Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00000___16aa3585e608cfa6dab62ee50047ffeb.pdf

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00001___f7a908b44cdbcb0b144267add684a5ae.pdfMechanicalShock

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00002___2e1d201a007d57d995ef01488ecfe092.pdfThis page intentionally left blank

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00003___dc5321678e83464ae6ae50c7c97b5bb1.pdfMechanical Vibration& Shock

MechanicalShockVolume II

Christian Lalanne

HPSHERMES PENTON SCIENCE

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00004___c460cd095754f79ede6b8804504aa66c.pdfFirst published in 1999 by Hermes Science Publications, ParisFirst published in English in 2002 by Hermes Penton Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, aspermitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,stored or transmitted, in any form or by any means, with the prior permission in writing of thepublishers, or in the case of reprographic reproduction in accordance with the terms and licences issuedby the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers atthe undermentioned address:

Hermes Penton Science120 Pentonville RoadLondon N1 9JN

Hermes Science Publications, 1999 English language edition Hermes Penton Ltd, 2002

The right of Christian Lalanne to be identified as the author of this work has been asserted by him inaccordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data

A CIP record for this book is available from the British Library.

ISBN 1 9039 9604 X

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynnwww.biddies,co.uk

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00005___4584356b8d29cf04e95ebb70cab53d3d.pdfContents

Introduction xi

List of symbols xiii

1 Shock analysis 11.1. Definitions 1

1.1.1. Shock 11.1.2. Transient signal 21.1.3. Jerk 21.1.4. Bump 21.1.5. Simple (or perfect) shock 31.1.6. Half-sine shock 31.1.7. Terminal peak saw tooth shock (TPS) or final peak

saw tooth shock (FPS) 31.1.8. Initial peak saw tooth shock (IPS) 31.1.9. Rectangular shock 31.1.10. Trapezoidal shock 31.1.11. Versed-sine (or haversine) shock 31.1.12. Decaying sinusoidal pulse 4

1.2. Analysis in the tune domain 41.3. Fourier transform 4

1.3.1. Definition 41.3.2. Reduced Fourier transform 61.3.3. Fourier transforms of simple shocks 7

1.3.3.1. Half-sine pulse 71.3.3.2. Versed-sine pulse 81.3.3.3. Terminal peak saw tooth pulse (TPS) 91.3.3.4. Initial peak saw tooth pulse (IPS) 101.3.3.5. Arbitrary triangular pulse 121.3.3.6. Rectangular pulse 14

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00006___0849f4f01e87a793a37757444596d15f.pdfvi Mechanical shock

1.3.3.7. Trapezoidal pulse 151.3.4. Importance of the Fourier transform 17

1.4 Practical calculations of the Fourier transform 181.4.1. General 181.4.2. Case: signal not yet digitized 181.4.3. Case: signal already digitized 20

2 Shock response spectra domains 232.1. Main principles 232.2. Response of a linear one-degree-of-freedom system 26

2.2.1. Shock defined by a force 262.2.2. Shock defined by an acceleration 272.2.3. Generalization 282.2.4. Response of a one-degree-of-freedom system

to simple shocks 332.3. Definitions 372.4. Standardized response spectra 402.5. Difference between shock response spectrum (SRS) and

extreme response spectrum (ERS) 472.6. Algorithms for calculation of the shock response spectrum 472.7. Subroutine for the calculation of the shock response spectrum 482.8. Choice of the digitization frequency of the signal 522.9. Example of use of shock response spectra 532.10. Use of shock response spectra for the study of systems

wi th se ver al degress of fr ee dom

3 Characteristics of shock response spectra 593.1. Shock response spectra domains 593.2. Characteristics of shock response spectra at low frequencies 60

3.2.1. General characteristics 603.2.2. Shocks with velocity changed from zero TT603.2.3. Shocks with AV = 0 and AD = 0 at end of pulse 693.2.4. Shocks with AV = 0 and AD = 0 at end of pulse 723.2.5. Notes on residual spectrum 74

3.3. Characteristics of shock response spectra at high frequencies 753.4. Damping influence 773.5. Choice of damping 783.6. Choice of frequency range 803.7. Charts 813.8. Relation of shock response spectrum to Fourier spectrum 81

3.8.1. Primary shock response spectrum and Fourier transform 813.8.2. Residual shock response spectrum and Fourier transform 833.8.3. Comparison of the relative severity of several shocks using

their Fourier spectra and their shock response spectra 863.9. Characteristics of shocks of pyrotechnic origin 88

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00007___c0499cef0119962b19c58693fb9ea126.pdfContents vii

3.10. Care to be taken in the calculation of spectra 903.10.1. Influence of background noise of the measuring equipment.. 903.10.2. Influence of zero shift 92

4 Development of shock test specifications 954.1. General 954.2. Simplification of the measured signal 964.3. Use of shock response spectra 98

4.3.1. Synthesis of spectra 984.3.2. Nature of the specification 994.3.3. Choice of shape 1004.3.4. Amplitude 1014.3.5. Duration 1014.3.6. Difficulties 105

4.4. Other methods 1074.4.1. Use of a swept sine 1074.4.2. Simulation of shock response spectra using a

fast swept sine 1084.4.3. Simulation by modulated random noise 1124.4.4. Simulation of a shock using random vibration 1134.4.5. Least favourable response technique 1144.4.6. Restitution of a shock response spectrum by

a series of modulated sine pulses 1154.5. Interest behind simulation of shocks on a shaker using a

shock spectrum 117

5. Kinematics of simple shocks 1215.1. General 1215.2. Half-sine pulse 121

5.2.1. Definition 1215.2.2. Shock motion study 122

5.2.2.1. General expressions 1225.2.2.2. Impulse mode 1245.2.2.3. Impact mode 126

5.3. Versed-sine pulse 1365.3.1. Definition 1365.3.2. Shock motion study 137

5.4. Rectangular pulse 1395.4.1. Definition 1395.4.2. Shock motion study 139

5.5. Terminal peak saw tooth pulse 1425.5.1. Definition 1425.5.2. Shock motion study 143

5.6. Initial peak saw tooth pulse 1455.6.1. Definition 145

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00008___2f672f00bc05cafa946a6190a57ac5f4.pdfviii Mechanical shock

5.6.2. Shock motion study 145

6 Standard shock machines 1496.1. Main types 1496.2. Impact shock machines 1516.3. High impact shock machines 160

6.3.1. Lightweight high impact shock machine 1606.3.2. Medium weight high impact shock machine 162

6.4. Pneumatic machines 1636.5. Specific test facilities 1646.6. Programmers 165

6.6.1. Half-sine pulse 1656.6.2. Terminal peak saw tooth shock pulse 1736.6.3. Rectangular pulse - trapezoidal pulse 1806.6.4. Universal shock programmer 181

6.6.4.1. Generating a half-sine shock pulse 1826.6.4.2. Generating a terminal peak saw tooth

shock pulse 1826.6.4.3. Trapezoidal shock pulse 1836.6.4.4. Limitations 183

7 Generation of shocks using shakers 1897.1. Principle behind the generation of a

simple shape signal versus time 1897.2. Main advantages of the generation of shock using shakers 1907.3. Limitations of electrodynamic shakers 191

7.3.1. Mechanical limitations 1917.3.2. Electronic limitations 193

7.4. The use of the electrohydraulic shakers 1937.5. Pre-and post-shocks 193

7.5.1. Requirements 1937.5.2. Pre- or post-shock 1957.5.3. Kinematics of the movement for symmetrical pre-

and post-shock 1987.5.3.1. Half-sine pulse 1987.5.3.2. TPS pulse 2067.5.3.3. Rectangular pulse 2077.5.3.4. IPS pulse 208

7.5.4. Kinematics of the movement for a pre-shockor a post-shock alone 208

7.5.5. Abacuses 2127.5.6. Influence of the shape of pre- and post-pulses 2137.5.7. Optimized pre- and post-shocks 216

7.6. Incidence of pre- and post-shocks on the quality of simulation 2207.6.1. General 220

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00009___d41911442eb5858f42e9bc2dd9422c4b.pdfContents ix

7.6.2. Influence of the pre- and post-shocks on the timehistory response of a one-degree-of-freedom system 220

7.6.3. Incidence on the shock response spectra 223

8 Simulation of pyroshocks 2278.1. Simulations using pyrotechnic facilities 2278.2. Simulation using metal to metal impact 2308.3. Simulation using electrodynamic shakers 2318.4. Simulation using conventional shock machines 232

9 Control of a shaker using a shock response spectrum 2359.1. Principle of control by a shock response spectrum 235

9.1.1. Problems 2359.1.2. Method of parallel 9.1.3. Current numerical methods 237

9.2. Decaying sinusoid 2399.2.1. Definition 2399.2.2. Response spectrum 2399.2.3. Velocity and displacement 2429.2.4. Constitution of the total signal 2439.2.5. Methods of signal compensation 2449.2.6. Iterations 250

9.3. D.L. Kern and C.D. Hayes' function 2519.3.1. Definition 2519.3.2. Velocity and displacement 252

9.4. ZERD function 2539.4.1. Definition 253

9.4.1.1. D.K. Fisher and M.R. Posehn expression 2539.4.1.2. D.O. Smallwood expression 254

9.4.2. Velocity and displacement 2559.4.3. Comparison of the ZERD waveform with a

standard decaying sinusoid 2579.4.4. Reduced response spectra 257

9.4.4.1. Influence of the damping n of the signal 2579.4.4.2. Influence of the Q factor 258

9.5. WAVSIN waveform 2599.5.1. Definition 2599.5.2. Velocity and displacement 2609.5.3. Response of a one-degree-of-freedom system 262

9.5.3.1. Relative response displacement 2639.5.3.2. Absolute response acceleration 265

9.5.4. Response spectrum 2659.5.5. Time history synthesis from shock spectrum 266

9.6. SHOC waveform 2679.6.1. Definition 267

filterss 236

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00010___024df847d2a6c54372e43dfa42fff615.pdfx Mechanical shock

9.6.2. Velocity and displacement 2709.6.3. Response spectrum 271

9.6.3.1. Influence of damping n of the signal 2719.6.3.2. Influence of the Q factor on the spectrum 272

9.6.4. Time history synthesis from shock spectrum 2739.7. Comparison of the WAVSIN, SHOC waveforms

and decaying sinusoid 2749.8. Use of a fast swept sine 2749.9. Problems encountered during the synthesis of the waveforms 2789.10. Criticism of control by a shock response spectrum 2809.11. Possible improvements 282

9.11.1. IBS proposal 2839.11.2. Specification of a complementary parameter 284

9.11.2.1. Rms duration of the shock 2849.11.2.2. Rms value of the signal 2869.11.2.3. Rms value in the frequency domain 2879.11.2.4. Histogram of the peaks of the signal 2889.11.2.5. Use of the fatigue damage spectrum 288

9.11.3. Remarks on the characteristics of response spectrum 2889.12. Estimate of the feasibility of a shock specified by its SRS 289

9.12.1 C.D. Robbins and E.P. Vaughan's method 2899.12.2. Evaluation of the necessary force, power and stroke 291

Appendix. Similitude in mechanics 297A1. Conservation of materials 297A2. Conservation of acceleration and stress 299

Mechanical shock tests: a brief historical background 301

Bibliography 303

Index 315

Synopsis of five volume series 319

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00011___e482d7f528d53bfabddbd26cfda30e38.pdfIntroduction

Transported or on-board equipment is very frequently subjected to mechanicalshocks in the course of its useful lifetime (material handling, transportation, etc.).This kind of environment, although of extremely short duration (from a fraction of amillisecond to a few dozen milliseconds) is often severe and cannot be neglected.

The initial work into shocks was carried out in the 1930s on earthquakes andtheir effect on buildings. This resulted in the notion of the shock response spectrum.Testing on equipment started during World War II. Methods continued to evolvewith the increase in power of exciters, making it possible to create synthetic shocks,and again in the 1970s, with the development of computerization, enabling tests tobe directly conducted on the exciter employing a shock response spectrum.

After a brief recapitulation of the shock shapes most widely used in tests and ofthe possibilities of Fourier analysis for studies taking account of the environment(Chapter 1), Chapter 2 presents the shock response spectrum with its numerousdefinitions and calculation methods.

Chapter 3 describes all the properties of the spectrum, showing that importantcharacteristics of the original signal can be drawn from it, such as its amplitude orthe velocity change associated with the movement during the shock.

The shock response spectrum is the ideal tool for drafting specifications. Chapter4 details the process which makes it possible to transform a set of shocks recorded inthe real environment into a specification of the same severity, and presents a fewother methods that have been proposed in the literature.

Knowledge of the kinematics of movement during a shock is essential to theunderstanding of the mechanism of shock machines and programmers. Chapter 5gives the expressions for velocity and displacement according to time for classicshocks, depending on whether they occur in impact or impulse mode.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00012___6bad5c28cfdf69754268db1bf3876d87.pdfxii Mechanical shock

Chapter 6 describes the principle of shock machines currently most widely usedin laboratories and their associated programmers. To reduce costs by restricting thenumber of changes in test facilities, specifications expressed in the form of a simpleshock (half-sine, rectangle, saw tooth with a final peak) can occasionally be testedusing an electrodynamic exciter. Chapter 7 sets out the problems encountered,stressing the limitations of such means, together with the consequences ofmodification, that have to be made to the shock profile, on the quality of thesimulation.

Pyrotechnic devices or equipment (cords, valves, etc.) are very frequently usedin satellite launchers due to the very high degree of accuracy that they provide inoperating sequences. Shocks induced in structures by explosive charges areextremely severe, with very specific characteristics. Their simulation in thelaboratory requires specific means, as described in Chapter 8.

Determining a simple shape shock of the same severity as a set of shocks, on thebasis of their response spectrum, is often a delicate operation. Thanks to progress incomputerization and control facilities, this difficulty can occasionally be overcomeby expressing the specification in the form of a response spectrum and bycontrolling the exciter directly from that spectrum. In practical terms, as the excitercan only be driven with a signal that is a function of time, the software of the controlrack determines a time signal with the same spectrum as the specification displayed.Chapter 9 describes the principles of the composition of the equivalent shock, givesthe shapes of the basic signals most often used, with their properties, andemphasizes the problems that can be encountered, both in the constitution of thesignal and with respect to the quality of the simulation obtained.

Containers must protect the equipment carried in them from various forms ofdisturbance related to handling and possible accidents. Tests designed to qualify orcertify containers include shocks mat are sometimes difficult, not to say impossible,to produce, given the combined weight of the container and its content. Onerelatively widely used possibility consists of performing shocks on scale models,with scale factors of the order of 4 or 5, for example. This same technique can beapplied, although less frequently, to certain vibration tests. At the end of thisvolume, the Appendix summarizes the laws of similarity adopted to define themodels and to interpret the test results.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00013___caa98eb1593c49e43bad9f8165a7015a.pdfList of symbols

The list below gives the most frequent definition of the main symbols used inthis book. Some of the symbols can have another meaning locally which will bedefined in the text to avoid any confusion.

amax Maximum value of a(t)a( t) Component of shock x( t)A

c Amplitude of compensationsignal

A(0) Indicial admittanceb parameter b of Basquin's

relation N ob = Cc Viscous damping constantC Basquin's law constant

(N ob = C)d(t) Displacement associated

with a(t)D Diameter of programmerD(f0) Fatigue damagee Neper's numberE Young's modulus or

energy of a shockERS Extreme response spectrumE(t) Function characteristic of

swept sinef Frequency of excitationf0 Natural frequency

F(t) External force applied tosystem

Prms Rms value of forceFm Maximum value of F(t)g Acceleration due to gravityh Interval (f/f0 )

or thickness of the targeth(t) Impulse responseH Drop heightHR Height of reboundH( ) Transfer functioni v=fIPS Initial peak saw tooth3(Q) Imaginary part of X(O)k Stiffness or coefficient of

uncertaintyK Constant of proportionality

of stress and deformation^nns Rms value of #(t)im Maximum of l(t)t(t) Generalized excitation

(displacement)t(t\ First derivative of ^(t)

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00014___a57a2fbaa66ad0e525ea37c533a12f91.pdfxiv Mechanical shock

"i(t) Second derivative of t(i)L LengthL(Q) Fourier transform of ^(t)m Massn Number of cycles undergone

by test-bar or materialN Number of cycles to failurep Laplace variable or

percentage of amplitude ofshock

q0 Value of q(0) for 0=0q0 Value of q(0) for 0=0q(0) Reduced responseq( 0) First derivative of q( 0)q( 0) Second derivative of q( 0)Q Q factor (quality factor)Q( p) Laplace transform of q (0)Re Yield stressRm Ultimate tensile strengthR(Q) Fourier transform of the

system response9?(Q) Real part of X(Q)s Standard deviationS AreaSRS Shock response spectrums( ) Power spectral densityt Timetd Decay time to zero of shocktj Fall durationtr Rise time of shocktR Duration of reboundT Vibration durationT0 Natural periodTPS Terminal peak saw toothu(t) Generalized responseu(t) First derivative of u(t)u(t) Second derivative of u(t)vf Velocity at end of shockVj Impact velocityVR Velocity of rebound

v(t) Velocity x(t) orvelocity associated with a(t)

v( ) Fourier transform of v(t)xm Maximum value of x(t)x(t) Absolute displacement of

the base of a one-degree-of-freedom system

x(t) Absolute velocity of the baseof a one-degree-of-reedomsystem

x(t) Absolute acceleration of thebase of a one-degree-of-freedom system

\ms Rms value of x(t)xm Maximum value of x(t)X

m Amplitude of Fouriertransform X(Q)

X(Q) Fourier transform of x(t)y(t) Absolute response of

displacement of mass of aone-degree-of-freedomsystem

y(t) Absolute response velocityof the mass of a one-degree-of-freedom system

y(t) Absolute responseacceleration of mass of aone-degree-of-freedomsystem

zm Maximum value of z(t)zs Maximum static relative

displacementzsup Largest value of z( t)z(t) Relative response

displacement of mass of aone-degree-of-freedomsystem with respect to itsbase

z(t) Relative response velocity

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00015___74a7eac0f5e526e77b68e6bc48165a7e.pdfz(t) Relative responseacceleration

a Coefficient of restitution5(t) Dirac delta functionAV Velocity change(ft Dimensionless product f0 T(j)(Q) PhaseTI Damping factor of damped

sinusoidTJC Relative damping of

compensation signalX.( ) Reduced excitationA(p) Laplace transform of A,( )K 3.14159265...6 Reduced time (co0 t)0d Reduced decay time

List of symbols xv

6m Reduced rise time60 Value of 0 for t = tp Densitya Stresscrcr Crushing stressam Maximum stressT Shock durationT! Pre-shock durationt2 Post-shock durationi nns Rms duration of a shockcoc Pulsation of compensation

signalco0 Natural pulsation (2 n f0)Q Pulsation of excitation

( 2 7 C f ) Damping factor

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00016___6726120b2e8008bb0e5905e0f944f1e3.pdfThis page intentionally left blank

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00017___d0f918353a83e22070c762d15b0f3dc7.pdfChapter 1

Shock analysis

1.1. Definitions

1.1.1. Shock

Shock is defined as a vibratory excitation with duration between once and twicethe natural period of the excited mechanical system.

Figure 1.1. Example of shock

Shock occurs when a force, a position, a velocity or an acceleration is abruptlymodified and creates a transient state in the system considered.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00018___1aaa6c7abbcf48356524a1bfe539f553.pdf2 Mechanical shock

The modification is normally regarded as abrupt if it occurs in a time periodwhich is short compared to the natural period concerned (AFNOR definition)[NOR].

1.1.2. Transient signal

This is a vibratory signal of short duration (of a fraction of second to a few tensof seconds), the mechanical shock, for example, in the air-braking phase on aircraftetc.

Figure 1.2. Example of transient signal

1.1.3. Jerk

A jerk is defined as the derivative of acceleration with respect to time. Thisparameter thus characterizes the rate of variation of acceleration with time.

1.1.4. Bump

A bump is a simple shock which is generally repeated many times when testing(AFNOR) [NOR].

Example

The GAM EG 13 (first part - Booklet 43 - Shocks) standard proposes a testcharacterized by a half-sine: 10 g, 16 ms, 3000 bumps (shocks) per axis, 3 bumps asecond [GAM86].

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00019___8e1deb4deafcd313ad552cd3c046206e.pdfShock analysis 3

1.1.5. Simple (orperfect) shockShock whose signal can be represented exactly in simple mathematical terms, for

example half-sine, triangular or rectangular shock.

1.1.6. Half-sine shockSimple shock for which the acceleration-time curve has the form of a half-period

(part positive or negative) of a sinusoid.

1.1.7. Terminal peak saw tooth shock (TPS) or final peak saw tooth shock (FPS)Simple shock for which the acceleration-time curve has the shape of a triangle

where acceleration increases linearly up to a maximum value and then instantlydecreases to zero.

1.1.8. Initial peak saw tooth shock (IPS)Simple shock for which the acceleration-time curve has the shape of a triangle

where acceleration instantaneously increases up to a maximum, and then decreasesto zero.

1.1.9. Rectangular shock

Simple shock for which the acceleration-time curve increases instantaneously upto a given value, remains constant throughout the signal and decreasesinstantaneously to zero.

1.1.10. Trapezoidal shock

Simple shock for which the acceleration-time curve grows linearly up to a givenvalue, remains constant during a certain time after which it decreases linearly tozero.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00020___2fbae356e492fdb9af0a07a4ed40e53a.pdf4 Mechanical shock

1.1.11. Versed-sine (or haversine) shock

Simple shock for which the acceleration-time curve has the form of one period ofthe curve representative of the function [1 - cos( )], with this period starting fromzero value of this function. It is thus a signal ranging between two minima.

1.1.12. Decaying sinusoidal pulse

A pulse comprised of a few periods of a damped sinusoid, characterized by theamplitude of the first peak, the frequency and damping:

This form is interesting, for it represents the impulse response of a one-degree-of-freedom system to a shock. It is also used to constitute a signal of a specifiedshock response spectrum (shaker control from a shock response spectrum).

1.2. Analysis in the time domain

A shock can be described in the time domain by the following parameters:

- the amplitude x(t);- duration t;- the form.

The physical parameter expressed in terms of time is, in a general way, anacceleration x(t), but can be also a velocity v(t), a displacement x(t) or a forceF(t).

In the first case, which we will particularly consider in this volume, the velocitychange corresponding to the shock movement is equal to

1.3. Fourier transform

1.3.1. Definition

The Fourier integral (or Fourier transform) of a function x(t) of the real variablet absolutely integrable is defined by

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00021___ba4414fb809ea4e67fa667113c2194e9.pdfShock analysis 5

The function X(Q) is in general complex and can be written, by separating thereal and imaginary parts ${(1) and 3(Q):

or

with

and

Thus is the Fourier spectrum of I the energy spectrum andis the phase.

The calculation of the Fourier transform is a one-to-one operation. By means ofthe inversion formula or Fourier reciprocity formula., it is shown that it is possible toexpress in a univocal way x(t) according to its Fourier transform X(Q) by therelation

(if the transform of Fourier X(Q) is itself an absolutely integrable function over allthe domain).

NOTES.I . For

dt

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00022___3f167c8ec3a60beb52fca60f838494f2.pdf6 Mechanical shock

The ordinate at f = 0 of the Fourier transform (amplitude) of a shock defined byan acceleration is equal to the velocity change AV associated with the shock (areaunder the curve x(t)).

2. The following definitions are also sometimes found [LAL 75]:

In this last case, the two expressions are formally symmetrical. The sign of theexponent of exponential is sometimes also selected to be positive in the expressionfor X(Q) and negative in that for x(t).

1.3.2. Reduced Fourier transformThe amplitude and the phase of the Fourier transform of a shock of given shape

can be plotted on axes where the product f T (T = shock duration) is plotted on theabscissa and on the ordinate, for the amplitude, the quantity A(f )/xm r .

In the following paragraph, we draw the Fourier spectrum by considering simpleshocks of unit duration (equivalent to the product ft) and of the amplitude unit. Itis easy, with this representation, to recalibrate the scales to determine the Fourierspectrum of a shock of the same form, but of arbitrary duration and amplitude.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00023___3b844f90e4efae6b3bcb61a7b4127776.pdfShock analysis 7

1.3.3. Fourier transforms of simple shocks

1.3.3.1. Half-sine pulse

Figure 1.3. Real and imaginary parts of the Fourier transform of a half-sine pulse

Amplitude [LAL 75]: Phase:

(k positive integer)

Imaginary part:Real part:

Figure 1.4. Amplitude and phase of the Fourier transform of a half-sine shock pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00024___24af34eb617774c118732764d638d19f.pdf8 Mechanical shock

1.3.3.2. Versed-sine pulse

Figure 1.5. Real and imaginary parts of the Fourier transform of a versed-sine shock pulse

Amplitude: Phase:

Imaginary part:Real part:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00025___a2ab2fb3809356552aca99760b9f680e.pdfShock analysis 9

Figure 1.6. Amplitude and phase of the Fourier transform of a versed-sine shockpulse

1.3.3.3. Terminal peak saw tooth pulse (TPS)

Amplitude:

Figure 1.7. Real and imaginary parts of the Fourier transform of a TPS shockpulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00026___752af10a9650204eb54c4f940a811ba4.pdf10 Mechanical shock

Phase:

Real part:

Imaginary part:

Figure 1.8. Amplitude and phase of the Fourier transform of a TPS shock pulse

1.3.3.4. Initial peak saw tooth pulse (IPS)

Amplitude:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00027___24f33307e2b106d1b279c763aa574be2.pdfPhase:

Shock analysis 11

Figure 1.9. Real and imaginary parts of the Fourier transform of an IPS shock pulse

Real part:

Figure 1.10. Amplitude and phase of the Fourier transform of an IPS shock pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00028___1d2e29edd50c3f116cce16ffff7c0ec9.pdf12 Mechanical shock

Imaginary part:

1.3.3.5. Arbitrary triangular pulse

If tr = the rise time and t^ = decay time.Amplitude:

Phase:

Real part:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00029___82303c9a892b79f68171a6f1655f0253.pdfShock analysis 13

Figure 1.11. Real and imaginary parts ofthe Fourier transform of a triangular

shock pulse

Figure 1.12. Real and imaginary parts ofthe Fourier transform of a triangular

shock pulse

Imaginary part:

Figure 1.13. Amplitude and phase of the Fourier transform of a triangular shock pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00030___414d5c36e271b89b1bec24bb5228b547.pdf14 Mechanical shock

Figure 1.14. Amplitude and phase of the Fourier transform of a triangular shock pulse

1.3.3.6. Rectangular pulse

Figure 1.15. Real and imaginary parts of the Fourier transform of arectangular shock pulse

Amplitude:

Phase:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00031___1452686e0b638037bee835989ff6e873.pdfReal part:

Imaginary part:

Shock analysis 15

Figure 1.16. Amplitude and phase of the Fourier transform of arectangular shock pulse

1.3.3.7. Trapezoidal pulse

Amplitude:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00032___ada469dc47a5ed3eb1c479442842674d.pdf16 Mechanical shock

Phase:

Real part:

Imaginary part:

Figure 1.17. Real and imaginary parts of the Fourier transform of atrapezoidal shock pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00033___cef8b46aa97abbf84f74c9ae3d828f66.pdfShock analysis 17

Figure 1.18. Amplitude and phase of the Fourier transform of atrapezoidal shock pulse

1.3.4. Importance of the Fourier transform

The Fourier spectrum contains all the information present in the original signal,in contrast, we will see, to the shock response spectrum (SRS).

It is shown that the Fourier spectrum R(Q) of the response at a point in astructure is the product of the Fourier spectrum X(Q) of the input shock and thetransfer function H(Q) of the structure:

R(Q) = H(Q) X(Q) [1-40]

The Fourier spectrum can thus be used to study the transmission of a shockthrough a structure, the movement resulting at a certain point being then describedby its Fourier spectrum. The response in the time domain can be also expressed froma convolution utilizing the 'input' shock signal according to the time and the impulseresponse of the mechanical system considered. An important property is used here:the Fourier transform of a convolution is equal to the scalar product of the Fouriertransforms of the two functions in the frequency domain.

It could be thought that this (relative) facility of change in domain (time frequency) and this convenient description of the input or of the response wouldmake the Fourier spectrum method one frequently used in the study of shock, inparticular for the writing of test specifications from experimental data.

These mathematical advantages, however, are seldom used within thisframework, because when one wants to compare two excitations, one runs upagainst the following problems:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00034___ca11b24fe33542d91cd59e9334ccffcc.pdf18 Mechanical shock

- The need to compare two functions. The Fourier variable is a complex quantitywhich thus requires two parameters for its complete description: the real part and theimaginary part (according to the frequency) or the amplitude and the phase. Thesecurves in general are very little smoothed and, except in obvious cases, it is difficultto decide on the relative severity of two shocks according to frequency when thespectrum overlap. In addition, the phase and the real and imaginary parts can takepositive and negative values and are thus not very easy to use to establish aspecification;

- The signal obtained by inverse transformation has in general a complex formimpossible to reproduce with the usual test facilities, except, with certain limitations,on electrodynamic shakers.

The Fourier transform is used neither for the development of specifications norfor the comparison of shocks. On the other hand, the one-to-one relation propertyand the input-response relation [1.40] make it a very interesting tool to controlshaker shock whilst calculating the electric signal by applying these means toreproduce with the specimen a given acceleration profile, after taking into accountthe transfer function of the installation.

1.4. Practical calculations of the Fourier transform

1.4.1. General

Among the various possibilities of calculation of the Fourier transform, the FastFourier Transform (FFT) algorithm of Cooley-Tukey [COO 65] is generally usedbecause of its speed (Volume 3). It must be noted that the result issuing from thisalgorithm must be multiplied by the duration of the analysed signal to obtain theFourier transform.

1.4.2. Case: signal not yet digitized

Let us consider an acceleration time history x(t) of duration T which one wishesto calculate the Fourier transform with nFT points (power of 2) until the frequencyfmax. According to the Shannon's theorem (Volume 3), it is enough that the signalis sampled with a frequency fsamp = 2 fmax, i.e. that the temporal step is equal to

The frequency interval is equal to

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00035___b16a103b1fc299c4a52475855bb293f8.pdfShock analysis 19

To be able to analyse the signal with a resolution equal to Af , it is necessary that itsduration is equal to

yielding the temporal step

If n is the total number of points describing the signal

and one must have

yielding

The duration T needed to be able to calculate the Fourier transform with the selectedconditions can be different to the duration T from the signal to analyse (for examplein the case of a shock). It cannot be smaller than T (if not shock shape would bemodified). Thus, if we set the condition T leads to i.e. to

If the calculation data (nFT and fmax) lead to a too large value of Af , it will benecessary to modify one of these two parameters to satisfy to the above condition.

If it is necessary that the duration T is larger than T, zeros must be added to thesignal to analyse between T and T, with the temporal step At.

The computing process is summarized in Table 1.1.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00036___163c547469c931cb07ca27b1a14d83b5.pdf20 Mechanical shock

Table 1.1. Computing process of a Fourier transform starting from a non-digitized signal

Data: - Characteristics of the signal to be analysed (shape, amplitude, duration) orone measured signal not yet digitized.

- The number of points of the Fourier transform (npj-) and its maximumfrequency (fmax).

Condition to avoid the aliasing phenomenon (Shannon'sTsamp. - fmax theorem) . If the measured signal can contain components at

frequencies higher than fmax, it must be filtered using alow-pass filter before digitalization. To take account of theslope of the filter beyond fmax, it is preferable to choosefsamp = 2.6 fmax (Volume 3).

At = Temporal step of the signal to be digitized (time interval2 fmax between two points of the signal).

fmax Frequency interval between two successive points of theAt = Fourier transform.

nFTn = 2 npr Number of points of the signal to be digitized.T = n At Total duration of the signal to be treated.

If T > t, zeros must added between T and T.If there are not enough points to represent correctly the signal between 0 and T,fmax must t>e increased.

f 1The condition Af = -^L < - must be satisfied (i.e. T > T ):

"FT T

- if fmax is imposed, take npj (power of 2) > T fmax.npr

- if n FT is imposed, choose fmax < .T

1.4.3. Case: signal already digitized

If the signal of duration T were already digitized with N points and a step 5T, thecalculation conditions of the transform are fixed:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00037___e75110935017fb8d8ca8511968c8ce0e.pdf(nearest power of 2)and

(which can thus result in not using the totality of the signal).

If however we want to choose a priori fmax and npj, the signal must beresampled and if required zeros must be added using the principles in Table 1.1.

Shock analysis 21

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00038___6c890cf017ccaca1db970c8aabe1c250.pdfThis page intentionally left blank

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00039___e615cce0e8fbf9a1beeded94086107d1.pdfChapter 2

Shock response spectra domains

2.1. Main principles

A shock is an excitation of short duration which induces transitory dynamicstress structures. These stresses are a function of:

- the characteristics of the shock (amplitude, duration and form);- the dynamic properties of the structure (resonance frequencies, Q factors).

The severity of a shock can thus be estimated only according to thecharacteristics of the system which undergoes it. The evaluation of this severityrequires in addition the knowledge of the mechanism leading to a degradation of thestructure. The two most common mechanisms are:

- The exceeding of a value threshold of the stress in a mechanical part, leading toeither a permanent deformation (acceptable or not) or a fracture, or at any rate, afunctional failure.

- If the shock is repeated many times (e.g. shock recorded on the landing gear ofan aircraft, operation of an electromechanical contactor, etc), the fatigue damageaccumulated in the structural elements can lead in the long term to fracture. We willdeal with this aspect later on.

The severity of a shock can be evaluated by calculating the stresses on amathematical or finite element model of the structure and, for example, comparisonwith the ultimate stress of the material. This is the method used to dimension thestructure. Generally, however, the problem is rather to evaluate the relative severityof several shocks (shocks measured in the real environment, measured shocks withrespect to standards, establishment of a specification etc). This comparison would be

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00040___653a9ec2c0a29404b76b60a8319d341f.pdf24 Mechanical shock

difficult to carry out if one used a fine model of the structure, and besides this is notalways available, in particular at the stage of the development of the specification ofdimensioning. One searches for a method of general nature, which leads to resultswhich can be extrapolated to any structure.

A solution was proposed by M.A. Biot [BIO 32] in 1932 in a thesis on the studyof the earthquakes effects on the buildings; this study was then generalized toanalysis of all kinds of shocks.

The study consists of applying the shock under consideration to a 'standard'mechanical system, which thus does not claim to be a model of the real structure,composed of a support and of N linear one-degree-of-freedom resonators,comprising each one a mass mi, a spring of stiffness kj and a damping device Cj,chosen such that the fraction of critical damping is the same for all N

resonators (Figure 2.1).

Figure 2.1. Model of the shock response spectrum (SRS)

When the support is subjected to the shock, each mass nij has a specific

movement response according to its natural frequency

chosen damping , while a stress GJ is induced in the elastic element.

and to the

The analysis consists of seeking the largest stress crmj observed at eachfrequency in each spring. A shock A is regarded as more severe than a shock B if itinduces in each resonator a larger extreme stress. One then carries out anextrapolation, which is certainly criticizable, by supposing that, if shock A is moresevere than shock B when it is applied to all the standard resonators, it is also more

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00041___de1e8e26d085a26c1474c480838e1ab6.pdfShock response spectra domains 25

severe with respect to an arbitrary real structure (which cannot be linear nor having asingle degree of freedom).

NOTE: A study was carried out in 1984 on a mechanical assembly composed of acircular plate on which one could place some masses and thus vary the number ofdegrees of freedom. The stresses generated by several shocks of the same spectra(in the frequency range including the principal resonance frequencies), but ofdifferent shapes [DEW 84], were measured and compared. One noted that for thisassembly whatever the number of degrees of freedom,

two pulses of simple form (with no velocity change) having the same spectruminduce similar stresses, the variation not exceeding approximately 20 %. It is thesame for two oscillatory shocks;

the relationship between the stresses measured for a simple shock and anoscillatory shock can reach 2.

These results were supplemented by numerical simulation intended to evaluatethe influence ofnon linearity. Even for very strong non-linearity, one did not notefor the cases considered, an important difference between the stresses induced bytwo shocks of the same spectrum, but of different form.

A complementary study was carried out by B.B. Petersen [PET 81] in order tocompare the stresses directly deduced from a shock response spectrum with thosegenerated on an electronics component by a half-sine shock envelop of a shockmeasured in the environment and by a shock of the same spectrum made up fromWA VSIN signals (Chapter 9) added with various delays. The variation between themaximum responses measured at five points in the equipment and the stressescalculated starting from the shock response spectra does not exceed a factor of 3 inspite of the important theoretical differences between the model of the responsespectrum and the real structure studied.

For applications deviating from the assumptions of definition of the shockresponse spectrum (linearity, only one degree of freedom), it is desirable to observea certain prudence if one wishes to estimate quantitatively the response of a systemstarting from the spectrum [BOR 89]. The response spectra are more often used tocompare the severity of several shocks.

It is known that the tension static diagram of many materials comprises a more orless linear arc on which the stress is proportional to the deformation. In dynamics,this proportionality can be allowed within certain limits for the peaks of thedeformation (Figure 2.2).

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00042___4d1c02f01568bd00e9960626e7387a98.pdf26 Mechanical shock

If mass-spring-damper system is supposed to be linear, it is then appropriate tocompare two shocks by the maximum response stress am they induce or by themaximum relative displacement zm that they generate, since:

Figure 2.2. Stress-strain curve

zm is a function only of the dynamic properties of the system, whereas am isalso a function, via K, of the properties of the materials which constitute it.

The curve giving the largest relative displacement zsup multiplied by oo0according to the natural frequency f0, for a given damping, is the shock responsespectrum (SRS). The first work defining these spectra was published in 1933 and1934 [BIO 33] [BIO 34], then in 1941 and 1943 [BIO 41] [BIO 43]. The shockresponse spectrum, then named the shock spectrum, was presented there in thecurrent form.

This spectrum was used in the field of environmental tests from 1940 to 1950:J.M. Frankland [FRA 42] in 1942, J.P. Walsh and R.E. Blake in 1948 [WAL 48],R.E. Mindlin [MIN 45]. Since then, there have beenmany works which used it astool of analysis and for simulation of shocks [HIE 74], [KEL 69], [MAR 87] and[MAT 77].

2.2. Response of a linear one-degree-of-freedom system

2.2.1. Shock defined by a force

Being given a mass-spring-damping system subjected to a force F(t) applied tothe mass, the differential equation of the movement is written as:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00043___d4c2815d2da769e351fc656af4e95b98.pdfShock response spectra domains 27

Figure 2.3. Linear one-degree-of-freedom system subjected to a force

where z(t) is the relative displacement of the mass m relative to the support inresponse to the shock F(t). This equation can be put in the form:

where

(damping factor) and

(natural pulsation).

2.2.2. Shock defined by an acceleration

Let us set as x(t) an acceleration applied to the base of a linear one-degree-of-freedom mechanical system, with y(t) the absolute acceleration response of the

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00044___3400612b75b242e95aa1265b02f2e0d1.pdf28 Mechanical shock

mass m and z(t) the relative displacement of the mass m with respect to the base.The equation of the movement is written as above:

Figure 2.4. Linear one-degree-of-freedom system subjected to acceleration

i.e.

or, while setting z(t) = y(t) - x(t):

2.2.3. Generalization

Comparison of the differential equations [2.3] and [2.8] shows that they are bothof the form

where /(t) and u(t) are generalized functions of the excitation and response.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00045___65d1f720cf410fed8d54c28b01d2b273.pdfShock response spectra domains 29

NOTE: The generalized equation [2.9] can be -written in the reduced form:

where

m = maximum of l(t)

Resolution

The differential equation [2.10] can be integrated by parts or by using theLaplace transformation. We obtain, for zero initial conditions, an integral calledDuhamel 's integral:

where variable of integration. In the generalized form, we deduce that

where a is an integration variable homogeneous with time. If the excitation is anacceleration of the support, the response relative displacement is given by:

and the absolute acceleration of the mass by:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00046___0476657a5fc616b7b92fc5205f711969.pdf30 Mechanical shock

Application

Let us consider a package intended to protect a material from mass m andcomprising a suspension made up of two elastic elements of stiffness k and twodampers of damping constant c.

Figure 2.5. Model of the package Figure 2.6. Equivalent model

We want to determine the movement of the mass m after free fall from a heightof h = 5 m, by supposing that there is no rebound of the package after the impacton the ground and that the external frame is not deformable (Figure 2.5). Thissystem is equivalent to the model in Figure 2.6. We have (Volume 1, Chapter 3):

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00047___7953946f481605192676216ac47bb727.pdfand

Shock response spectra domains 31

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00048___c51cdd34a2c2dfe26b9ef2ed8d922bb8.pdf32 Mechanical shock

where

and

With the chosen numerical values, it becomes:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00049___cacc1ae14eed493afbca9cc84494df80.pdfShock response spectra domains 33

From this it is easy to deduce the velocity z(t) and the acceleration z(t)from successive derivations of this expression. The first term corresponds to thestatic deformation of the suspension under load of 100 kg.

2.2.4. Response of a one-degree-of-freedom system to simple shocks

Half-sine pulse

Versed-sine pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00050___8cfa97cee046ceac466feb3b34ac153f.pdf34 Mechanical shock

Rectangular pulse

Initial peak saw tooth pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00051___42e037c9ea5987b695eeaf8e8b5ecfcc.pdfShock response spectra domains 35

Terminal peak saw tooth pulse

Arbitrary triangular pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00052___f0b83fad2e720f3b78b1fed1572f5f00.pdf36 Mechanical shock

Trapezoidal pulse

Figure 2.7. Trapezoidal shock pulse

where

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00053___986c586ba66347e4555481686ac6c91e.pdfShock response spectra domains 37

For an isosceles trapezoid, we set 6r = 00 - 6d . If the rise and decay each havea duration equal to 10% of the total duration of the trapezoid, we have

Absolute acceleration shock response spectrum

In the most usual cases where the excitation is defined by an absoluteacceleration of the support or by a force applied directly to the mass, the response ofthe system can be characterized by the absolute acceleration of the mass (whichcould be measured using an accelerometer fixed to this mass): the response spectrumis then called the absolute acceleration shock response spectrum. This spectrum canbe useful when absolute acceleration is the parameter easiest to compare with acharacteristic value (study of the effects of a shock on a man, comparison with thespecification of an electronics component etc).

Relative displacement shock spectrum

In similar cases, we often calculate the relative displacement of the mass withrespect to the base of the system, displacement which is proportional to the stresscreated in the spring (since the system is regarded as linear). In practice, one ingeneral expresses in ordinates the quantity co0 zsup called the equivalent staticacceleration. This product has the dimension of an acceleration, but does notrepresent the acceleration of the mass, except when damping is zero; this term isthen strictly equal to the absolute acceleration of the mass. However, when dampingis close to the current values observed in mechanics, and in particular when

24 = 0.05, one can assimilate as a first approximation co0 zsup to the absoluteacceleration ysup of the mass m [LAL 75].

2.3. Definitions

Response spectrum

A curve representative of the variations of the largest response of a linear one-degree-of-freedom system subjected to a mechanical excitation, plotted against itsnatural frequency f0 = for a given value of its damping ratio.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00054___5db30507554a0352010bfc7b146e4cb8.pdf38 Mechanical shock

Very often in practice, it is the stress (and thus the relative displacement) whichseems the most interesting parameter, the spectrum being primarily used to study thebehaviour of a structure, to compare the severity of several shocks (the stress createdis a good indicator), to write test specifications (it is also a good comparisonbetween the real environment and the test environment) or to dimension asuspension (relative displacement and stress are then useful).

The quantity co0 zsu is termed pseudo-acceleration. In the same way, one termspseudo-velocity the product o)0 zsup.

2The spectrum giving co0 zsup versus the natural frequency is named the relative

displacement shock spectrum.

In each of these two important categories, the response spectrum can be definedin various ways according to how the largest response at a given frequency ischaracterized.

Primary positive shock response spectrum or initial positive shock responsespectrum

The highest positive response observed during the shock.

Primary (or initial) negative shock response spectrum

The highest negative response observed during the shock.

Secondary (or residual shock) response spectrum

The largest response observed after the end of the shock. Here also, the spectrumcan be positive or negative.

Positive (or maximum positive) shock response spectrum

The largest positive response due to the shock, without reference to the durationof the shock. It is thus about the envelope of the positive primary and residualspectra.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00055___6fcd61c695b0f6cfe9ebcde6931111bd.pdfShock response spectra domains 39

Negative (or maximum negative) shock response spectrum

The largest negative response due to the shock, without reference to the durationof the shock. It is in a similar way the envelope of the negative primary and residualspectra.

Example

Maximax shock response spectrum

Envelope of the absolute values of the positive and negative spectra.

Which spectrum is the best? The damage is supposed proportional to the largestvalue of the response, i.e. to the amplitude of the spectrum at the frequencyconsidered, and it is of little importance for the system whether this maximumzm takes place during or after the shock. The most interesting spectra are thus thepositive and negative spectra, which are most frequently used in practice, with themaximax spectrum. The distinction between positive and negative spectra must bemade each time the system, if disymmetrical, behaves differently, for example underdifferent tension and compression. It is, however, useful to know these variousdefinitions so as to be able to correctly interpret the curves published.

Figure 2.8. Shock response spectra of a rectangular shock pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00056___c946b3ae87b814129180f4f084098f59.pdf40 Mechanical shock

2.4. Standardized response spectra

For a given shock, the spectra plotted for various values of the duration and theamplitude are homothetical. It is thus interesting, for simple shocks to have astandardized or reduced spectrum plotted in dimensionless co-ordinates, whileplotting on the abscissa the product f0 t (instead of f0) or co0 t and on the ordinatethe spectrum/shock pulse amplitude ratio co0 zm /xm , which, in practice, amountsto tracing the spectrum of a shock of duration equal to 1 s and amplitude 1 m/s2.

Figure 2.9. Standardized SRS of a half-sine pulse

These standardized spectra can be used for two purposes:- plotting of the spectrum of a shock of the same form, but of arbitrary amplitude

and duration;- investigating the characteristics of a simple shock of which the spectrum

envelope is a given spectrum (resulting from measurements from the realenvironment).

The following figures give the spectra of reduced shocks for various pulse forms,unit amplitude and unit duration, for several values of damping. To obtain thespectrum of a particular shock of arbitrary amplitude xm and duration T (differentfrom 1) from these spectra, it is enough to regraduate the scales as follows:

- for the amplitude; by multiply the reduced values by xm;

- for the abscissae, replace each value (= f0 T ) by f0 =

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00057___6f70409132ea963e0fc0c484a802d11f.pdfShock response spectra domains 41

We will see later on how these spectra can be used for the calculation of testspecifications.

Half-sine pulse

Figure 2.11. Standardized primary and residual relative displacement SRSof a half-sine pulse

Figure 2.10. Standardized positive and negative relative displacement SRSof a half-sine pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00058___72f8601132d7bda2f7971ce850db18b0.pdf42 Mechanical shock

Figure 2.12. Standardized positive and negative absolute acceleration SRSof a half-sine pulse

Versed-sine pulse

Figure 2.13. Standardized positive and negative relative displacement SRSof a versed-sine pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00059___42ae9ecd44fc8320ed9bc28e769c1f09.pdfShock response spectra domains 43

Terminal peak saw tooth pulse

Figure 2.14. Standardized primary and residual relative displacement SRSof a versed-sine pulse

Figure 2.15. Standardized positive and negative relative displacement SRSof a TPS pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00060___5570801387d250d918723481a704055b.pdf44 Mechanical shock

Figure 2.16. Standardized primary and residual relative displacement SRSof a TPS pulse

Figure 2.17. Standardized positive and negative relative displacement SRSof a TPS pulse with zero decay time

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00061___92bb813f2cb5fc519358824940a56536.pdfShock response spectra domains 45

Initial peak saw tooth pulse

Figure 2.19. Standardized primary and residual relative displacement SRSof an IPS pulse

Figure 2.18. Standardized positive and negative relative displacements SRSof an IPS pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00062___63541cc175232503410b1660476004d1.pdf46 Mechanical shock

Figure 2.20. Standardized positive and negative relative displacement SRSof an IPS with zero rise time

Rectangular pulse

Figure 2.21. Standardized positive and negative relative displacement SRSof a rectangular pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00063___c8a5fe0e439a854f109bd562effa9582.pdfShock response spectra domains 47

Trapezoidal pulse

Figure 2.22. Standardized positive and negative relative displacement SRSof a trapezoidal pulse

2.5. Difference between shock response spectrum (SRS) and extreme responsespectrum (ERS)

A spectrum known as of extreme response spectrum (ERS) and comparable withthe shock response spectrum (SRS) is often used for the study of vibrations (Volume5). This spectrum gives the largest response of a linear single-degree-of-freedomsystem according to its natural frequency, for a given Q factor, when it is subjectedto the vibration under investigation. In the case of the vibrations, of long duration,this response takes place during the vibration: the ERS is thus a primary spectrum.In the case of shocks, we in general calculate the highest response, which takes placeduring or after the shock.

2.6. Algorithms for calculation of the shock response spectrum

Various algorithms have been developed to solve the second order differentialequation [2.9] ([COL 90], [COX 83], [DOK 89], [GAB 80], [GRI 96], [HAL 91],[HUG 83a], [IRV 86], [MER 91], [MER 93], [OHA 62], [SEI 91] and [SMA 81]).One which leads to the most reliable results is that of F. W. Cox [COX 83] (Section2.7.).

Although these calculations are a priori relatively simple, the round robins thatwere carried out ([BOZ 97] [CHA 94]) showed differences in the results, ascribable

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00064___f9fd7fd5ca05d505635a5d26c11a7786.pdf48 Mechanical shock

sometimes to the algorithms themselves, but also to the use or programming errorsof the software.

2.7. Subroutine for the calculation of the shock response spectrum

The following procedure is used to calculate the response of a linear single-degree-of-freedom system as well as the largest and smallest values after the shock(points of the positive and negative SRS, primary and residual, displacementsrelative and absolute accelerations). The parameters transmitted to the procedure arethe number of points defining the shock, the natural pulsation of the system and itsQ factor, the temporal step (presumably constant) of the signal and the array of theamplitudes of the signal. This procedure can be also used to calculate the response ofa one-degree-of-freedom system to an arbitrary excitation, and in particular to arandom vibration (where one is only interested in the primary response).

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00065___451e7fe81e470b42630fb24fbf249765.pdfShock response spectra domains 49

Procedure for the calculation of a point of the SRS at frequency f0 (GFA-BASIC)

From F. W. Cox [COX 83]

PROCEDURE S_R_S(npts_signal%,wO,Q_factor,dt,VAR xppO)LOCAL i%,a,a1 ,a2,b,b1,b2,c,c1,c2,d,d2,e,s,u,v,wdt,w02,w02dtLOCAL p1d,p2d,p1a,p2a,pd,pa,wtd,wta,sd,cd,ud,vd,ed,sa,ca,ua,va,ea' npts_signal% = Number of points of definition of the shock versus time' xpp(npts_signal%) = Array of the amplitudes of the shock pulse' dt= Temporal step' wO= Undamped natural pulsation (2*PI*fO)' Initialization and preparation of calculationspsi=l/2/Q_factor // Damping ratiow=wO*SQR(l-psiA2) // Damped natural pulsationd=2*psi*wOd2=d/2wdt=w*dte=EXP(-d2*dt)s=e*SIN(wdt)c=e*COS(wdt)u=w*c-d2*sv=-w*s-d2*cw02=wOA2w02dt=w02*dt1 Calculation of the primary SRS' Initialization of the parameters

srcajprim_min=lE100 // Negative primary SRS (absolute acceleration)srca_prim_max=-srcajprim_mm // Positive primary SRS (absolute acceleration)srcd_prim_min=srca_prim_min // Negative primary SRS (relative displacement)srcdjprim_max=-srcd_prim_min // Positive primary SRS (relative displacement)displacement_z=0 // Relative displacement of the mass under the shockvelocity_zp=0 // Relative velocity of the mass' Calculation of the sup. and inf. responses during the shock at the frequency fO

FOR i%=2 TO npts_signal%a=(xpp(i%-1 )-xpp(i%))/w02dtb=(-xpp(i%-1 )-d*a)/w02c2=displacement_z-bc 1 =(d2*c2-t-velocity_zp-a)/wdisplacement_z=s*cl+c*c2+a*dt+bvelocity_zp=u * c1 + v* c2+aresponsedjprim=-displacement_z*w02 // Relative displac. during shock x

square of the pulsation

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00066___0c28bfcef998754d12e93153ea76187b.pdf50 Mechanical shocki

responsea_prim=-d*velocity_zp-displacement_z*w02 // Absolute responseaccel. during the shock

' Positive primary SRS of absolute accelerationssrcaj3rim_max=ABS(MAX(srca_prim_max,responsea_prim))

' Negative primary SRS of absolute accelerationssrca_prim_min=MIN(srca_prim_min,responsea_prim)

' Positive primary SRS of the relative displacementssrcd_prim_max=ABS(MAX(srcd_prim_max,responsed_prim))

' Negative primary SRS of the relative displacementssrcdjprim_min=MrN(srcd_prim_min,responsed_prim)

NEXT i%' Calculation of the residual SRS1 Initial conditions for the residual response = Conditions at the end of the shocksrca_res_max=responsea_prim // Positive residual SRS of absolute accelerationssrcajres_min=responseajprim //Negative residual SRS of absolute accelerationssrcd_res_max=responsedjprim // Positive residual SRS of the relativedisplacementssrcd_res_min=responsed_prim // Negative residual SRS of the relative

displacements' Calculation of the phase angle of the first peak of the residual relative

displacementc 1 =(d2 *displacement_z+velocity_zp)/wc2=displacement_zal=-w*c2-d2*cla2=w*cl-d2*c2pld=-alp2d=a2IFpld=0pd=PI/2*SGN(p2d)ELSEpd=ATN(p2d/pld)ENDIFIF pd>=0wtd=pdELSEwtd=PI+pdENDIF' Calculation of the phase angle of the first peak of residual absolute

accelerationbla=-w*a2-d2*alb2a=w*al-d2*a2pla=-d*bla-al*w02p2a=d*b2a+a2*w02IFpla=0pa=PI/2*SGN(p2a)

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00067___1b87f8efdaaca9f23f9edadd9807322c.pdfShock response spectra domains 51

ELSEpa=ATN(p2a/pla)ENDIFIFpa>=0wta^a

ELSEwta=PI+paENDIFFOR i%=l TO 2 // Calculation of the sup. and inf. values after the shock at the

frequency fO' Residual relative displacement

sd=SIN(wtd)cd=COS(wtd)ud=w*cd-d2*sdvd=-w*sd-d2*cded=EXP(-d2*wtd/w)displacementd_z=ed*(sd*c 1 +cd*c2)velocityd_zp=ed*(ud*c 1 +vd*c2)

' Residual absolute accelerationsa=SIN(wta)ca=COS(wta)ua=w*ca-d2*sava=-w*sa-d2*caea=EXP(-d2*wta/w)displacementa_z=ea*(sa*c 1 +ca*c2)velocitya_zp=ea*(ua*c 1 +va*c2)1 Residual SRSsrcd_res=-displacementd_z*w02 // SRS of the relative displacementssrca_res=-d*velocitya_zp-displacementa_z*w02 // SRS of absolute

accelerationssrcd_res_max=MAX(srcd_res_max,srcd_res) // Positive residual SRS of the

relative displacementssrcd_res_min=MIN(srcd_res_min,srcd_res)//Negative residual SRS of the

relative displacementssrca_res_max=MAX(srca_res_max,srca_res) // Positive residual SRS of the

absolute accelerationssrca_res_min=MIN(srca_res_min,srca_res) // Negative residual SRS of the

absolute accelerationswtd=wtd+PIwta=\vta+PI

NEXT i%srcdj)os=MAX(srcd_prim_max,srcd_res_max) // Positive SRS of the relative

displacementssrcd_neg=MrN(srcdjprim_min,srcd_res_min) // Negative SRS of the relativedisplacements

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00068___da91dec273bf4cef57882ad9a37fb572.pdf52 Mechanical shock

srcd_maximax=MAX(srcd_pos,ABS(srcd_neg)) // Maximax SRS of the relativedisplacementssrcajpos=MAX(srca_prim_max,srca_res_max) // Positive SRS of absoluteaccelerationssrca_neg=MIN(srca_prim_min,srca_res_min) //Negative SRS of absolute

accelerationssrca_maximax=MAX(srcajpos,ABS(srca_neg)) // Maximax SRS of absolute

accelerationsRETURN

2.8. Choice of the digitization frequency of the signal

The frequency of digitalization of the signal has an influence on the calculatedresponse spectrum. If this frequency is too small:

-The spectrum of a shock with zero velocity change can be false at lowfrequency, digitalization leading artificially to a difference between the positive andnegative areas under the shock pulse, i.e. to an apparent velocity change that is notzero and thus leading to an incorrect slope in this range. Correct restitution of thevelocity change (error of about 1% for example) can require, according to the shapeof the shock, up to 70 points per cycle.

- The spectrum can be erroneous at high frequencies. The error is here related tothe detection of the largest peak of the response, which occurs throughout shock(primary spectrum). Figure 2.23 shows the error made in the stringent case morewhen the points surrounding the peak are symmetrical with respect to the peak.

If we set

it can be shown that, in this case, the error made according to the sampling factorSF is equal to [SIN 81] [WIS 83]

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00069___f956692dea5f6955e2df94ff44d78e4e.pdfShock response spectra domains 53

Figure 2.23. Error made in measuring the Figure 2.24. Error made in measuring theamplitude of the peak amplitude of the peak plotted against

sampling factor

The sampling frequency must be higher than 16 times the maximum frequencyof the spectrum so that the error made at high frequency is lower than 2% (23 timesthe maximum frequency for an error lower than 1%).

The rule of thumb often used to specify a sampling factor equal 10 can lead to anerror of about 5%.

The method proposing a parabolic interpolation between the points to evaluatethe value of the maximum does not lead to better results.

2.9. Example of use of shock response spectra

Let us consider as an example the case of a package intended to limit to 100 m/s2acceleration on the transported equipment of mass m when the package itself issubjected to a half-sine shock of amplitude 300 m/s2 and of duration 6 ms. One inaddition imposes a maximum displacement of the equipment in the package (underthe effect of the shock) equal to e = 4 cm (to prevent that the equipment cominginto contact with the wall of the package).

It is supposed that the system made up by the mass m of the equipment and thesuspension is comparable to a one-degree-of-freedom system with a Q factor equalto Q = 5. We want to determine the stiffness k of the suspension to satisfy theserequirements when the mass m is equal to 50 kg.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00070___028fb9f9daa81c34619dce8b1baace0a.pdf54 Mechanical shock

Figure 2.25. Model of the package

Figures 2.26 and 2.27 show the response spectrum of the half-sine shock pulsebeing considered, plotted between 1 and 50 Hz for a damping of , = 0.10(= 1/2 Q). The curve of Figure 2.26 gives zsu on the ordinate (maximum relativedisplacement of the mass, calculated by dividing the ordinate of the spectrumODO zsup by co0). The spectrum of Figure 2.27 represents the usual curveG)0 zsup(f0). We could also have used a logarithmic four coordinate spectrum tohandle just one curve.

Figure 2.26. Limitation in displacement Figure 2.27. Limitation in acceleration

Figure 2.26 shows that to limit the displacement of the equipment to 4 cm, thenatural frequency of the system must be higher or equal to 4 Hz. The limitation ofacceleration on the equipment with 100 m/s2 also imposes f0 < 16 Hz (Figure 2.27).The range acceptable for the natural frequency is thus 4 Hz < f0 < 16 Hz.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00071___fe1e7b6b5564ce502953491a5739b719.pdfShock response spectra domains 55

2.10. Use of shock response spectra for the study of systems with severaldegrees of freedom

By definition, the response spectrum gives the largest value of the response of alinear single-degree-of-freedom system subjected to a shock. If the real structure iscomparable to such a system, the SRS can be used to evaluate this response directly.This approximation is often possible, with the displacement response being mainlydue to the first mode. In general, however, the structure comprises several modeswhich are simultaneously excited by the shock. The response of the structureconsists of the algebraic sum of the responses of each excited mode.

One can read on the SRS the maximum response of each one of these modes, butone does not have any information concerning the moment of occurrence of thesemaxima. The phase relationships between the various modes are not preserved andthe exact way in which the modes are combined cannot be known simply. Inaddition, the SRS is plotted for a given constant damping over all the frequencyrange, whereas this damping varies from one mode to another in the structure. Withrigour, it thus appears difficult to use a SRS to evaluate the response of a systempresenting more than one mode. But it happens that this is the only possible means.The problem is then to know how to combine these 'elementary' responses so as toobtain the total response and to determine, if need be, any suitable participationfactors dependent on the distribution of the masses of the structure, of the shapes ofthe modes etc.

Let us consider a non-linear system with n degrees of freedom; its response to ashock can be written as:

Knowing that

we deduce that

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00072___f6cdc8f0455c893a3775e4205751a0c3.pdf56 Mechanical shock

where

n = total number of modesan = modal participation factor for the mode nhn(t)= impulse response of mode nx(t) = excitation (shock)

(j) - modal vector of the systema = variable of integration

If one mode (m) is dominant, this relation is simplified according to

The value of the SRS to the mode m is equal to

The maximum of the response z(t) in this particular case is thus

When there are several modes, several proposals have been made to limit thevalue of the total response of the mass j of the one of the degrees of freedom startingfrom the values read on the SRS as follows.

A first method was proposed in 1934 per H. Benioff [BEN 34], consisting simplyof adding the values with the maxima of the responses of each mode, without regardto the phase.

A very conservative value was suggested by M.A. Biot [BIO 41] in 1941 for theprediction of the responses of buildings to earthquakes, equal to the sum of theabsolute values of the maximum modal responses:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00073___e937af6510f7c4aff89bf307eccc5cdb.pdfShock response spectra domains 57

The result was considered sufficiently precise for this application [RID 69]. As it isnot very probable that the values of the maximum responses take place all at thesame moment with the same sign, the real maximum response is lower than the sumof the absolute values. This method gives an upper limit of the response and thus hasa practical advantage: the errors are always on the side of safety. However, itsometimes leads to excessive safety factors [SHE 66].

In 1958, S. Rubin [RUB 58] made a study of undamped two-degrees-of-freedomsystems in order to compare the maximum responses to a half-sine shock calculatedby the method of modal superposition and the real maximum responses. This tsudyshowed that one could obtain an upper limit of the maximum response of thestructure by a summation of the maximum responses of each mode and that, in themajority of the practical problems, the distribution of the modal frequencies and theshape of the excitation are such that the possible error remains probably lower than10%. The errors are largest when the modal frequencies are in different areas of theSRS, for example, if a mode is in the impulse domain and the other in the staticdomain.

If the fundamental frequency of the structure is sufficiently high, Y.C. Fung andM.V. Barton [FUN 58] considered that a better approximation of the response isobtained by making the algebraic sum of the maximum responses of the individualmodes:

Clough proposed in 1955, in the study of earthquakes, either to add to theresponse of the first mode a fixed percentage of the responses of the other modes, orto increase the response of the first mode by a constant percentage.

The problem can be approached differently starting from an idea drawn fromprobability theory. Although the values of the response peaks of each individualmode taking place at different instants of time cannot, in a strict sense, being treatedin purely statistical terms, Rosenblueth suggested combining the responses of themodes by taking the square root of the sum of the squares to obtain an estimate ofthe most probable value [MER 62].

This criterion, used again in 1965 by F.E. Ostrem and M.L. Rumerman [OST 65]in 1955 [RID 69], gives values of the total response lower than the sum of theabsolute values and provides a more realistic evaluation of the average conditions.

This idea can be improved by considering the average of the sum of the absolutevalues and the square root of the sum of the squares (JEN 1958). One can alsochoose to define positive and negative limiting values starting from a system of

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00074___6b29749700605cabd0ba63d5b4050de0.pdf58 Mechanical shock

weighted averages. For example, the relative displacement response of the mass j isestimated by

where the terms are the absolute values of the maximum responses of eachmode and p is a weighting factor [MER 62].

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00075___d8bdcf730655788343fbe123591ec67c.pdfChapter 3

Characteristics of shock response spectra

3.1. Shock response spectra domains

Three domains can be schematically distinguished in shock spectra:- An impulse domain at low frequencies, in which the amplitude of the spectrum

(and thus of the response) is lower than the amplitude of the shock. The shock hereis of very short duration with respect to the natural period of the system. The systemreduces the effects of the shock. The characteristics of the spectra in this domain willbe detailed in Section 3.2.

- A static domain in the range of the high frequencies, where the positivespectrum tends towards the amplitude of the shock whatever the damping. Alloccurs here as if the excitation were a static acceleration (or a very slowly varyingacceleration), the natural period of the system being small compared with theduration of the shock. This does not apply to rectangular shocks or to the shockswith zero rise time. The real shocks having necessarily a rise time different fromzero, this restriction remains theoretical.

- An intermediate domain, in which there is dynamic amplification of the effectsof the shock, the natural period of the system being close to the duration of theshock. This amplification, more or less significant depending on the shape of theshock and the damping of the system, does not exceed 1.77 for shocks of traditionalsimple shape (half-sine, versed-sine, terminal peak saw tooth (TPS)). Much largervalues are reached in the case of oscillatory shocks, made up, for example, by a fewperiods of a sinusoid.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00076___07d24d462a217ad1a0a997f17659477e.pdfTTTTT

3.2. Characteristics of shock response spectra at low frequencies

3.2.1. General characteristics

In this impulse region

- The form of the shock has little influence on the amplitude of the spectrum. Wewill see below that only (for a given damping) the velocity change AV associatedwith the shock, equal to the algebraic surface under the curve x(t), is important.

-The positive and negative spectra are in general the residual spectra (it isnecessary sometimes that the frequency of spectrum is very small, and there can beexceptions for certain long shocks in particular). They are nearly symmetrical solong as damping is small.

2-The response (pseudo-acceleration co0 zsup or absolute acceleration ysup) is

lower than the amplitude of the excitation. There is an 'attenuation'. It is thus in thisimpulse region that it would be advisable to choose the natural frequency of anisolation system to the shock, from which we can deduce the stiffness envisaged ofthe insulating material:

(with m being the mass of the material to be protected).- The curvature of the spectrum always cancels at the origin (f0 = 0 Hz)

[FUN 57].

The characteristics of the SRS are often better demonstrated by a logarithmicchart or a four coordinate representation.

3.2.2. Shocks with velocity changed from zeroFor the shocks simple in shaoe , the residual spectrum is larger than the

primary spectrum at low frequencies.

For an arbitrary damping it can be shown that the impulse response is given by

where z(t) is maximum for t such as 0, i.e. fort such that

echanical shock

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00077___75f37e3b53407745f40ac65af346a442.pdfCharacteristics of shock response spectra 61

The tangent at the origin of the spectrum plotted for zero damping in linearscales has a slope proportional to the velocity change AVcorresponding to theshock pulse.

If damping is small, this relation is approximate.

yielding

The SRS is thus equal at low frequencies to

sin(arctan

i.e.

1 and the slope tends towards AV. The slope p of thespectrum at the origin is then equal to:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00078___891f9a8d23567f7ba2d168b15d21d01c.pdf62 Mechanical shock

Example

Half-sine shock pulse 100 m/s2, 10 ms, positive SRS (relative displacements).

The slope of the spectrum at the origin is equal to (Figure 3.1):

a value to be compared with the surface under the half-sine shock pulse:

Figure 3.1. Slope of the SRS at the origin

With the pseudovelocity plotted against to0, the spectrum is defined by

yielding

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00079___79a39137b39d1e91e96e09b7ce8b4038.pdfCharacteristics of shock response spectra 63

When O>Q tends towards zero, co0 zsup tends towards the constant valueAV cp(^). Figure 3.2 shows the variations of cp(^) versus .

Example

IPS shock pulse 100 m/s2, 10 ms),

Pseudovelocity calculated starting from the positive SRS (Figure 3.3).

Figure 3.3. Pseudovelocity SRS of a TPS shock pulse

It is seen that the pseudovelocity spectrum plotted for = 0 tends towards 0.5at low frequencies (area under TPS shock pulse).

Figure 3.2. Variations of the function 0 zsup) decreases at

low frequencies with a slope equal to 1, i.e., on a logarithmic scale, with a slope of6dB/octave( = 0).

The impulse absolute response of a linear one-degree-of-freedom system is givenby (Volume 1, relation 3.85):

where

If damping is zero,

The 'input' impulse can be represented in the form

as long as The response which results

The maximum of the displacement takes place during the residual response, for

yielding the shock response spectrum

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00081___653a43afd5875346de736534196373d4.pdfCharacteristics of shock response spectra 65

and

Figure 3.4. TPS shock pulse

A curve defined by a relation of the form y = a f is represented by a line ofslope n on a logarithmic grid:

The slope can be expressed by a number N of dB/octave according to

The undamped shock response spectrum plotted on a log-log grid thus has aslope at the origin equal to 1, i. e. 6 dB/octave.

Terminal peak saw tooth pulse 10 ms, 100 m/s2

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00082___5bb9643b189888609a30b76ff39b01b4.pdf66 Mechanical shock

Figure 3.5. Residual positive SRS (relative displacements) of a TPS shock pulse

The primary positive SRS o>0 zsu_ always has a slope equal to 2 (12 dB/octave)(example Figure 3.6) [SMA 85].

Figure 3.6. Primary positive SRS of a half-sine shock pulse

The relative displacement zsup tends towards a constant value z0 = xm equal tothe absolute displacement of the support during the application of the shock pulse(Figure 3.7). At low resonance frequencies, the equipment is not directly sensitive toaccelerations, but to displacement:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00083___eca57725c79b115acb4b00e77fcad241.pdfCharacteristics of shock response spectra 67

Figure 3.7. Behaviour of a resonator at very low resonance frequency

The system works as soft suspension which attenuates accelerations with largedisplacements [SNO 68].

This property can be demonstrated by considering the relative displacementresponse of a linear one-degree-of-freedom system given by Duhamel's equation(Volume 1, Chapter 2):

After integration by parts we obtain

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00084___80b2faf81940a77e2f77d0777682872d.pdf68 Mechanical shock

The mass m of an infinitely flexible oscillator and therefore of infinite naturalperiod (f0 = 0), does not move in the absolute reference axes. The spectrum of therelative displacement thus has as an asymptotic value the maximum value of theabsolute displacement of the base.

Example

Figure 3.8 shows the primary positive SRS zsup(f0) of a shock of half-sineshape 100 m/s2,10 ms plotted for = 0 between 0.01 and 100 Hz.

Figure 3.8. Primary positive SRS of a half-sine (relative displacements)

The maximum displacement xm under shock calculated from the expressionx(t) for the acceleration pulse is equal to:

The SRS tends towards this value when

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00085___e8a63ffc96575da6ef3bcb9d19ad2a2b.pdfCharacteristics of shock response spectra 69

For shocks of simple shape, the instant of time t at which the first peak of the

response takes place tends towards , tends towards zero [FUN 57].

The primary positive spectrum of pseudovelocities has, a slope of 6 dB/octave atthe low frequencies.

Figure 3.9. Primary positive SRS of a TPS pulse (four coordinate grid)

Example

3.2.3. Shocks for AV = 0 and AD * 0 at end of pulse

In this case, for , = 0 :

- The Fourier transform of the velocity for f = 0, V(0), is equal to

Since acceleration is the first derivative of velocity, the residual spectrum isequal to co0 AD for low values of co0. The undamped residual shock responsespectrum thus has a slope equal to 2 (i.e. 12 dB/octave) in this range.

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00086___3254a6a179d28f82fe94e362dd59376d.pdf70 Mechanical shock

Example

Shock consisted by one sinusoid period of amplitude 100 m/s2 and duration 10

Figure 3.10. Residual positive SRS of a 'sine 1 period' shockpulse

- The primary relative displacement (positive or negative, according to the formof the shock) zsup tends towards a constant value equal to xm, absolutedisplacement corresponding to the acceleration pulse x(t) defining the shock:

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00087___2064696cb3e65aa8c9efa31ab7487ded.pdfCharacteristics of shock response spectra 71

ExampleLet us consider a terminal peak saw tooth pulse of amplitude 100 m/s2 and

duration 10 ms with a symmetrical rectangular pre- and post-shock of amplitude10 m/s2. The shock has a maximum displacement given by (Chapter 7):

At the end of the shock, there is no change in velocity, but the residualdisplacement is equal to

Using the numerical data of this example, we obtain

xm = -4.428 mm

We find this value of xm on the primary negative spectrum of this shock(Figure 3.11). In addition,

Figure 3.11. Primary negative SRS (displacements) of a TPS pulsewith rectangular pre- and post-shocks

^residua] = -0-9576 10"4 mm

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00088___16e48f9c54adb125ebae87cb9df67878.pdf72 Mechanical shock

3.2.4. Shocks with AV = 0 and AD = 0 at end of pulse

For oscillatory type shocks, we note the existence of the following regions [SMA85] (Figure 3.12):

-just below the principal frequency of the shock, the spectrum has, on alogarithmic scale, a slope characterized by the primary response (about 3);

- when the frequency of spectrum decreases, its slope tends towards a smallervalue of 2;

- when the natural frequency decreases further, one observes a slope equal to 1(6 dB/octave) (residual spectrum). In a general way, all the shocks, whatever theirform, have a spectrum of slope of 1 on a logarithmic scale if the frequency is rathersmall.

The primary negative SRS o0 zsup has a slope of 12 dB/octave; the relativedisplacement zsup tends towards the absolute displacement xm associated with theshock movement x(t).

Figure 3.12. Shock response spectrum (relative displacements) of a ZERD pulse

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00089___e84a3c087669b9e07b4bfeaf9a9c9c13.pdfCharacteristics of shock response spectra 73

Examples

Figure 3.13. Primary negative SRS of a half-sine pulsewith half-sine pre- and post-shocks

Figure 3.14. Primary negative SRS (displacements) of a half-sine pulsewith half-sine pre- and post-shocks

Mechanical_Vibrations_and_Shocks_Mechanical_Shock_v._2_/190399604X/files/00090___7264495ad80e4c2451943d15c361abf5.pdf74 Mechanical shock

If the velocity change and the variation in displacement are zero the end of theshock, but if the integral of the displacement has a non zero value AD, theundamped residual spectrum is given by [SMA 85]

Figure 3.15. Residual positive SRS of a half-sine pulsewith half-sine pre-