a f i e r h e a d

S h a - e r ‘Ieid bc t pattern

FIGURE 14.21. A push-bar coupler with 11 bolts to increase the stiffness.

The dynamic load shown by Eq. 14.17 is greater than the total preload on the bolts shown in Eq. 14.15; this means the bolt preload is not high enough to prevent a resonant condition from occurring at about 1310 Hz, as shown by Eq. 14.12. A push-bar coupling could be designed to pick up 11 bolts on the shaker head as shown in Fig. 14.21. (For maximum rigidity, the slider plate should be welded to the push-bar coupler.) All bolted interfaces should load the bolts in tension and not in shear, to obtain the maximum effective stiffness. Bolted interfaces loaded in shear can slip and slide with respect to each other at high frequencies. Even if several large press-fit dowels are used in the interface to prevent relative motion, it can still occur at high G loads. Therefore. the bolts should be loaded in tension for the best results. 14.1 5

SLIDER PLATE LONGITUDINAL RESONANCE

A slider plate can develop resonances during vibration in the longitudinal direction if the plate is long enough. The natural frequency for a structure under these conditions can be determined from Eq. 14.1. Since the width of the plate is not important during longitudinal vibration, consider a unit width of 1.0 in. and a length of 30.0 in. (Fig. 14.22). The following information is

FIGURE 14.22. A unit width o n a n oil-film slider plate (dimensions in inches).

365

ACCELERATION FORCE CAPABILITY OF SHAKER

then required to determine the resonant frequency: 1.0 in.* (cross-sectional area) 10.5 x l o 6 lb/in.* (modulus for aluminum) = 386 in/s2 (gravity) W = (1)(1)(30)(0.1) = 3.0 lb L = 30 in. A

=

E g

=

Substituting into Eq. 14.1, the longitudinal natural frequency becomes (1.0)( 10.5 X 106)(3.86 X l o 2 )

(3 .o>(30)

f, = 1680 HZ

(14.18)

This represents the natural frequency of the bare slider plate. If the masses of the fixture and electronic box are included, the natural frequency will be lower. A more comprehensive analysis of the slider plate, vibration fixture, electronic box, push-bar coupling, and shaker head could treat this combination as a multiple spring-mass system or as a distributed system using finite element modeling methods (FEM) with a computer. 14.16 ACCELERATION FORCE CAPABILITY OF SHAKER

Vibration machines are rated in peak force (pounds) based on sinusoidal vibration, which is determined from the maximum allowable power that can be dissipated in the machine. In order to calculate the maximum G force possible in a system, the force rating of the machine must be known along with the entire weight of the moving mass. The equation then becomes

Gpeak

=

rated force shaker head + slider + specimen

(14.19)

The peak G-force capability for the system shown in Section 14.14 can be determined for the MB C-126 vibration machine, which has a peak rating of 9000 force pounds. Substituting Eqs. 14.9 and 14.10 into Eq. 14.19, the peak G-force capability becomes

9000 Gpeak =

100 + 300

= 22.5

(14.20)

366

14.17

VIBRATION FIXTURES AND VIBRATION TESTING

POSITIONING THE SERVO-CONTROL ACCELEROMETER

The G force transmitted from the shaker head to the vibration test specimen can be controlled directly from the shaker head or from an accelerometer placed near the test specimen. When the G force is controlled from the shaker head, a constant G force can be maintained on the shaker head over the entire frequency band. If a resonance should develop in the slider plate or the vibration fixture, the G forces will build up at resonance so that the input to the test specimen will be far greater than the output of the shaker head. At frequencies above the resonance the opposite situation occurs, and the input to the test specimen will be far less than the output of the shaker head. When the G force controlled from an accelerometer placed near the test specimen, the vibration machine can vary the output of the shaker head to hold a constant G force on the control accelerometer over the entire frequency band. If the control accelerometer is placed on a resonant structure, the output of the shaker head will be reduced when the structure passes through its resonance. At frequencies above the resonance, the output of the shaker head will be increased to hold a constant G force on the control accelerometer. Standard piezoelectric accelerometers, the same types that are generally used to monitor and record acceleration data, are normally used as servocontrol accelerometers. These small devices are usually equipped with screw studs to fasten them to the test equipment. Many testing laboratories use adhesives such as dental cement, Eastman 910 quick-drying cement, and even double-backed tape to mount accelerometers. These methods all work quite well on accelerometers that are used to monitor acceleration data, but they should never be used to fasten a control accelerometer to a vibrating system. The servo-control accelerometer actually controls the G-force output of the shaker head. If the servo-control accelerometer should fall off during the vibration test, the shaker head will lose the feedback provided by this accelerometer. This results in a rapid buildup of the acceleration force until the limit displacement switch of the shaker head cuts off the power. By this time the damage has already been done: very high G forces can be developed very rapidly. Once an expensive electronic system has been subjected to acceleration loads that far exceed its specifications, there is no good way to determine the actual and potential damage. In most cases where this has happened, and it has happened at many places, the electronic equipment has to be rebuilt because the customer will not accept an item that may fail shortly after it is put into service. Since the position of the servo-control accelerometer can determine the actual G force received by the tested specimen, the location selected for this accelerometer can be quite important to the success of a vibration test. A resonant fixture on a slider plate is shown in Fig. 14.23.

MORE ACCURATE METHOD FOR ESTIMATING TRANSMISSIBILITY Q IN STRUCTURES

367

Resonant fixture

,X\\\\K

Electronic box

\'~\\\,,\\\\'~\\\\'~\ \ \ \ \ \\ \\ \\ \'\\\\\\\\,

FIGURE 14.23. A resonant fixture on an oil-film slider plate.

If the top of the resonant fixture has a transmissibility of 10, then it will see 20 G at its resonant peak when the control accelerometer is held at 2 G. This means the electronic box will probably see an average of about 11 G at the fixture resonance point instead of the required input of 2 G. If the control accelerometer is mounted at the top of the resonant fixture, the bottom of the fixture will see only 0.2 G at the resonant peak. This means that the electronic box will probably see an average of about 1.1 G at the fixture resonance point instead of the required input of 2 G. The best solution to this problem, of course, is to design resonance-free fixtures. Since this is not always possible, a compromise is the only possibility, so the control accelerometer will probably be placed halfway up the fixture.

14.18 MORE ACCURATE METHOD FOR ESTIMATING THE TRANSMISSIBILITY Q IN STRUCTURES

Damping in a vibrating structure plays an important part in establishing the magnitude of the transmissibility Q that will be developed when the structure is excited at its natural frequency. Dynamic coupling and phasing relationships can also play an important part. When the damping is increased, more kinetic energy is converted into heat so there is less energy available to do work on the structure, which results in a decreased transmissibility. When the damping is decreased, less kinetic energy is converted into heat so there is more energy available to do work on the structure, which results in an increased transmissibility. This can also be related to the dynamic displacements and stresses. Higher values produce more damping, which decreases

368

VIBRATION FIXTURES AND VIBRATION TESTING

the transmissibility. Lower values produce less damping, which increases the transmissibility. The input acceleration G levels play an important part because high input acceleration G levels result in high stresses and high damping, which reduces the transmissibility. Low input acceleration G levels have the opposite effect. An examination of Eq. 2.30 shows that when the G level is held constant, the displacement Yo is inversely related to the square of the frequency. When the frequency is doubled, the displacement is reduced by a factor of 4. Reducing the displacement reduces the stress, which decreases the damping so the transmissibility is increased. The above information leads to two very important conclusions: 1. Increasing the natural frequency will increase the transmissibility. 2. Increasing the input acceleration G level will decrease the transmissiblilty.

The above conclusions were combined with several years of accumulated test data on different types of electronic support structures that were broken down into three basic groups: beams supporting some electronics, PCBs with some perimeter supports, and small enclosed electronic boxes. The data were plotted on log-log paper to see if some general type of straight-line curve, similar to a S-N curve, could be developed. After churning a large number of approximations, the general equation shown below was finally selected: (14.21)

where A A A

1.0 for beam-type structures = 0.50 for plug-in PCBs or perimeter-supported PCBs = 0.25 for small electronic chassis or electronic boxes f, = natural frequency (Hz) G,, = sinusoidal vibration input acceleration G level (dimensionless) =

The above equation relates to electronic assemblies, so a beam structure is expected to hold several electronic components with some interconnecting wires or cables. The PCB are expected to be well populated with an assortment of electronic components. The small chassis is expected to be between about 8 and 30 in. in its longest dimension, with a bolted cover to provide access to various types of electronic components such as PCBs, harnesses, cables, and connectors. Sample Problem-Transmissibility

Expected for a Plug-in PCB

Determine the approximate transmissibility expected for a plug-in PCB with a natural frequency of 280 Hz using peak sine vibration input G levels of 0.2 G, 2 G , 5 G , and 8 G.

CROSS-COUPLING EFFECTS IN VIBRATION TEST FIXTURES

Solution. Use Eq. 14.21 with the value A input level.

[

=

1

280 Q=O.50 (0,210.6

369

0.50 starting with the 0.2 G

0.76

=

75’4

Peak Sine Input Level Gin

Approximate Q Expected

0.2 2.0 5.0 8.0

75.4 26.4 17.4 14.0

(14.22)

The above data correlate fairly well with vibration test data performed on a wide variety of PCBs. Many PCB sample problems in this book use an approximate transmissibility Q value based on the square root of the natural frequency, which results in a value of 16.7. This is close to the value for the 5-G peak sinusoidal vibration input level shown above.

VIBRATION TESTING CASE HISTORIES 14.19 CROSS-COUPLING EFFECTS IN VIBRATION TEST FIXTURES

The best way to start this section is to describe the events associated with the vibration testing of a small electronics module. An electrodynamic shaker was set up to vibrate in the vertical direction. The vibration test fixture was in the shape of a cube frame that was assembled by welding together the ends of 12 aluminum bars each measuring 1.0 in. square and 9.0 in. long as shown in Fig. 14.24. The fixture was bolted to the top of the shaker head and the small

Frame vibration fixture

9.0

I l,O-ln, square aluminum bars

FIGURE 14.24. Poor vibration fixture using 1.0 in. square aluminum bars welded together.

370

VIBRATION FIXTURES AND VIBRATION TESTING

electronics module was bolted to the top of the fixture. A small accelerometer was fastened to the top of the fixture, adjacent to the test module, to control a 5.0-G peak sinusoidal vibration input level in the vertical direction. The electronics module being tested was rated at 25 G peak. The sine sweep started at 25 Hz with the module operating electrically. Somewhere in the area of about 200 Hz the module failed. Everyone was upset because they did not expect any problems with a 5-G test on a module that was rated at 25 G. A call was placed to the module vendor complaining about the false advertised rating of 25 G. The module vendor again verified and guaranteed the 25-G rating and suggested they hire someone to look at their problem. An outside consultant came in and requested that two additional accelerometers be mounted in the two horizontal axes to monitor the magnitude of the cross coupling. The program managers denied the request saying there was no horizontal cross coupling on the fixture. The consultant asked them if they had test data that verified the lack of horizontal cross coupling. The program managers replied that cross-coupling test date were not necessary and that it was a waste of time. They claimed that when the system is being vibrated in the vertical direction it can only move in the vertical direction and no other direction was possible. The consultant now insisted that two more horizontally oriented accelerometers be attached to the fixture to examine cross-coupling responses. Again the program managers denied the request. This standoff went on for about an hour until the program managers allowed two more accelerometers to be mounted on the fixture in the horizontal direction. The test data showed the horizontal response was 104 G peak for a 5-G peak vertical input. It was obvious to the consultant that the frame fixture had a low natural frequency in the horizontal direction. The consultant recommended the frame fixture be replaced with a small solid aluminum block fixture that would have a natural frequency well above 2000 Hz, and the problem was solved. Nature tends to be very efficient in our world of dynamics. The theory of deformation says that structures under load will deform in a pattern that keeps the strain energy at a minimum. Another way of saying the same thing is that any time a structure is being vibrated, it will respond by moving in the direction that generates the minimum strain energy, which always results in the lowest natural frequency. The direction of maximum response is often in a direction that is different from the input direction. PROGRESSIVE VIBRATION SHEAR FAILURES IN BOLTED STRUCTURES 14.20

A rugged wheel axle on a piece of farm machinery was bolted to a rigid frame with many large bolts, as shown in Fig. 14.25. Every few years the wheel axle would become loose because many of the large bolts had sheared off. The farmer kept replacing the bolts when they broke to keep the machinery

VIBRATION PUSH-BAR COUPLERS WITH BOLTS LOADED IN SHEAR

371

Bolts loading in shear Vibration

FIGURE 14.25. Progressive shear in bolted joints subjected to alternating dynamic loads.

working. A dynamic stress analysis was made of the bolts using data from accelerometers that were attached to the axle. The analysis assumed that half of the bolts would carry the alternating shear loads. This analysis showed the bolts should not fail so the farmer thought the bolts were defective. Higherquality and higher-strength bolts were installed. They lasted a little longer but they also failed. Bolted joints require large holes in the mated parts to allow easy assembly for mass produced parts with standard machining tolerances. A close examination of bolted joints under an alternating load shows that the bolted part will move slightly during vibration since friction is trying to prevent relative motion between the two bolted parts. Friction and vibration are like oil and water-they do not mix. The bolted part will move until one bolt picks up the load. The first bolt that picks up the load will become the pivot point and the part will continue to move until the second bolt picks up the load. This loading will alternate between the two bolts until one of the bolts fractures so another bolt picks up the load. This process will be repeated again and again until the bolted assembly falls apart. Only two bolts will cany the alternating load in a bolted assembly when loose tolerances are used in mass produced machine parts, no matter how many bolts are used. If bucked rivets are used instead of bolts, the rivets expand to fill the holes so relative motion between the riveted parts is not possible. In this case all of the rivets will carry some part of the alternating load. If rivets cannot be used, then two (or more) large press-fit dowel pins should be added with a shear area large enough to carry the alternating shear loads. The standard practice is that bolts should be loaded in tension and dowel pins should be loaded in shear.

4.21 VIBRATION PUSH-BAR COUPLERS WITH BOLTS LOADED IN SHEAR

Engineers with limited vibration testing experience are often requested to design special fixtures for testing electronic equipment. Bolted joints are often employed because they are quick and easy to install and remove. A

372

VIBRATION FIXTURES AND VIBRATION TESTING Shaker head

Bolts loaded in shear

Vibration direction

Bolts loading in tension

FIGURE 14.26. Poor arrangement for oil-film slider vibration fixture using a push-bar coupler with attachment bolts loaded in shear.

typical installation of this type for an aluminum oil-film slider plate and a bolted coupler is shown in Fig. 14.26. When the test specimen is heavy, the vibration machine cannot transfer high acceleration G levels to the slider plate at high frequency because there is too much slippage at the slider plate bolted interface. This type of installation relies on friction at the bolted interface to transfer the load from the shaker to the slider plate. This system works well for forcing frequencies below about 800 Hz. With a shaker forcing frequency of about 1000 Hz and a 10-G peak sinusoidal vibration input, only about 2 G peak is transferred to the slider plate because of the slippage at the bolted joint. When clearances are provided in the fixture bolt holes to allow easy assembly, the rigid connection from the shaker to the slider plate is lost because the interface friction at the bolted joint cannot carry the high forces without slipping. The bolt tightening torque can be increased, or a couple of extra bolts can be added, or larger bolts can be used, but these changes will not improve the high-frequency testing performance very much. In the previous section it was pointed out that bolts should be loaded in tension and dowel pins should be loaded in shear, so dowel pins are proposed. Three 1.0-in. diameter holes are drilled and reamed for closely machined press-fit dowel pins. Three hard steel dowel pins are placed in a cold chamber before they are press-fit into the fixture to ensure a good tight fit as the pins return to room temperature and expand. The test engineers feel confident that the dowel pin fixture will solve their high-frequency testing problem. The tests are resumed with a shaker forcing frequency of 1000 Hz and a 10-G peak sinusoidal vibration input. The slider plate accelerometer only shows a value of 3 G peak instead of the expected 10 G peak. The test engineers think the slider plate accelerometer is defective. It is not defective. What happened? The hard steel dowel pin acting against the softer aluminum causes the aluminum to brinell locally at the corners in about 4 seconds, as shown in Fig. 14.27. With 1000 cycles per second, in 4 seconds there are 4000 stress reversals, which loosens the dowel pins. One good way to solve this problem is to weld a slider plate to a push-bar coupler, which

BOLTING PCB CENTERS TOGETHER TO IMPROVE THEIR VIBRATION FATIGUE LIFE

373

Brinelling corners

FIGURE 14.27. Loss of effectiveness for press-fit dowel pins in push-bar couplers d u e to rapid brinelling of the softer aluminum corners during high-frequency vibration.

Shaker head

i

1

1 --&e\ l-dfl Push-bar coupler

Bolts loaded in tension

FIGURE 14.28. Good arrangement for vibration fixture with push-bar coupler welded to slider plate and many mounting bolts loaded in tension.

has all of the attachment bolts to the shaker head loaded in tension as shown in Fig. 14.28.

14.22 BOLTING PCB CENTERS TOGETHER TO IMPROVE THEIR VIBRATION FATIGUE LIFE

Plug-in type PCBs usually have relatively low natural frequencies because their centers are unsupported so they tend to “oil can.” As a result, the transmissibility Q values increase, producing more rapid component lead

374

VIBRATION FIXTURES AND VIBRATION TESTING Perimeter flange

Piug-in Cover

"'\

Vi bration

direction

,

I,

,

lj

V

I

0.25-in. diameter steel rod

Spacer between each PCB

through every PCB

'Slider

piate

FIGURE 14.29. Long steel rod through the center of every plug-in PCB for improved support to increase PCB natural frequency. It did not work. High dynamic loads destroyed the chassis.

wire and solder joint fatigue failures. When vibration tests are run on individual plug-in PCBs that have their centers supported, there is a big increase in the natural frequency and a big increase in the fatigue life. A prototype aluminum dip-brazed chassis with 0.090-in. thick walls was fabiicated to hold 20 dummy plug-in type PCBs, weighing 1.0 lb each, to study this idea. The chassis used a 0.25-in. diameter steel rod, threaded at both ends, that passed through a hole in the center of each PCB. Hollow aluminum cylinder tube spacers with close tolerances were placed between every PCB so the centers of every PBC would be locked together when the steel bar was bolted and clamped at both ends of the chassis as shown in Fig. 14.29. In order to avoid any possible vibration problems, a rugged reinforcing rib 1.0 in. long by 0.38 in. thick was dip-brazed around the perimeter of the chassis. The steel rod through the chassis was placed under, but close to, the reinforcing perimeter rib. Doublers were used at both ends of the chassis to make sure the 1.0-in. long rib would carry, the dynamic load instead of the thinner 0.090-in. thick aluminum wall. The assembled chassis with the 20 dummy PCBs was instrumented with several small accelerometers and bolted to an oil-film slider plate, which was bolted to the head of an electrodynamic shaker for a 5.0-G peak sinusoidal vibration test from 20 to 2000 Hz. The vibration input was along the longitudinal axis of the chassis, perpendicular to the plane of the PCBs. The forcing frequency was increased slowly while several X-Y plotters printed the responses of several accelerometers mounted within the test chassis. As the forcing frequency approached the 425-Hz natural frequency of the chassis, there was a sharp increase in the noise level similar to a railroad train passing through the test area. This was quickly followed by what can only be described as an explosion, as the rugged dip-brazed electronic chassis with the stiff perimeter reinforcing rib literally tore itself apart.

VIBRATION FAILURES CAUSED BY CARELESS MANUFACTURING METHODS

375

Hindsight is always very clear. It should have been obvious that when all of the PCBs were locked together they would all have the same natural frequency. Unless someone has had some personal testing experience, it is very difficult to imagine the amount of kinetic energy that is available when 20 lb of mass goes into a resonant condition. With only a 5-G peak sinusoidal vibration input acceleration level, a 20 lb mass with a transmissibility Q of about 20 will produce a dynamic load of 20 X 20 X 5 = 2000 lb acting on two reinforcing ribs, front and rear, This results in a dynamic force of 1000 lb on one rib. With a chassis width of 8.0 in. the approximate bending moment on one reinforcing rib will be 500 X 4 = 2000 lb ‘in. The moment of inertia for one rib will be 0.0316 in.4. The dynamic bending stress on one rib will be approximately

Mc

sb -- - =I

(2000) (1.0/2) 0.0316

= 31,645

lb/in.’

(14.23)

The approximate fatigue life of the 6061 T-4 aluminum dip-brazed chassis can be obtained from Eq. 8.2 using an ultimate tensile strength of 36,000 for the dip-brazed aluminum and a fatigue exponent of 6.4 as shown below:

zr

1

36,000 6.4 N , = N 2 - =(1000) 31,645

( ]

(

=

2282 cycles to fail

(14.24)

The time to fail can be estimated from the chassis natural frequency of 425 Hz as follows: Time to fail

2282 cycles to fail =

(425 cycles/s) (60 s/min)

= 0.089 minute = 5.4 seconds

(14.25)

14.23 VIBRATION FAILURES CAUSED BY CARELESS MANUFACTURING METHODS

A small shaft on a spinning gyro for an inertial navigation system was failing during a flight acceptance test of 3.5 G RMS. A dynamic stress and fatigue life analysis of the shaft, including a generous stress concentration factor showed that it should not fail. The shaft was fabricated by machining the end of a 0.25-in. diameter dowel rod down to a smaller diameter of 0.065 in., with a very small radius at the step in the shaft. The failures occurred at the step in the shaft. There were some questions as to just how the machining operation was being performed. If a lathe with a single cutter was being used, there was a possibility that the cutting action on a small diameter could generate a bending stress high enough to initiate a small crack. This crack

376

VIBRATION FIXTURES A N D VIBRATION TESTING

might then propagate during the vibration acceptance test, producing the observed series of failures. A visit was made to the manufacturing area and a discussion was held with the shop supervisor. The shop supervisor was very upset by the idea that he would allow such a machining operation to be performed in his machine shop. The supervisor claimed that he had more knowledge of fatigue fractures than most of the engineers working on the program. He insisted that all of his workers were instructed to use three cutters on the shaft, spaced at 120" apart so there would be no possibility of any crack initiation in the shaft. His parting words were that the engineers had a lot of nerve to blame their poor designs on his careless manufacturing methods. The engineering dynamic loads and stresses were checked and rechecked with the same conclusion. The vibration level was not high enough to cause a shaft failure. Another visit was made to the manufacturing area, but instead of talking to the supervisor again the individual machining areas were inspected. Sure enough, in one of the areas it was obvious that the machinist was using only a single cutter on the shaft. The supervisor was then called over to examine the machining setup. His face turned white when he saw that only one cutter was being used to machine the shaft. When he asked the machinist why a single cutter was being used when the use of three cutters was standard practice, the machinist said that it took too long to set up and cut with three cutters so the daily quota could not be met. The single cutter was much faster, and it was obvious to the machinist that the single cutter did the job just as well as the three cutters. The above circumstances are a classic case of giving instructions with no follow-up to make sure the instructions are being followed. Some supervisors are too lazy to follow up on their instructions while other supervisors feel embarrassed when they have to verify that their instructions are being followed. In any case these people will always assume that they know what is going on in their areas, when they really do not know. They often pass on incorrect information in critical investigations, which makes it very difficult to get good data that can be used to make intelligent decisions. 14.24 ALLEGED VIBRATION FAILURE THAT WAS REALLY CAUSED BY DROPPING A LARGE CHASSIS

A group of four mechanical engineers sought out the advice of a vibration specialist regarding what they called a vibration failure in a large expensive inertial navigation cast aluminum chassis. The 0.25-in. thick casting had a deep V shaped crack where each leg of the V shape was about 2 in. long. The crack penetration was about 0.50 in. deep into the chassis so the thick walls of the chassis were clearly visible in both legs of the V crack. The vibration specialist took one look at the deep concentrated crack penetration in the thick casting and declared that the crack was not a vibration failure. The crack was caused by a high impact, probably from dropping the chassis onto a

METHODS

FOR INCREASING THE VIBRATION AND SHOCK CAPABILITY

377

hard floor, The four engineers insisted that they had proof that the chassis was not dropped. When they were asked what their proof was, they replied that they asked everyone on the program if the chassis was dropped and everyone said no. Since the chassis had just completed a vibration test, they concluded that the only possible cause of the crack had to be the vibration. The vibration specialist suggested that no one in their right mind would ever admit to dropping a chassis that was worth at least $150,000. The sad part of this incident is that four graduate mechanical engineers could not tell the difference between an obvious impact crack and a vibration-induced crack. The lesson learned in this incident is that people will lie when they are afraid of losing their jobs. Again it shows that it is often very difficult to obtain the correct information concerning failures, especially when people are afraid they may lose their jobs.

14.25 METHODS FOR INCREASING THE VIBRATION AND SHOCK CAPABILITY ON EXISTING SYSTEMS

Military procurement agencies recently decided to scrap many popular documents that related to the procurement of military electronic equipment because of increased expenses. They have turned to best commercial practice electronic equipment to reduce these costs. Commercial electronic equipment does not have the required high-temperature reliability normally needed for military programs. This means that better cooling must be supplied for the commercial equipment when it is used in military programs. This can often be achieved by using a more powerful fan for improved cooling, or by adding larger heat exchanges, or by adding larger convection cooling fins. Improving the vibration and shock capability on a commercial program, for possible use in a military program, will usually require some structural adjustments to existing PCBs and to existing chassis enclosures. The PCBs can often be made more vibration and shock resistant by semi-encapsulating the PCBs with a semisoft silicone or some type of foam. This will increase the fatigue life but it will make repairs very difficult. Snubbers have been used successfully in converting commercial hardware for use in military programs. Snubbers work by striking each other when the natural frequency of the PCB is excited. This impacting action significantly reduces the dynamic displacement, thus improving the fatigue life. Epoxy glass rods about 0.25 in. in diameter work very well when they are cemented near the centers of adjacent PCBs with a small clearance between each pair of snubbers. It may be difficult to find good open spaces between the components near the center of the PCB. Good snubber locations can often be found by making transparent copies of each side of each PCB. These can then be grouped together in the proper sequence and held up to a light to find the best locations for snubber pairs.

378

VIBRATION FIXTURES AND VIBRATION TESTING

One quick-fix approach that has been used to improve the vibration and shock capability of an electronic chassis full of plug-in PCBs is to fill the chassis with small rubber balls or small hollow lightweight spheres similar to table tennis (Ping-Pong) balls. When the chassis is full of these spheres the PCBs cannot deflect very much so the vibration performance and the shock performance are substantially improved. This system will work very well in a chassis that is cooled by conduction from the PCBs to the side walls of the chassis. This system is also very easy to repair. The chassis cover is unbolted and the balls are spilled out and saved. This gives free access to the PCBs for any required maintenance. The balls are then replaced and the cover is replaced to complete the maintenance or repair. Sometimes the chassis structure itself requires some reinforcement. This can often be done with the screw-and-glue method. Reinforcing angles or doubler side panels can be glued to critical sections of the chassis, adding small screws if necessary for electrical continuity or for extra reinforcement. These types of quick fixes work very well when only a few systems have to be reinforced. When several hundred systems have to be modified, it may be cheaper to redesign the entire system for the new applications.

Environmental Stress Screening for Electronic Equipment (ESSEE)

15.1

INTRODUCTION

A couple of decades ago manufacturers of electronic equipment used burn-in procedures to eliminate defective electronic components. New electronic components were placed in a chamber set at some elevated temperature between 55°C and 71"C, and electrically operated for several days. The components that survived this process were used in the electronic systems that were sold to the consumers. This practice eliminated many weak components, which resulted in a more reliable product. As the electronic products became more complex and more sophisticated, the burn-in procedures did not work as well and the failure rates began to increase once again. Some new screening processes and procedures were required. A 2.2-G peak sinusoidal vibration stimulus was added to the high-temperature soak condition, using a low input frequency that did not excite any natural frequencies within the electronic system being screened. Eventually this procedure added broadband random vibration and temperature cycling to further improve the screening effectiveness. This vibration and temperature screen was often called a shake-and-bake test. This upset many people who insisted that the screen was an extension of the manufacturing process, and not a test. The argument given was that failures are not desired in a test, but failures are desired in the screen. The purpose of the screening is to get rid of the infant mortality groups of early failures, as shown in Fig. 15.1. The latent manufacturing defects are precipitated into hard electronic failures so they can be repaired at the factory, before the product is delivered to the consumer. The consumer is very happy with the more reliable product. The news of the good quality product spreads rapidly so more are sold and everyone is happy.

15.2

ENVIRONMENTAL STRESS SCREENING PHILOSOPHY

Environmental stress screening can be defined in many different ways, depending on the objectives of the program. One simple definition relates to 379

380

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

Time

FIGURE 15.1. Bathtub reliability curve showing infant mortality, useful life, and

wear-out.

its purpose, which is to provide a cost-effective method for improving the reliability of the electronic equipment. Another popular definition states that the purpose of the screen is to stimulate the electronics in some way that will force hidden flaws into hard failures, so the product can be repaired at the factory before it is shipped, so the consumer receives a reliable product. The idea behind the screen is that hidden defects and flaws are related to the manufacturing and the assembly processes and procedures, and not to the basic design of the product. However, experience with various types of screens has shown that they can reveal poor manufacturing practices as well as poor design practices. A careful examination of the screening data will often reveal failure patterns on component parts: improper processing procedures may show up, circuit design errors may become obvious, or careless manufacturing methods may be revealed. A large number of failures during the screening process, or very few failures during the screen, may be a clue that the screening process itself is deficient. A very effective document on the stress screening of electronic equipment can be found in the Navy’s NAVMAT-P-9492, issued in May 1979 under the title of Navy Manufacturing Screening Program [471. The final objective of the screening process is to eliminate its requirement. The screening process should eventually enable the engineering and manufacturing groups to constantly improve their techniques. The constant improvement process in the design, the hardware, and the manufacturing quality should eventually reach a point where a high quality is achieved with a low cost, so the screening is no longer required. If there is no continuous improvement in the hardware quality over time. it should become obvious that the screening process is not effective and requires a change, otherwise it is a never-ending exercise that ends up costing the company extra money.

SCREENING ENVIRONMENTS

381

15.3 SCREENING ENVIRONMENTS

The most common screens are usually based on the actual operating environments. The most popular screens are thermal cycling and some form of vibration. Experience has shown that some form of thermal cycling screen and vibration screen can be effective for revealing manufacturing flaws. Screens are also effective for electronics that are designed to work in very quiet and stable air-conditioned offices, where thermal cycling and vibration conditions are never expected. Military electronics experience has shown that thermal cycling screens are more effective than vibration screens in revealing manufacturing flaws. However, the use of both types of screens is strongly recommended for improved results. These can be done separately, or they can be combined and done simultaneously, electrically operating or nonelectrically operating. It is easier to observe the failures when the system is operating electrically. However, this screen method is more expensive. A few successful screens have used shock pulses and some screens have used acoustic noise. Time is money, so there is a strong desire to reduce the screening time to save money. Some early thermal cycling screens used thermal shocking to reduce the time it takes to ramp the temperature up and down. Two different chambers were set next to each other, either vertically or horizontally. One chamber was set at + 100°C and the other chamber was set at - 40°C. A piston was used to carry the electronic unit from chamber to chamber in about 5 seconds. The dwell time at each extreme temperature was based on the size of the electronic unit. This method was scrapped. Many more failures were produced in the screen than were ever experienced in the field. The ramp rate for the temperature cycle was investigated and tested in an attempt to find a rate that provided good reliability results in the field with an acceptable cost basis. Ramp rates from 2°C per minute to 30°C per minute were investigated. The ramp rate favored was 20°C per minute for seven thermal cycles from -55°C to +71"C, with power on during the increasing temperature cycle and power off during the decreasing temperature cycle. A few companies have found their screens work best with a temperature ramp rate of 2°C per minute. Most companies appear to be using ramp rates between about 10°C and 20°C per minute. The larger, more massive electronic components and the PCB itself should use thermocouples to find the approximate time it takes to reach a stable condition at the high and low temperatures. This information will be helpful in establishing the dwell time at the high and low temperatures. Different forms of vibration exposure have been very successful in finding latent defects in electronic systems. Sinusoidal vibration sweeping from 20 Hz to 2000 Hz to 20 Hz at a rate of half an octave per minute has some benefits. Broadband random vibration for several minutes covering a frequency range from 20 to 2000 Hz appears to be the most effective form of vibration based on the successful screening data obtained from many different companies. Vibration is usually applied for a period of about 5 minutes in each of three

382

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

major axes, for a total of 15 minutes. When all the PCBs are installed in the same direction, then the vibration is typically applied for 10 minutes along the axis perpendicular to the plane of the PCBs [501. When the PCBs are installed in two different directions, then simultaneous shaking along two different axes perpendicular to the PCBs is often used. The typical installation uses two different shakers with an oil-film slider plate shaking in the horizontal plane set 90” to each other. This is an expensive operation. The lower-cost compressed-air shakers with piston-driven thumper vibrators shake along all three axes and also provide rotation about all three axes, simultaneously exciting six degrees of freedom. The thumpers usually have a provision for performing thermal cycling in combination with the vibration. This is a big advantage for screening. The disadvantage is that there are no peak-notch filters available to flatten out any high resonant transmissibility spikes that have the potential for damaging good electronic components. A single electrodynamic shaker excites only one axis at a time. This is a disadvantage. These shaker systems often have the peak-notch filters, which is an advantage. A popular random vibration profile is shown in Fig. 15.2. Military programs have been successful using an input acceleration level of about 6.0 G RMS on small electronic boxes, where the power spectral density (PSD) is 0.04 G’/Hz from 80 to 350 Hz. Small is defined here as having a maximum dimension of about 24 in. Random vibration can be very dangerous, especially on large boxes or cabinets, if it is not used properly. A good understanding of structural dynamics and cumulative fatigue damage is required to ensure a successful program. High vibration acceleration levels even for short periods of time can generate a large amount of damage. Broadband random vibration can be developed in several different ways. The most accurate control method is with an electrodynamic shaker, which is very expensive. Some electrodynamic vibration machines can use closed-loop tapes to perform a lower-cost random vibration. Several companies offer low-cost pneumatic piston thumper quasirandom vibration machines that are combined with thermal cycling capability packaged in a relatively small enclosure.

i

0.04 G2/Hz

20

80

3 50

2000

FIGURE 15.2. Proposed ESS random vibration PSD profile from NAVMAT-P-9492.

383

THINGS AN ACCEPTABLE SCREEN ARE NOT EXPECTED TO DO

Random vibration levels of 6.0 G RMS can be very damaging on large cabinets. This can be demonstrated by comparing the U.S. Navy shock requirement MIL-S-901 and vibration MIL-STD-I 67 requirements for a 6-ft cabinet weighing 500 lb. The Navy uses a 3000-lb swinging hammer to impact a fixture that supports the cabinet for a shock test. This produces an input shock level of about 150 G. The electronic equipment inside the cabinet usually sees a level of about 200 G. These cabinets almost always pass this shock test. However, the same cabinet very seldom passes the 1.0-G peak input sinusoidal vibration test for a period of about 1 minute, so a waiver is often requested and granted. This means that a 1.0-G peak input sine vibration test for 1 minute generates more damage than a 150-G peak input shock test. The purpose of this comparison is to demonstrate that vibration can excite natural frequencies, which can be very damaging to large structures. A good preliminaly rule to follow for a random vibration screen is that the longest dimension ( L ) of an electronic box, in feet, multiplied by the G RMS input level should not exceed a value of about 6.0, as follows:

( L ) x ( G RMS) 5 6.0

(15.1)

This means that a 6-ft tall electronic cabinet should not be exposed to a random vibration screening level of more than about 1.0 G RMS without extensive proof-testing. A 1.0-ft long electronic box should not be exposed to a screening level of more than about 6.0 G RMS without extensive prooftesting. A 3.0-ft electronic box should not be exposed to a random vibration screening level of more than about 2.0 G RMS without extensive prooftesting.

15.4 THINGS AN ACCEPTABLE SCREEN ARE EXPECTED TO DO

1. Provide a cost-effective screening process. 2. Prescipitate hidden flaws rapidly so they can be detected. 3. Not use up too much life of the system. 4. Not exceed the design capability of the system. 5. Precipitate a large percentage of the hidden flaws. 6. Increase the reliability of the hardware in the field. 7. Provide some means for measuring the effectiveness of the screen. 15.5 THINGS AN ACCEPTABLE SCREEN ARE NOT EXPECTED TO DO

1. Generate high damage levels that cause good parts to fail. 2. Generate high damage levels that produce additional flaws.

384

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

3. Become the basis for establishing the design requirements. 4. Try to use the measured operating environments for the screen. 5 . Use up more than about 5 1 0 % of the effective fatigue life.

15.6 TO SCREEN OR NOT TO SCREEN, THAT IS THE PROBLEM

Many companies do not believe that the screening process is effective in improving the reliability of their products in the field. In fact, they have data that show that the screening process actually reduces the reliability of their products in the field. An equally large number of companies believe the screening process substantially improves the reliability of their products in the field, and they have actual case histories that support their positions. Both groups are right in their conclusions. The success of any screening program must be tailored to the specific requirements for that program. Every company wants to save money. Some companies treat a screening program like a pair of stretch socks, where one size fits all. Their sister company across the river spent a lot of time and money establishing their screen, which appears to be very successful. Why not simply use their successful screen and save a lot of time and money? This screen is implemented and it turns out to be a disaster. This is strong evidence in the minds of the people responsible for the poor screen that screens do not work. There is strong evidence from hundreds of companies, however, that screening does work [48].

15.7 PREPARATIONS PRIOR TO THE START OF A SCREENING PROGRAM

Electronic products are often like children, where everyone has a different personality. When they are not treated properly they put up a fuss and they misbehave. There must be an understanding of the character of the child or the character of the product, in order to come up with recommendations for an effective screen. The physical properties of the electronic hardware should be known before any screening program is implemented. This involves an understanding of the thermal, dynamic, and fatigue properties of the individual electronic components and materials used in the system. This is usually done in two ways, by analysis and by test. Analysis alone may not be accurate enough to establish an effective screen, depending on the knowledge and experience of the analyst. Testing alone may not be enough, depending again on the knowledge and experience of the testing personnel. Another factor here is the type of equipment that is available. Temperature cycling chambers, automatic temperature recorders, electrodynamic vibration machines with random capability, oil-film slider plates, many small accelerometers,

PREPARATIONS PRIOR TO THE START OF A SCREENING PROGRAM

385

many recording channels, and strobe lights are required to perform an adequate screen. All this equipment is very expensive. Many small organizations do not have the funds available. Some may already have vibration machines with sinusoidal vibration capability only. Or they may already have the lower-cost pneumatic thumper type of shakers. Companies with no thermal or vibration capability often use outside vendor subcontractors with these facilities to perform their thermal and vibration screens. Vibration screens can be very dangerous and can cause the most problems if they are not carefully controlled. The natural frequencies and transmissibility Q values of various structural elements can be obtained by making a sinusoidal vibration survey. Real hardware or mock-up hardware can be used but it must be well instrumented with small accelerometers. An input level of about 2.0 G peak should be used to obtain good test data. A sweep rate of about 1 octave per minute should be used, from 10 Hz to 2000 Hz back to 10 Hz. The sweep should be made in both directions, up and down, to make sure they look similar. If the Q plots are sharply different, it is a sign of a nonlinear system. The higher Q values should be used to evaluate the random vibration response. This type of information must be known or obtained before a good screen can be recommended. Another piece of information that is very valuable to have before the screen is recommended is the fragility level of the electronic hardware. This can be obtained by analysis or by test. The best data are obtained by testing real hardware or mock-up hardware. The hardware must be well instrumented with small accelerometers and tested with increasing vibration acceleration G levels, using sinusoidal resonant dwells or broadband random vibration, until a failure occurs. If the system is not working electrically, it may be difficult to know if or when a failure has occurred. The test may have to be stopped every few minutes to check various cables, connectors, lead wires, and solder joints for failures. Plug-in types PCBs have high transmissibility Q values at their center with high failure rates, so these areas should be checked carefully, especially if there are large components surface mounted or through-hole mounted near the center of the PCB. When a failure is discovered it should be recorded, the test should be stopped, and repairs should be made. The test should then be continued using increasing vibration acceleration G levels until another failure occurs. This process of increasing testing G levels, failure, repair, and retest should be repeated until it is obvious that most of the life has been used up [49]. Some companies with random vibration capability in electrodynamic shakers use the notching filter capabilities to reduce the effective transmissibility Q levels at the centers of the most critical plug-in PCBs from values of about 18 down to values of about 2. This reduces the large differences in the acceleration levels developed at the center of the PCB compared to the edges of the PCB. This process flattens the response acceleration levels across the face of the PCB, resulting in a better screen, according to some ESS personnel. The other side of this coin is that the real world does not have

386

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

peak-notch filters, so the centers of the plug-in PCBs in the real world will always have higher acceleration levels than the edges of the PCBs. But this is another area where there are many opposing opinions as to whether the actual operating environment should have any influence on the screening methods proposed. It would be nice if the budget could support a repeat of the vibration fragility tests on a couple of different samples because the failures are always different on different samples. This is due to the differences in material properties and differences in the manufacturing tolerances on the physical dimensions of the parts being tested. See Chapter 13, Section 13.8, for more on the sensitivity analysis, which shows how small tolerance differences can turn into big differences in the fatigue life in the lead wires and solder joints. Engineers should be thankful if they are allowed the luxury of even one prototype test model. The attitude in many organizations is that prototype test models are no longer necessary because all of the required information is in their computers. So, when a test model is available, it is important to make careful notes of the physical dimension for future use to help define internal forces and stresses as well as the required tolerances on manufactured parts. Prototype or production hardware is also recommended for performing thermal cycling tests to understand the failure mechanisms and the fatigue properties of the lead wires and solder joints for different thermal cycling rates and different maximum and minimum temperatures. Solder joint failures in thermal cycling tests are not as dramatic as in vibration tests. Thermal cycling damage is accumulated very slowly while vibration damage is accumulated very fast. Good test results can be obtained using mock-up solder joints that are produced by the same methods that will be used in the production hardware, The approximate number of thermal cycles desired in a screening program is often based on the number of component parts in the system as shown in Fig. 15.3. This shows only about 8 to 10 thermal cycles are adequate for screening a system with 4000 component parts. Research shows that many military suppliers use 10 to 20 thermal cycles for large complex electronic assemblies and 20 to 40 thermal cycles for individual electronic component parts. A very successful solder joint fragility test, or proof of life test, was developed for one of the new Air Force fighter electronic systems. This test was run on a series of PCBs with a variety of surface mounted and throughhole mounted components. The purpose of these tests was to prove that the PCB design and manufacturing methods could provide a service life with at least a 10,000-hour failure-free operating period (FFOP). The system was designed to provide a FFOP of 20,000 hours, which means that a safety factor of 2.0 was used on the design life to ensure it could meet the original operational life requirements. The manufacturing group used the production vapor phase soldering methods to assemble the test PCBs, so the test results would accurately reflect production methods. A total of 1000 thermal cycles were used in the test. Two different test groups were used with 500 thermal

THERMAL CYCLING, RANDOM VIBRATION, ELECTRICAL OPERATION

W

4-

P

387

o"T----7

0.6

Complexity of equipment

0.4

0.1 1

2

3

4

5

7

6

8

9 1 0 1 1 1 2

Temperature cycles

FIGURE 15.3. Proposed number of thermal cycles based on the complexity of the

equipment.

cycles each. The first group was 500 cycles from -50°C to +50"C and the second group was 500 cycles from 0°C to 100°C. This covered the complete temperature range of 150°C without extreme temperature changes. Both groups used a temperature ramp rate of 20°C per minute, and a dwell period of 30 minutes at each extreme temperature. The units were not working electrically. The definition of a solder joint failure was any visible crack using a 35 power microscope. It was not possible to examine the solder joints while they were in the temperature cycling ovens. Every 25 cycles that test PCBs were removed from the oven at room temperature and examined with a microscope. Several people were used to perform the visual examinations because after about 150 thermal cycles the solder becomes wavy. It is very difficult to distinguish the difference between a crack and a wave with a shadow, because a shadow on a wave looks like a crack. Tests were instrumented with thermocouples to record the temperature change rate for different critical components and PCBs. This information was used as a guide for selecting the temperature cycling rates and peak temperature for the proposed thermal cycling screening program.

+

15.8 COMBINED THERMAL CYCLING, RANDOM VIBRATION, AND ELECTRICAL OPERATION

Experience and research on stress screening show that combining random vibration, thermal cycling, and electrical operation all at the same time precipitates the most latent defects, but at a high relative cost. The pneumatic thumper piston shakers, with their integral thermal cycling enclosures, are very convenient for this type of screen. The equipment cost is low but the

388

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

time and the personnel requirement costs are high. Many different technical groups are required to coordinate, control, and record the screening results under these conditions. The typical screen starts with an electrical checkout of the system. This process very seldom goes smoothly due to problems with the large electrical control consoles usually required to operate the system electrically. Moving large electronic control consoles across rough floors to screening rooms in remote areas often causes internal electrical problems due to vibration that can shake wires loose, loosen screws, and disengage electrical connectors. The electrical engineers that oversee and monitor the electrical system very seldom know anything about the design of the test control consoles. These are two different functions that are normally performed by two different groups. Everyone associated with the screening then sits down and waits for the test control console engineer to show up to repair the test equipment. People become bored so they go back to their offices, or take a coffee break. Another two hours are lost trying to get the group together again. These are real problems that waste a lot of time and money. It is necessary to coordinate the work effort so these types of problems are solved in advance. This is another reason that many organizations choose not to operate their systems electrically, because of the problems associated with this function. The normal screening procedure using combined thermal cycling and vibration environments is to run about half of the required thermal cycles with no vibration. This is followed by 10 to 15 minutes of combined random vibration with the thermal cycling, then finishing up the remaining thermal cycles with no vibration. The combined thermal cycling and vibration is easy to run when the environmental screening system is located in a single pneumatic piston thumper enclosure. The random vibration is often split up so some of it is imposed at the high-temperature part of the thermal cycle and some of it is imposed at the low-temperature part of the thermal cycle. The combined thermal cycle and vibration screen is more time consuming to run and more expensive with an electrodynamic shaker system because the screening sample has to be installed, removed, and installed again. The electronic system screen will still start with about half of the required number of thermal cycles run in a special thermal chamber. The electronic system is then removed from the thermal cycling chamber and mounted on the vibration shaker. A special plywood box is often placed over the electronic assembly. This box has hoses attached to a local or to a portable thermal cycling chamber. The ambient air around the electronic system can then be temperature cycled while the electronic assembly is exposed to the random vibration at the same time for 10 to 15 minutes. The electronic assembly is then removed from the vibration shaker and placed back into the thermal cycling chamber to complete the remaining thermal cycles. Everyone prays that the electronic system will not be dropped and damaged during the process of installing, removing, and transporting it from one laboratory to another for the various parts of the screen. This has happened before and it will probably happen again somewhere. Several companies manufacture

IMPORTANCE OF THE SCREENING ENVIRONMENT SEQUENCE

389

temperature cycling chambers that sit on the top of an electrodynamic vibration machine for combining vibration with thermal cycling. These assemblies usually permit the vibration machine to move up and down only along the vertical axis. These combination chambers are not normally used for vibration along the horizontal axis, where an oil-film slider plate must be used for a large electronic system. It is easy to see why very few companies want to bother with combining thermal cycling, vibration, and electrical operation at the same time. Even when screening data show this method provides the best results, the complex procedure, the personnel required, and the time involved to achieve these reliability improvements are often not cost effective. 15.9 SEPARATE THERMAL CYCLING, RANDOM VIBRATION, AND ELECTRICAL OPERATION

Separate screening environments with thermal cycling, random vibration, and electrical checkout are the choice of most organizations. Separate screening environments may not be as good as combined environments, but they are much easier and much less expensive to implement. The normal procedure is to check the system first electrically to verify that it is working properly. The system is then subjected to about half of the required thermal cycles not electrically operating. The system is checked electrically after the initial thermal cycling to again verify that it is working properly. Any intermittent failures that occurred in the initial thermal cycling phase will probably not be noticed at this point. It is strongly recommended that during the electrical checkout after thermal screening that the system be dynamically excited somehow, to possibly reveal any intermittent failures that do not show up in a static condition. The easiest type of dynamic excitement that can be used is to rapidly and firmly rap the electronic system in different areas with the eraser part of an ordinary pencil, while the system is being checked electrically. This may not be a well-controlled dynamic condition, but it often reveals intermittent failures. The next phase involves the random vibration exposure for 10 to 15 minutes along one or more axes. Many companies choose to run this phase without any electrical operation. However, this is a very good place to electrically operate the system to watch for any intermittent failures. Hard failures can be observed under any static electrical checkout, but intermittent failures normally require some type of dynamic stimulus to uncover a fault. The electronic system is now returned to complete the remaining thermal cycles without electrical operation. 15.10 IMPORTANCE OF THE SCREENING ENVIRONMENT SEQUENCE

The sequence of the screen does not appear to have a significant effect on the overall results, according to the data obtained from many organizations

390

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

involved in screening programs. It does not appear to matter a great deal if the thermal cycling is done first or last, or if the random vibration is done at the beginning, in the middle, or at the end of the screening program. Are there differences in the results when there are differences in the screening sequence? The answer is yes, but they are not dramatic. The important requirement is to impose some type of thermal cycling and some type of broadband random vibration. These two environments have demonstrated a very high success rate for rapidly uncovering latent defects in production electronic equipment. This applies to commercial, industrial, or military equipment. The number of thermal cycles, the temperature ramp rates, and the random vibration all have to be tailored to fit each type of electronic unit. This is not like a pair of stretch socks, where one size fits all. A screen for an individual component part is bound to be substantially different from a screen for a large sophisticated electronic assembly. The success or failure of any screening program is decided by the failure rate of the production units in the field. A high field failure rate is usually an indication of a poor screening program. It is also a sign that some changes must be made in the program to reduce the field failure rate. A low field failure rate is usually a sign of a good screening program. 15.11 HOW DAMAGE CAN BE DEVELOPED IN A THERMAL CYCLING SCREEN

Differences in the thermal coefficient of expansion (TCE) between the PCB and the various components mounted on the PCB can produce deflection differences during thermal cycling conditions. These deflection differences may result in high stresses in the component body, the lead wires, the solder joints in the lead wires, and the cables and harnesses. These factors depend on the geometry of the different types of leaded or leadless components that can be surface mounted or through-hole mounted on the PCB, the strain relief in the cables and harnesses, as well as the thermal cycling conditions. Solder can cause a lot of problems during thermal cycling conditions. This is due to its elastoplastic and creep properties at temperatures near 100°C. Displace a cantilevered bar of solder rapidly through an amplitude A at 100°C and assume it generates an initial stress of about 1000 Ib/in.*. Hold that amplitude A constant for awhile. The solder will start to creep and relieve the initial stress. In a period of about two hours the solder stress will be close to zero. Now bring the bar of solder rapidly back to its original starting position through the same amplitude A. The solder stress level will now be 1000 Ib/in.’ compared to the zero stress at the original starting position. If the rapid displacement of the solder bar is continued in the opposite direction through a negative displacement amplitude A , the solder stress will be 2000 lb/in.’. The creep and stress-relieving properties of solder in this case resulted in the doubling of the solder stress. When solder is

HOW DAMAGE CAN BE DEVELOPED IN A THERMAL CYCLING SCREEN

391

displaced and held in the displaced condition for long periods at high temperatures, the creep effect can relax the stresses down to a near-zero condition. When the solder is then displaced in the opposite direction through the same amplitude, the solder stress can be doubled. Higher stress levels will shorten the effective fatigue life of the solder. Solder has a short-term ultimate strength of about 6500 lb/in.* at room temperature. Extensive testing and analysis experience with solder creep has shown that it is a good practice to keep the solder joint stress down to a value of about 400 lb/in.’ to ensure a long failure-free operating life in thermal cycling conditions [54]. The increased solder stress conditions due to creep described above can occur during thermal cycling events at the high point of a long temperature dwell, when there is a temperature reversal of a similar magnitude. High solder joint stresses can reduce the expected fatigue life of the solder very rapidly. When creep in the solder joint can relax the original stress level 100% to a near-zero condition, a thermal cycling event can increase the original stress level in the solder 100%. When the creep in the solder joint can relax the original stress level 5096, a thermal cycling event can increase the original solder joint stress 50%. When the creep in the solder joint can relax the original stress level 25%, a thermal cycling event can increase the original solder joint stress 25%. When there is 0% creep in the solder joint and there is 0% stress relaxing, a thermal cycling event will increase the original solder joint stress 0%. The standard method for evaluating the fatigue life of a structure is to examine the single-amplitude stress compared to the number of cycles required to produce a failure. In a typical sine wave the single amplitude refers to the displacement from zero to the peak. This is the same as half of the double amplitude, which is half of the peak-to-peak amplitude. The same philosophy is applied to thermal cycling. The single-amplitude stress will then be related to the single-amplitude temperature cycle. This is half of the double amplitude or half of the peak-to-peak temperature cycle. These analysis methods are based on linear structures. But solder is not a linear material: it can creep extensively, especially at high temperatures above about 100°C. Therefore, the methods shown above are intended to provide a more accurate method for evaluating the approximate fatigue life of solder when some creep is included. There are many arguments related to the analysis methods for evaluating solder joint strains and stresses. Should the peak-to-peak temperature range be used in the analysis of a thermal cycling event or should half the peak-to-peak temperature cycle range be used for the analysis? In a rapid temperature cycle event, where the solder does not have a chance to creep, the solder should be treated like a linear material. In this case one-half of the peak-to-peak temperature cycle range should be used. In a condition where there is extensive creep in a thermal cycling event, the peak-to-peak temperature range should be used for the analysis. When there is no information

392

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

available as to the use of the electronic equipment, or when a conservative design is desired, the peak-to-peak temperature cycle range might be considered. If the information on solder creep presented above appears to be too complex, there is nothing wrong with performing a conservative analysis by using the double amplitude or the peak-to-peak temperature cycling range. This is equivalent to using a safety factor of 2.0 on a rapid temperature cycle where little or no creep will occur in the solder joints. The fatigue exponent (b) for solder in thermal cycling conditions is about 2.5. When a safety factor of 2.0 is used in the fatigue life calculation, it will reduce the calculated fatigue life by a factor of about 5.6. 15.12 ESTIMATING THE AMOUNT OF FATIGUE LIFE USED UP IN A RANDOM VIBRATION SCREEN

Random vibration for a period of about 10 minutes along one axis has proved to be an effective screening tool for locating latent defects. It is not as effective as imposing several thermal cycles on the electronic assembly, but it does help to find additional potential failures. Random vibration is effective because it is broadband so it can excite many different natural frequencies and higher harmonics at the same time in the chassis support structure, in the various internal PCBs, and in the interconnecting cables and harnesses. Natural frequencies of small (less than 30 in.) electronic chassis with PCBs typically have natural frequencies over 100 Hz. This means that every second of vibration the structure accumulates 100 stress reversals. Every stress reversal uses up some part of the fatigue life of the system. When all of the life is used up the system will fail. The most critical structural elements will fail first. A chain is only as strong as its weakest link, so the weakest links will fail first. It is impossible to estimate how much fatigue life is being used up in a vibration screen, without knowing the actual fatigue life of the weakest link of the system. Once the approximate fatigue life of the weakest link is a particular electronic design has been established, the question becomes how much of that fatigue life should be used up to obtain a cost-effective screen? This is a tough question to answer, because everyone seems to have a different idea or feeling in this area. Some people feel it is acceptable to use up about 5% of the fatigue life of an electronic system to achieve an acceptable screen. They may be reminded that this situation is similar to the purchase of a new car with a 100,000-mile warranty. Their new car has 5000 miles showing on the odometer, which was required to achieve the 100,000-mile warranty. This represents only about 5% of life used up, but the idea of a new car with 5000 miles now appears to be different because the idea is unacceptable. Surveys made among engineers on the subject of how much fatigue life should be used up in an acceptable

ESTIMATING FATIGUE LIFE USED UP IN A RANDOM VIBRATION SCREEN

393

screen, varied from zero fatigue life used up to a maximum of 10% of the fatigue life used up. Another question that comes up concerns the disposition of an electronic system that keeps failing in successive screens, until about 25% of the life is used up. Should this system be shipped? Should the customer be informed that 25% of the life was used up? What happens if the customer rejects the system, should it be scrapped? The answers to these questions depend on the price of the electronic system. If the system cost is less than about $50, it will be scrapped. If the system cost is more than about $1000 it will be shipped even if an analysis shows that about 50% of the life has been used up. This is really a very simple problem. The typical procedure for the expensive electronic system will be a meeting with the company vice president, who happens to be an accountant or an attorney or a business major who has a good knowledge of business but not engineering. The vice president will arrange a meeting with the engineer who calculated the 25% or 50% life number and request a copy of the calculations. The meeting will be very short. The vice president will glance at the detailed analysis and quickly thumb through the pages pretending to understand what is being read. After less than one minute, the vice president will challenge the analysis. The engineer will be asked to positively guarantee the accuracy of the analysis on that one system without any additional testing. Fatigue has such a wide amount of scatter that it is impossible to calculate the precise amount of fatigue life used up in one particular system. It is possible to estimate the fatigue life probability on a statistical basis when many systems are involved, but not on a single system. The engineer has to answer that the accuracy cannot positively be guaranteed on that single system without any additional testing. The vice president will claim the engineer’s analysis is flawed and the engineer will be directed to have the system shipped. This has happened many times and will happen again and again because no one want to lose any money when hard evidence is lacking. Two giant aerospace companies were recently involved in an ESS dispute where a $150,000 electronic assembly failed the random vibration ESS environment about 20 times. It finally passes and was shipped to the customer who rejected the unit because it had too much life used up. The system manufacturer claimed that an analysis showed there was enough life left to complete the equivalent of about five missions. The customer did not trust the analysis and demanded more proof that the system had a very high probability of performing its required mission. The manufacturer was not able to provide any dat that would satisfy the customer. There was a standoff and neither side would give in. Finally, the decision was made to split the cost of the unit equally between the two companies and to impose the operational random vibration environment continuously, with the unit operating electrically, until it failed. Either by luck or by good engineering calculations, the fatigue life test showed the system did have enough life left

394

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

to complete five full missions. Both companies agreed to use the results of this destructive test to settle any future problems associated with excessive ESS failures. Many companies try to avoid the problem of using up too much fatigue life by reducing the vibration and thermal cycling exposure after each failureand-repair cycle. The vibration acceleration G RMS level may be reduced 50% or the vibration duration may be reduced 50% after each failure-and-repair cycle. This reduction ratio is often applied to the number of thermal cycles required after each failure-and-repair cycle. Methods for calculating the approximate vibration fatigue life can be obtained from a database developed from a combination of extensive vibration tests and finite element methods (FEMs). Several prototype models should be used to run vibration tests with increasing acceleration levels until failures occur. One model is not enough because there is so much scatter to the fatigue test data that substantial errors could occur. The test units should be fabricated by the manufacturing group, using production-type manufacturing methods. These prototype models must be carefully instrumented with many small accelerometers properly placed in critical areas. Sinusoidal vibration sweeps should be made first to find the natural frequencies and the transmissibility Q values in critical areas. A strobe light should be used to examine the mode shapes at the various natural frequencies to obtain a better understanding of the failure mechanisms. The next step is to run broadband random vibration using a white noise (flat-top PSD) starting with an input level of about 5 G RMS. Hold this level for about 20 minutes. If there are no failures, increase the input level in increments of 2 G RMS and hold each level for about 20 minutes until a failure is observed. Record, repair, and photograph each failure and continue increasing the G RMS until there is virtually no more life left in the system. Perform a computergenerated structural FEM analysis of the system using the observed and recorded average test fatigue life for various structural members to obtain physical constants from the FEM analysis. This type of data can be used to calculate the approximate fatigue life of similar structural members in other electronic systems subjected to similar random vibration conditions. The methods for evaluating the fatigue life characteristics of an electronic assembly described above will work well. The problem will be to convince the program manager or the vice president of engineering to provide the funds, the personnel, and the time to run such a series of tests. The typical answer to the request for funds to run such a program is a flat no. The reason given is that the company has provided the engineers with millions of dollars worth of fast new computers with software to perform these tasks so building and testing prototype models is no longer necessary. After spending all that money, many company executives do not want to hear any requests for prototype model building and testing because, in their minds, it would mean that the money was wasted.

ESTIMATING FATIGUE LIFE USED UP IN A RANDOM VIBRATION SCREEN

395

Experience has shown that the PCBs are the most critical parts of the assembly, so they will often fail first. Before any type of vibration or shock dynamic analysis can be made, or before any ESS program can be recommended, it is important to know how the chassis assembly was designed. Random vibration is broadband so it can excite every natural frequency of the chassis and the PCBs within the frequency band. This means that it may be possible for a chassis resonance to couple with and amplify a PCB resonance, which can result in a very rapid PCB failure. The dynamic coupling properties between the chassis and the various PCBs must be known before an individual PCB dynamic analysis can be made. However, if the octave rule has been followed, the dynamic coupling between the chassis and the PCBs will be sharply reduced, so the coupling effects can be ignored with very little error. Octave means to double. The natural frequency of the chassis and the PCB must be separated by a ratio of more than 2 : 1. In addition, the weight of any one PCB must be less than 10% of the weight of the chassis. With these two conditions, the dynamic coupling between the chassis and the PCBs will be small. Therefore, the dynamic input to the PCBs will be approximately the same as the dynamic input to the chassis. Sample Problem-Fatigue Thermal Cycling Screen

Life Used Up in a Vibration and

Find the approximate amount of fatigue life used up in a random vibration and thermal cycling screen of an electronic system that has several 5.0 X 7.0 X 0.070-in. thick plug-in epoxy fiberglass PCBs mounted in a chassis. The proposed vibration screen is 5.0 minutes per axis for three axes, for a total of 15 minutes with a white-noise PSD input of 0.060 G2/Hz. The octave rule was followed so the dynamic input to the PCBs will be about the same as the input to the chassis. The proposed thermal cycling screen is 15 cycles from 88°C to - 36°C at a rate of 20°C per minute with a 1.0-hour dwell period at each temperature extreme. One of the most critical components for vibration and thermal cycling is expected to be a 1.2-in. long hybrid shown in Fig. 15.4. The component is mounted at the center of the PCB parallel to the 5.0-in. edge. The lead wires extend from the bottom surface of the component, which can be through-hole or surface mounted on the PCB.

+

Solution A Fatigue Life Used Up in the Raridom Vibration Screen. The natural frequency needed for the PCB to achieve a fatigue life of about 20 million stress cycles in the lead wires and solder joints can be obtained from Chapter 9 random vibration as shown below:

fd =

[

2 9 . 4 C h r d m 0.00022 B

)'"

(15.2)

396

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

-I

l-

I

0.070in

PCB FIGURE 15.4. Differences in the PCB and component thermal coefficients of expansion (TCE) force the lead wires to bend in thermal cycling environments.

where C = 1.26 (component type lead wires extending from bottom surface) h = 0.070 in. (PCB thickness) Y = 1.0 (relative position factor, for component mounted at center of PCB) P = 0.060 G’/Hz (PSD input to PCB) L = 1.20 in. (length of critical hybrid component) B = 5.0 in. (length of PCB edge parallel to length of critical component)

/ (29.4)(1.26)(0.070)(1.0)J(

~ / 2 ) ( 0 . 0 6 ) (1.

The approximate fatigue life of the component lead wires and solder joints will be about Life

20 x 10‘ cycles to fail =

(208 cycles/s) (3600 s/h)

= 26.7

hours to fail

(15.4)

A 15-minute (0.25-hour) random vibration screen will use up the following amount of life:

Life used up

=

0.25 hour 26.7 hours

= 0.0094 = 0.94%

(15.5)

The random vibration screen will use up about 5 % of the life if it is imposed five times.

ESTIMATING FATIGUE LIFE USED UP IN A RANDOM VIBRATION SCREEN

397

Solution 8 Thermal Cycling Screen Effects on the Lead Wires. Thermal expansion differences X between the component and the PCB will take place with respect to centroid of the component. The maximum distance between the lead wires will be the diagonal dimension of the component body. The temperature difference used to find the expansion difference will be from the middle (or neutral temperature) point to the maximum and minimum temperatures. This will be half of the peak-to-peak temperature range. Note that the expansion difference X determines the relative bending displacement and the bending stress in the lead wire. It also determines the shear stress in the solder joint when there are no solder creep effects. Solder normally will not creep during a very rapid thermal cycle. At a temperature of 125°C and an initial solder stress of 3000 1b/im2, the creep effects in solder will reduce the stress level about 50% in about 45 seconds. In a thermal cycling event, this can increase the effective joint stress by 50%. When the solder creep effects reduce the stress level 25%, a thermal cycling event can increase the effective solder joint stress by 25%. Solder creep can increase the effective solder joint stress and cause more rapid fatigue failures. The solder creep factor is based on the stress level, the temperature, and the dwell time at the high temperature. The subscripts P and C refer to the PCB and the component, respectively.

X = ( a P- a,-)L(A t ) where a p = 15.0 x aC = 6.0 x

id(

( 15.6)

in./in./"C (TCE of epoxy fiberglass in X-Y plane) in./in./"C (TCE of ceramic component body)

1.20)2+ (0.50)2 = 0.65 in. (diagonal body length from centroid to end) A t = +[(88) - (-3611 = 62°C (half the peak-to-peak temperature range) L

=

X = (15.0 - 6.0) = 3.627 X

X

10-6(0.65)(62)

l o w 4in.

(relative displacement)

(15.7)

The component lead wires are very compliant compared to the stiffness of the PCB and the electronic component body. Therefore, all of the deflection can be assumed to take place in the bending of the lead wires with very little error. The end lead wires will experience the most bending because they are the farthest away from the centroid. The length of the lead wire does not stop at the interface of the component body or at the interface of the PCB. FEM analysis and test data show the effective length of a wire in bending extends about one diameter into the component body and about one diameter into the PCB solder joint for a through-hole mounting condition. The bending deflection of the lead wire can be obtained by considering the wire

398

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

to be a uniform beam fixed at both ends, with lateral motion possible. The deflection equation can be obtained from a structural handbook as shown below:

(15 -8) where X = 3.627 X in. (see Eq. 15.7) E , = 20 X lo6 lb/in.2 (kovar wire modulus of elasticity) d, = 0.018 in. (wire diameter) i n 4 (wire moment I , = (~r/64)(d,)~ = ( ~ / 6 4 ) ( 0 . 0 1 8 )=~ 5.15 x of inertia) L , = 0.060 + 2(0.018) = 0.096 in. (effective length of wire in bending includes one wire diameter into component and one into the PCB)

P,

(12)(20 x 106)(5.15x 10-9)(3.627 x =

=

( 0.096)3

0.507 lb (15.9)

The bending moment in the wire can be obtained from the sum of the moments:

M,=--

P,L,

-

2

(0.507)(0.096) 0.0243 Ibsin. 2

(15.10)

The bending stress in the wire can be obtained from a structural handbook as follows:

su

M,C, ,=--

-

I,

(0.0243) (0.018/2) 5.15 x i o p 9

=

42,466 lb/in.*

(15.11)

The thermal cycle fatigue life of the kovar wire can be obtained from Fig. 3.1 using the fatigue equation. N,

=N,

[2

=

(1000)

[ 42,4661 84,000

= 78,691 wire

thermal cycles to fail (15.12)

The electronic system will never be able to accumulate this many thermal cycles in its life, so the probability of a lead wire failure is small, as long as there are no sharp and deep cuts or scratches in the wire to produce a high stress concentration.

ESTIMATING FATIGUE LIFE USED UP IN A RANDOM VIBRATION SCREEN

399

Solder shear tearout

FIGURE 15.5. Shear tearout stress pattern in a through-hold solder joint produced by lead wire bending during thermal cycling.

Solution C Thermal Cycling Screen Effects on the Solder Joints. The overturning moment in the lead wire that forces it to bend will produce a shear tearout stress in the solder joint, as shown in Fig. 15.5. The magnitude of this stress can be obtained from the following [54]: (15.13) where M ,

lb .in. (overturning moment in solder joint; see Eq. 15.10) h , = 0.070 in. (solder joint height assumed to be the same as the PCB thickness) d, = 0.018 in. (wire diameter) d P T H= 0.030 in. (plated through-hole diameter) d,, = (d, + d,,, )/2 = (0.018 + 0.030)/2 = 0.024 in. (average solder shear area diameter) A , = (7r/4)(da, = ( ~ / 4 > ( 0 . 0 2 4 )=~4.52 X i n 2 (solder shear tearout area) = 0.0243

>’

SST =

0.0243 (0.070)(4.52 X lo-‘)

= 768

Ib/in.*

(nominal shear tearout stress) (15.14)

400

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

I

9-

!

a-

I

Solder

-

Initial stress, 6 89 MN/m2 (1000 PSI! I

I

!

1

The solder is expected to creep and strain relieve itself at a temperature of 88°C. The amount of creep after one hour is expected to be about 25% as shown in Fig. 15.6. The thermal cycling event is expected to increase the shear tearout stress level 25% as shown below. Adjusted S,,

=

1.25(768)

=

960 Ib/in.’

(15.15)

The approximate thermal cycle fatigue life of the solder joint can be obtained from Fig. 3.2 with the fatigue equation. N2 = N ,

(2] ’

=

(80.000)

[ 960 1 200

2.5

=

1585 solder thermal cycles to fail (15.16)

The amount of fatigue life that would be used up by 15 thermal cycles will be as follows: Life used up

=

15 cycles 1585 cycles to fail

= 0.0095 =

0.9572

(15.17)

The thermal cycling screen will use up about 5% of the life if it is imposed five times.

-

BIBLIOGRAPHY

1. Stephen H. Crandall, Random Vibration, Technology Press, Wiley, New York, 1958. 2. Charles E . Crede, Vibration and Shock Isolation, Wiley, New York, 1957.

3. Edward J. Lunney and Charles E. Crede, The Establishment of Vibration and

Shock Tests for Airborne Electronics, Wright Air Development Center, Technical Report 57-75, January 1958, ASTIA Document AD142349. 4. Arthur W. Leissa, Vibration of Plates, National Aeronuatics and Space Administration, 1969. 5. Norman E . Lee, Mechanical Engineering as Applied to Militaly Electronic Equipment, Coles Signal Laboratory, Red Bank, NJ, June 22, 1949. 6. Geoffrey W. A. Dumrner and Norman B. Griffin, Environmental Testing Tech-

niques for Electronics and Materials. 7. Charles E. Crede and Edward J. Lunney, Establishment of Vibration and Shock Tests for Missile Electronics as Derived from the Measured Environment, Wright Air Development Center Technical Report 56-503, December 1, 1956, ASTIA Document AD118133.

8. Rough road tests, performed at A.C. Spark Plug, Division of General Motors, Milwaukee, WI, July 15, 1959. 9. W. C. Stewart, “Determining Bolt Tension,” Mach. Des. Mug., November 1955. 10. R. W. Dicely and H. J. Long, “Torque Tension Charts for Selection and Application of Socket H e a d Cap Screws,” Mach. Des. Mag., September 5, 1957. 11. Kent’s Mechanical Engineer’s Handbook, Design and Production Volume, Wiley, New York, 1950. 12. B. Saelman, “Calculating Tearout Strength for Cantilevered Beams,” Much. Des. Mag., January 1954. 13. E. F. Bruhn, Analysis and Design of Aircraft Structures, Tristate Offset Co., Cincinnati, OH, 1952.

14. M. R. Achter, Effects of High Vacuum on Mechanical Properties, US. Naval Research Laboratory, Washington, DC, 1961. 15. McClintock and Argon, Mechanical Behavior of Materials, Addison-Wesley, Reading, MA, 1966. 16. A. M. Freudenthal, Fatigue in Aircraft Structures, Academic Press, New York, 1956. 17. Marks Handbook, McGraw-Hill, New York, 1951. 18. R. E . Peterson, Stress Concentration Design Factors, Wiley, New York, 1959. 19. R. J. Roark, Formulas for Stress and Strain, McGraw-Hill, New York, 1943.

401

402

BIBLIOGRAPHY

20. S. L. Hoyt, Metals and Alloys Data Book, Reinhold, New York, 1943. 21 C. Lipson and R. Juvinal, Handbook of Stress and Strength, Macmillan, New York, 1963. 22. H. J. Grover, S. A. Gordon, and L. R . Jackson, Fatigue of Metals and Structures, Bureau of Aeronautics, Department of the Navy, 1954. 23. M. A . Miner. “Cumulative Damage in Fatigue,” J . Appl. Mech., 12, September 1945. 24. Reynolds Metals Co., Structural Aluminum Design Handbook, 1968. 25. F. B. Stulen, H . N. Cummings, and W. C. Schulte, “ A Design Guide Preventing Fatigue Failures,”Mach. Des. Mag., Part 5. June 22, 1961. 26. F. R. Shanley, Strength of Materials, McGraw-Hill, New York, 1957. 27. MIL-Handbook-SA, Metallic Materials and Elements for Aerospace Vehicle Structures. Department of Defense, Washington, D C . 28. J. W. S. Rayleigh, The Theoy of Sound, Dover Publications, Mineola, N Y , 1945. 29. S. Timoshenko and S. W. Krieger. Theor?,of Plates and Shells, McGraw-Hill, New York, 1959. 30. R. W. Little, Master‘s Thesis, University of Wisconsin, 1959. 31. G . B. Warburton, “ T h e Vibrations of Rectangular Plates,” Proc. Insr. Mech. Eng., 168 (121, 1954.

32. F. J. Stanek, Uniformb Loaded Square Plate with No Lateral or Tangential Edge Displacements, Ph.D. Thesis, University of Illinois, 1956. 33. P. A. Laura and B. F. Saffel, Jr., .‘Study of Small Amplitude Vibrations of Clamped Rectangular Plates Using Polynomial Approximations,” J . Acoust. SOC. Amer., 41(4), 1967. 34. M. Vet, “Vibration Analysis of Thin Rectangular Plates,” Mach. Des. Mag.. April 13. 1967. 35. Road Shock and Vibration Environment for a Series of Wheeled and Track-Laying Vehicles, Report DPS-999, March-June, 1963. 36. Navships 900, 185, A Design of Shock and Resistant Vibration Electronic Equipment

for Shipboard Use. 37. D. S . Steinberg, “Circuit Components vs. G Forces,“ Mach. Des. Mag., October 14. 1971. 38. Steinberg & Associates, Assessment of Vibration on Avionic Design, prepared for Universal Energy Systems, Dayton. OH, August 1984. 39. R . N. Wild, Some Fatigue Properties of Solders and Solder Joints, IBM Report 74Z00044S, July 1974.

40. Werner Engelmaier, Effects of Power Cycling on Leadless Chip Carrier Mounting

Reliabiliv and Technology, Bell Laboratories, November 1982. 41. S. 0. Rice, “Mathematical Analysis of R a n d o m Noise,” Bell Sys. Tech. J., July 1944. 42. Raymond D. Mindlin, “Dynamics of Package Cushioning,” Bell Sys. Tech. J., pp. 353-461, July-October 1945. 43. D. S. Steinberg. “Snubbers Calm PCB Vibration,” Mach. Des. Mag., March 24. 1977.

BIBLIOGRAPHY

403

44. M. McWhirter, “Shock Machines and Shock Test Specifications,” Sandia Corp. Lecture Series, 1970. 45. J. Sugamele, Evaluation of Acoustic Environmental Effects on Flight Electronic Equipment, Boeing Airplane Co., 1966. 46. R. W. Hess, R. W. Fralich, and H. H . Hubbard, Studies of Structural Failures Due to Acoustic Loading, NACA T.N. 4050, 1957. 47. NAVMAT P-9492, Nu y Manufacturing Screening Program, Department of Navy, May 1979. 48. Institute of Environmental Science, 940 E. Northwest Highway, Mt. Prospect, IL 60056. 49. W. Silver, Proposed Recommended Practice in Applying Broadband Vibration Screening to Electronic Hardware, IES Shock and Vibration Committee, February 1981. 50. A. J. Curtis and R . D. McKain, A Quantitative Method of Tailoring Input Spectra for Random Vibration Screens, Hughes Aircraft Co., June 1987. . September 25, 51. Martin Dynamics Manual, Section 4, “Vibration of B e a m s , ” ~ 102, 1957. 52. R . A. Wilk, “ W e a r of Connector Contacts Exposed to Relative Motion,” ZES Proceedings, Random Vibration Seminar, Los Angeles, March 25-26, 1982. 53. MIL-HDBK-304, Militaly Standardization Handbook Package Cushioning Design, Department of Defense, Washington, DC, November 1964. 54. D. S. Steinberg, Cooling Techniques for Electronic Equipment, Second Edition, Wiley, New York, 1991.

INDEX

Index Terms

Links

A Acceleration, relation to frequency sample problem, transformer bracket Accelerometer positioning

26 31 106

370

106

366

367

370 servo control

367

Acoustic noise

234

air jets

237

cumulative damage

244

damage to panels

236

fatigue life

240

generating methods

236

membranes

242

microphonic effects

235

nonlinear stress

242

one-third octave bands

238

sample problem, thin chassis panel

240

siren

237

sound acceleration spectral density

245

sound pressure level

234

sound pressure spectral density

236

Air transport rack (ATR) Alignment pins, electronic box

12 301

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Amplification

31

shock

258

vibration

261

32

Aspect ratio, side panel

317

AVIP

338

B Beams

19

37

56

bending displacements

20

28

56

bending moments

29

bending stresses

30

beryllium

61

352

cantilever, see Cantilevered beams chart, natural frequencies

63

clamped (or fixed)

62

natural frequency equations

62

nonuniform cross section

64

polynomial deflection equations

62

sample problem, beam natural frequency

60

simply supported

27

trigonometric deflection equations

57

Bending moments sample problem, beam bending moments

57

29

30

210

267

Bents, see Frames bents and arcs Bolted covers sample problem, bolted covers on a chassis Bolted efficiency factors

306 305 137

Bolts, see Screws

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Buckling, critical stress

318

sample problem, chassis wall in bending

316

sample problem, chassis wall in shear

324

C Cantilevered beams

10

19

63

208

nonuniform cross sections

64

sample problem, stepped beams

66

Castigliano’s strain energy theory

90

frames and bents

80 302

chassis mount

302

overturning moments

309

Chassis

300

bolted covers

305

critical panel buckling stress

318

dynamic coupling with PCB

227

sample problem, chassis coupling with PCB natural frequency sample problem, nonuniform chassis

303

309

305

226 307

311

327

68

response to random vibration

206

torsional mode

309

sample problem, chassis torsion frequency

68

80

circular arcs

Center of gravity

62

311

torsional stresses

321

transmissibility Q,

314

trigonometric load distribution

312

368

sample problem, trigonometric load on

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Chassis (Cont.) trigonometric load distribution (Cont.) a chassis uniform load distribution sample problem, uniform load on a chassis Circuit boards

bolted installation component mounting

315 328 328 103

107

116

122

127

145

105

107

145 49 336

conformal coatings

104

105

337

damping

159

162

163

desired natural frequency

172

216

217

120

127

116

122

141

260 random vibration, sample problem

218

shock, sample problem

262

sinusoidal vibration, sample problem

174

different types of supports

111 146

displacements

111 130

dynamic stress sample problem, PCB dynamic stress

131 131

edge conditions

112

117

electrical connectors

105

223

223

224

epoxy fiberglass boards

121

131

fatigue life

174

219

174

219

sample problem, connector fatigue life

sample problem, PCB fatigue life

134

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Circuit boards (Cont.) heat sinks

103

105

kinetic energy in bending

113

119

laminated construction

103

105

loose edge guides

104

154

sample problem, loose edge guides

330

Miner’s damage ratio

212

341

multilayer

103

110

mounting methods

104

107

natural frequencies for different supports

127

145

circular plate

148

octagonal plate

146

rectangular plate

127

triangular plate

147

sample problem, relays plug-in types

110

182

manufacturing tolerances

octave rule

110

150

146

154

156 103

120

154

156

159

300

374

385

120

132

strain energy in bending

112

119

stiffening ribs

132

sample problem, natural frequency

sample problem, natural frequency with ribs stiffening ribs fastened with screws

132 137

sample problem, natural frequency ribs with screws surface mounted components testing to destruction with vibration

137 39

49

385

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Circuit boards (Cont.) through-hole mounted components

49

97

sample problem, relative PCB displacement

50

torsion with stiffening ribs

135

transmissibility Q

108

109

173

385

effects of input acceleration G level wedge clamps sample problem, PCB with wedge clamps Circular arcs

368 177 179 90

clamped ends, strain energy analysis

92

displacements with a concentrated load

94

hinged ends, strain energy analysis

90

spring rate with a concentrated load

92

strain relief, component lead wire

95

sample problem, offset fatigue life

110

97

Cold plates

11

103

110

Component mounting

75

97

170

173

176

216

101

174

208

212

219

260 fatigue life

location effects

175

orientation effects

176

size effects

173

surface mount

39

49

through-hole mount

49

97

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Composite structures

Links 69

72

133

305

71

72

74

71

73

74

circuit boards

133

138

electronic chassis

305

Computer applications

144

Conformal coatings

337

Connectors, electrical

105

223

edge support properties

106

107

effects of acceleration G levels

108

effects of random vibration

223

fatigue life

224

beams, stiffness sample problem, natural frequency

172

105

216

224

sample problem, random vibration fatigue life 224 fretting corrosion

223

intermittent electrical signals

223

Coupled vibration modes

6

37

308

311

sample problem, chassis and PCB

226

bending and torsion

311

226

sample problem, chassis in bending and torsion definition test for

308 37 308

Covers on chassis bolted efficiency factors

305 137

305

sample problem, natural frequency with bolted covers

306

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Cumulative fatigue, see Fatigue, cumulative damage Curves

32

152

258

283

153

shock amplification, single degree-of-freedom

258

shock amplification, two degree-of-freedom

283

284

velocity shock, shipping crates

285

287

288

17

30

32

134

258

268

281

367

as a function of acceleration G level

34

108

368

as a function of frequency

33

35

105

108

368

108

368

30

34

constrained layer

163

164

conversion of kinetic energy into heat

268

276

26

159

electronic chassis

361

368

low levels

267

367

ratio

31

33

relation to transmissibility

33

108

viscous

30

vibration amplification

32

vibration transmissibility

32

D Damping

circuit boards coefficients

effects of displacement and stress

348

374

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Decibel

Links 197

234

acoustic noise

234

235

random vibration

197

198

3

4

18

20

26

28

111

116

159

191

209

217

254

261

27

56

58

62

64

266

sample problem, concentrated load on a beam 19

27

265

27

28

26

29

173

218

254

261

111

116

122

126

173

217

256

261

Degrees of freedom Displacement

beams

bending

19 57

dynamic

plates

261 relation to frequency shock

26 254 266

sample problem, pulse shock vibration

sample problem, beam displacement Dunkerley coupling equation

262

265

20

26

28

111

116

173

217

261

27

266

311

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Dynamic bending stresses

29

132

211

243

267

316

29

209

267

26

29

173

218

254

261

27

56

58

62

64

266

circuit boards

172

217

261

chassis

313

321

Dynamic loads

29

132

211

243

267

316

235

239

241

209

267

beams sample problem, simply supported beam circuit boards sample problem, dynamic load on a PCB chassis

29 132 131 317

sample problem, dynamic stress on a chassis 311 Dynamic displacements

beams

acoustic noise

sample problem, acoustic pressure on a panel 235 beams

29

bolts

360

chassis

317

sample problem, dynamic loads on a chassis

311

circuit boards

132

electrical connectors

223

electrical lead wires

49

fasteners

360

pyrotechnic shock

290

328

292

296

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Dynamic loads (Cont.) random vibration

193

197

199

207

209

216

223

227

shaker head

361

sine vibration

170

311

323

328 spring members

18

38

transformer bracket

27

265

velocity shock

285

Edge guides, loose

104

154

Electrical connectors

105

223

E

Electrical wire harness Electronic chassis

11 300

area moment of inertia

305

bolted cover

305

sample problem, chassis with bolted covers

182

305

buckling panels

317

center of gravity mount

302

coupling bending and torsion

311

high center of gravity mount

301

mass moment of inertia

310

natural frequency

307

octave rule

150

perimeter stiffening rib

374

shear flow

321

shear panel

324

303

309

308

322

324

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Electronic chassis (Cont.) simply supported ends

302

single jacking screw

301

tall narrow cross section

303

testing to destruction

385

torsional resonance

309

Electronic components

cementing to circuit boards

308

39

76

170

176

217

220

379

391

396

338

dynamic loads

40

46

lead wires

39

94

97

40

49

53

97

396

5

398

43

48

54

174

212

214

392

396

forces and stresses

41

46

sample problem, wire fatigue life

43

54

174

212

214

396

solder joint stresses

44

100

399

spring rates

87

92

94

98

100

396 as beam members

effective length fatigue life

strain relief mounting considerations

97 76

97

170

173

176

216

260 This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Electronic components (Cont.) mounting considerations (Cont.) location on circuit board

176

orientation on circuit board

176

surface mounting

39

49

through hole mounting

49

76

97

49

76

97

relative motion on vibrating circuit boards

216 sample problem, wire forces due to relative motion vibration testing to failure End support conditions

beams

circuit boards

circular arcs

50

97

385 39

94

97

111

127

130

147

396

19

27

63

68

111

120

128

147

90

92

95

94

97

57

127

97 component lead wires

39 396

electronic chassis

68

301

309

frames and bents

78

81

84

rectangular plates

111

120

127

346

354

358

367

369

372

146 vibration fixtures

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Endurance limit

Links 42

44

48

211 aluminum alloys

211

electrical lead wires

42

48

fatigue life relations

39

42

45

167

168

174

44

48

137

305

219 relation to ultimate strength

42 211

Environmental stress screening bathtub reliability curve

379 380

combined thermal, vibration, electrical operation 387 damage potential

390

effects of temperature on solder creep

400

estimating fatigue life used up in screen

392

NAVMAT-P-9492

380

philosophy

379

random vibration environments

382

sample problem, fatigue life

395

screening preparations

384

separate thermal, vibration and electrical operation 389 thermal environments

381

F Fasteners

7 350

cut threads or rolled threads loosening progressive shear failures

353 8 370

371

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Fasteners (Cont.) sample problem, preload torque on a screw

352

selection criteria

8

9

350

torque requirements

9

43

48

54

99

174

166

212

392

396

AVIP

338

341

cumulative damage

168

169

vibration fixtures Fatigue life

350

343

344 electrical lead wires

343

sample problem, fatigue in lead wires

168

solder joints

344

169

cycle ratio

343

344

damage

212

341

definition

39

endurance limit

42

48

life

39

330

connectors, fretting fatigue

223

224

effects of manufacturing tolerances

330

estimating, methods

340

341

212

341

Miner’s cumulative damage ratio sample problem, estimating the fatigue life S–N curves

aluminum

212 42

44

167

211

48

211

copper wires

48

kovar wires

42

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Index Terms

Links

Fatigue life (Cont.) S–N curves (Cont.) fatigue exponents

167

sample problem, fatigue life of kovar wires

40

sample problem, solder shear fatigue life

43

solder

44

Flanges on chassis panels

305

Foam rubber supports for PCB

107

test data, PCB with foam rubber supports Forced vibrations amplification curve

106 30 32

circuit boards

106

counterweights

356

dynamic shift in the center of gravity

364

dynamic similarity

356

enclosed chassis

304

resonant vibration fixtures

348

rocking modes

354

single degree-of-freedom with damping

31

transmissibility curve

32

Frames bents and arcs

75

circular arcs

90

displacement with concentrated load

90

365

305

93

component lead wires

97

deflections

80

82

83

87

92

94

96 fixed ends circular arcs, strain energy

83 90

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Frames bents and arcs (Cont.) fixed ends (Cont.) rectangular bent strain energy

81

wire arc strain relief, strain energy

95

hinged ends

78

circular arc strain energy

90

rectangular bent strain energy

81

rectangular bent superposition

78

Free body diagram

Free vibrations

31

38

79

81

84

88

17

with damping

30

multiple mass system, no damping

25

36

359

4

21

24

56

59

112

117

123

beams

56

59

definitions

56

deflections

56

62

111

116

122

124

111

116

122

124

126

pendulum rotating vector

26

Fretting corrosion, see Wear, connectors Fuse, mechanical

358

G Geometric boundary conditions

125 plates

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Geometric boundary conditions (Cont.) slope

57

112

117

123 Geometric stress concentrations G forces

beams sample problem, G levels on beams circuit boards

167

168

10

13

15

108

131

140

174

208

218

253

258

262

27

208

265

28

265

13

108

171

174

214

215

131

174

262

300

310

311

315

320

328

312

321

328

362

365

369

372

374

260 sample problem, G levels on PCBs electronic chassis

sample problem, G levels on chassis relation to frequency

26

trucks

15

vibration fixtures

H Half power points

186

Harmonic modes

5

Heat exchangers

11

Hertz, definition

2

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

I Inertia

area moment

20

23

28

60

74

123

138

209

265

20

28

60

74

123

138

265 average boxes beams

64

68

305 20

28

209

265

72

73

74

forces

11

209

374

lumped masses

23

226

265

composite

74

310 mass moment

23

310

plates

123

138

polar

23

310

57

59

112

57

59

112

114

119

beams

57

59

plates

112

114

Integration methods random vibration PSD area sample problem, area under PSD curve shock pulse area strain energy

Isolators chassis

198 199 253

119

275 277

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Isolators (Cont.) heat development

276

sample problem, isolators for vibration and shock 278 shock

280

vibration

278

wire cable considerations

11

J J factor, torsion rectangular chassis round shaft sample problem, torsion J factor for a chassis

21

22

310 22 310

K K factor, buckling

318

bending

317

sample problem, buckling, chassis side walls

318

shear

324

325

Kinetic energy

58

113

beams

58

uniform loads

119

58

circuit boards

113

119

circular arcs

91

93

frames and bents

81

84

multiple mass systems

36

one mass system

18

relation to work

18

torsion system

21

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Index Terms

Links

L Leadless ceramic chip carrier (LCCC) LCCC component factor for PCB vibration Lead wires, electrical

component factor for PCB vibration effects of PCB displacements

217 217 40

49

76

97

170

173

216

221

217 76

97

170

173 fatigue life

97

174

219

force developed

76

95

99

170

216

83

87

92

94

96

99

spring rate

100 strain relief sample problem, wire strain relief Line replaceable unit (LRU) Load path through chassis Lock washers Loose PCB edge guides sample problem, PCB with loose edge guides Lugs, chassis mount

97 97 12 300

301

9 182 187 229

302

303

19

21

24

36

37

172

183

208

309

19

208

304 Lumped mass systems

beams

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Lumped mass systems (Cont.) chassis

309

313

321

chassis and PCB

172

215

226

PCB

183

215

sample problem, coupling chassis and PCB kinetic energy

226 18

21

36

330

332

334

M Manufacturing problems

338 AVIP program

339

effects of tolerances

331

failure free operating period (FFOP)

345

MTBF

343

sample problem, effects on PCB frequency

332

Margin of safety, chassis

319

buckling due to bending

317

combined bending and shear

325

shear

324

torsion

320

Membranes, acoustic noise

237

242

MIL specifications

175

185

Miner’s cumulative fatigue damage

212

341

definition

212

electrical lead wires

343

344

fatigue cycle ratio

212

341

random vibration

212

251

398

sample problem, fatigue life using Miner’s method

212

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Mode shapes Moment of inertia, area

beams

nonuniform, average

Links 5 20

23

28

60

74

123

138

209

265

20

28

74

209

265

64

68

bolted efficiency factor

138

electronic chassis

305

sample problem, chassis with bolted covers electrical lead wires sample problem, wire displacement stresses ribs on PCB sample problem, ribs on a PCB torsion, thin wall Moment of inertia, mass electronic chassis

304 40 50

97

133

138

133

138

310 23 310

base mount

310

center of gravity mount

327

round flat disk sample problem, disk and shaft torsion Mounting lugs on a chassis

310

22

23

22

23

229

302

19

22

25

28

35

37

83

87

N Natural frequency, simple systems

beams, nonuniform

64

beams, uniform

63

bents and frames

80

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Natural frequency, simple systems (Cont.) circuit boards

103

120

127

145

154

156

159

300

374

385 bolted ribs

138

change in edge condition

106

circular shape, various supports

148

octagon shape, various supports

146

polynomial functions

116

rectangular shape

127

ribs in two directions

141

triangular shape, various supports

147

trigonometric functions

111

circular arcs

90

fixed ends

92

hinged ends

90

108

145

92

composite beams

69

damped vibration

17

definition

17

effects on transmissibility Q

31

172

368

369

300

303

electronic chassis

94

33

173

307

309 bending mode

303

center of gravity mount

302

326

coupled mode

308

311

high center of gravity mount

308

309

torsion mode

309

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Natural frequency, simple systems (Cont.) fundamental

17

Rayleigh method

56

57

58

59

111

114

116 cantilever beams clamped rectangular plates clamped uniform beams supported rectangular plates supported uniform beams

62 122 62 111 57

static displacement method

19

torsion system

21

NAVMAT-P-9492 environmental stress screening

116

301

309

152

154

379

combining thermal, vibration and electrical operation

387

estimating amount of fatigue life used up

392

preparations prior to the screen

384

proposed number of thermal cycles

387

random vibration power spectral density curve sample problem, fatigue life used up Nonuniform cross sections sample problem, average moment of inertia

382 395 64 68

O Octave rule, chassis with PCBs

150 225

forward rule

155

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Index Terms

Links

Octave rule, chassis with PCBs (Cont.) natural frequency ratio sample problem, relay chattering problem

152

153

156

potential problems with reverse rule

155

156

reverse rule

155

transmissibility ratio

152

153

weight ratio

152

153

Oil film slider plates

355

356

361

364 bolts loaded in shear

350

371

372

bolts loaded in tension

352

361

364

dowel pin applications

372

373

2

18

99

100

103

120

154

156

159

300

374

385

34

172

368

369

P Periodic motion Plated through holes Plates, natural frequency

effects on transmissibility

circle, hexagon, rectangle, triangle

145

polynomial series equation derivation

116

sample problem, plate natural frequency

120

trigonometric series equation derivation

111

with bolted ribs

138

with rigidly attached ribs

133

173

sample problem, PCB natural frequency with ribs

133

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Power spectral density (PSD)

192

195

effects of PSD levels on acceleration G levels

207

222

effects of transmissibility

207

210

finding PSD and frequency break points

200

sample problem, finding PSD break points input area under PSD curves

201 195

196

sample problem, finding area under PSD curve196 integration methods for finding areas

198

integration methods for finding PSD values

201

PSD response values

207

Preloaded bolts sample problem, torque required for bolt preload Printed circuit boards (PCBs)

351

352

361

352

353

103

106

108

111

116

127

217

261

218

261

145 fretting corrosion problems on plug-in connectors 223 sample problem, connector fatigue life

224

maximum allowable dynamic displacement

172

random vibration

217

shock

261

sinusoidal vibration

172

minimum desired natural frequency

173

random vibration equation derivation

218

sample problem, random natural frequency

218

sample problem, shock natural frequency

262

sample problem, sinusoidal natural frequency 174 shock frequency equation derivation

261

sinusoidal vibration equation derivation

173

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Probability distribution functions Gaussian instantaneous distribution

Links 202

205

202

quick method, random fatigue life

213

sample problem, random fatigue life

208

three band technique random fatigue life

204

Rayleigh peak distribution

205

Push bar couplers, vibration testing

355

Pyrotechnic shock

289

364

372

346

348

353

357

367

acceleration levels

290

effects on electronic component

296

hybrid design

297

shock response spectrum

292

sample problem, response of chassis and PCB 292 simulations with hammer impacting a plate,

291

typical frequencies and acceleration G levels

290

Q Qualification tests

design equipment to survive 5 qualification tests

224

R Rack, air transport (ATR)

12

301

chassis mount

301

326

Random vibration

188

190

bending stress

209

cumulative damage

212

damage

213

desired natural frequency

218

210

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Index Terms

Links

Random vibration (Cont.) failure modes

188

189

fatigue

211

input curves

194

195

life

212

214

392

395 multiple degree-of-freedom

224

octave rule

225

PCB as a single degree-of-freedom system

215

positive zero crossings

231

sample problem, positive zero crossings quick respond analysis method sample problem, quick response analysis

233 213 214

response characteristics

189

single degree-of-freedom response

224

three band life technique

204

sample problem, three band life analysis time history Rotating vector describing vibration Rotational vibration modes

Rough road tests Rubber, PCB support

225

225

208 204 3 21

301

309

321

308

15 107

S Screws

7

138

350

352 selection criteria

8

cover attachment

305

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Index Terms

Links

Screws (Cont.) effective length

351

efficiency factors

137

friction

352

loosening

352

preload effects

351

spring rates

351

torque and preload relations

352

352

9

Shaker head force

362

365

Shear flow, torsion

321

322

Ship and submarine vibration shock, velocity change

350

8

lubrication

torque values, recommended

138

13 252

254

285

248

258

296

affects on components

291

296

amplification

258

259

chassis response

269

282

cushioning material

270

287

desired PCB resonant frequency

260

261

286 Shock

sample problem, PCB resonant frequency

262

equivalent shock pulse

269

full rebound

252

half sine pulse

253

area under half sine pulse

258

253

hammer impact

291

isolators

275

sample problem, selecting shock isolators

294

278

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Index Terms

Links

Shock (Cont.) mounts for ships

13

nonlinear

286

octave rule

284

PCB response

291

pulse shock

250

253

258

259 sample problem, full rebound pyrotechnic sample problem, chassis and PCB

258 289 292

response spectrum

251

square wave

259

stresses

267

triangular pulse

250

two degree-of-freedom

282

response curves for different damping

283

velocity shock

285

zero rebound

253

Simple harmonic motion Sinusoidal vibration allowable PCB displacement component mounting

290

259

284

2

18

166

171

172 76

97

173

176

damage estimate

166

167

desired PCB natural frequency

173

170

sample problem, desired PCB natural frequency lead wire strain relief sample problem, strain relief fatigue life

174 94

96

97

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Index Terms

Links

Sinusoidal vibration (Cont.) loose PCB edge guides reasons to avoid loose PCB guides

182

183

182

sample problem, PCB frequency with loose guides qualification tests

sine sweep through a resonance sample problem, sweep through a resonance wedge clamps on PCBs

185 346

348

357

367

353

186 186 177

sample problem, natural frequency, wedge clamps wedge clamp efficiency factor Slider plates, oil film

Suspension systems Solder

179 178 355

356

361

367

372

373

44

104

387

395

44

100

358 43 399

alternating stress

44

bonding PCB stiffening ribs

104

component size, effects

396

creep rate

400

fatigue

43 390

physical properties

44

400

plated through holes

43

99

100

sample problem, solder stress

47

101

399

shear tearout stress

43

399

surface mounted components

217

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Index Terms

Links

Solder (Cont.) temperature effects Springs chassis torsion spring rate

400 23 310

coil springs

25

damping

31

deflection

26

dynamic loads

18

36

38

effective, for bolts

351

352

nonlinear effects

352

362

parallel

24

PCBs

183

preload effects

352

series snubbers

24 164

static deflection

28

strain energy

18

torsion

21

work

18

Square plates

6

129

240

19

20

27

28

78

209

133

138

Static deflection

36

265 Stepped beams Stiffeners

64 107 141

Strobe light Superposition, bent sample problem, bent deflection

106 78 78

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Index Terms Surface mounted devices

Links 217

desired natural frequency

173

Suspension vibration systems

358

Sweep through a resonance

186

218

261

138

350

T Tee sections

132

beams

350

stiffening ribs

132

138

141

345

346

354

355

357

358

Test fixtures

bolted assemblies bolt torque

350 9

352

dynamic shift in center of gravity (CG)

354

355

fastening accelerometers

360

366

367

mechanical fuse

358

oil film slider

355

356

367

364

372

354

360

21

22

301

308

321

326

closed sections

301

308

combined with bending

308

311

372 positioning accelerometers

366

preloaded bolts

352

push bar couplers

355 373

rocking modes

353

summary for good design

357

suspension systems

358

Torsion

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Index Terms

Links

Torsion (Cont.) electronic chassis

301

308

321

322

mass moment of inertia

23

310

327

rectangular cross section

310 21

22

309

133

138

141

resonant frequency ribs

309

single degree-of-freedom

21

spring rate

23

310

strain energy

21

58

80

82

83

90

112

118

Transformer bracket

27

265

Transmissibility

28

35

41

368 beams

28

41

circuit boards

35

104

108

368 damping relations

30

dynamic loads

46

311

314

368

function of frequency

28

34

general equation

33

368

mathematical model

31

single degree-of-freedom

32

transformer bracket

47

electronic chassis

vibration fixture

348

41

370

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Index Terms Trigonometric functions

Links 3

26

56

62

111

125

140

177

312

beams

56

58

62

boxes

308

312

327

62

combined with polynomial

62

displacements

56

58

dynamic loads

14

29

kinetic energy

40

58

natural frequency

58

60

63

108

120

127

111

125

177

111

116

122

125

130

177

Rayleigh method

56

139

140

slopes

57

59

112

18

22

57

80

83

90

94

112

21

135

321

1

17

150

188

300

346

106

360

367

145 plates and PCBs displacements

117 strain energy

Torsion natural frequency

309

Trunion, vibration machine

347

V Vibration

accelerometers

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Index Terms

Links

Vibration (Cont.) airplanes and missiles

10

automobiles, trucks and trains

156

bending coupled with torsion

308

counter weights

356

die bond wires

297

fixtures

347

design summary

357

wood laminations

348

forced

30

free

17

harmonic modes isolators

300

5 275

277

301

sample problem, selecting vibration isolators 278 linear systems mechanical fuse

2 358

modes

5

nodes

5

nonlinear systems

342

nose cones

356

oil film slider

355

351

356

364

168

175

189

224

250

367 qualification tests

random representation

188 3

servo control accelerometer

366

effects of location

367

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Index Terms

Links

Vibration (Cont.) shaker machine

347

bolt patterns

361

mounting

347

ships and submarines sinusoidal sources

13 166 1

suspension systems

358

unsymmetrical fixtures

354

variable cross section

364

64

beams

64

electronic chassis

68

66

67

W Wear, connectors

223

224

Wedge clamps

177

179

percent fixity Work

178 21

22

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