Mechanical Vibrations and Shocks: Mechanical Shock v. 2 (Mechanical vibration & shock)

Mechanical Shock This page intentionally left blank Mechanical Vibration & Shock Mechanical Shock Volume II Chris...

Author: Christian Lalanne

432 downloads 2653 Views 9MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Mechanical Shock

This page intentionally left blank

Mechanical Vibration & Shock

Mechanical Shock Volume II

Christian Lalanne

HPS

HERMES PENTON SCIENCE

First published in 1999 by Hermes Science Publications, Paris First 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, as permitted 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 the publishers, or in the case of reprographic reproduction in accordance with the terms and licences issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: Hermes Penton Science 120 Pentonville Road London 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 in accordance 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 Lynn www.biddies,co.uk

Contents

Introduction List of symbols 1 Shock analysis 1.1. Definitions 1.1.1. Shock 1.1.2. Transient signal 1.1.3. Jerk 1.1.4. Bump 1.1.5. Simple (or perfect) shock 1.1.6. Half-sine shock 1.1.7. Terminal peak saw tooth shock (TPS) or final peak saw tooth shock (FPS) 1.1.8. Initial peak saw tooth shock (IPS) 1.1.9. Rectangular shock 1.1.10. Trapezoidal shock 1.1.11. Versed-sine (or haversine) shock 1.1.12. Decaying sinusoidal pulse 1.2. Analysis in the tune domain 1.3. Fourier transform 1.3.1. Definition 1.3.2. Reduced Fourier transform 1.3.3. Fourier transforms of simple shocks 1.3.3.1. Half-sine pulse 1.3.3.2. Versed-sine pulse 1.3.3.3. Terminal peak saw tooth pulse (TPS) 1.3.3.4. Initial peak saw tooth pulse (IPS) 1.3.3.5. Arbitrary triangular pulse 1.3.3.6. Rectangular pulse

xi xiii 1 1 1 2 2 2 3 3 3 3 3 3 3 4 4 4 4 6 7 7 8 9 10 12 14

vi

Mechanical shock

1.4

2

1.3.3.7. Trapezoidal pulse 1.3.4. Importance of the Fourier transform Practical calculations of the Fourier transform 1.4.1. General 1.4.2. Case: signal not yet digitized 1.4.3. Case: signal already digitized

Shock response spectra domains 2.1. Main principles 2.2. Response of a linear one-degree-of-freedom system 2.2.1. Shock defined by a force 2.2.2. Shock defined by an acceleration 2.2.3. Generalization 2.2.4. Response of a one-degree-of-freedom system to simple shocks 2.3. Definitions 2.4. Standardized response spectra 2.5. Difference between shock response spectrum (SRS) and extreme response spectrum (ERS) 2.6. Algorithms for calculation of the shock response spectrum 2.7. Subroutine for the calculation of the shock response spectrum 2.8. Choice of the digitization frequency of the signal 2.9. Example of use of shock response spectra 2.10. 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 3.1. Shock response spectra domains 3.2. Characteristics of shock response spectra at low frequencies 3.2.1. General characteristics 3.2.2. Shocks with velocity changed from zero 3.2.3. Shocks with AV = 0 and AD = 0 at end of pulse 3.2.4. Shocks with AV = 0 and AD = 0 at end of pulse 3.2.5. Notes on residual spectrum 3.3. Characteristics of shock response spectra at high frequencies 3.4. Damping influence 3.5. Choice of damping 3.6. Choice of frequency range 3.7. Charts 3.8. Relation of shock response spectrum to Fourier spectrum 3.8.1. Primary shock response spectrum and Fourier transform 3.8.2. Residual shock response spectrum and Fourier transform 3.8.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra 3.9. Characteristics of shocks of pyrotechnic origin

15 17 18 18 18 20 23 23 26 26 27 28 33 37 40 47 47 48 52 53

59 59 60 60 TT60 69 72 74 75 77 78 80 81 81 81 83 86 88

Contents

3.10. Care to be taken in the calculation of spectra 3.10.1. Influence of background noise of the measuring equipment.. 3.10.2. Influence of zero shift 4 Development of shock test specifications 4.1. General 4.2. Simplification of the measured signal 4.3. Use of shock response spectra 4.3.1. Synthesis of spectra 4.3.2. Nature of the specification 4.3.3. Choice of shape 4.3.4. Amplitude 4.3.5. Duration 4.3.6. Difficulties 4.4. Other methods 4.4.1. Use of a swept sine 4.4.2. Simulation of shock response spectra using a fast swept sine 4.4.3. Simulation by modulated random noise 4.4.4. Simulation of a shock using random vibration 4.4.5. Least favourable response technique 4.4.6. Restitution of a shock response spectrum by a series of modulated sine pulses 4.5. Interest behind simulation of shocks on a shaker using a shock spectrum 5. Kinematics of simple shocks 5.1. General 5.2. Half-sine pulse 5.2.1. Definition 5.2.2. Shock motion study 5.2.2.1. General expressions 5.2.2.2. Impulse mode 5.2.2.3. Impact mode 5.3. Versed-sine pulse 5.3.1. Definition 5.3.2. Shock motion study 5.4. Rectangular pulse 5.4.1. Definition 5.4.2. Shock motion study 5.5. Terminal peak saw tooth pulse 5.5.1. Definition 5.5.2. Shock motion study 5.6. Initial peak saw tooth pulse 5.6.1. Definition

vii

90 90 92 95 95 96 98 98 99 100 101 101 105 107 107 108 112 113 114 115 117 121 121 121 121 122 122 124 126 136 136 137 139 139 139 142 142 143 145 145

viii

Mechanical shock

5.6.2.

Shock motion study

6 Standard shock machines 6.1. Main types 6.2. Impact shock machines 6.3. High impact shock machines 6.3.1. Lightweight high impact shock machine 6.3.2. Medium weight high impact shock machine 6.4. Pneumatic machines 6.5. Specific test facilities 6.6. Programmers 6.6.1. Half-sine pulse 6.6.2. Terminal peak saw tooth shock pulse 6.6.3. Rectangular pulse - trapezoidal pulse 6.6.4. Universal shock programmer 6.6.4.1. Generating a half-sine shock pulse 6.6.4.2. Generating a terminal peak saw tooth shock pulse 6.6.4.3. Trapezoidal shock pulse 6.6.4.4. Limitations 7 Generation of shocks using shakers 7.1. Principle behind the generation of a simple shape signal versus time 7.2. Main advantages of the generation of shock using shakers 7.3. Limitations of electrodynamic shakers 7.3.1. Mechanical limitations 7.3.2. Electronic limitations 7.4. The use of the electrohydraulic shakers 7.5. Pre-and post-shocks 7.5.1. Requirements 7.5.2. Pre- or post-shock 7.5.3. Kinematics of the movement for symmetrical preand post-shock 7.5.3.1. Half-sine pulse 7.5.3.2. TPS pulse 7.5.3.3. Rectangular pulse 7.5.3.4. IPS pulse 7.5.4. Kinematics of the movement for a pre-shock or a post-shock alone 7.5.5. Abacuses 7.5.6. Influence of the shape of pre- and post-pulses 7.5.7. Optimized pre- and post-shocks 7.6. Incidence of pre- and post-shocks on the quality of simulation 7.6.1. General

145 149 149 151 160 160 162 163 164 165 165 173 180 181 182 182 183 183 189 189 190 191 191 193 193 193 193 195 198 198 206 207 208 208 212 213 216 220 220

Contents

7.6.2. 7.6.3.

Influence of the pre- and post-shocks on the time history response of a one-degree-of-freedom system Incidence on the shock response spectra

ix

220 223

8 Simulation of pyroshocks 8.1. Simulations using pyrotechnic facilities 8.2. Simulation using metal to metal impact 8.3. Simulation using electrodynamic shakers 8.4. Simulation using conventional shock machines

227 227 230 231 232

9 Control of a shaker using a shock response spectrum 9.1. Principle of control by a shock response spectrum 9.1.1. Problems 9.1.2. Method of parallel filterss 9.1.3. Current numerical methods 9.2. Decaying sinusoid 9.2.1. Definition 9.2.2. Response spectrum 9.2.3. Velocity and displacement 9.2.4. Constitution of the total signal 9.2.5. Methods of signal compensation 9.2.6. Iterations 9.3. D.L. Kern and C.D. Hayes' function 9.3.1. Definition 9.3.2. Velocity and displacement 9.4. ZERD function 9.4.1. Definition 9.4.1.1. D.K. Fisher and M.R. Posehn expression 9.4.1.2. D.O. Smallwood expression 9.4.2. Velocity and displacement 9.4.3. Comparison of the ZERD waveform with a standard decaying sinusoid 9.4.4. Reduced response spectra 9.4.4.1. Influence of the damping n of the signal 9.4.4.2. Influence of the Q factor 9.5. WAVSIN waveform 9.5.1. Definition 9.5.2. Velocity and displacement 9.5.3. Response of a one-degree-of-freedom system 9.5.3.1. Relative response displacement 9.5.3.2. Absolute response acceleration 9.5.4. Response spectrum 9.5.5. Time history synthesis from shock spectrum 9.6. SHOC waveform 9.6.1. Definition

235 235 235 236

237 239 239 239 242 243 244 250 251 251 252 253 253 253 254 255 257 257 257 258 259 259 260 262 263 265 265 266 267 267

x

Mechanical shock

9.6.2. 9.6.3.

Velocity and displacement Response spectrum 9.6.3.1. Influence of damping n of the signal 9.6.3.2. Influence of the Q factor on the spectrum 9.6.4. Time history synthesis from shock spectrum 9.7. Comparison of the WAVSIN, SHOC waveforms and decaying sinusoid 9.8. Use of a fast swept sine 9.9. Problems encountered during the synthesis of the waveforms 9.10. Criticism of control by a shock response spectrum 9.11. Possible improvements 9.11.1. IBS proposal 9.11.2. Specification of a complementary parameter 9.11.2.1. Rms duration of the shock 9.11.2.2. Rms value of the signal 9.11.2.3. Rms value in the frequency domain 9.11.2.4. Histogram of the peaks of the signal 9.11.2.5. Use of the fatigue damage spectrum 9.11.3. Remarks on the characteristics of response spectrum 9.12. Estimate of the feasibility of a shock specified by its SRS 9.12.1 C.D. Robbins and E.P. Vaughan's method 9.12.2. Evaluation of the necessary force, power and stroke

270 271 271 272 273 274 274 278 280 282 283 284 284 286 287 288 288 288 289 289 291

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

297 297 299

Mechanical shock tests: a brief historical background

301

Bibliography

303

Index

315

Synopsis of five volume series

319

Introduction

Transported or on-board equipment is very frequently subjected to mechanical shocks in the course of its useful lifetime (material handling, transportation, etc.). This kind of environment, although of extremely short duration (from a fraction of a millisecond 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 and their effect on buildings. This resulted in the notion of the shock response spectrum. Testing on equipment started during World War II. Methods continued to evolve with 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 to be directly conducted on the exciter employing a shock response spectrum. After a brief recapitulation of the shock shapes most widely used in tests and of the possibilities of Fourier analysis for studies taking account of the environment (Chapter 1), Chapter 2 presents the shock response spectrum with its numerous definitions and calculation methods. Chapter 3 describes all the properties of the spectrum, showing that important characteristics of the original signal can be drawn from it, such as its amplitude or the velocity change associated with the movement during the shock. The shock response spectrum is the ideal tool for drafting specifications. Chapter 4 details the process which makes it possible to transform a set of shocks recorded in the real environment into a specification of the same severity, and presents a few other methods that have been proposed in the literature. Knowledge of the kinematics of movement during a shock is essential to the understanding of the mechanism of shock machines and programmers. Chapter 5 gives the expressions for velocity and displacement according to time for classic shocks, depending on whether they occur in impact or impulse mode.

xii

Mechanical shock

Chapter 6 describes the principle of shock machines currently most widely used in laboratories and their associated programmers. To reduce costs by restricting the number of changes in test facilities, specifications expressed in the form of a simple shock (half-sine, rectangle, saw tooth with a final peak) can occasionally be tested using an electrodynamic exciter. Chapter 7 sets out the problems encountered, stressing the limitations of such means, together with the consequences of modification, that have to be made to the shock profile, on the quality of the simulation. Pyrotechnic devices or equipment (cords, valves, etc.) are very frequently used in satellite launchers due to the very high degree of accuracy that they provide in operating sequences. Shocks induced in structures by explosive charges are extremely severe, with very specific characteristics. Their simulation in the laboratory requires specific means, as described in Chapter 8. Determining a simple shape shock of the same severity as a set of shocks, on the basis of their response spectrum, is often a delicate operation. Thanks to progress in computerization and control facilities, this difficulty can occasionally be overcome by expressing the specification in the form of a response spectrum and by controlling the exciter directly from that spectrum. In practical terms, as the exciter can only be driven with a signal that is a function of time, the software of the control rack determines a time signal with the same spectrum as the specification displayed. Chapter 9 describes the principles of the composition of the equivalent shock, gives the shapes of the basic signals most often used, with their properties, and emphasizes the problems that can be encountered, both in the constitution of the signal and with respect to the quality of the simulation obtained. Containers must protect the equipment carried in them from various forms of disturbance related to handling and possible accidents. Tests designed to qualify or certify containers include shocks mat are sometimes difficult, not to say impossible, to produce, given the combined weight of the container and its content. One relatively 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 be applied, although less frequently, to certain vibration tests. At the end of this volume, the Appendix summarizes the laws of similarity adopted to define the models and to interpret the test results.

List of symbols

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

max a( t) Ac

Maximum value of a(t) Component of shock x( t) Amplitude of compensation signal A(0) Indicial admittance b parameter b of Basquin's relation N ob = C c Viscous damping constant C Basquin's law constant (N ob = C) d(t) Displacement associated with a(t) D Diameter of programmer D(f 0 ) Fatigue damage e Neper's number E Young's modulus or energy of a shock ERS Extreme response spectrum E(t) Function characteristic of swept sine f Frequency of excitation f0 Natural frequency

F(t) Prms Fm g h

External force applied to system Rms value of force Maximum value of F(t)

h(t) H HR

Acceleration due to gravity Interval (f/f 0 ) or thickness of the target Impulse response Drop height Height of rebound

H( )

Transfer function

i

v=f

IPS 3(Q) k K ^nns im t(t) t(t\

Initial peak saw tooth Imaginary part of X(O) Stiffness or coefficient of uncertainty Constant of proportionality of stress and deformation Rms value of #(t) Maximum of l(t) Generalized excitation (displacement) First derivative of ^(t)

xiv

Mechanical shock

"i(t) L L(Q) m n N p q0 q0 q(0) q( 0) q( 0) Q Q( p) Re Rm R(Q) 9?(Q) s S SRS s( ) t td tj tr tR T T0 TPS u(t) u(t) u(t) vf Vj VR

Second derivative of t(i) Length Fourier transform of ^(t) Mass Number of cycles undergone by test-bar or material Number of cycles to failure Laplace variable or percentage of amplitude of shock Value of q(0) for 0=0 Value of q(0) for 0=0 Reduced response First derivative of q( 0) Second derivative of q( 0) Q factor (quality factor) Laplace transform of q (0) Yield stress Ultimate tensile strength Fourier transform of the system response Real part of X(Q) Standard deviation Area Shock response spectrum Power spectral density Time Decay time to zero of shock Fall duration Rise time of shock Duration of rebound Vibration duration Natural period Terminal peak saw tooth Generalized response First derivative of u(t) Second derivative of u(t) Velocity at end of shock Impact velocity Velocity of rebound

v(t) v( ) xm x(t) x(t) x(t) \ms xm Xm X(Q) y(t)

y(t) y(t)

zm zs zsup z(t)

z(t)

Velocity x(t) or velocity associated with a(t) Fourier transform of v(t) Maximum value of x(t) Absolute displacement of the base of a one-degree-offreedom system Absolute velocity of the base of a one-degree-of-£reedom system Absolute acceleration of the base of a one-degree-offreedom system Rms value of x(t) Maximum value of x(t) Amplitude of Fourier transform X(Q) Fourier transform of x(t) Absolute response of displacement of mass of a one-degree-of-freedom system Absolute response velocity of the mass of a one-degreeof-freedom system Absolute response acceleration of mass of a one-degree-of-freedom system Maximum value of z(t) Maximum static relative displacement Largest value of z( t) Relative response displacement of mass of a one-degree-of-freedom system with respect to its base Relative response velocity

List of symbols

z(t)

Relative response acceleration

a 5(t) AV (ft (j)(Q) TI

Coefficient of restitution Dirac delta function Velocity change Dimensionless product f0 T Phase Damping factor of damped sinusoid Relative damping of compensation signal Reduced excitation Laplace transform of A,( ) 3.14159265... Reduced time (co0 t) Reduced decay time

TJC X.( ) A(p) K 6 0d

6m 60 p a crcr am T T! t2 i nns coc co0 Q

Reduced rise time Value of 0 for t = t Density Stress Crushing stress Maximum stress Shock duration Pre-shock duration Post-shock duration Rms duration of a shock Pulsation of compensation signal Natural pulsation (2 n f0) Pulsation of excitation (27Cf)

£

Damping factor

xv


Chapter 1

Shock analysis

1.1. Definitions 1.1.1. Shock Shock is defined as a vibratory excitation with duration between once and twice the 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 abruptly modified and creates a transient state in the system considered.

2

Mechanical shock

The modification is normally regarded as abrupt if it occurs in a time period which 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 tens of seconds), the mechanical shock, for example, in the air-braking phase on aircraft etc.

Figure 1.2. Example of transient signal 1.1.3. Jerk A jerk is defined as the derivative of acceleration with respect to time. This parameter 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 test characterized by a half-sine: 10 g, 16 ms, 3000 bumps (shocks) per axis, 3 bumps a second [GAM86].

Shock analysis

3

1.1.5. Simple (orperfect) shock Shock whose signal can be represented exactly in simple mathematical terms, for example half-sine, triangular or rectangular shock. 1.1.6. Half-sine shock Simple 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 instantly decreases 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 decreases to zero. 1.1.9. Rectangular shock Simple shock for which the acceleration-time curve increases instantaneously up to a given value, remains constant throughout the signal and decreases instantaneously to zero. 1.1.10. Trapezoidal shock Simple shock for which the acceleration-time curve grows linearly up to a given value, remains constant during a certain time after which it decreases linearly to zero.

4

Mechanical shock

1.1.11. Versed-sine (or haversine) shock Simple shock for which the acceleration-time curve has the form of one period of the curve representative of the function [1 - cos( )], with this period starting from zero 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 the amplitude of the first peak, the frequency and damping:

This form is interesting, for it represents the impulse response of a one-degreeof-freedom system to a shock. It is also used to constitute a signal of a specified shock 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, an acceleration x(t), but can be also a velocity v(t), a displacement x(t) or a force F(t). In the first case, which we will particularly consider in this volume, the velocity change 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 variable t absolutely integrable is defined by

Shock analysis

5

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

or

with

and

Thus is the Fourier spectrum of is the phase.

I the energy spectrum and

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

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

NOTES. I. For

dt

6

Mechanical shock

The ordinate at f = 0 of the Fourier transform (amplitude) of a shock defined by an acceleration is equal to the velocity change AV associated with the shock (area under 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 the exponent of exponential is sometimes also selected to be positive in the expression for X(Q) and negative in that for x(t).

1.3.2. Reduced Fourier transform The 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 the abscissa and on the ordinate, for the amplitude, the quantity A(f )/xm r . In the following paragraph, we draw the Fourier spectrum by considering simple shocks of unit duration (equivalent to the product ft) and of the amplitude unit. It is easy, with this representation, to recalibrate the scales to determine the Fourier spectrum of a shock of the same form, but of arbitrary duration and amplitude.

Shock analysis 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) Real part:

Imaginary part:

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

7

8

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:

Real part:

Imaginary part:

Shock analysis

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

9

10

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:

Shock analysis

Phase:

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

11

12

Mechanical shock

Imaginary part:

1.3.3.5. Arbitrary triangular pulse

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

Phase:

Real part:

Shock analysis

Figure 1.11. Real and imaginary parts of the Fourier transform of a triangular shock pulse

13

Figure 1.12. Real and imaginary parts of the Fourier transform of a triangular shock pulse

Imaginary part:

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

14

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 a rectangular shock pulse

Amplitude:

Phase:

Shock analysis Real part:

Imaginary part:

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

1.3.3.7. Trapezoidal pulse Amplitude:

15

16

Mechanical shock Phase:

Real part:

Imaginary part:

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

Shock analysis

17

Figure 1.18. Amplitude and phase of the Fourier transform of a trapezoidal 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 a structure is the product of the Fourier spectrum X(Q) of the input shock and the transfer 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 shock through a structure, the movement resulting at a certain point being then described by its Fourier spectrum. The response in the time domain can be also expressed from a convolution utilizing the 'input' shock signal according to the time and the impulse response 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 Fourier transforms 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 would make the Fourier spectrum method one frequently used in the study of shock, in particular for the writing of test specifications from experimental data. These mathematical advantages, however, are seldom used within this framework, because when one wants to compare two excitations, one runs up against the following problems:

18

Mechanical shock

- The need to compare two functions. The Fourier variable is a complex quantity which thus requires two parameters for its complete description: the real part and the imaginary part (according to the frequency) or the amplitude and the phase. These curves in general are very little smoothed and, except in obvious cases, it is difficult to decide on the relative severity of two shocks according to frequency when the spectrum overlap. In addition, the phase and the real and imaginary parts can take positive and negative values and are thus not very easy to use to establish a specification; - The signal obtained by inverse transformation has in general a complex form impossible 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 nor for the comparison of shocks. On the other hand, the one-to-one relation property and the input-response relation [1.40] make it a very interesting tool to control shaker shock whilst calculating the electric signal by applying these means to reproduce with the specimen a given acceleration profile, after taking into account the 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 Fast Fourier Transform (FFT) algorithm of Cooley-Tukey [COO 65] is generally used because of its speed (Volume 3). It must be noted that the result issuing from this algorithm must be multiplied by the duration of the analysed signal to obtain the Fourier transform. 1.4.2. Case: signal not yet digitized Let us consider an acceleration time history x(t) of duration T which one wishes to calculate the Fourier transform with nFT points (power of 2) until the frequency f max . According to the Shannon's theorem (Volume 3), it is enough that the signal is sampled with a frequency fsamp = 2 f max , i.e. that the temporal step is equal to

The frequency interval is equal to

Shock analysis

19

To be able to analyse the signal with a resolution equal to Af , it is necessary that its duration 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 selected conditions can be different to the duration T from the signal to analyse (for example in the case of a shock). It cannot be smaller than T (if not shock shape would be modified). Thus, if we set

the condition

T leads to

i.e. to

If the calculation data (n FT and f max ) lead to a too large value of Af , it will be necessary 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 the signal to analyse between T and T, with the temporal step At. The computing process is summarized in Table 1.1.

20

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) or one measured signal not yet digitized. - The number of points of the Fourier transform (npj-) and its maximum frequency (f max ). T

samp. -

fmax

Condition to avoid the aliasing phenomenon (Shannon's theorem) . If the measured signal can contain components at frequencies higher than f max , it must be filtered using a low-pass filter before digitalization. To take account of the slope of the filter beyond f max , it is preferable to choose fsamp = 2.6 fmax (Volume 3).

At = 2 fmax fmax At =

n

Temporal step of the signal to be digitized (time interval between two points of the signal). Frequency interval between two successive points of the Fourier transform.

FT

n = 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 1 The condition Af = -^L < - must be satisfied (i.e. T > T ): T "FT - if fmax is imposed, take npj (power of 2) > T f max . 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, the calculation conditions of the transform are fixed:

Shock analysis

(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 be resampled and if required zeros must be added using the principles in Table 1.1.

21


Chapter 2

Shock response spectra domains

2.1. Main principles A shock is an excitation of short duration which induces transitory dynamic stress 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 the characteristics of the system which undergoes it. The evaluation of this severity requires in addition the knowledge of the mechanism leading to a degradation of the structure. The two most common mechanisms are: - The exceeding of a value threshold of the stress in a mechanical part, leading to either a permanent deformation (acceptable or not) or a fracture, or at any rate, a functional failure. - If the shock is repeated many times (e.g. shock recorded on the landing gear of an aircraft, operation of an electromechanical contactor, etc), the fatigue damage accumulated in the structural elements can lead in the long term to fracture. We will deal with this aspect later on. The severity of a shock can be evaluated by calculating the stresses on a mathematical or finite element model of the structure and, for example, comparison with the ultimate stress of the material. This is the method used to dimension the structure. Generally, however, the problem is rather to evaluate the relative severity of several shocks (shocks measured in the real environment, measured shocks with respect to standards, establishment of a specification etc). This comparison would be

24

Mechanical shock

difficult to carry out if one used a fine model of the structure, and besides this is not always available, in particular at the stage of the development of the specification of dimensioning. One searches for a method of general nature, which leads to results which can be extrapolated to any structure. A solution was proposed by M.A. Biot [BIO 32] in 1932 in a thesis on the study of the earthquakes effects on the buildings; this study was then generalized to analysis 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

and to the

chosen damping £, while a stress GJ is induced in the elastic element. The analysis consists of seeking the largest stress cr mj observed at each frequency in each spring. A shock A is regarded as more severe than a shock B if it induces in each resonator a larger extreme stress. One then carries out an extrapolation, which is certainly criticizable, by supposing that, if shock A is more severe than shock B when it is applied to all the standard resonators, it is also more


25

severe with respect to an arbitrary real structure (which cannot be linear nor having a single degree of freedom). NOTE: A study was carried out in 1984 on a mechanical assembly composed of a circular plate on which one could place some masses and thus vary the number of degrees of freedom. The stresses generated by several shocks of the same spectra (in the frequency range including the principal resonance frequencies), but of different shapes [DEW 84], were measured and compared. One noted that for this assembly whatever the number of degrees of freedom, — two pulses of simple form (with no velocity change) having the same spectrum induce similar stresses, the variation not exceeding approximately 20 %. It is the same for two oscillatory shocks; — the relationship between the stresses measured for a simple shock and an oscillatory shock can reach 2. These results were supplemented by numerical simulation intended to evaluate the influence ofnon linearity. Even for very strong non-linearity, one did not note for the cases considered, an important difference between the stresses induced by two shocks of the same spectrum, but of different form. A complementary study was carried out by B.B. Petersen [PET 81] in order to compare the stresses directly deduced from a shock response spectrum with those generated on an electronics component by a half-sine shock envelop of a shock measured in the environment and by a shock of the same spectrum made up from WA VSIN signals (Chapter 9) added with various delays. The variation between the maximum responses measured at five points in the equipment and the stresses calculated starting from the shock response spectra does not exceed a factor of 3 in spite of the important theoretical differences between the model of the response spectrum and the real structure studied. For applications deviating from the assumptions of definition of the shock response spectrum (linearity, only one degree of freedom), it is desirable to observe a certain prudence if one wishes to estimate quantitatively the response of a system starting from the spectrum [BOR 89]. The response spectra are more often used to compare the severity of several shocks. It is known that the tension static diagram of many materials comprises a more or less 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 the deformation (Figure 2.2).

26

Mechanical shock

If mass-spring-damper system is supposed to be linear, it is then appropriate to compare two shocks by the maximum response stress am they induce or by the maximum 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 is also a function, via K, of the properties of the materials which constitute it. The curve giving the largest relative displacement zsup multiplied by oo0 according to the natural frequency f0, for a given £ damping, is the shock response spectrum (SRS). The first work defining these spectra was published in 1933 and 1934 [BIO 33] [BIO 34], then in 1941 and 1943 [BIO 41] [BIO 43]. The shock response spectrum, then named the shock spectrum, was presented there in the current 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 as tool 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 to the mass, the differential equation of the movement is written as:


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 in response 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-offreedom mechanical system, with y(t) the absolute acceleration response of the

28

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 both of the form

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


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 the Laplace transformation. We obtain, for zero initial conditions, an integral called Duhamel '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 an acceleration of the support, the response relative displacement is given by:

and the absolute acceleration of the mass by:

30

Mechanical shock

Application Let us consider a package intended to protect a material from mass m and comprising a suspension made up of two elastic elements of stiffness k and two dampers 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 height of h = 5 m, by supposing that there is no rebound of the package after the impact on the ground and that the external frame is not deformable (Figure 2.5). This system is equivalent to the model in Figure 2.6. We have (Volume 1, Chapter 3):


and

31

32

Mechanical shock

where

and

With the chosen numerical values, it becomes:


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 the static 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

33

34

Mechanical shock

Rectangular pulse

Initial peak saw tooth pulse


Terminal peak saw tooth pulse

Arbitrary triangular pulse

35

36

Mechanical shock

Trapezoidal pulse

Figure 2.7. Trapezoidal shock pulse

where


37

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

2.3. Definitions Response spectrum A curve representative of the variations of the largest response of a linear onedegree-of-freedom system subjected to a mechanical excitation, plotted against its natural frequency f0 =

for a given value of its damping ratio.

Absolute acceleration shock response spectrum In the most usual cases where the excitation is defined by an absolute acceleration of the support or by a force applied directly to the mass, the response of the system can be characterized by the absolute acceleration of the mass (which could be measured using an accelerometer fixed to this mass): the response spectrum is then called the absolute acceleration shock response spectrum. This spectrum can be useful when absolute acceleration is the parameter easiest to compare with a characteristic value (study of the effects of a shock on a man, comparison with the specification of an electronics component etc). Relative displacement shock spectrum In similar cases, we often calculate the relative displacement of the mass with respect to the base of the system, displacement which is proportional to the stress created in the spring (since the system is regarded as linear). In practice, one in general expresses in ordinates the quantity co0 zsup called the equivalent static acceleration. This product has the dimension of an acceleration, but does not represent the acceleration of the mass, except when damping is zero; this term is then strictly equal to the absolute acceleration of the mass. However, when damping is close to the current values observed in mechanics, and in particular when 2 4 = 0.05, one can assimilate as a first approximation co0 zsup to the absolute acceleration ysup of the mass m [LAL 75].

38

Mechanical shock

Very often in practice, it is the stress (and thus the relative displacement) which seems the most interesting parameter, the spectrum being primarily used to study the behaviour of a structure, to compare the severity of several shocks (the stress created is a good indicator), to write test specifications (it is also a good comparison between the real environment and the test environment) or to dimension a suspension (relative displacement and stress are then useful). The quantity co0 zsu is termed pseudo-acceleration. In the same way, one terms pseudo-velocity the product o)0 zsup. 2 The 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 defined in various ways according to how the largest response at a given frequency is characterized.

Primary positive shock response spectrum or initial positive shock response spectrum 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 spectrum can be positive or negative.

Positive (or maximum positive) shock response spectrum The largest positive response due to the shock, without reference to the duration of the shock. It is thus about the envelope of the positive primary and residual spectra.


39

Negative (or maximum negative) shock response spectrum The largest negative response due to the shock, without reference to the duration of the shock. It is in a similar way the envelope of the negative primary and residual spectra.

Example

Figure 2.8. Shock response spectra of a rectangular shock pulse

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 largest value of the response, i.e. to the amplitude of the spectrum at the frequency considered, and it is of little importance for the system whether this maximum zm takes place during or after the shock. The most interesting spectra are thus the positive and negative spectra, which are most frequently used in practice, with the maximax spectrum. The distinction between positive and negative spectra must be made each time the system, if disymmetrical, behaves differently, for example under different tension and compression. It is, however, useful to know these various definitions so as to be able to correctly interpret the curves published.

40

Mechanical shock

2.4. Standardized response spectra For a given shock, the spectra plotted for various values of the duration and the amplitude are homothetical. It is thus interesting, for simple shocks to have a standardized or reduced spectrum plotted in dimensionless co-ordinates, while plotting on the abscissa the product f0 t (instead of f0) or co0 t and on the ordinate the spectrum/shock pulse amplitude ratio co0 zm /xm , which, in practice, amounts to 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 real environment). 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 the spectrum of a particular shock of arbitrary amplitude xm and duration T (different from 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 =


41

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

Half-sine pulse

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

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

42

Mechanical shock

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

Versed-sine pulse

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


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

Terminal peak saw tooth pulse

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

43

44

Mechanical shock

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

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

Shock response spectra domains Initial peak saw tooth pulse

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

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

45

46

Mechanical shock

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

Rectangular pulse

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


47

Trapezoidal pulse

Figure 2.22. Standardized positive and negative relative displacement SRS of a trapezoidal pulse 2.5. Difference between shock response spectrum (SRS) and extreme response spectrum (ERS) A spectrum known as of extreme response spectrum (ERS) and comparable with the shock response spectrum (SRS) is often used for the study of vibrations (Volume 5). This spectrum gives the largest response of a linear single-degree-of-freedom system according to its natural frequency, for a given Q factor, when it is subjected to 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 place during or after the shock.

2.6. Algorithms for calculation of the shock response spectrum Various algorithms have been developed to solve the second order differential equation [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] (Section 2.7.). Although these calculations are a priori relatively simple, the round robins that were carried out ([BOZ 97] [CHA 94]) showed differences in the results, ascribable

48

Mechanical shock

sometimes to the algorithms themselves, but also to the use or programming errors of 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 singledegree-of-freedom system as well as the largest and smallest values after the shock (points of the positive and negative SRS, primary and residual, displacements relative and absolute accelerations). The parameters transmitted to the procedure are the number of points defining the shock, the natural pulsation of the system and its Q factor, the temporal step (presumably constant) of the signal and the array of the amplitudes of the signal. This procedure can be also used to calculate the response of a one-degree-of-freedom system to an arbitrary excitation, and in particular to a random vibration (where one is only interested in the primary response).


49

Procedure for the calculation of a point of the SRS at frequency f0 (GFABASIC) 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,w02dt LOCAL 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 calculations psi=l/2/Q_factor // Damping ratio w=wO*SQR(l-psiA2) // Damped natural pulsation d=2*psi*wO d2=d/2 wdt=w*dt e=EXP(-d2*dt) s=e*SIN(wdt) c=e*COS(wdt) u=w*c-d2*s v=-w*s-d2*c w02=wOA2 w02dt=w02*dt 1 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 shock velocity_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%))/w02dt b=(-xpp(i%-1 )-d*a)/w02 c2=displacement_z-b c 1 =(d2*c2-t-velocity_zp-a)/w displacement_z=s*cl+c*c2+a*dt+b velocity_zp=u * c1 + v* c2+a responsedjprim=-displacement_z*w02 // Relative displac. during shock x square of the pulsation

50

Mechanical shock i

responsea_prim=-d*velocity_zp-displacement_z*w02 // Absolute response accel. during the shock ' Positive primary SRS of absolute accelerations srcaj3rim_max=ABS(MAX(srca_prim_max,responsea_prim)) ' Negative primary SRS of absolute accelerations srca_prim_min=MIN(srca_prim_min,responsea_prim) ' Positive primary SRS of the relative displacements srcd_prim_max=ABS(MAX(srcd_prim_max,responsed_prim)) ' Negative primary SRS of the relative displacements srcdjprim_min=MrN(srcd_prim_min,responsed_prim) NEXT i% ' Calculation of the residual SRS 1 Initial conditions for the residual response = Conditions at the end of the shock srca_res_max=responsea_prim // Positive residual SRS of absolute accelerations srcajres_min=responseajprim //Negative residual SRS of absolute accelerations srcd_res_max=responsedjprim // Positive residual SRS of the relative displacements srcd_res_min=responsed_prim // Negative residual SRS of the relative displacements ' Calculation of the phase angle of the first peak of the residual relative displacement c 1 =(d2 *displacement_z+velocity_zp)/w c2=displacement_z al=-w*c2-d2*cl a2=w*cl-d2*c2 pld=-al p2d=a2 IFpld=0 pd=PI/2*SGN(p2d) ELSE pd=ATN(p2d/pld) ENDIF IF pd>=0 wtd=pd ELSE wtd=PI+pd ENDIF ' Calculation of the phase angle of the first peak of residual absolute acceleration bla=-w*a2-d2*al b2a=w*al-d2*a2 pla=-d*bla-al*w02 p2a=d*b2a+a2*w02 IFpla=0 pa=PI/2*SGN(p2a)

Shock response spectra domains ELSE pa=ATN(p2a/pla) ENDIF IFpa>=0 wta^a ELSE wta=PI+pa ENDIF FOR 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*sd vd=-w*sd-d2*cd ed=EXP(-d2*wtd/w) displacementd_z=ed*(sd*c 1 +cd*c2) velocityd_zp=ed*(ud*c 1 +vd*c2) ' Residual absolute acceleration sa=SIN(wta) ca=COS(wta) ua=w*ca-d2*sa va=-w*sa-d2*ca ea=EXP(-d2*wta/w) displacementa_z=ea*(sa*c 1 +ca*c2) velocitya_zp=ea*(ua*c 1 +va*c2) 1 Residual SRS srcd_res=-displacementd_z*w02 // SRS of the relative displacements srca_res=-d*velocitya_zp-displacementa_z*w02 // SRS of absolute accelerations srcd_res_max=MAX(srcd_res_max,srcd_res) // Positive residual SRS of the relative displacements srcd_res_min=MIN(srcd_res_min,srcd_res)//Negative residual SRS of the relative displacements srca_res_max=MAX(srca_res_max,srca_res) // Positive residual SRS of the absolute accelerations srca_res_min=MIN(srca_res_min,srca_res) // Negative residual SRS of the absolute accelerations wtd=wtd+PI wta=\vta+PI NEXT i% srcdj)os=MAX(srcd_prim_max,srcd_res_max) // Positive SRS of the relative displacements srcd_neg=MrN(srcdjprim_min,srcd_res_min) // Negative SRS of the relative displacements

51

52

Mechanical shock

srcd_maximax=MAX(srcd_pos,ABS(srcd_neg)) // Maximax SRS of the relative displacements srcajpos=MAX(srca_prim_max,srca_res_max) // Positive SRS of absolute accelerations srca_neg=MIN(srca_prim_min,srca_res_min) //Negative SRS of absolute accelerations srca_maximax=MAX(srcajpos,ABS(srca_neg)) // Maximax SRS of absolute accelerations RETURN

2.8. Choice of the digitization frequency of the signal The frequency of digitalization of the signal has an influence on the calculated response spectrum. If this frequency is too small: -The spectrum of a shock with zero velocity change can be false at low frequency, digitalization leading artificially to a difference between the positive and negative areas under the shock pulse, i.e. to an apparent velocity change that is not zero and thus leading to an incorrect slope in this range. Correct restitution of the velocity change (error of about 1% for example) can require, according to the shape of the shock, up to 70 points per cycle. - The spectrum can be erroneous at high frequencies. The error is here related to the 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 more when 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 factor SF is equal to [SIN 81] [WIS 83]


Figure 2.23. Error made in measuring the amplitude of the peak

53

Figure 2.24. Error made in measuring the amplitude of the peak plotted against sampling factor

The sampling frequency must be higher than 16 times the maximum frequency of the spectrum so that the error made at high frequency is lower than 2% (23 times the maximum frequency for an error lower than 1%). The rule of thumb often used to specify a sampling factor equal 10 can lead to an error of about 5%. The method proposing a parabolic interpolation between the points to evaluate the 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/s2 acceleration on the transported equipment of mass m when the package itself is subjected to a half-sine shock of amplitude 300 m/s2 and of duration 6 ms. One in addition imposes a maximum displacement of the equipment in the package (under the effect of the shock) equal to e = 4 cm (to prevent that the equipment coming into contact with the wall of the package). It is supposed that the system made up by the mass m of the equipment and the suspension is comparable to a one-degree-of-freedom system with a Q factor equal to Q = 5. We want to determine the stiffness k of the suspension to satisfy these requirements when the mass m is equal to 50 kg.

54

Mechanical shock

Figure 2.25. Model of the package

Figures 2.26 and 2.27 show the response spectrum of the half-sine shock pulse being 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 relative displacement of the mass, calculated by dividing the ordinate of the spectrum ODO zsup by co0). The spectrum of Figure 2.27 represents the usual curve G)0 z sup (f 0 ). We could also have used a logarithmic four coordinate spectrum to handle 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, the natural frequency of the system must be higher or equal to 4 Hz. The limitation of acceleration 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.


55

Knowing that

we deduce that

2.10. Use of shock response spectra for the study of systems with several degrees of freedom By definition, the response spectrum gives the largest value of the response of a linear single-degree-of-freedom system subjected to a shock. If the real structure is comparable to such a system, the SRS can be used to evaluate this response directly. This approximation is often possible, with the displacement response being mainly due to the first mode. In general, however, the structure comprises several modes which are simultaneously excited by the shock. The response of the structure consists 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, but one does not have any information concerning the moment of occurrence of these maxima. The phase relationships between the various modes are not preserved and the exact way in which the modes are combined cannot be known simply. In addition, the SRS is plotted for a given constant damping over all the frequency range, whereas this damping varies from one mode to another in the structure. With rigour, it thus appears difficult to use a SRS to evaluate the response of a system presenting 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 to obtain the total response and to determine, if need be, any suitable participation factors dependent on the distribution of the masses of the structure, of the shapes of the modes etc. Let us consider a non-linear system with n degrees of freedom; its response to a shock can be written as:

56

Mechanical shock

where n = total number of modes an = modal participation factor for the mode n h n (t)= impulse response of mode n x(t) = excitation (shock) (j)

- modal vector of the system

a = 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 the value of the total response of the mass j of the one of the degrees of freedom starting from the values read on the SRS as follows. A first method was proposed in 1934 per H. Benioff [BEN 34], consisting simply of adding the values with the maxima of the responses of each mode, without regard to the phase. A very conservative value was suggested by M.A. Biot [BIO 41] in 1941 for the prediction of the responses of buildings to earthquakes, equal to the sum of the absolute values of the maximum modal responses:


57

The result was considered sufficiently precise for this application [RID 69]. As it is not very probable that the values of the maximum responses take place all at the same moment with the same sign, the real maximum response is lower than the sum of the absolute values. This method gives an upper limit of the response and thus has a practical advantage: the errors are always on the side of safety. However, it sometimes leads to excessive safety factors [SHE 66]. In 1958, S. Rubin [RUB 58] made a study of undamped two-degrees-of-freedom systems in order to compare the maximum responses to a half-sine shock calculated by the method of modal superposition and the real maximum responses. This tsudy showed that one could obtain an upper limit of the maximum response of the structure by a summation of the maximum responses of each mode and that, in the majority of the practical problems, the distribution of the modal frequencies and the shape of the excitation are such that the possible error remains probably lower than 10%. The errors are largest when the modal frequencies are in different areas of the SRS, for example, if a mode is in the impulse domain and the other in the static domain. If the fundamental frequency of the structure is sufficiently high, Y.C. Fung and M.V. Barton [FUN 58] considered that a better approximation of the response is obtained by making the algebraic sum of the maximum responses of the individual modes:

Clough proposed in 1955, in the study of earthquakes, either to add to the response of the first mode a fixed percentage of the responses of the other modes, or to increase the response of the first mode by a constant percentage. The problem can be approached differently starting from an idea drawn from probability theory. Although the values of the response peaks of each individual mode taking place at different instants of time cannot, in a strict sense, being treated in purely statistical terms, Rosenblueth suggested combining the responses of the modes by taking the square root of the sum of the squares to obtain an estimate of the 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 the absolute 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 absolute values and the square root of the sum of the squares (JEN 1958). One can also choose to define positive and negative limiting values starting from a system of

58

Mechanical shock

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

where the terms

are the absolute values of the maximum responses of each

mode and p is a weighting factor [MER 62].

Chapter 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 here is of very short duration with respect to the natural period of the system. The system reduces the effects of the shock. The characteristics of the spectra in this domain will be detailed in Section 3.2. - A static domain in the range of the high frequencies, where the positive spectrum tends towards the amplitude of the shock whatever the damping. All occurs here as if the excitation were a static acceleration (or a very slowly varying acceleration), the natural period of the system being small compared with the duration of the shock. This does not apply to rectangular shocks or to the shocks with zero rise time. The real shocks having necessarily a rise time different from zero, this restriction remains theoretical. - An intermediate domain, in which there is dynamic amplification of the effects of the shock, the natural period of the system being close to the duration of the shock. This amplification, more or less significant depending on the shape of the shock and the damping of the system, does not exceed 1.77 for shocks of traditional simple shape (half-sine, versed-sine, terminal peak saw tooth (TPS)). Much larger values are reached in the case of oscillatory shocks, made up, for example, by a few periods of a sinusoid.

TTTTT echanical shock

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. We will see below that only (for a given damping) the velocity change AV associated with 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 is necessary sometimes that the frequency of spectrum is very small, and there can be exceptions for certain long shocks in particular). They are nearly symmetrical so long as damping is small. 2 -The response (pseudo-acceleration co0 zsup or absolute acceleration y sup ) is lower than the amplitude of the excitation. There is an 'attenuation'. It is thus in this impulse region that it would be advisable to choose the natural frequency of an isolation system to the shock, from which we can deduce the stiffness envisaged of the 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 logarithmic chart or a four coordinate representation.

3.2.2. Shocks with velocity changed from zero For the shocks simple in shaoe primary spectrum at low frequencies.

, the residual spectrum is larger than the

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


61

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 the spectrum at the origin is then equal to:

The tangent at the origin of the spectrum plotted for zero damping in linear scales has a slope proportional to the velocity change AVcorresponding to the shock pulse. If damping is small, this relation is approximate.

62

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):

yielding

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


63

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

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 of 6dB/octave(£ = 0).

The impulse absolute response of a linear one-degree-of-freedom system is given by (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


65

and

A curve defined by a relation of the form y = a f slope n on a logarithmic grid:

is represented by a line of

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 a slope at the origin equal to 1, i. e. 6 dB/octave.

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

Figure 3.4. TPS shock pulse

66

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 to the 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 to accelerations, but to displacement:


67

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

The system works as soft suspension which attenuates accelerations with large displacements [SNO 68]. This property can be demonstrated by considering the relative displacement response of a linear one-degree-of-freedom system given by Duhamel's equation (Volume 1, Chapter 2):

After integration by parts we obtain

68

Mechanical shock

The mass m of an infinitely flexible oscillator and therefore of infinite natural period (f0 = 0), does not move in the absolute reference axes. The spectrum of the relative displacement thus has as an asymptotic value the maximum value of the absolute displacement of the base. Example Figure 3.8 shows the primary positive SRS z sup (f 0 ) of a shock of half-sine shape 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 expression x(t) for the acceleration pulse is equal to:

The SRS tends towards this value when


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 at the low frequencies. Example

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

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 is equal to co0 AD for low values of co0. The undamped residual shock response spectrum thus has a slope equal to 2 (i.e. 12 dB/octave) in this range.

70

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 form of the shock) zsup tends towards a constant value equal to xm, absolute displacement corresponding to the acceleration pulse x(t) defining the shock:


71

Example Let 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 amplitude 10 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 residual displacement 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, ^residua] = -0-9576 10"4 mm

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

72

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 [SMA 85] (Figure 3.12): -just below the principal frequency of the shock, the spectrum has, on a logarithmic scale, a slope characterized by the primary response (about 3); - when the frequency of spectrum decreases, its slope tends towards a smaller value 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 their form, have a spectrum of slope of 1 on a logarithmic scale if the frequency is rather small.

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

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


Examples

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

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

73

74

Mechanical shock

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

for small values of co0 (slope of 18 dB/octave). Example

Figure 3.15. Residual positive SRS of a half-sine pulse with half-sine pre- and post-shocks

3.2.5. Notes on residual spectrum Spectrum of absolute displacements When co0 is sufficiently small, the residual spectrum of an excitation x(t) is identical to the corresponding displacement spectrum in one of the following ways [FUN 61]:

a) b) c) However, contrary to the case (c) above, if


75

but if there exists more than one value t_ of time in the interval 0 x(t) dt = 0, then the residual spectrum is equal to

which

while

the spectrum of the displacements is equal to the largest values of [FUN 61].

If AV and AD are zero at the end of the shock, the response spectrum of the absolute displacement is equal to 2 x(t) where x(t) is the residual displacement of the base. If x(t) = 0, the spectrum is equal to the largest of the two quantities where t = tp is the time when the integral is cancelled. The absolute displacement of response is not limited if the input shock is such that AV * 0. Relative displacement When (00 is sufficiently small, the residual spectrum and the spectrum of the displacements are identical in the following cases: a) if X(T) * 0 at the end of the shock, b) if X(T) = 0, but x(t) is maximum with t = T . If not, the residual spectrum is equal to X(T), while the spectrum of the displacements is equal to the largest absolute value of x(t).

3.3. Characteristics of shock response spectra at high frequencies The response can be written, according to the relation [2.16]

while setting

76

Mechanical shock

We want to show that

Let us set

Integrating by parts:

w(t) tends towards

such that

when co0 tends towards infinity. Let us show that

. constant.

If the function x(t) is continuous, the quantity


77

tends towards zero as u tends towards zero. There thus exists r\ e [o, t] such that and we have

The function x(t) is continuous and therefore limited at

when, for

, and we have

Thus for

At high frequencies, o>0 z(t) thus tends towards x(t) and, consequently, the shock response spectrum tends towards xm, a maximum x(t).

3.4. Damping influence Damping has little influence in the static region. Whatever its value, the spectrum tends towards the amplitude of the signal depending on time. This property is checked for all the shapes of shocks, except for the rectangular theoretical shock

78

Mechanical shock

which, according to damping, tends towards a value ranging between one and twice the amplitude of the shock. In the impulse domain and especially in the intermediate domain, the spectrum has a lower amplitude when the clamping is greater. This phenomenon is not great for shocks with velocity change and for normal damping (0.01 to 0.1 approximately). It is marked more for oscillatory type shocks (decaying sine for example) at frequencies close to the frequency of the signal. The peak of the spectrum here has an amplitude which is a function of the number of alternations of the signal and of the selected damping. 3.5. Choice of damping The choice of damping should be carried out according to the structure subjected to the shock under consideration. When this is not known, or studies are being carried out with a view to comparison with other already calculated spectra, the outcome is that one plots the shock response spectra with a relative damping equal to 0.05 (i.e. Q = 10). It is about an average value for the majority of structures. Unless otherwise specified, as noted on the curve, it is the value chosen conventionally. With the spectra varying relatively little with damping (with the reservations of the preceding paragraph), this choice is often not very important. To limit possible errors, the selected value should, however, be systematically noted on the diagram.

NOTE. In practice, the most frequent range of variation of the Q factor of the structures lies between approximately 5 and 50. There is no exact relation which makes it possible to obtain a shock response spectrum of given Q factor starting from a spectrum of the same signal calculated with another Q factor. M.B. Grath and W.F. Bangs [GRA 72] proposed an empirical method deduced from an analysis of spectra of pyrotechnic shocks to carry out this transformation. It is based on curves giving, depending on Q, a correction factor, amplitude ratio of the spectrum for Q factor with the value of this spectrum for Q = 10 (Figure 3.16). The first curve relates to the peak of the spectrum, the second the standard point (non-peak data). The comparison of these two curves confirms the greatest sensitivity of the peak to the choice ofQ factor. These results are compatible with those of a similar study carried out by W.P. Rader and W.F. Bangs [RAD 70],which did not however distinguish between the peaks and the other values.


79

Figure 3.16. SRS correction factor of the SRS versus Q factor

To take account of the dispersion of the results observed during the establishment of these curves and to ensure reliability, the authors calculated the standard deviation associated -with the correction factor (in a particular case, a point on the spectrum plotted for Q = 20; the distribution of the correction factor is not normal, but near to a Beta or type I Pearson law). Table 3.1. Standard deviation of the correction factor

Q 5 10 20 30 40 50

Standard points 0.085 0 0.10 0.15 0.19 0.21

Peaks 0.10 0 0.15 0.24 0.30 0.34

The results show that the average is conservative 65% of time,and the average plus one standard deviation 93%. They also indicate that modifying the amplitude of the spectrum to take account of the value of Q factor is not sufficient for fatigue analysis. The correction factor being determined, they proposed to calculate the number of equivalent cycles in this transformation using the relation developed by J.D. Crum and R.L. Grant [CRU 70] (cf. Section 4.4.2.) giving the expression for the

80

Mechanical shock

response (OQ z(t) depending on the time during its establishment under a sine wave excitation as:

(where N = number of cycles carried out at time t).

Figure 3.17. SRS correction factor versus Q factor This relation, standardized by dividing it by the amount obtained for the particular case where Q = 10, is used to plot the curves of Figure 3.17 which make it possible to readN, for a correction factor and given Q. They are not reliable for Q < 10, the relation [3.23] being correct only for low damping.

3.6. Choice of frequency range It is customary to choose as the frequency range: - either the interval in which the resonance frequencies of the structure studied are likely to be found; -or the range including the important frequencies contained in the shock (in particular in the case of pyrotechnic shocks).


81

3.7. Charts There are two spectral charts: - representation (x, y), the showing value of the spectrum versus the frequency (linear or logarithmic scales); -the four coordinate nomographic representation (four coordinate spectrum). One notes here on the abscissae the frequency

on the ordinates the

pseudovelocity co0 zm and, at two axes at 45° to the two first, the maximum relative displacement zm and the pseudo-acceleration co0 zm. This representation is interesting for it makes it possible to directly read the amplitude of the shock at the high frequencies and, at low frequencies, the velocity change associated with the shock (or if AV = 0 the displacement).

Figure 3.18. Four coordinate diagram

3.8. Relation of shock response spectrum to Fourier spectrum 3.8.1. Primary shock response spectrum and Fourier transform The response u(t) of a linear undamped one-degree-of-freedom system to a generalized excitation ^(t) is written [LAL 75] (Volume 1, Chapter 2):

82

Mechanical shock

We suppose here that t is lower than T.

which expression is of the form

with

where C and S are functions of time t. u(t) can be still written:

with

The function

is at a maximum when its derivative is zero

This yields the maximum absolute value of u(t)

where the index P indicates that it is about the primary spectrum. However, where the Fourier transform of £(t), calculated as if the shock were non-zero only between times 0 and t with co0 the pulsation is written as

and has as an amplitude under the following conditions:


Comparison of the expressions of

83

shows that

In a system of dimensionless coordinates, with

The primary spectrum of shock is thus identical to the amplitude of the reduced Fourier spectrum, calculated for t < T [CA V 64]. The phase L

of the Fourier spectrum is such that

However, the phase <j>p is given by [3.28]

where k is a positive integer or zero. For an undamped system, the primary positive shock spectrum and the Fourier spectrum between 0 and t are thus related in phase and amplitude. 3.8.2. Residual shock response spectrum and Fourier transform The response can be written, whatever value of t

84

Mechanical shock

which is of the form B1 sin co0t + B2 cos 0 can take an arbitrary value since the simple mechanical system is not yet chosen, equal in particular to Q. We thus obtain the relation

The phase is given by

71

71

Only the values of <j>L € (-—,+ —) will be considered. Comparison of <j)R and (|>L 2 2 show that

For an undamped system, the Fourier spectrum and the residual positive shock spectrum are related in amplitude and phase [CA V 64]. NOTE: If the excitation is an acceleration,

and if, in addition,

the Fourier transform of x (t), we have [GER 66], [NAS 65]:

86

Mechanical shock

yielding

with VR (oo) being the pseudovelocity spectrum. The dimension of |L(p)| is that of the variable of excitation ^t) multiplied by time. The quantity Q |L(Q)| is thus that of l ( i ) . If the expression of l{t) is standardized by dividing it by its maximum value lm , it becomes, in dimensionless form

With this representation, the Fourier spectrum of the signal identical to its residual shock spectrum

for zero damping [SUT 68].

3.8.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra Let us consider the Fourier spectra (amplitude) of two shocks, one being an isosceles triangle shape and the other TPS (Figure 3.19), like their positive shock response spectra, for zero damping (Figure 3.20).


87

Figure 3.19. Comparison of the Fourier transform amplitudes of a TPS pulse and an isosceles triangle pulse

Figure 3.20. Comparison of the positive SRS of a TPS pulse and an isosceles triangle pulse

It is noted that the Fourier spectra and shock response spectra of the two impulses have the same relative position as long as the frequency remains lower than f =1.25 Hz, the range for which the shock response spectrum is none other than the residual spectrum, directly related to the Fourier spectrum. On the contrary, for f > 1.25 Hz, the TPS pulse has a larger Fourier spectrum, whereas the SRS (primary spectrum) of the isosceles triangle pulse is always in the form of the envelope.

88

Mechanical shock

The Fourier spectrum thus gives only one partial image of the severity of a shock by considering only its effects after the end of the shock (and without taking damping into account).

3.9. Characteristics of shocks of pyrotechnic origin The aerospace industry uses many pyrotechnic devices such as explosive bolts, squib valves, jet cord, pin pushers etc. During their operation these devices generate shocks which are characterized by very strong acceleration levels at very high frequencies which can be sometimes dangerous for the structures, but especially for the electric and electronic components involved. These shocks were neglected until about 1960 approximately but it was estimated that, in spite of their high amplitude, they were of much too short duration to damage the materials. Some incidents concerning missiles called into question this postulate. An investigation by C. Moening [MOE 86] showed that the failures observed on the American launchers between 1960 and 1986 can be categorized as follows: - due to vibrations: 3; - due to pyroshocks: 63. One could be tempted to explain this distribution by the greater severity of the latter environment. The Moening study shows that it was not the reason, the causes being: - the partial difficulty in evaluating these shocks a priori; - more especially the lack of consideration of these excitations during design, and the absence of rigorous test specifications.

Figure 3.21. Example ofapyroshock


89

Such shocks have the following general characteristics: - the levels of acceleration are very important; the shock amplitude is not simply related to the quantity of explosive used [HUG 83b]. Reducing the load does not reduce the consequent shock. The quantity of metal cut by a jet cord is, for example, a more significant factor; - the signals assume an oscillatory shape; - in the near-field, close to the source (material within about 15 cm of point of detonation of the device, or about 7 cm for less intense pyrotechnic devices), the effects of the shocks are primarily related to the propagation of a stress wave in the material; - the shock is then propagated whilst attenuating in the structure. The mid-field (material within about 15 cm and 60 cm for intense pyrotechnic devices, between 3 cm and 15 cm for less intense devices) from, which the effects of this wave are not yet negligible and combine with a damped oscillatory response of the structure at its frequencies of resonance, is to be distinguished from the far-field, where only this last effect persists; - the shocks have very close components according to three axes; their positive and negative response spectra are curves that are coarsely symmetrical with respect to the axis of the frequencies. They begin at zero frequency with a very small slope at the origin, grow with the frequency until a maximum located at some kHz, even a few tens of kHz, is reached and then tend according to the rule towards the amplitude of the temporal signal. Due to their contents at high frequencies, such shocks can damage electric or electronic components; - the a priori estimate of the shock levels is neither easy nor precise. These characteristics make them difficult to measure, requiring sensors that are able to accept amplitudes of 100,000 g, frequencies being able to exceed 100 kHz, with important transverse components. They are also difficult to simulate. The dispersions observed in the response spectra of shocks measured under comparable conditions are often important (3 dB with more than 8 dB compared to the average value, according to the authors [SMI 84] [SMI 86]), The reasons for this dispersion are in general related to inadequate instrumentation and the conditions of measurement [SMI 86]: - fixing the sensors on the structure using insulated studs or wedge which act like mechanical filters; - zero shift, due to the fact that high accelerations make the crystal of the accelerometer work in a temporarily non-linear field. This shift can affect the calculation of the shock response spectrum (cf. Section 3.10.2.); - saturation of the amplifiers;

90

Mechanical shock

- resonance of the sensors. With correct instrumentation, the results of measurements carried out under the same conditions are actually very close. The spectrum does not vary with the tolerances of manufacture and the assembly tolerances.

3.10. Care to be taken in the calculation of spectra 3.10.1. Influence of background noise of the measuring equipment The measuring equipment is gauged according to the foreseeable amplitude of the shock to be measured. When the shock characteristics are unknown, the rule is to use a large effective range in order not to saturate the conditioning module. Even if the signal to noise ratio is acceptable, the incidence of the background noise is not always negligible and can lead to errors of the calculated spectra and the specifications which are extracted from it. Its principal effect is to increase the spectra artificially (positive and negative), increasing with the frequency and Q factor. Example

Figure 3.22. TPSpulse with noise (rms value equal to one-tenth amplitude of the shock)

Figure 3.23 shows the positive and negative spectra of a TPS shock (100 m/s2, 25 ms) plotted in the absence of noise for an Q factor successively equal to 10 and 50, as well as the spectra (calculated in the same conditions) of a shock (Figure 3.22) composed of mis TPS pulse to which is added a random noise of rms value 10 m/s2 (one tenth of the shock amplitude).


91

Figure 3.23. Positive and negative SRS of the TPS pulse and with noise

Due to its random nature, it is practically impossible to remove the noise of the measured signal to extract the shock alone from it. Techniques, however have been developed to try to correct the signal by cutting off the Fourier transform of the noise from that of the total signal (subtraction of the modules, conservation of the phase of the total signal) [CAI 94].

92

Mechanical shock

3.10.2. Influence of zero shift One very often observes a continuous component superimposed on the shock signal on the recordings, the most frequent origin being the presence of a transverse high level component which disturbs the operation of the sensor. If this component is not removed from the signal before calculation of the spectra, it can it also lead to considerable errors [BAG 89] [BEL 88]. When this continuous component has constant amplitude, the signal treated is in fact a rectangle modulated by the true signal. It is not thus surprising to find on the spectrum of this composite signal the characteristics, more or less marked, of the spectra of a rectangular shock. The effect is particularly important for oscillatory type shocks (with zero or very small velocity change) such as, for example, shocks of pyrotechnic origin. In this last case, the direct component has as a consequence a modification of the spectrum at low frequencies which results in [LAL 92a]: - the disappearance of the quasi-symmetry of the positive and negative spectra characteristic of this type of shocks; - appearance of more or less clear lobes in the negative spectrum, similar to those of a pure rectangular shock. Example

Figure 3.24. 24. pyrotechnic shock with zero shift


93

The example treated is that of a pyrotechnic shock on which one artificially added a continuous component (Figure 3.24). Figure 3.25 shows the variation generated at low frequencies for a zero shift of about 5%. The influence of the amplitude of the shift on the shape of the spectrum (presence of lobes) is shown in Figure 3.26.

Figure 3.25. Positive and negative SRS of the centered and non-centered shocks

94

Mechanical shock

Figure 3.26. Zero shift influence on positive and negative SRS

Under certain conditions, one can try to center a signal presenting a zero shift constant or variable according to time, by addition of a signal of the same shape as this shift and of opposite sign [SMI 85]. This correction is always a delicate operation which supposes that only the average value was affected during the disturbance of measurement. In particular one should ensure that the signal is not saturated.

Chapter 4

Development of shock test specifications

4.1. General The first tests of the behaviour of materials in response to shocks were carried out in 1917 by the American Navy [PUS 77] and [WEL 46]. The most significant development started at the time of the World War II with the development of specific free fall or pendular hammer machines. The specifications are related to the type of machine and its adjustments (drop height, material constituting the programmer, mass of the hammer). Given certain precautions, this process ensures a great uniformity of the tests. The demonstration is based on the fact that the materials, having undergone this test successfully resist well the real environment which the test claims to simulate. It is necessary to be certain that the severity of the real shocks does not change from one project to another. It is to be feared that the material thus designed is more fashioned to resist the specified shock on the machine than the shock to which it will be really subjected in service. Very quickly specifications appeared imposing contractually the shape of acceleration signals, their amplitude and duration. In the mid-1950s, taking into account the development of electrodynamic exciters for vibration tests, and the interest in producing mechanical shocks, the same methods were developed (it was that time that simulation vibrations by random vibrations under test real conditions were started). This testing on a shaker, when possible, indeed presents a certain number of advantages [COT 66]; vibration and impact tests on the same device, the possibility of carrying out shocks of very diverse shapes, etc.

96

Mechanical shock

In addition the shock response spectrum became the tool selected for the comparison of the severity of several shocks and for the development of specifications, the stages being in this last case the following: - calculation of shock spectra of transient signals of the real environment; - plotting of the envelope of these spectra; - searching for a signal of simple shape (half-sine, saw tooth etc) of which the spectrum is close to the spectrum envelope. This operation is generally delicate and cannot be carried out without requiring an over-test or an under-test in certain frequency bands. In the years 1963/1975 the development of computers gave way to a method, consisting of giving directly the shock spectrum to be realized on the control system of the shaker. Taking into account the transfer function of the test machine (with the test item), the software then generates on the input of the test item a signal versus time which has the desired shock spectrum. This makes it possible to avoid the last stage of the process. The shocks measured in the real environment are in general complex in shape; they are difficult to describe simply and impossible to reproduce accurately on the usual shock machines. These machines can generate only simple shape shocks such as rectangle, half-sine, terminal peak saw tooth pulses. Several methods have been proposed to transform the real signal into a specification of this nature. 4.2. Simplification of the measured signal This method consists of extracting the first peak, the duration being defined by time when the signal x(t) is cancelled for the first time, or extraction of the highest peak.

Figure 4.1. Taking into account the largest peak


97

The shock test specification is then described in the form of an impulse of amplitude equal to that of the chosen peak in the measured signal, of duration equal to the half-period thus defined and whose shape can vary, while approaching as early as possible that of the first peak (Figure 4.1). The choice can be guided by the use of an abacus making it possible to check that the profile of the shock pulse remains within the tolerances of one of the standardized forms [KIR 69]. Another method consists of measuring the velocity change associated with the shock pulse by integration of the function x(t) during the half-cycle with greater amplitude. The shape of the shock is selected arbitrarily. The amplitude and the duration are fixed in order to preserve the velocity change [KIR 69] (Figure 4.2).

Figure 4.2. Specification with same velocity change

The transformation of a complex shock environment into a simple shape shock, realizable in the laboratory, is under these conditions an operation which utilizes in an important way the judgement of the operator. It is rare, in practice, that the shocks observed are simple, with a form easy to approach, and it is necessary to avoid falling into the trap of over-simplification.

Figure 4.3. Difficulty of transformation of real shockpulses

98

Mechanical shock

In the example in Figure 4.3, the half-sine signal can be a correct approximation of the relatively "clean" shock 1; but the real shock 2, which contains several positive and negative peaks, cannot be simulated by just one unidirectional wave. It is difficult to give a general empirical rule to ensure the quality of simulation in laboratory carried out according to this process and the experimental quality is important. It is not shown that the criterion of equivalence chosen to transform the complex signal to a simple shape shock is valid. It is undoubtedly the most serious defect. This method lends itself little to statistical analysis which would be possible if one had several measurements of a particular event and which would make it possible to establish a specification covering the real environment with a given probability. In the same way, it is difficult to determine a shock enveloping various shocks measured in the life profile of the material.

4.3. Use of shock response spectra 4.3.1. Synthesis of spectra The most complex case is where the real environment, described by curves of acceleration against time, is supposed to be composed of p different events (handling shock, inter-stage cutting shock on a satellite launcher), with each one of these events itself being characterized by ri successive measurements. These ri measurements allow a statistical description of each event. The folowing procedure consists for each one: - To calculate the shock response spectrum of each signal recorded with the damping factor of the principal mode of the structure if this value is known, if not with the conventional value 0.05. In the same way, the frequency band of analysis will have to envelop the principal resonance frequencies of the structure (known or foreseeable frequencies). - If the number of measurements is sufficient, to calculate the mean spectrum m (mean of the points at each frequency) as well as the standard deviation spectrum, then the standard deviation/mean ratio, according to the frequency; if it is insufficient, to make the envelope of the spectra. - To apply to the mean spectrum or the mean spectrum + 3 standard deviations a statistical uncertainty coefficient k, calculated for a probability of tolerated maximum failure (cf Volume 5), or contractual (if one uses the envelope).


99

Each event thus being synthesized in only one spectrum, one proceeds to an envelope of all the spectra obtained to deduce a spectrum from it covering the totality of the shocks of the life profile. After multiplication by a test factor (Volume 5), this spectrum will be used as reference 'real environment' for the determination of the specification.

Table 4.1. Process of developing a specification from real shocks measurements Event #1 Handling shock r1 measured data

Mean and standard deviation spectra or envelope

Calculation of

r1 S.RS.

Event #2 Landing shock T2 measured data

Calculation of

Event #p Ignition shock r_ measured data

Calculation of

T2 S.RS.

rp S.R.S.

-

k (m + 3 s) or k env.

Mean and standard k (m + 3 s) deviation spectra or k env. or envelope Mean and standard deviation spectra or envelope

k (m + 3 s) or k env.

—

Envelope X

Envelope

Test factor

The reference spectrum can consist of the positive and negative spectra or the envelope of their absolute value (maximax spectrum). In this last case, the specification will have to be applied according to the two corresponding half-axes of the test item.

4.3.2. Nature of the specification According to the characteristics of the spectrum and available means, the specification can be expressed in the form of: -A simple shape signal according to time realizable on the usual shock machines (half-sine, T.P.S., rectangular pulse). There is an infinity of shocks having a given response spectrum. The fact that this transformation is universal is related to its very great loss of information, since one retains only the largest value of the response according to time to constitute the SRS at each natural frequency. One can thus try to find a shock of simple form, to which the spectrum is closed to the reference spectrum, characterized by its form, its amplitude and its duration. It is in

100

Mechanical shock

general desirable that the positive and negative spectra of the specification respectively cover the positive and negative spectra of the field environment. If this condition cannot be obtained by application of only one shock (particular shape of the spectra, limitations of the facilities), the specification will be made up of two shocks, one on each half-axis. The envelope must be approaching the real environment as well as possible, if possible on all the spectrum in the frequency band retained for the analysis, if not in a frequency band surrounding the resonance frequencies of the test item (if they are known). - A shock response spectrum. In this last case, the specification is directly the reference SRS.

4.3.3. Choice of shape The choice of the shape of the shock is carried out by comparison of the shapes of the positive and negative spectra of the real environment with those of the spectra of the usual shocks of simple shape (half-sine, TPS, rectangle) (Figure 4.4).

Figure 4.4. Shapes of the SRS of the realizable shocks on the usual machines


101

If these positive and negative spectra are nearly symmetrical, one will retain a terminal peak saw tooth, whilst remembering, however, that the shock which will be really applied to the tested equipment will have a non-zero decay time so that its negative spectrum will tend towards zero at very high frequencies. This disadvantage is not necessarily onerous, if for example a preliminary study could show that the resonance frequencies of the test item are in the frequency band where the spectrum of the specified shock envelops the real environment. If only the positive spectrum is important, one will choose any form, the selection criterion being the facility for realization, or the ratio between the amplitude of the first peak of the spectrum and the value of the spectrum at high frequencies: approximately 1.65 for the half-sine pulse (Q = 10), 1.18 for the terminal peak saw tooth pulse, and no peak for the rectangular pulse.

4.3.4. Amplitude The amplitude of the shock is obtained by plotting the horizontal straight line which closely envelops the positive reference SRS at high frequency.

Figure 4.5. Determination of the amplitude of the specification

This line cuts the y-axis at a point which gives the amplitude sought (one uses here the property of the spectra at high frequencies, which tends in this zone towards the amplitude of the signal in the time domain).

4.3.5. Duration The shock duration is given by the coincidence of a particular point of the reference spectrum and the reduced spectrum of the simple shock selected above (Figure 4.6).

102

Mechanical shock

Figure 4.6. Determination of the shock duration

One in general considers the abscissa f01 of the first point which reaches the value of the asymptote at the high frequencies (amplitude of shock). Table 4.2 joins together some values of this abscissa for the most usual simple shocks according to the Q factor [LAL 78].

Table 4.2. Values of the dimensionless frequency corresponding to the first passage of the SRS by the amplitude unit f 01

£

Half-sine

Versed-sine

2 3

0.2500

0.413

0.542

IPS /

0.1667

0.358

0.465

0.564

0.219

4

0.1250

0.333

0.431

0.499

0.205

Q

Rectangle 0.248

5

0.1000

0.319

0.412

0.468

0.197

6 7 8 9 10 15 20 25 30 35 40 45 50

0.0833

0.310

0.400

0.449

0.192

0.0714 0.0625

0.304 0.293

0.392

0.437

0.188

0.385

0.427

0.185

0.0556 0.0500

0.295

0.381 0.377

0.421

0.183

0.415

0.181

00

0.293

0.0333

0.284

0.365

0.400

0.176

0.0250

0.280

0.360

0.392

0.174

0.0200

0.277

0.357

0.388

0.173

0.0167

0.276

0.354

0.385

0.172

0.0143

0.275

0.353

0.383

0.171

0.0125

0.274

0.0111 0.0100

0.273 0.272

0.352 0.351

0.382 0.380

0.170 0.170

0.0000

0.267

0.350

0.379

0.170

0.344

0.371

0.167


103

NOTES: 1. If the calculated duration must be rounded (in milliseconds), the higher value should always be considered, so that the spectrum of the specified shock remains always higher or equal to the reference spectrum. 2. It is in general difficult to carry out shocks of duration lower than 2 ms on standard shock machines. This difficulty can be circumvented for very light equipment with a specific assembly associated with the shock machine (dual mass shock amplifier, Section 6.2). One will validate the specification by checking that the positive and negative spectra of the shock thus determined are well enveloped by the respective reference spectra and one will verify, if the resonance frequencies of the test item are known, that one does not over-test exaggeratedly at these frequencies.

Example Let us consider the positive and negative spectra characterizing the real environment plotted (Figure 4.7) (result of a synthesis).

Figure 4.7. SRS of the field environment

104

Mechanical shock

It is noted that the negative spectrum preserves a significant level in all the frequency domain (the beginning of the spectrum being excluded). The most suitable simple shock shape is the terminal peak saw tooth. The amplitude of the shock is obtained by reading the ordinate of a straight line enveloping the positive spectrum at high frequencies (340 m/s2). The duration is deduced from the point of intersection of this horizontal line with the curve (point of lower frequency), which has as an abscissa equal to 49.5 Hz (Figure 4.8). One could also consider the point of intersection of this horizontal line with the tangent at the origin.

Figure 4.8. Abscissa of the first passage by the unit amplitude

One reads on the dimensionless spectrum of a TPS pulse (same damping ratio) the abscissa of this point: f0 T = 0.415, yielding, so that f0 = 49.5 Hz

The duration of the shock will thus be (rounding up)

which slightly moves the spectrum towards the left and makes it possible to bettei cover the low frequencies. Figure 4.9 shows the spectra of the environment and those of the TPS pulse thus determined.


105

Figure 4.9. SRS of the specification and of the real environment

NOTE: In practice, it is only at this stage that the test factor can be applied to the shock amplitude.

4.3.6. Difficulties This method leads easily to a specification when the positive spectrum of reference increases regularly from the low frequencies to a peak value not exceeding approximately 1.7 times the value of the spectrum at the highest frequencies, and then decreases until it is approximately constant at high frequencies. This shape is easy to envelop since it corresponds to the shape of the spectra of normal simple shocks.

Figure 4.10. Case of a SRS presenting an important peak

106

Mechanical shock

In practice it can happen that the first peak of the reference spectrum is much larger, that this spectrum has several peaks, and that it is almost tangential to the frequency axis at the low frequencies etc. In the first case (Figure 4.10), a conservative method consists of enveloping the whole of the reference spectrum. After choosing the shape as previously, one notes the coordinates of a particular point, for example: the amplitude Sp of the peak and its abscissa fp.

Figure 4.11. Coordinates of the peak of the dimensionless SRS of the selected shock

On the dimensionless positive spectrum of the selected signal, plotted with the same damping ratio, one reads the coordinates of the first peak: oz m depends only on the values of Q and N (for xm fixed). Being given a shock measured in the real environment, J.D. Crum and R.L. Grant [CRU 70] plotted the ratio of the response spectra calculated for Q = 25 and Q = 5 versus frequency f0. Their study, carried out on a great number of shocks, shows that this ratio varies little in general around a value a. The specification is obtained by plotting a horizontal linear envelope of each spectrum (in the ratio a). In sinusoidal mode, the ratio

is, for Q given, only a function of N. With a

swept sine excitation, one obtains a spectrum of constant amplitude if the number of cycles AN carried out between the half-power points is independent of the natural frequency f0, i.e. if the sweeping is hyperbolic. J.D. Crum and R.L. Grant expressed their results according to the parameter N'= Q AN.

110

Mechanical shock

If the sweep rate were weak, the ratio would be equal to 5 or 25 according to choice of Q (whatever, the sweep mode). To obtain spectra in the ratio a (in general lower than 5), a fast sweep should be used therefore. The hyperbolic swept sine is defined as follows, starting from a curve giving the ratio to responses for Q = 25 and Q = 5 versus N' and of

versus N'.

-The desired ratio a allows one to define N'= N'0 and N'0 gives using the two preceding curves. - Knowing the envelope spectrum w02 zm specified for Q = 5, one deduces from it the necessary amplitude xm. - The authors have given for an empirical rule the sweep starting from a frequency f1 lower by 25% than the lowest frequency of the spectrum of the specified shock and finishing at a frequency f2 higher by 25% than the highest frequency of the specified spectrum. The excitation is thus defined by:

with:

if the sweep is at increasing frequencies, or by:

for a sweep at decreasing frequencies. - The sweep duration is given by:

The durations obtained are between a few hundreds of milliseconds and several seconds.


111

It is possible to modulate the amplitude xm according to the frequency to satisfy a specification which would not be a horizontal line and to vary N'0 to better follow the variations of the ratio a of the spectra calculated for Q = 25 and Q = 5 [CRU 70] [ROU 74]. The formulation of Routree and Freberg is more general. It is based on the relations:

The modifiable parameters are a, (3, f0, R and y where: - a is the initial value of A(t) (with t = 0); -(3 characterizes the variations of the amplitude A(t) according to time (or according to f); - f(t) is the instantaneous frequency, equal to f0 for t = 0; - R and y characterize the variations oft versus time. If y = 0, the law f(t) is linear, with a sweep rate equal to R. If y = 1, sweep is exponential, such that f = e

Rt

If y = 2, sweep is hyperbolic (as in the assumptions of Crum and Grant)

Advantages These methods: - produce shocks pulses well adapted for the reproduction on a shaker;

112

Mechanical shock

- allow the simulation of a spectrum simultaneously for two values of the Q factor.

Drawbacks These methods lead to shock pulses which do not resemble the real environment at all These techniques were developed to simulate spectra which can be represented by a straight line on log log scales and they adapt badly to spectra with nother shapes.

4.4.3. Simulation by modulated random noise It was recognized that the shocks measured in the seism domain have a random nature. This is why many proposals [BAR 73], [LEV 71] were made to seek a random process which, after multiplication by an adequate window, provides a shock comparable with this type of shock. The aim is to determine a wave form showing the same statistical characteristics as the signal measured [SMA 74a], [SMA 75]. This wave form is made up of a nonstationary modulated random noise having the same response spectrum as the seismic shock to be simulated. It is, however, important to note that this type of method allows reproduction of a specified shock spectrum only in one probabilistic sense. L.L.Bucciarelli and J.Askinazi [BUG 73] proposed using an excitation of this nature to simulate pyrotechnic shocks with an exponential window of the form:

where g(t) is a deterministic function of the time, which characterizes the transitory nature of the phenomenon

and n(t) is a stationary broad band noise process with average zero and power spectral density Sn(Q).


113

Being given a whole set of measurements of the shock, one seeks to determineSn(Q) and the time constant (3 to obtain the best possible simulation. The function Sn(o) is calculated from:

where E[X(Q) X*(Q)] is the mean value of the squares of the amplitudes of the Fourier spectra of the shocks measured. The constant B must be selected to be lower than the smallest interesting frequency of the shock response spectrum. N.C. Tsai [TSA 72] was based on the following process: - choice of a sample of signal x(t); - calculation of the shock response spectrum of this sample; - being given a white noise n(t), addition of energy to the signal by addition of sinusoids to n(t) in the ranges where the shock spectrum is small; in the ranges where the shock spectrum is large, filtering of n(t) with a filter attenuating a narrow band (—I I—I L); - calculation of the shock spectrum of the modified signal n(t); and repetition of the process until reaching the desired shock spectrum. Although interesting, this technique is not the subject of marketed software and is thus not used in the laboratory. NOTE: J.F. Unruth [UNR 82] suggested simulating the seisms while controlling the shock spectrum, the signal reconstituted being obtained by synthesis from the sum of pseudo-random noises into 1/6 octave. Each component of narrow band noise is the weighted sum of 20 cosine functions out of phase whose frequencies are uniformly distributed in the band considered. The relative phases have a random distribution in the interval [0, n].

4.4.4. Simulation of a shock using random vibration The probability that a maximum of w02 z(t) is lower than w02 zm over the duration T is equal to 1 - Pp(w02 zm) with PP being the distribution function of the peaks of the response. The number of cycles to be applied during the test is equal approximately to f0 T. If these peaks are supposed independent, the probability PT that all the

114

Mechanical shock

0

"7

maxima of w02 z(t) are lower than w02 zm is then

I

/

"7

1-Pp(w02

\ rO

zm)

. The

probability that a maximum of O>Q z(t) is higher than w02 zm is thus equal to

i4-4fo-FUse of a narrow band random vibration A narrow band random vibration can be applied to the material at a single frequency or several frequencies simultaneously. This process has some advantages [KER 84]: - the number of cycles exceeding a given level can be limited; - several resonances can be excited simultaneously; - amplification at resonance is reduced compared to the slow swept sine (the response varies as /Q instead of Q). But the nature of the vibration does not make it possible to ensure the reproducibility of the test.

4.4.5. Least favourable response technique Basic assumption It is supposed that the Fourier spectrum (amplitude) is specified, which is equivalent to specifying the undamped residual shock spectrum (Section 3.8.2.). It is shown that if the transfer function between the input and the response of the test item (and not that of the shaker) can be characterized by:

then the peak response of the structure will be maximized by the input [SMA 74a] [SMA 75]:

where Xe(Q) is the module of the specified Fourier transform and


115

The calculation of the above expressions is relatively easy today. The phase angle 6 of the transfer function is measured using a test. With this function and the specified module X e (Q), one calculates the input x(t). The method supposes simply that the studied system is linear with a critical response well defined. There is no assumption on the number of degrees of freedom or on damping. It guarantees that the largest possible response peak will be reached, in practice, at about 1 to 2.5 times the response with the real shock (guarantee of a conservative test) [SMA 72] [WIT 74]. The techniques of the shock spectrum cannot give this insurance for systems to several degrees of freedom. The method requires important calculations and thus numerical means. An alternative can be found in supposing that H(Q) = 1 and to calculate the input to be applied to the specimen so that:

With Xe(Q) being a real positive function and x(t)a real even function. An input thus defined will resemble a SHOC waveform (Chapter 9). This input is independent of the characteristics of the test item and thus eliminates the need for defining the transfer function H(Q). The only necessary parameter is the module of the Fourier transform (or the undamped residual shock spectrum). A series of tests showed that this approach is reasonable [SMA 72].

4.4.6. Restitution of a shock response spectrum by a series of modulated sine pulses This method, suggested by D.L. Kern and C.D. Beam [KER 84], consists of applying a series of modulated sine wave shocks sequentially. The retained waveform resembles the response versus time of the mass of a one-degree-offreedom system base-excited when it is subjected to an exponentially decayed sine wave excitation; it has as an approximate equation

elsewhere where Q = 2 n f T] = damping of the signal A = Q e T| xm xm = amplitude of x(t) e = Neper number

116

Mechanical shock

Figure 4.13. Shock waveform (D.L Kern and CD. Hayes)

The choice of r\ must meet two criteria: - to be close to 0.05, a value characteristic of many complex structures; - to allow that the maximum of x(t) (the largest peak) takes place at the same time as the peak of the envelope of x(t).

Figure 4.14. Coincidence of the peaks of the signal and its envelope

The interesting point of this approach, which takes again a proposal of J.T.Howlett and D.J.Martin [HOW 68] containing purely sinusoidal impulses, is in the facility of determination of the characteristics of each sinusoid, since each one of them is considered separately, contrary to the case of a control per spectrum (Chapter 9). The shocks are easy to create and to realize.


117

The adjustable parameters are the amplitude and possibly the number of cycles. The number of frequencies is selected so that the point of intersection of the spectra of two adjacent signals is not lower by more than 3 dB than the amplitude of the peak of the spectrum (plotted for a damping equal to 0.05). Like the slowly swept sine, this method does not make it possible to excite all resonances simultaneously. We will see in Chapter 9 how this waveform can be used to constitute a complex drive signal restoring the whole of the spectrum.

4.5. Interest behind simulation of shocks on a shaker using a shock spectrum The data of a shock specification for a response spectrum has several advantages: - the response spectrum should be more easily exploitable for dimensioning of the structure than the signal x(t) itself; - this spectrum can result directly from measurements of the real environment and does not require, at the design stage to proceed to an often delicate equivalence with a signal of simple shape; - the spectrum can be treated in a statistical way if one has several measurements of the same phenomenon, it can be the envelope of several different transitory events and can be increased by a uncertainty coefficient; -the reference most commonly allowed to judge quality of the shock simulation is comparison of the response spectra of the specification with the shock carried out. In a complementary way, when the shock tests can be carried out using a shaker, one can have direct control from a response spectrum: - The search for a simple form shock of a given spectrum compatible with the usual test facilities is not always a simple operation, according to the shape of the reference spectrum resulting from measurements of the real environment. - The shapes of the specified spectra can be very varied, contrary to those of the spectra of the usual shocks (half-sine, triangles, rectangles etc) carried out on the shock machines. One can therefore improve the quality of simulation and reproduce shocks difficult to simulate with the usual means (case of the pyroshocks for example) [GAL 73] and [ROT 72]. - Taking into account the oscillatory nature of the elementary signals used, the positive and negative spectra are very close, which makes a reversal of the test item [PAI 64] useless.

118

Mechanical shock

- In theory, simple shape shocks created on a shock machine are reproducible, which makes it possible to expect uniform tests from one laboratory to another. In practice, one was obliged to define tolerances on the shapes of the signals to take account of the distortions really measured and difficult to avoid. The limits are rather broad (+15%) and can result, however, in accepting two shocks included within these limits likely to have very different effects (which one can evaluate with the shock spectra) [FAG 67].

Figure 4.15. Nominal half-sine and its tolerances

Figure 4.16. Shock located between the tolerances

Figure 4.17. SRS of the nominal half-sine and the tolerance limits

Figures 4.15 and 4.17 show as an example a nominal half-sine (100 m/s2, 10 ms) and its tolerance limits, as well as the shock spectra of the nominal shock and each lower and upper limit. Figure 4.16 represents a shock made up of the sum of the nominal half-sine and of a sinusoid of amplitude 15 m/s2 and frequency 2500 Hz.


119

The spectrum of this signal is superimposed on the spectra of the tolerance limits in Figure 4.18. Although this composite signal remains within the tolerances, it is noted that it has a spectrum very different from the spectra of the tolerance limits for small £, in a frequency band around 250 Hz and mat the negative spectra of the tolerance limits intersect and thus do not delimit a well defined domain [LAL 72].

Figure 4.18. SRS of the shock of Figure 4.16 and of the tolerance limits

With mis some practical advantages are added: - sequence of the shock and vibration tests without disassembly and with the same test fixture (saving of time and money); - maintenance of the test item with its normal orientation during the test.

120

Mechanical shock

These the last two points are not, however, specific with spectrum control, but more generally relate to the use of a shaker. Control by the spectrum, however, increases the capacities of simulation because of the possibility of the choice of the shape of the elementary waveforms and of their variety.

Chapter 5

Kinematics of simple shocks

5.1. General The shock test is in general specified by an acceleration varying with time. This profile of acceleration can be obtained with various velocity and displacement profiles depending on the initial velocity of the table supporting the specimen, leading theoretically to various types of programme. All shock test facilities are in other respects limited in respect of force (i.e. in acceleration, taking into account the mass of the whole of the moving element, made up of the table, the test fixure, the armature assembly in the case of a shaker and the specimen), velocity and displacement. It is thus useful to study the kinematics of the principal shock pulses carried out classically on the machines, namely the half-sine (or versed-sine), the terminal peak saw tooth and the rectangle (or the trapezoid).

5.2. Half-sine pulse 5.2.1. Definition The excitation, zero for t < 0 and t > T can be written in the interval (0, T), in the form

122

Mechanical shock

where xm is the amplitude of the shock and T its duration. The pulsation is equal to This expression becomes, in generalized form, l(t) = lm sin Q t.

According to the type of excitation, l ( t ) is then a force

an acceleration

In reduced (dimensionless) form, and with the notations used in preceding chapters, the definition of shock can be

Note that h

Figure 5.1. Half-sine shock

5.2.2. Shock motion study 5.2.2.1. General expressions The motion study during the application of the shock is useful for the choice of the programmer and the test facility which will make it possible to carry out the


123

specification. We will limit ourselves, in what follows, to the most general case where the shock is defined by an acceleration pulse x(t) [LAL 75]. With the signal of acceleration

corresponds

by

integration

to

the

instantaneous

velocity

constant. Let us suppose that at the initial moment t = 0, the velocity is equal to Vj:

The constant is thus equal to

and the velocity to

At the moment t = T of the end of the shock, the velocity vf has as an expression

i.e., since Q t = n,

The body subjected to this shock thus undergoes a velocity change

It is the area delimited by the curve x(t) and the time axis between 0 and T.

124

Mechanical shock

Figure 5.2. Velocity change of a half-sine

The displacement is calculated by a second integration; we will take for initial conditions t = 0, x = 0 , a s i t i s practically always the case in these problems. This yields

To further the study of this movement x(t), it is preferable to particularize the test conditions. Two cases arise; the velocity vi being able to be: - either zero before the beginning of the shock: the object subjected to the shock, initially at rest, comes under the effect of the impulse a velocity vf = AV; - or arbitrary nonzero: the specimen has a velocity which varies during the shock duration T from a value Vi to a value vf for t = T; it is then said that there is impact NOTE: This refers mostly to shocks obtained on shock machines. This classification can be open to confusion insofar as the shocks can be carried out on exciters with a pre-shock and/or post-shock which communicates to the carriage (table, fixture and test item) a velocity before the application of the shock itself (we will see in Chapter 7 the need for a pre-shock and/or a post-shock to cancel the table velocity at the end of movement). 5.2.2.2. Impulse mode Since Vi = 0


125

The velocity increases without changing sign from 0 to vf. In the interval (0, T), the displacement is thus at a maximum for t = T:

It is the area under the curve v(t) in (0, T). Equations [5.4], [5.10] and [5.11] describe the three curves x(t), v(t) and x(t) in this interval.

126

Mechanical shock Table 5.1. Kinematics of a half-sine shock generated by an impulse

Velocity

Acceleration

Maximum at

Displacement

Maximum at

Maximum at

Zero slope at

Zero slope at equal to

Inflection point at

and for

Inflection point at

5.2.2.3. Impact mode General case The initial velocity Vi is arbitrary, zero here. The body subjected to the shock arrives on the target with the velocity v i ,touches the target (which has a programmer intended to shape the acceleration x(t) according to a half-sine) between time t = 0 and t = T. Several cases can arise. At time t = t at the end of the shock, the velocity vf can be: - either zero (no rebound);


127

-or arbitrary, different from zero. It is said there is rebound with velocity V R (= v f ). We suppose that the movement is carried out along only one axis, the velocity having a different direction from the velocity of impact. The velocity change is equal, in absolute terms, to AV = VR - Vi . The most general case is where VR is arbitrary:

with

coefficient of restitution). The velocity change AV, equal to makes it possible to calculate Vi:

i.e., in algebraic value, and by definition

The velocity v(t), given by [5.6], is thus written:

i.e., since QT = n:

and the displacement:

To facilitate the study, we will consider some particular cases where the rebound velocity is zero, where it is equal (and opposite) to the impact velocity and finally Vi

where it is equal to -—. 2

128

Mechanical shock

Impact without rebound The rebound velocity is zero (a = 0). The mobile arrives on the target with velocity Vi at time t = 0, undergoes the shock x(t) for time i and stops at t = T.

and

The maximum displacement xm throughout the shock takes place here also for t = t since v(t) passes from -v4 to 0 continuously, without a change in sign. Moreover, since VR = 0, x remains equal to xm for t > t.

This value of x(t) is equal to the area under the curve v(t) delimited by the curve (between 0 and T) and the two axes of coordinates.


129

Table 5.2. Kinematics of a half-sine shock carried out by impact without rebound

Acceleration

Maximum at

Velocity

Displacement^

Maximum at

Zero at

Zero slope when

Inflection point at

and

Zero slope at

and equal

Inflection point at

Velocity of rebound equal and opposite to the velocity of impact (perfect rebound) After impact, the specimen sets out again in the opposite direction with a velocity equal to the initial velocity (a = 1 and VR = -Vi). It then becomes

and

130

Mechanical shock

The velocity varies from Vj to VR = -Vj when t varies from 0 to T. Let us take again the general expressions [5.18] and [5.19] for v(t) and x(t) and set a = 1:

and, since x = 0 with t = 0 by assumption

Table 5.3. Kinematics of a half-sine shock carried out by impact with perfect rebound

Acceleration

Velocity

Displacement


131

dx

The displacement is maximum for t = tm

corresponding to — = 0, so that dt

T

If K = 0, tm = —. The maximum displacement xm thus has as a value 2

In the case of a perfect rebound (VR = -Vi), the amplitude xm of the displacement is smaller by a factor n than if VR = 0. It is pointed out that the amplitude xm is none other than the area ranging between the curve v(t) and the T

two axes of coordinates, in the time interval (0, —): 2

Velocity of rebound equal and opposed to half of the impact velocity 1 In this case, a = —. The mobile arrives at the programmer with a velocity vi5 2 meets it at time t = 0, undergoes the impact for the length of time T, rebounds and v i sets out again in the opposite direction with a velocity VR = -—: 2

132

Mechanical shock

Let us set a = — in the general expressions [5.18] and [5.19] of v(t) and x(t); it 2 then becomes

and

The maximum displacement takes place when v(t) = 0, i.e. when t = tm such that

We will take, in (0, T),

This value of

yielding

lies between the two values

The hatched area under the curve v(t) is equal to xm.

and


133

Table 5.4. Kinematics of a half-sine shock caused by impact with 50% rebound velocity

Velocity

Acceleration

Maximum at

Displacement

Maximum at

Zero at

Zero slope when

and

Zero slope at equal to

in

and to

in

Inflection point at

Summary chart — remarks on the general case of an arbitrary rebound velocity

All these results are brought together in Table 5.5, One can note that: - the maximum displacements required in the case of impulse and the case of impact without rebound are equal; -the maximum displacement in the case of a 100% rebound velocity is smaller by a factor TC; the energy spent by the corresponding shock machine will thus be smaller [WHI].

134

Mechanical shock

Locus of the maxima The velocity of rebound is, in the general case, a fraction of the velocity of impact:

However

or

and

or, since

x(t) is at a maximum when v(t) = 0, i.e. when is positive when 0 < t < t and since

for

Thus

The locus of maxima, given by the parametric representation t m (a), xm (a), can be expressed according to a relation x m (t m ) while eliminating a between the two relations:


135

The locus of the maxima is an arc of the curve representative of this function in T

the interval — < tm < T . 2

Table 5.5. Summary of the conditions for the realization of a half-sine shock

Impulse

Impact without rebound

Impact with perfect rebound

Impact with rebound to 50% of the initial velocity

136

Mechanical shock

5.3. Versed-sine pulse 5.3.1. Definition The versed-sine* (or haversine** ) shape consists of an arc of sinusoid ranging between two successive minima.

Figure 5.3. Haversine shock pulse

It can be represented by

for elsewhere Generalized form

We set here

One minus Cosine One half of one minus Cosine

Reduced form

for

for

elsewhere

elsewhere

Kinematics of simple shocks a General expressions

(it is supposed that x(o) = 0). Impulse mode

v

Table 5.6. Velocity and displacement for carrying out a versed-sine shock pulse Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement

137

138

Mechanical shock

(by preserving the notation VR = -a vi). Table 5.6 gives the expressions for the velocity and the displacement using the same assumptions as for the half-sine pulse.

Table 5.7. Summary of the conditions for the realization of a haversine shock pulse

Impulse





5.4. Rectangular pulse 5.4.1. Definition

Figure 5.4. Rectangular shock pulse

for elsewhere

Generalized form

for elsewhere Reduced form

for elsewhere

5.4.2. Shock motion study General expressions

139

140

Mechanical shock

Impact

Impulse

Table 5.8. Velocity and displacement: rectangular shock pulse

Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement


141

Table 5.9. Summary of the conditions for the realization of a rectangular shock pulse

Impulse




142

Mechanical shock

5.5. Terminal peak saw tooth pulse 5.5.1. Definition

Figure 5.5. Terminal peak saw tooth pulse



for elsewhere

Kinematics of simple shocks 5.5.2. Shock motion study General expressions

Impulse

Impact

Table 5.110. Velocity and displacement to carry out a TPS shock pulse Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement

143

144

Mechanical shock Table 5.11. Summary of the conditions for the realization of a TPS shock

Impulse




5.6. Initial peak saw tooth pulse 5.6.1. Definition

Figure 5.6. IPS shock pulse



for elsewhere

5.6.2. Shock motion study General expressions

146

Mechanical shock

Impact

Impulse

Table 5.12. )Velocity and displacement needed to carry out an IPS shock pulse Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement


147

Table 5.13. Summary of the conditions for the realization of an IPS shock

Impulse




Whatever the shape of the shock, perfect rebound leads to the smallest displacement (and to the lowest drop height). With traditional shock machines, this

148

Mechanical shock

cannot be really exploited, since one is not able to choose the kinematics of the shock.

Chapter 6

Standard shock machines

6.1. Main types The first specific machines developed at the time of World War II belong to two categories: -Pendular type machines, equipped with a hammer which, after falling in a circular motion, strike a steel plate which is fixed to the specimen (high-impact machine) [CON 51] [CON 52] [VIG 61a]. The first of these machines was manufactured in England in 1939 to test the light equipment which was subjected, on naval ships, to shocks produced by underwater explosions (mines, torpedoes). Several models were developed in the United States and in Europe to produce shocks on equipment of more substantial mass. These machines are still used (cf. Figure 6.1). - Sand drop machines are made up of a table sliding on two vertical guide columns and free falling into a sand box, characteristics of the shock obtained being a function of the shape and the number of wooden wedges fixed under the table, as well as the granularity of sand (cf. Figure 6.2) [BRO 61] [LAZ 67] [VIG 61b]. NOTES: An alternative to this machine which was used simply comprised a wooden table supporting the specimen, under which a series of wooden wedges was fixed. The table was released from a given height, without guidance, and impacted the sand in the box.

150

Mechanical shock

Figure 6.1. Sand-drop shock testing machine

Figure 6.2. Sand-drop impact simulator


151

The test facilities now used are classified as follows: -Free fall machines, derived from the sand-drop machines, the impact being made on a programmer adapted to the shape of the specified shock (elastomer discs, conical or cylindrical lead pellets, pneumatic programmers etc). To increase the impact velocity, which is limited by the drop height, i.e. by the height of the guide columns, the fall can be accelerated by the use of bungee cords. - Pneumatic machines, the velocity being derived from a pneumatic actuator. - Electrodynamic exciters, the shock being specified either by the shape of a temporal signal, its amplitude and its duration, or by a shock response spectrum. - Exotic machines, designed to carry out non-realizable shocks by the preceding methods, generally because their amplitude and duration characteristics are not compatible with the performances from these means, because the desired shapes, not being normal, are not possible with the programmers delivered by the manufacturers. A shock machine, whatever its standard, is primarily a device allowing modification over a short time period of the velocity of the material to be tested. Two principal categories are usually distinguished: - "impulse" machines, which increase the velocity of the test item during the shock. The initial velocity is in general zero. The air gun, which creates the shock during the setting of velocity in the tube, is an example; - "impact" machines, which decrease the velocity of the test item throughout the shock and/or which change its direction. 6.2. Impact shock machines Most machines with free or accelerated drops belong to this last category. The machine itself allows the setting of velocity of the test item. The shock is carried out by impact on a programmer which formats the acceleration of braking according to the desired shape. The impact can be without rebound when the velocity is zero at the end of the shock, or with rebound when the velocity changes sign during the movement. The laboratory machines of this type consists of two vertical guide rods on which the table carrying test item (Figure 6.3) slides. The impact velocity is obtained by gravity, after the dropping of the table from a certain height or using bungee cords allowing one to obtain a larger impact velocity.

152

Mechanical shock

Figure 6.3. Elements of a shock test machine

In all cases, whatever the method for realization of the shock, it is useful to consider the complete movement of the test item between the moment when its velocity starts to take a nonzero value and that where it again becomes equal to zero. One thus always observes the presence of a pre-shock and/or a post-shock. Let us consider a free fall shock machine for which the friction of the shock table on the guidance system can be neglected. The necessary drop height to obtain the desired impact velocity vi' is given by:

if M = the mass of the moving assembly of the machine (table, fixture, programmer) m = mass of the test item g = acceleration of gravity (9.81 m/s2). yielding


153

These machines are limited by the possible drop height, i.e. by the height of the columns and the height of the test item when the machine is provided with a gantry. It is difficult to increase the height of the machine due to overcrowding and problems with guiding the table. One can increase, however, the impact velocity using a force complementary to gravity by means of bungee cords tended before the test and exerting a force generally directed downwards. The acceleration produced by the cords is in general much higher than gravity which then becomes negligible. This idea was used to design horizontal [LON 63] or vertical machines [LAV 69] [MAR 65], this last configuration being less cumbersome.

Figure 6.4. Use of elastic cords

The Collins machine is an example. Its principle of operation is illustrated in Figure 6.4. The table is guided by two vertical columns in order to ensure a good position of the test item at impact. When the carriage is accelerated by elastic cords, the force applied to the table is due to gravity and to the action of these cords. One has then, if Th is the tension of the elastic cord at the instant of dropping and Tj the tension of the cord at the time of the impact:

154 Mechanical shock

(neglecting the kinetic energy of the elastic cords).

Figure 6.5. Principle of operation using elastic cords

Figure 6.6. Principle behindpendular shock test machine

If machine is of the pendular type, the impact velocity is obtained from

i.e.

where L is the length of the arm of the pendulum and a is the angle of drop.


155

During impact, the velocity of the table changes quickly and forces of great amplitude appear between the table and machine bases. To generate a shock of a given shape, it is necessary to control the amplitude of the force throughout the stroke during its velocity change. This is carried out using a shock programmer. Universal shock test machine

Figure 6.7. MRL universal shock test machine (impact mode)

Figure 6.8. MRL universal shock test machine (impulse mode)

The MRL Company (Monterey Research Laboratory) markets a machine allowing the carrying out of shocks according to two modes: impulse and impact [BRE 66]. In the two test configurations, the test item is installed on the upper face of the table. The table is guided by two rods which are fixed at a vertical frame.

156

Mechanical shock

To carry out a test according to the impact mode (general case), one raises the table by the height required by means of a hoist attached to the top of the frame, by the intermediary assembly for raising and dropping (Figure 6.7). By opening the blocking system in a high position, the table falls under the effect of gravity or owing to the relaxation of elastic cords if the fall is accelerated. After rebound on the programmer, the table is again blocked to avoid a second impact. NOTE: A specific device has been developed in order to make it possible to test relatively small specimens with very short duration high acceleration pulses (up to 100,000 g, 0.05 ms) on shock machines which would not otherwise be capable of generating these pulses. This shock amplifier ("Dual mass shock amplifier", marketed by MRL) consists of a secondary shock table (receiving the specimen) and a massive base which is bolted to the top of the carriage of the shock machine. When the main table impacts and rebounds from the programmer on the base of the machine (shock duration of about 6 ms), the secondary table, initially maintained above its base by elastic shock cords, continues downward, stretching the shock cords. The secondary table impacts on an high density felt programmer placed at the base of the shock amplifier, the generating the high acceleration shock. The impulse mode shocks (Figure 6.8) are obtained while placing the table on the piston of the programmer (used for the realization of initial peak saw tooth shock pulses). The piston of this hydropneumatic programmer propels the table upward according to an appropriate force profile to produce the specified acceleration signal. The table is stopped in its stroke to prevent its falling down a second time on the programmer. Pre- and post-shocks The realization of shocks on free or accelerated fall machines imposes de facto pre- shocks and/or post-shocks, the existence of which the user is not always aware, but which can modify the shock severity at low frequencies (Section 7.6). The movement of shock starts with dropping the table from the necessary height to produce the specified shock and finishes with stopping the table after rebound on the programmer. The pre-shock takes place during the fall of the table, the post-shock during its rebound. Freefall Let us set a as the rate of rebound (coefficient of restitution) of the programmer. If AV is the velocity change necessary to carry out the specified shock (AV = J x(t) dt), the carriage rebound velocity and the carriage impact velocity are

Standard shock machines es

related by [5.15] AV = v R - v i and v, =

AV

157

. One deduces from this the

1 + ct necessary drop height

where g = acceleration due to gravity.

Figure 6.9. Movement of the table

The movement of the table of the machine since the moment of its realease until impact is given by

yielding, at impact, the instant of time

where tjis the duration of the pre-shock, which has as an amplitude -g. Since the rebound velocity is equal, in absolute terms, to VR = a vi? the rebound of the carriage assembly occurs until a height HR is reached so that

158

Mechanical shock

and it lasts

The whole of the movement thus has the characteristics summarized in Figure 6.10.

Figure 6.10. Shock performed

A cceleratedfall Let us set m as the total impacting mass (table + fixture + test item), and k as the stiffness of the elastic cords.

Figure 6.11. Movement of the table during accelerated fall

Standard shock machines The differential equation of the movement

has as a solution

where

yielding

At impact, z = 0 and t = t; such that:

In addition

The impact velocity is equal to

yielding

and the duration of the pre-shock is

159

160

Mechanical shock

After the shock, the rebound is carried out with velocity VR = a v, . We have in the same way

and

6.3. High impact shock machines 6.3.1. Lightweight high impact shock machine This machine was developed in 1939 to simulate the effects of underwater explosions (mines) on the equipment onboard military ships. Such explosions, which occur basically large distances from the ships, create shocks which are propagated in all the structures. The high impact shock machine was reproduced in the United States in 1940 for use with light equipment; a third machine was built in 1942 for heavier equipment of masses ranging between 100 and 2500 kg [VIG 6la] (Section 6.3.2).


161

Figure 6.12. High impact shock machine for lightweight equipment

The procedure consisted not of specifying a shock response spectrum or a simple shape shock, but rather of the machine being used, the method of assembly, the adjustment of the machine etc. The machine consists of a welded frame of standard steel sections, of two hammers, one sliding vertically, the other describing an arc of a circle in a vertical plane, according to a pendular motion (Figure 6.12). A target plate carrying the test item can be placed to receive one or the other of the hammers. The combination of the two movements and the two positions of the target makes it possible to deliver shocks according to three perpendicular directions without disassembling the test item. Each hammer weighs approximately 200 kg and can fall a maximum height of 1.50 m [CON 52]. The target is a plate of steel of 86 cm x 122 cm x 1.6 cm, reinforced and stiffened on its back face by I-beams. In each of the three impact positions of the hammer, the target plate is assembled on springs in order to absorb the energy of the hammer with a limited displacement (38 mm to the maximum). Rebound of the hammer is prevented. Several intermediate standardized plates simulate various conditions of assembly of the equipment on board. These plates are inserted between the target and the equipment tested to provide certain insulation at the time of impact and to restore a shock considered comparable with the real shock.

162

Mechanical shock

The mass of the equipment tested on this machine should not exceed 100 kg. For fixed test conditions (direction of impact, equipment mass, intermediate plate), the shape of the shock obtained is not very sensitive to the drop height. The duration of the produced shocks is about 1ms and the amplitudes range between 5000 and 10000 m/s5. 6.3.2. Medium weight high impact shock machine This machine was designed to test equipment whose mass, including the fixure, is less than 2500 kg (Figure 6.13). It consists of a hammer weighing 1360 kg which swings through an arc of a circle at an angle greater than 180° and coming to strike an anvil at its lower face. Under the impact, this anvil, fixed under the table carrying the test item, moves vertically upwards. The movement of this unit is limited to approximately 8 cm at the top and 4 cm at the bottom ([CON 51], [LAZ67], [VIG 47] and [VIG 61b] by stops which stop it and reverse its movement. The equipment being tested is fixed on the table via a group of steel channel beams (and not directly to the rigid anvil structure), so that the natural frequency of the test item on this support metal structure is about 60 Hz. The shocks obtained are similar to those produced with the machine for light equipment. It is difficult to accept a specification which would impose a maximum acceleration. It is easier 'to control' starting from a velocity change, the function drop height of the hammer and total mass of the moving assembly (anvil, fixture and test item) [LAZ 67].

Figure 6.13. High impact machine for medium weight equipment

Standard shock machineses

163

The shocks carried out on all these facilities are not very reproducible, sensitive to the ageing of the machine and the assembly (the results can differ after dismantling and reassembling the equipment on the machine under identical conditions, in particular at high frequencies) [VIG 6la]. These machines can also be used to generate simple shape shocks such as halfsine or T.P.S. pulses [VIG 63], while inserting between the hammer and the anvil carrying the test item either an elastic or plastic material. One thus obtains durations of about 10 ms at 20 ms for the half-sine pulse and 10 ms for the TPS pulse.

6.4. Pneumatic machines Pneumatic machines in general consist of a cylinder separated in two parts by a plate bored to let pass the rod of a piston located lower down (Figure 6.14). The rod crosses the higher cylinder, comes out of the cylinder and supports a table receiving the test item.

Figure 6.14. The principle of pneumatic machines

164

Mechanical shock

The surface of the piston subjected to the pressure is different according to whether it is on the higher face or the lower face, as long as it is supported in the higher position on the Teflon seat [THO 64]. Initially, the moving piston, rod and table rose by filling the lower cylinder (reference pressure). The higher chamber is then inflated to a pressure of approximately five times the reference pressure. When the force exerted on the higher face of the piston exceeds the force induced by the pressure of reference, the piston releases. The useful surface area of the higher face increases quickly and the piston is subjected in a very short time to a significant force exerted towards the bottom. It involves the table which compresses the programmers (elastomers, lead cones etc) placed on the top of the body of the jack. This machine is assembled on four rubber bladders filled with air to uncouple it from the floor of the building. The body of the machine is used as solid mass of reaction. The interest behind this lies in its performance and its compactness.

6.5. Specific test facilities When the impact velocity of standard machines is insufficient, one can use other means to obtain the desired velocity: - Drop testers, equipped for example with two vertical (or inclined) guide cables [LAL 75], [WHI], [WHI 63]. The drop height can reach a few tens of metres. It is wise to make sure that the guidance is correct and in particular, that friction is negligible. It is also desirable to measure the impact velocity (photo-electric cells or any other device). - Gas guns, which initially use the expansion of a gas (often air) under pressure in a tank to propel a projectile carrying the test item towards a target equipped with a programmer fixed at the extremity of a gun on a solid reaction mass [LAZ 67], [LAL 75], [WHI], [WHI 63] and [YAR 65]. One finds the impact mode to be as above. It is necessary that the shock created at the time of the velocity setting in the gun is of low amplitude with regard to the specified shock carried out at the time of the impact. Another operating mode consists of using the phase of the velocity setting to program the specified shock, the projectile then being braked at the end of the gun by a pneumatic device, with a small acceleration with respect to the principal shock. A major disadvantage of guns is related to the difficulty of handling cables instrumentation, which must be wound or unreeled in the gun, in order to follow the movement of the projectile. -Inclined-plane impact testers [LAZ 67], [VIG 61b]. These were especially conceived to simulate shocks undergone during too severe handling operations or in


165

trains. They are made up primarily ofa carriage on which the test item is fixed, travelling on an inclined rail and coming to run up against a wooden barrier.

Figure 6.15. Inclined plane impact tester (CONBUR tester)

The shape of the shock can be modified y using elastomeric "bumpers' or springs. Tests of this type are often named 'CONBUR tests'.

6.6. Programmers We will describe only the most-frequently used programmers used to carry out half-sine, terminal peak saw tooth and trapezoid shock pulses. 6.6.1. Half-sine pulse These shocks are obtained using an elastic material interposed between the table and the solid mass reaction.

Shock duration The shock duration is calculated by supposing that the table and the programmer, for this length of time, constitute a linear mass-spring system with only one-degreeof-freedom. The differential equation of the movement can be written

where m = mass of the moving assembly (table + fixture + test item) k = stiffness constant of the programmer i.e.

166

Mechanical shock

The solution of this equation is a sinusoid of period T =

2n

. It is valid only

co0

during the elastomeric material compression and its relaxation, so long as there is contact between the table and the programmer, i.e. during a half-period. If t is the shock duration, we thus have

This expression shows that, theoretically, the duration can be regarded as a function alone of the mass m and of the stiffness of the target. It is in particular independent of the impact velocity. The mass m and the duration i being known, we deduce from it the stiffness constant k of the target:

Maximum deformation of the programmer If Vj is the impact velocity of the table and xm the maximum deformation of the programmer during the shock, it becomes, by equalizing the kinetic loss of energy and the deformation energy during the compression of the programmer

yielding

Shock amplitude k

From [6.25], one has, in absolute terms, m xm = k xm, yielding xm = xm — m

and, according to [6.30]


167

where the impact velocity YJ is equal to

with g = acceleration of gravity (1 g = 9.81 m/s2) H = drop height This relation, established theoretically for perfect rebound, remains usable in practice as long as the rebound velocity remains higher than approximately 50% of the impact velocity. Having determined k from m and T, it is enough to act on the impact velocity, i.e. on the drop height, to obtain the required shock amplitude. Characteristics of the target For a cylindrical programmer, we have

where S and L are respectively the cross-section and the height of the programmer and where E is Young's modulus of material in compression. Depending on the materials available, i.e. possible values of E, one chooses the values of L and S which lead to a realizable programmer (by avoiding too large a height to diameter ratio to eliminate the risks from buckling). When the table has a large surface, it is possible to place four programmers to distribute the effort. The cross-section of each programmer is then calculated starting from the value of S determined above and divided by 4. The elasticity modulus which intervenes here is the dynamic modulus, which is in general larger than the static modulus. This divergence is mainly a function of the type of elastomeric material used, although other factors such as the configuration, the deformation and the load can have an effect The ratio dynamic modulus Ed to static modulus Es ranges in general between 1 and 2. It can exceed 2 in certain cases [LAZ 67]. The greatest values of this ratio are observed with most damped materials. For materials such as rubber and Neoprene, it is close to unity.

168

Mechanical shock

Figure 6.16. High frequencies at impact

Figure 6.17. Impact module with conical impact face (open module)

If the surface of impact is plane, a wave created at the time of the impact is propagated in the cylinder and makes several up and down excursions. From it at the beginning of the signal the appearance of a high frequency oscillation which distorts the desired half-sine pulse results. To avoid this phenomenon, the front face of the programmer is designed to be slightly conical, in order to insert the load material gradually (open module). The shock thus created is between a half-sine and a versed-sine pulse.

Propagation time of the shock wave So that the target can be regarded a simple spring and not as a system with distributed constants, it is necessary that the propagation time of the shock wave through the target is weak with respect to the duration T of the shock. If a is the velocity of the sound in material constituting the target and h its height, this condition is written:

i.e., since a

(E = Young's modulus, p= density) and


If the mass of the target is equal to Mc = hSp necessary then that

169

(S = cross-section), it is

i.e.

Rebound The coefficient of restitution is a function of the material. The smallest rebounds are obtained with the elastic materials that are most strongly damped. The metal springs have small damping and thus produce significant rates of rebound, often about 75%. The elastomers vary greatly, with the rate of rebound which can be located as being between 0 and 75% of the drop height. The coefficient of restitution is also a function of the configuration and of the deformation of elastic material. The targets which are made up of very soft material, presenting great deformations, lead in general to significant rebounds, whereas the elastomeric materials, which are stiff and thin, are calculated to become deformed only by a few hundredths of millimetre, and produce only very little rebound [LAZ 67]. A not very substantial rebound can mean that the material of the programmer reacts during the impact like a viscoelastic material, the table taking a rebound velocity higher than the relaxation velocity of the material [BRO 63]. To create a perfectly half-sine shock pulse with this type of programmer, one needs a perfect rebound, with a rebound velocity equal to the impact velocity. It is necessary thus that damping is zero. The shock pulse obtained under these conditions is symmetrical. When the rate of rebound decreases, the return of acceleration to zero (relaxation) is faster than the rise of acceleration.

170

Mechanical shock

Figure 6.18. Distortsion of the half-sine pulse related to the damping of the material

A good empirical rule is to limit the maximum dynamic deformation of the programmer from 10 to 15% of its initial thickness. If this limit is exceeded, the shape obtained risks non-linear tendencies.


171

Example

Realization of a half-sine shock 300 m/s2, 10 ms. It is supposed that the mass of the moving assembly (table + fixture + test item) is equal to 600 kg. The elastomeric programmers often have a coefficient of restitution k (VR = -k Vj) of about 50 %. From [6.28]

The impact velocity is calculated from [6.31]:

2

which leads to the drop height H =

V:

_3

« 47 10

m. During the impact, the 2g elastomeric target will be deformed to a height equal to [6.30]:

The velocity change during the shock is equal to 2 2 ~> AV = - xm t = — 300 10 ^ » 1.91 m/s. It is checked that AV = 2 v{. 7i 7i with L being the height of the target, its diameter D is calculated from ES . u L

f\

")

If the target is an elastomer of hardness 60 Shore, then E » 4 10 N/m yielding, if L = 0.1 m, D « 0.137 m. It remains to check that the stress in material does not exceed the acceptable value.

172

Mechanical shock

NOTES: 1. The relations [6.27] and [6.31] were established by supposing that the material of the target is perfectly elastic and that the rebound is perfect. If it is not the case, these relations give only one approximation of xm and T (or k and vi). In difficult cases, it is undoubtedly quicker to carry out a first test, to measure the values of xm and i obtained, then to correct k and Vj using

i.e., according to the drop height

Index 1 corresponds to the first shock carried out, index 2 with the required shock. These relationship remain usable as long as there is a certain rebound and as long as the shock remains symmetrical. It is unfortunately difficult to maintain the same shape of the shock when one tries to modify its amplitude and its duration. Thus, when a rubber target is deformed by more than 30% approximately its length at rest, its characteristic force-displacement becomes non-linear, which leads to a distortion of the profile of the shock [BRO 63] [WHI] [WHI63]. 2. For a confined material (liquid for example), we have k =

^dv Sp

V (Edv = bulk dynamic modulus, V = volume of the liquid contained and Sp = effective area of the piston compressing the liquid).

The manufacturers provide cylindrical modules made up of an elastomer sandwiched between two metal plates. The programmer is composed of a stacked modules of various stiffnesses (Figure 6.19). It is enough for a relatively low number of different modules to cover a broad range of shock durations by combinations of these elements [BRE 67], [BRO 66a], [BRO 66b] and [GRA 66].

Standard shock machines es

173

Figure 6.19. Distribution of the modules (half-sine shock pulse)

The modules are in general distributed between the bottom of the table and the top of the solid mass of reaction to regularly distribute the load at the time of the shock in the lower part of the table. One thus avoids exciting its bending mode at lower frequency and amplifying the vibrations due to resonance of the table. The programmers for very short duration shock are made up of a high-strength thermoplastic material and with a large numbers of modules. The selected plastic is highly resilient and very hard. It is used within its yield stress and can thus be useful almost indefinitely. Reproducibility is very good. The programmer is composed of a cylinder of this material stuck on a plane circular plate screwed to the lower part of the table of the shock machine.

6.6.2. Terminal peak saw tooth shock pulse Programmers using crushable materials We showed that, at the time of a shock by impact without rebound, the deflection varies according to time according to the law

which can be written, since x(t) = xm — and F(t) = -m x(t),

174

Mechanical shock

To generate a terminal peak saw tooth shock pulse, any target made up of an inelastic material (crushable material) with a curve dynamic deflection-load which follows a cubic law is thus appropriate [WHI]. To obtain a perfect TPS shock pulse, it is necessary that

and, by integration

Knowing that |F = m x(t) = acr S(t), it becomes

where S(t) = surface of the programmer in contact with the table at time t. acr = crush stress of material constituting the target. The law S(t) is thus relatively complicated. If we set SQ =

—, we can write: °cr

and


Example Mass of the unit table + fixture + test item: 400 kg Maximum acceleration: 500 m/s2 Shock duration: 10 ms acr = 760 kg/cm2 = 760 104 kg/cm2

yielding

and

Figure 6.20 shows the variations of S(x) with x.

Figure 6.20. Evolution of the impact area according to the crushed length

175

176

Mechanical shock

It is supposed that S = A, x (A, = constant) and we have at time t

yielding

Let us set

This gives

which has as a solution x(t)= x m sinh£2t. Differentiating twice, we have successively x(t)= QxmcoshQt and x(t) = Q x m sinhQt. With this assumption, the rise curve is not perfectly linear. In practice, the approximation is sufficient. So that F(t)or x(t) presents a constant slope, it is thus enough that the cross-section of the programmer increases linearly according to the distance to its top (point of impact), i.e. to define a cone. For t = 0, x = 0 and x = Vj = AV . For t = T, X(T) = xm, yielding

i.e.

These relations make it possible in theory to determine the characteristics of the target. The calculations are however complex, with Q being related to X. Although it is possible to determine by calculation the required load deformation characteristic, according to a particular law of acceleration, it is very difficult to use this information in practice. The difficulty rests in the determination of the form of the programmer and the characteristic of the dynamic crushing to produce a given


177

shock. For each machine and each shock, it is necessary to carry out preliminary tests to check that the programmer is well calculated. The programmers are destroyed with each test. It is thus a relatively expensive method. One prefers to use, if possible, a universal programmer (Section 6.6.4). The material generally used is lead or honeycomb. The cones can be calculated as follows: - crushed length:

yielding the height of the cone h > 1.2 xm (to allow material to become deformed to the necessary height); - force maximum:

yielding the cross-section Sm of the cone at height xm:

When all the kinetic energy of the table is dissipated by crushing of lead, acceleration decreases to zero. The shock machine must have a very rigid solid mass of reaction, so that the time of decay to zero is not too long and satisfies the specification. The speed of this decay to zero is a function of the mass of reaction and of the mass of the table: if the solid mass of reaction has a not negligible elasticity, this time, already non-zero because of the imperfections inherent in the programmer, can become too long and unacceptable. For lead, the order of magnitude of acr is 760 kg/cm2 (7.6 107 N/m2 = 76 MPa). The range of possible durations lies between 2 and 20 ms approximately.

Penetration of a steel punch in a lead block Another method of generating a terminal using the penetration of a punch of required lead. The punch is fixed under the table of solid reaction mass. The velocity setting of

peak saw tooth shock pulse consists of form in a deformable material such as the machine, the block of lead on the the table is obtained, for example, by

178

Mechanical shock

free fall [BOC 70], [BRO 66a] and [ROS 70]. The duration and the amplitude of the shock are functions of the impact velocity and the point angle of the cone.

Figure 6.21. Realization of a TPS shock by punching of a lead block

Figure 6.22. Penetration of the steel punch in a lead block

The force which tends to slow down the table during the penetration of the conical punch in the lead is proportional to the greatest section S(x) which is penetrated, at distance x from the point. If cp is the point angle of the cone

yielding, in a simplified way, if m is the total mass of the moving assembly, by equalizing the inertia and braking forces in lead

with a being a constant function of the crush stress of lead (by supposing that only this parameter intervenes and that the other phenomena such as steel-lead friction are negligible). Let us set a

If v is the carriage velocity at the time t and Vj the impact velocity, this relation can be written


179

yielding

The constant of integration b is calculated starting from the initial conditions: for x = 0, v = Vi yielding

Let us write [6.45] in the form

it becomes by integration:

If we set

and

we obtain

Acceleration then results from [6.44]:

We have in addition v = vi yi - y . The velocity of the table is cancelled when all its kinetic energy is dissipated by the plastic deformation of lead. Then, y = 1 and

180

Mechanical shock

Knowing that vi = <J2 g H (H = drop height),

the shock duration is not very sensitive to the drop height. From these expressions, we can establish the relations:

and

In addition,

As an indication, this method allows us to carry out shocks of a few hundreds to a few thousands of grams, with durations from 4 to 10 ms approximately (for a mass m equal to 25 kg).

6.6.3. Rectangular pulse - trapezoidal pulse This test is carried out by impact. A cylindrical programmer consists of a material which is crushed with constant force (lead, honeycomb) or using the universal programmer. In the first case, the characteristics of the programmer can be calculated as follows:


181

- the cross-section is given according to the shock amplitude to be realized using the relation

yielding

-starting from the dynamics of the impact without rebound, the length of crushing is equal to

and that of the programmer must be at least equal to 1.4 xm, in order to allow a correct crushing of the matter with constant force. One can say that the shock amplitude is controlled by the cross-section of the programmer, the crush stress of material and the mass of the total carriage mass. The duration is affected only by the impact velocity. For this pulse shape also, it is possible to use the penetration of a rigid punch in a crushable material such as lead. The two methods produce relatively disturbed signals, because of impact between two plane surfaces. They are adapted only for shocks of short duration, because of the limits of deformation. A long duration indeed requires a plastic deformation over a big length; but it is difficult to maintain constant the force of resistance on such a stroke. The honeycombs lend themselves better to the realization of a long duration [GRA 66]. One could also use the shearing of a lead plate. 6.6.4. Universal shock programmer The MTS Monterey programmer known as universal can be used to produce half-sine, TPS and trapezoidal shock pulses after various adjustments. This programmer consists of a cylinder fixed under the table of the machine, filled with a gas under pressure and in the lower part of a piston, a rod and a head (Figure 6.23).

182

Mechanical shock

6.6.4.1. Generating a half-sine shock pulse The chamber is put under sufficient pressure so that, during the shock, the piston cannot move (Figure 6.23). The shock pulse is thus formatted only by the compression of the stacking of elastomeric cylinders (modular programmers) placed under the piston head. One is thus brought back to the case of Section 6.6.1.

Figure 6.23. Universal programmer MTS (half-sine and rectangle pulse configuration)

Figure 6.24. Universal programmer MTS (TPS pulse configuration)

6.6.4.2. Generating a terminal peak saw tooth shock pulse The gas pressure (nitrogen) in the cylinder is selected so that, after compression of elastomer during duration T, the piston, assembled in the cylinder as indicated in Figure 6.24, suddenly is released for a force corresponding to the required maximum acceleration xm. The pressure which was exerted before separation over the whole area of the piston applies only after separation to one area equal to that of the rod, producing a negligible resistant force.


183

Acceleration thus passes very quickly from xm to zero. The rise phase is not perfectly linear, but corresponds rather to an arc of versed-sine (since if the pressure were sufficiently strong, one would obtain a versed-sine by compression of the elastomer alone).

Figure 6.25. Realization of a TPS shock pulse

6.6.4.3. Trapezoidal shock pulse The assembly here is the same as that of the half-sine pulse (Figure 6.23). At the time of the impact, there is: - compression of the elastomer until the force exerted on the piston balances the compressive force produced by nitrogen. This phase gives the first part (rise) of the trapezoid; - up and down displacement of the piston in the part of the cylinder of smaller diameter, approximately with constant force (since volume varies little). This phase corresponds to the horizontal part of the trapezoid; - relaxation of elastomer: decay to zero acceleration. The rise and decay parts are not perfectly linear for the same reason as in the case of the TPS pulse. 6.6.4.4. Limitations Limitations of the shock machines The limitations are often represented graphically by straight lines plotted in logarithmic scales delimiting the domain of realizable shocks (amplitude, duration). The shock machine is limited by [IMP]: - the allowable maximum force on the table. To carry out a shock of amplitude xm, the force generated on the table, given by

184

Mechanical shock

must be lower or equal to the acceptable maximum force F,^. Knowing the total carriage mass, the relation [6.55] allows calculation of the possible maximum acceleration under the test conditions:

This limitation is represented on the abacus by a horizontal line xm = constant;

Figure 6.26. Abacus of the limitations of a shock machine

-the maximum free fall height H or the maximum impact velocity, i.e. the velocity change AV of the shock pulse. If VR is the rebound velocity, equal to a percentage a of the impact velocity, we have

yielding

where a is a function of the shape of the shock and of the type of programmer used. In practice, there are losses of energy by friction during the fall and especially in the programmer during the realization of the shock. To take account of these losses is difficult to calculate analytically and so one can set;


185

[6.58] where P takes into account at the same time losses of energy and rebound. As an example, the manufacturer of machine IMPAC 60 x 60 (MRL) gives, according to the type of programmers [IMP]: Table 6.1. Loss coefficient (3 Programmer

Value of P

Elastomer (half-sine pulse)

0.556

Lead (rectangle pulse)

0.2338 1.544

Lead (TPS pulse)

Figure 6.27. Drop height necessary to obtain a given velocity change

The limitation related to the drop height can be represented by parallel straight lines on a diagram giving the velocity change AV as a function of the drop height in logarithmic scales. The velocity change being, for all simple shocks, proportional to the product xm T , we have

186

Mechanical shock

yielding, while setting a =

[6.59] Table 6.2. Amplitude x duration limitation Programmer

Waveform Half-sine TPS Rectangle

(x (m/s) v m m T) 'max

Elastomer

17.7

Lead cone

10.8

Universal programmer

7.0

Universal programmer

9.2

On logarithmic scales (xm,T), the limitation relating to the velocity change is represented by parallel inclined straight lines (Figure 6.26). Limitations of programmers Elastomeric materials are used to generate shocks of -half-sine shape (or versed-sine with a conical frontal module to avoid the presence of high frequencies); - TPS and rectangular shapes, in association with a universal programmer. Elastomer programmers are limited by the allowable maximum force, a function of Young's modulus and their dimensions (Figure 6.26) [JOU 79]. This limitation is in fact related to the need to maintain the stress lower than the yield stress of material, so that the target can be regarded as a pure stiffness. The maximum stress a max developed in the target at the time of the shock can be expressed according to Young's modulus E, to the maximum deformation xm and to the thickness h of the target according to

with, for an impact with perfect rebound, the elastic ultimate stress

. It is necessary that, if Re is


187

i.e.

Exa mple MR1. IMP AC 60 x 60 shock machine Table 6.3. Examples of the characteristics of half-sine programmers Maximum force (kN) Type

Colour

Hard

Diameter 150.5 mm

Diameter 295 mm

Red

667

2 224

Mean

Blue

445

1201

Soft

Green

111

333

Taking into account the mass of the carriage assembly, this limitation can be transformed into maximum acceleration (Fm = m x m ). Thus, without a load, with a programmer made out of a hard elastomer with diameter 295 mm and a table mass of 3000 g, we have xm « 740 m/s2. With four programmers used simultaneously, maximum acceleration is naturally multiplied by four. This limitation is represented on the abacus of Figure 6.26 by the straight lines of greater slope. The universal programmer is limited [MRL]: - by the acceptable maximum force; -by the stroke of the piston: the relations established in the preceding paragraphs, for each waveform, show that displacement during the shock is always proportional to the product xm T (Figure 6.28).

188

Mechanical shock

(

Figure 6.28. Stroke limitation of universal programmers

This information is provided by the manufacturer. In short, the domain of the realizable shock pulses is limited on this diagram by straight lines representative of the following conditions:

Table 6.4. Summary of limitations on the domain of realizable shock pulses xm = constant

Acceptable force on the table or on the universal programmer

xm T = constant

Drop height (AV)

xm I2 = constant

Piston stroke of the universal programmer

xm t4 = constant

Acceptable force for elastomers

Chapter 7

Generation of shocks using shakers

In about the mid 1950s with the development of electrodynamic exciters for the realization of vibration tests, the need for a realization of shocks on this facility was quickly felt. This simulation on a shaker, when possible, indeed presents a certain number of advantages [COT 66].

7.1. Principle behind the generation of a simple shape signal versus time The objective is to carry out on the shaker a shock of simple shape (half-sine, triangle, rectangle etc) of given amplitude and duration similar to that made on the normal shock machines. This technique was mainly developed during the years 1955-1965 [WEL 61]. The transfer function between the electric signal of the control applied to the coil and acceleration to the input of the test item is not constant. It is thus necessary to calculate the signal of control according to this transfer function and the signal to be realized. One of the first methods used consisted in compensating for the system using analogue filters gauged in order to obtain a transfer function equal to H-1 (Q) (if H(Q) is the transfer function of the shaker-test item unit). The compensation must relate at the same time to the amplitude and the phase [SMA 74a]. One of the difficulties of this approach resides in the time and work needed to compensate for the system, with, in addition, a not always satisfactory result obtained.

190

Mechanical shock

The digital methods seemed to be much better. The process is as follows [FAV 69], [MAG 71]: - measurement of the transfer function of the installation (including the fixture and the test item) using a calibration signal; - calculation of the Fourier transform of the signal specified at the input of the test item; - by division of this transform by the transfer function, calculation of the Fourier transform of the signal of control; - calculation of the control signal vs time, by inverse transformation. Transfer function The measurement of the transfer function of the installation can be made using a calibration signal of the shock type, random vibration or sometimes fast swept sine [FAV 74]. In all cases, the procedure consists of measurement and calculation of the signal of control to -n dB (-12, -9, -6 and/or -3). The specified level is applied only after several adjustments on a lower level. These adjustments are necessary because of the sensitivity of the transfer function to the amplitude of the signal (nonlinearities). The development can be carried out using a dummy item representative of the mass of the specimen. However, and in particular if the mass of the test specimen is significant (with respect to that of the moving element), it is definitely preferable to use the real test item or a model with dynamic behaviour very near it. If random vibration is used as the calibration signal, its rms value is calculated in order to be lower than the amplitude of the shock (but not too distant in order to avoid the effects of any non-linearities). This type of signal can result in application to the test item of many substantial peaks of acceleration compared with the shock itself.

7.2. Main advantages of the generation of shock using shakers The realization of the shocks on shakers has very interesting advantages: - possibility of obtaining very diverse shocks shapes; - use of the same means for the tests with vibrations and shocks, without disassembly (saving of time) and with the same fixtures [HAY 63], [WEL 61]; - possibility of a better simulation of the real environment, in particular by direct reproduction of a signal of measured acceleration (or of a given shock spectrum);


191

- better reproducibility than on the traditional shock machines; - very easy realization of the test on two directions of an axis; - saves using a shock machine. In practice, however, one is rather quickly limited by the possibilities of the exciters which therefore do not make it possible to generalize their use for shock simulation.

7.3. Limitations of electrodynamic shakers 7.3.1. Mechanical limitations Electrodynamic shakers have limited performances in the following fields [MIL 64], [MAG 72]: - The maximum stroke of the coil-table unit (according to the machines being used, 25.4 or 50.8 mm peak to peak). Motion study of the coil-table assembly during the usual simple form shocks (half-sine, terminal peak saw tooth, rectangle) show that the displacement is always carried out the same side compared with the equilibrium position (rest) of the coil. It is thus possible to improve the performances for shock generation by shifting this rest position from the central value towards one of the extreme values [CLA 66] [MIL 64] [SMA 73].

Figure 7.1. Displacement of the coil of the shaker

- The maximum velocity [YOU 64]: 1.5 to 2 m/s in sine mode (in shock, one can admit a larger velocity with non-transistorized amplifiers (electronic tubes), because these amplifiers can generally accept a very short overvoltage). During the movement of the moving element in the air-gap of the magnetic coils, there is an electromotive force produced which is opposed to the voltage supply. The velocity must thus have a value such as this emf is lower than the acceptable maximum

192

Mechanical shock

output voltage of the amplifier. The velocity must in addition be zero at the end of the shock movement [GAL 73], [SMA 73]. - Maximum acceleration, related to the maximum force. The limits of velocity, displacement and force are not affected by the mass of the specimen. J.M. McClanahan and J.R. Fagan [CLA 65] consider that the realizable maxima shock levels are approximately 20% below the vibratory limit levels in velocity and in displacement. The majority of the authors agree that the limits in force are, for the shocks, larger than those indicated by the manufacturer (in sine mode). The determination of the maximum force and the maximum velocity is based, in vibration, on considerations of fatigue of the shaker mechanical assembly. Since the number of shocks which the shaker will carry out is very much lower than the number of cycles of vibrations than it will undergo during its life, the parameter maximum force can be, for the shock applications, increased considerably. Another reasoning consist of considering the acceptable maximum force, given by the manufacturer in random vibration mode, expressed by its rms value. Knowing that, one can observe random peaks being able to reach 4.5 times this value (limitation of control system), one can admit the same limitation in shock mode. One finds other values in the literature, such as: - < 4 times the maximum force in sine mode, with the proviso of not exceeding 300 g on the armature assembly [HUG]; - more than 8 times the maximum force in sine mode in certain cases (very short shocks, 0.4 ms for example) [GAL 66]. W.B. Keegan [KEE 73] and D.J. Dinicola [DIN 64] give a factor of about 10 for the shocks of duration lower than 5 ms. The limitation can also be due to: - The resonance of the moving element (a few thousands Hertz). Although it is kept to the maximum by design, the resonance of this element can be excited in the presence of signals with very short rise time. -The resistance of the material. Very great accelerations can involve a separation of the coil of the moving component.


193

7.3.2. Electronic limitations 1. Limitation of the output voltage of the amplifier [SMA 74a] which limits coil velocity. 2. Limitation of the acceptable maximum current in the amplifier, related to the acceptable maximum force (i.e. with acceleration). 3. Limitation of the bandwidth of the amplifier. 4. Limitation in power, which relates to the shock duration (and the maximum displacement) for a given mass. Current transistor amplifiers make it possible to increase the low frequency bandwidth, but do not handle even short overtensions well and thus are limited in mode shock [MIL 64].

7.4. The use of the electrohydraulic shakers Shocks are realizable on the electrohydraulic exciters, but with additional stresses: - contrary to the case of the electrodynamic shakers, one cannot obtain via these means shocks of amplitude larger than realizable accelerations in the steady mode; - the hydraulic vibration machines are in addition strongly non-linear [FAV 74].

7.5. Pre- and post-shocks 7.5.1. Requirements The velocity change

shock duration) associated with

shocks of simple shape (half-sine, rectangle, terminal peak saw tooth etc) is different from zero. At the end of the shock, the velocity of the table of the shaker must however be zero. It is thus necessary to devise a method to satisfy this need. One way of bringing back the variation of velocity associated with the shock to zero can be the addition of a negative acceleration to the principal signal so that the area under the pulse has the same value on the side of positive accelerations and the side of negative accelerations. Various solutions are possible a priori: - a pre-shock alone; - a post-shock alone;

194

Mechanical shock

- pre- and post-shocks, possibly of equal durations.

Figure 7.2. Possibilities for pre- and post-shocks positioning Another parameter is the shape of these pre- and post-shocks, the most used shapes being the triangle, the half-sine and the rectangle.

Figure 7.3. Shapes of pre- and post-shock pulses

Due to discontinuities at the ends of the pulse, the rectangular compensation is seldom satisfactory [SMA 85]. One often prefers a versed-sine applied to all the signal (Hann window) which has the advantages of being zero and smoothed at the ends (first zero derivative) and to present symmetrical pre- and post-shocks. In all the cases, the amplitude of pre- and post-shocks must remain small with respect to that of the principal shock (preferably lower than approximately 10%), in order not to deform too much the temporal signal and consequently, the shock spectrum. For a given shape of pre- and post-shocks, this choice thus imposes the duration.


195

7.5.2. Pre- or post-shock As an example, treated below is the case of a terminal peak saw tooth shock pulse (amplitude unit, duration equal to 1) with rectangular pre- and/or post-shocks (ratio p of the absolute values of the pre- and post-shocks amplitude and of the principal shock amplitude equal to 0.1).

Figure 7.4. Terminal peak saw tooth with rectangular pre- and post-shocks

Figure 7.4 shows the signal as a function of time. The selected parameter is the duration il of the pre-shock.

Figure 7.5. Influence of pre-shock duration on velocity during the shock

It is important to check that the velocity is always zero at the beginning and the end of the shock (Figure 7.5). Between these two limits, the velocity remains positive

196

Mechanical shock

when there is only one post-shock (TJ = 0) and negative for a pre-shock alone (T! = 5.05).

Figure 7.6. Influence of pre-shock duration on displacement during the shock

Figure 7.6 shows the displacement corresponding to this movement for the same values of the duration tj of the pre-shock between TJ = 0 and TJ = 5.05 s. Figure 7.7 shows that the residual displacement at the end of the shock is zero for T! » 2.4 s. The largest displacement during shock, envelope of the residual displacement and of the maximum displacement, is given according to TJ in Figure 7.8 (absolute values). This displacement has a minimum at t1 « 2s.

Figure 7.7. Influence of the pre-shock duration on the residual displacement

Figure 7.8. Influence of the pre-shock duration on the maximum displacement


197

Figure 7.9. TPS pulse with pre-shock alone (1) and post-shock alone (2)

If we compare the kinematics of the movements now corresponding to the realization of a TPS shock with only one pre-shock (1) and only one post-shock (2), we note, from Figures 7.9 to 7.11, that [YOU 64]: - the peak amplitude of the velocity is (in absolute value) identical; - in (2), the acceleration peak takes place when the velocity is very large. It is thus necessary to be able to provide the maximum force when the velocity is significant [MIL 64]; - in (1) to the contrary, the velocity is at a maximum when acceleration is zero.

Figure 7.10. Velocity curve with pre-shock alone (1) and post-shock alone (2)

Figure 7.11. Displacement curve with preshock alone (I) and post-shock alone (2)

Solution (1), which requires a less powerful power amplifier, thus seems preferable to (2). The use of symmetrical pre- and post-shocks is however better, because of a certain number of additional advantages [MAG 72]:

198

Mechanical shock

- the final displacement is minimal. If the specified shock is symmetrical (with respect to the vertical line

this residual displacement is zero [YOU 64];

- for the same duration x of the specified shock and for the same value of maximum velocity, the possible maximum level of acceleration is twice as big; - the maximum force is provided at the moment when acceleration is maximum, i.e. when the velocity is zero (one will be able to thus have the maximum current). The solution with symmetrical pre-post-shocks requires minimal electric power.

Figure 7.12. Kinematics of the movement with pre-shock alone (I), symmetrical preand post-shocks (2) and post-shock alone (3)

7.5.3. Kinematics of the movement for symmetrical pre- and post-shock 7.5.3.1. Half-sine pulse Half-sine pulse with half-sine pre- and post-shocks Duration of pre- and post-shocks [LAL 83]:


199

Figure 7.13. Half-sine -with half-sine symmetrical pre- and post-shocks The following relations give the expressions of the acceleration, the velocity and the displacement as a function of time in each interval of definition of the signal.

Velocity

Displacement

200

Mechanical shock

Half-sine pulse with triangular pre- and post-shocks

Figure 7.14. Half-sine with triangular symmetrical pre- and post-shocks

Duration of pre- and post-shocks


201

Rise time of pre-shock

(by supposing that the slope of the segment joining the top of the triangle to the foot of the half-sine is equal to the slope of the half-sine at its origin).

202

Mechanical shock

Generation of shocks using shakers Half-sine pulse with rectangular pre- and post-shocks Duration of pre- and post-shocks

Figure 7.15. Half-sine with rectangular symmetrical pre- and post-shocks

203

204

Mechanical shock


205

[7.43]

[7.44]

The expressions of the largest velocity during the movement and those of the maximum and residual displacements are brought together in Table 7.1.

Table 7.1. Half-sine ~ maximum velocity and displacement - residual displacement Symmetrical pre- and post-shocks Half-sine

Maximum velocity

Maximum displacement

Residual displacement

Half-sine

Triangles

Rectangles

Similar expressions can be established for the other shock shapes (TPS, rectangle, IPS pulses). The results appear in Tables 7.2 to 7.4.

206

Mechanical shock

7.5.3.2. TPS pulse Table 7.2. TPS pulse - maximum velocity and displacement — residual displacement Symmetrical pre- and post-shocks Half-sine

Triangles

Rectangles

TPS

Durations of pre- and postshocks

Maximum velocity Maximum displacement Residual displacement

1. TJ is the total duration of the pre-shock (or post-shock if they are equal). 12 is tne duration of the first part of the pre-shock when it is composed of two straight-line segments (or of the last part of the post-shock). 13 is the total duration of the post-shock when it is different from Tj.


207

7.5.3.3. Rectangular pulse Table 7.3. Rectangular pulse — maximum velocity and displacement - residual displacement Symmetrical pre- and post-shocks Half-sine Rectangle

Durations of pre-and postshocks Maximum velocity Maximum displacement Residual displacement

Triangles

Rectangles

208

Mechanical shock

7.5.3.4. IPS pulse Table 7.4. IPS pulse - maximum velocity and displacement - residual displacement Symmetrical pre and post-shocks Half-sine

Triangles

Rectangles

IPS

Durations of pre- and postshocks

Maximum velocity Maximum displacement Residual displacement

7.5.4. Kinematics of the movement for a pre-shock or a post-shock alone In the case of a pre-shock or a post-shock alone, the maximum velocity, equal to the velocity change AV related to the shock, takes place at the time of transition between the compensation signal and the shock itself. The displacement starts from zero and reaches its largest value at the end of the movement (without changing sign). Like the velocity, it is negative with a pre-shock and positive with a postshock. Tables 7.5-7.8 bring together the expressions for this displacement according to the shape of the principal shock and of that of the compensation signal, with the same notations and conventions as those in the preceding paragraphs.


209

Table 7.5. Half-sine withpre- or post-shock only - maximum velocity and displacement — residual displacement Pre-shock or post-shock only Half-sine

Half-sine

Triangle

Duration of pre-shock or post-shock

Maximum velocity Residual displacement

Pre-shock

Post-shock:

Rectangle

210

Mechanical shock Table 7.6. TPS with pre- or post-shock only - maximum velocity and displacement - residual displacement Pre-shock or post-shock only TPS

Half-sine

Rectangle

Triangle

Pre-shock:

Pre-shock:

Post-shock:

Post-shock:


Maximum velocity

Post-shock:

Pre-shock: Pre-shock:


Pre-shock:

Post-shock: Post-shock:

Pre-shock:

Post-shock:

Generation of shocks using shakers Table 7.7. Rectangular pulse withpre- or post-shock only — maximum velocity and displacement — residual displacement Pre-shock or post-shock only Rectangle

Half-sine

Triangle

Duration of e pre-shock or post-shock Maximum velocity Residual displacement

Pre-shock:

Post-shock:

Rectangle

211

212

Mechanical shock Table 7.8. IPS pulse with pre- or post-shock only - maximum velocity and displacement — residual displacement Pre-shock or post-shock only IPS

Half-sine


Maximum velocity


Rectangle

Triangle

Pre-shock:

Pre-shock:

Post-shock:

Post-shock:

Pre-shock:

Post-shock:

Pre-shock:

Pre-shock:

Post-shock:

Post-shock:

Pre-shock:

Post-shock:

7.5.5. Abacuses For a given shock and for given pre- and post-shocks shapes, we can calculate, by integration of the expressions of the acceleration, the velocity and the displacement as a function of time, as well as the maximum values of these parameters, in order to compare them with the characteristics of the facilities. This work was carried out for pre- and post-shocks - respectively half-sine, triangular and rectangular [LAL 83] in order to establish abacuses allowing quick

evaluation of the possibility of realization of a specified shock on a given test facility (characterized by its limits of velocity and of displacement). These abacuses are made up of straight line segments on logarithmic scales (Figure 7.16): -AA', corresponding to the limitation of velocity: the condition vm < VL (V L = acceptable maximum velocity on the facility considered) results in a relationship of the form xmt < constant (independent of p); - CC, DD, etc, greater slope corresponding to the limitation in displacement for various values of p (p = 0.05, 0.10, 0.25, 0.50 and 1.00). A particular shock will be thus realizable on shaker only if the point of coordinates T, xm (duration and amplitude of the shock considered) is located under these lines, this useful domain increasing when p increases.

Figure 7.16. Abacus of the realization domain of a shock

7.5.6. Influence of the shape of pre- and post-pulses The analysis of the velocity and the displacement varying with time associated with some simple shape shocks shows that [LAL 83]: - For all the shocks having a vertical axis of symmetry, the residual displacement is zero. - For a shock of given amplitude, duration and shape, the maximum displacement during movement is most important with half-sine pre- and postshocks. It is weaker in the case of the triangles, then rectangles.

214

Mechanical shock

- Triangular pre-post-shocks lead to the greatest duration of signal, the rectangle giving the smallest duration. Under these criteria of displacement and duration, it is thus preferable to use rectangular or triangular pre- and post-shocks. The rectangle has however the disadvantage of having slope discontinuities which make its reproduction difficult and which in addition can excite resonances at high frequencies. It seems, however, interesting to try to approach this form [MIL 64]. - The maximum displacement decreases, as one might expect, when p increases. It seems, however, hazardous to retain values higher than 0.10 (although possible with certain control systems), the total shock communicated to the specimen being then too deformed compared with the specification, which results in response spectra appreciably different from those of the pure shocks [FRA 77].

Example Half-sine shock with half-sine symmetrical pre- and post-shocks. Electrodynamic shaker

Figure 7.17. Half-sine pulse with half-sine symmetrical pre- and post-shocks


Figure 7.18. Acceleration, velocity and displacement during the shock of Figure 7.17

Figure 7.19. Abacus for a half-sine shock with half-sine pre- and post-shocks

215

216

Mechanical shock

Figure 7.20. Comparison between maximum displacements obtained with the typical shape of pre- and post-shocks

7.5.7. Optimized pre- and post-shocks At the time of the realization of a shock on shaker, the displacement starts from the equilibrium position, passes through a maximum, then returns to the initial position. One uses in fact only half of the available stroke. For better use of the capacities of the machine, we saw (Figure 7.1) that it is possible to shift the zero position of the table. Another method was developed [FAN 81] in order to fulfill the following objectives: - to take into account the tolerances on the shape of the signal allowed by the standards (R.T. Fandrich refers to standard MIL-STD 8IOC); - to best use the possibilities of the shaker. The solution suggested consists of defining: 1. A pre-shock made up of the first two terms of the development in a Fourier series of a rectangular pulse (with coefficients modified after a parametric analysis), of the form:


217

Figure 7.21. Optimizedpre-shock

The rectangular shape is preferred for the reasons already mentioned, the choice of only the first two terms of the development in series being intended to avoid the disadvantages related to slope discontinuities. The pre-shock consists of one period of this signal, each half-period having a different amplitude: - positive arch:

- negative arch:

where xm is the amplitude of the shock to be realized (in m/s2) and f is the fundamental frequency of the signal, estimated from the relationship [7.45] where g = 9.81 m/s2. This expression is calculated by setting the maximum displacement during the pre-shock lower than the possible maximum displacement on the shaker (for example 1.27 cm). This maximum displacement takes place at the end of the first arch, comparable at first approximation with a rectangle. duration, the maximum displacement is equal to yielding, if f

is its

218 if

Mechanical shock m

The total duration of the pre-shock is thus equal to

The factor of 0.05 corresponds to the tolerance limit of the quoted standard before the principal shock (5%). The constant 0.24 is the reduced amplitude of the first arch, the real amplitude for a shock of maximum value xm being equal to 0.24 (0.046 x m ). The second arch has unit amplitude. The table being, before the test, in equilibrium in a median position, the objective of this pre-shock is two fold: - to give to the velocity, just before the principal shock, a value close to one of the two limits of the shaker, so that during the shock, the velocity can use all the range of variation permitted by the machine (Figure 7.22);

Figure 7.22. Velocity during the optimizedpre-shock and the shock

Figure 7.23. Acceleration, velocity and displacement during the pre-shock


219

- to place, in the same way, the table as close as possible to one of the thrusts so that the moving element can move during the shock in all the space between the two thrusts (limitation in displacement equal, according to the machines, to 2.54 or 5.08 cm).

Figure 7.24. Acceleration, velocity and displacement during thepre-shock and the shock (half-sine) 2. A post-shock composed of one period of a signal of the shape K ty sin(2 7i fj t) where the constants K, y and fj are evaluated in order to cancel the acceleration, the velocity and the displacement at the end of the movement of the table.

Figure 7.25. Overall movement for a half-sine shock

220

Mechanical shock

The frequency and the exponent are selected in order to respect the ratio of the velocity to the displacement at the end of the principal shock. The amplitude of the post-shock is adjusted to obtain the desired velocity change. Figure 7.25 shows the total signal obtained in the case of a principal shock halfsine 30 g, 11 ms. This methodology has been improved to provide a more general solution [LAX 01]. 7.6. Incidence of pre- and post-shocks on the quality of simulation 7.6.1. General The specification of shock is in general expressed in the form of a signal varying with time (half-sine, triangle etc). We saw the need for an addition of pre- and/or post-shock to cancel the velocity at the end of the shock when it is carried out on an electrodynamic shaker. There is no difference in principle between the realization of a shock by impact after free or accelerated fall and the realization of a shock on a shaker. On a shock machine of the impact type, the test item and the table have zero velocity at the beginning of the test. The free or accelerated fall corresponds to the pre-shock phase. The rebound, if it exists, corresponds to the post-shock. The practical difference between the two methods lies in the characteristics of shape, duration and amplitude of pre- and post-shocks. In the case of impact, the duration of these signals is in general longer than in the case of the shocks on the shaker, so that the influence on the response appears for systems of lower natural frequency.

7.6.2. Influence of the pre- and post-shocks on the time history response of a onedegree-of-freedom system To highlight the problems, we treated the case of a specification which can be realized on a shaker or on a drop table, applied to a material protected by a suspension with a 5 Hz natural frequency and with a Q factor equal to 10.

Nominal shock - half-sine pulse

ms.

Generation of shocks using shakers 221

Shock on shaker - identical pre-shock and post-shock; - half-sine shape; - amplitu - duration such that:

Shock by impact -freefall

- shock with rebound to 5 - velocity of impact: - velocity of rebou

- drop height

222

Mechanical shock The duration of the fall is tj where

Duration of the rebound

Figure 7.26. Influence of the realization mode of a half-sine shock on the response of a one-degree-of-freedom system

Figure 7.26 shows the response a>0 z(t) of a one-degree-of-freedom system ( f 0 = 5 H z , £ = 0.05): - for z0 = z0 = 0 (conditions of the response spectrum); - in the case of a shock with impact; - in the case of a shock on shaker. We observed in this example the differences between the theoretical response at 5 Hz and the responses actually obtained on the shaker and shock machine. According to the test facility used, the shock applied can under-test or over-test the


223

material. For the estimate of shock severity one must take account of the whole of the signal of acceleration.

7.6.3. Incidence on the shock response spectra In Figure 7.27, for £ = 0.05 , is the response spectrum of: -the nominal shock, calculated under the usual conditions of the spectra (z0 = z0 = 0); - the realizable shock on shaker, with its pre- and post-shocks, - the realizable shock by impact, taking of account of the fall and rebound phases.

Figure 7.27. Influence of the realization mode of a half-sine shock on the SRS One notes in this example that for: - f 30 Hz, all the spectra are superimposed. This result appears logical when we remember that the slope of the shock spectrum at the origin is, for zero damping, proportional to the velocity change associated with the shock. The compensation signal added to bring back to zero the velocity change thus makes the slope of the spectrum at the origin zero. In addition, the response spectrum of the compensated signal can be larger than the spectrum of the theoretical signal close to the frequency corresponding to the inverse of the duration of the compensation signal. It is thus advisable to make sure that the

224

Mechanical shock

variations observed are not in a range which includes the resonance frequencies of the test item. This example was treated for a shock on shaker carried out with symmetrical pre- and post-shocks. Let us consider the case where only one pre-shock or one postshock is used. Figure 7.28 shows the response spectra of: - the nominal signal (half-sine, 500 m/s2, 10 ms); - a shock on a shaker with only one post-shock (half-sine, p = 0.1) to cancel the velocity change; - a shock on a shaker with a pre-shock alone; - a shock on a shaker with identical pre- and post-shocks.

Figure 7.28. Influence of the distribution of pre- and post-shocks on the SRS of a half-sine shock

It is noted that: -the variation between the spectra decreases when pre-shock or post-shock alone is used. The duration of the signal of compensation being then larger, the spectrum is deformed at a lower frequency than in the case of symmetrical pre- and post-shocks; - the pre-shock alone can be preferred with the post-shock, but the difference is weak. On the other hand, the use of symmetrical pre- and post-shocks has the already quoted well-known advantages.


225

NOTE: In the case of heavy resonant test items, or those assembled in suspension, there can be a coupling between the suspended mass m and the mass M of the coiltable -fixture unit, with resulting modification of the natural frequency according to the rule:

Figure 7.29 shows the variations of 0 /f 0 according to the ratio m / M. For m close to M, the frequency f0 can increase by a factor of about 1.4. The stress undergone by the system is therefore not as required.

Figure 7.29. Evolution of the natural frequency in the event of coupling


Chapter 8

Simulation of pyroshocks

Many works have been published on the characterization, measurement and simulation of shocks of pyrotechnic origin (generated by bolt cutters, explosive valves, separation nuts, etc) [ZIM 93]. The test facilities suggested are many, ranging from traditional machines to very exotic means. The tendency today is to consider that the best simulation of shocks measured in near-field (cf. Section 3.9) can be obtained only by subjecting the material to the shock produced by the real device (which poses the problem of the application of an uncertainty factor to cover the variability of this shock). For shocks in the mid-field , simulation can be carried out either using real the pyrotechnic source and a particular mechanical assembly or using specific equipment using explosives, or by impacting metal to metal if the structural response is more important. In the far-field, when the real shock is practically made up only of the response of the structures, a simulation on a shaker is possible (when use of this method allow).

8.1. Simulations using pyrotechnic facilities If one seeks to carry out shocks close to those experienced in the real environment, the best simulation should be the generation of shocks of a comparable nature on the material concerned. The simplest solution consists of making functional real pyrotechnic devices on real structures. Simulation is perfect but [CON 76], [LUH 76]:

228

Mechanical shock - It can be expensive and destructive.

-One cannot apply an uncertainty factor without being likely to create unrealistic local damage (a larger load, which requires an often expensive modification of the devices can be much more destructive). To avoid this problem, an expensive solution consists of carrying out several tests in a statistical matter. One often prefers to carry out a simulation on a reusable assembly, the excitation still being pyrotechnic in nature. Several devices have been designed. Some examples of which are described below: 1. A test facility made up of a cylindrical structure [IKO 64] which comprises a 'consumable' sleeve cut out for the test by an explosive cord (Figure 8.1). Preliminary tests are carried out to calibrate 'the facility' while acting on the linear charge of the explosive cord and/or the distance between the equipment to be tested (fixed on the structure as in the real case if possible) and the explosive cord.

Figure 8.1. Barrel tester for pyroshock simulation

2. For a large-sized structure subjected to this type of shock, one in general prefers to make the real pyrotechnical systems placed on the structure as they could under operating conditions. The problem of the absence of the uncertainty factor for the qualification tests remains. 3. D.E. White, R.L. Shipman and W.L. Harlvey proposed placing a greater number of small explosive charges near the equipment to be tested on the structure, in 'flowers pots'. The number of pots to be used axis depends on the amplitude of the shock, of the size of the equipment and of the local geometry of the structure. They are manufactured in a stainless steel pipe which is 10cm in height, 5 cm in interior diameter, 15 cm in diameter external and welded to approximately 13 mm steel base plates [CAR 77], [WHI65].


229

Figure 8.2. 'Flower pot 'provided with an explosive charge

A number of preliminary shots, reduced as a result of experience one acquires from experiment, are necessary to obtain the desired shock. The shape of the shock can be modified within certain limits by use of damping devices, placing the pot more or less close to the equipment, or by putting suitable padding in the pot. If, for example, one puts sand on the charge in the pot, one transmits to the structure more low frequency energy and the shape of the spectrum is more regular and smoother. One can also place a crushable material between the flower pot and the structure in order to absorb the high frequencies. When the explosive charge necessary is substantial, this process can lead to notable permanent deformations of the structure. The transmitted shock then has an amplitude lower than that sought and, to compensate, one can be tempted to use a larger charge with the following shooting. To avoid entering this vicious circle, it is preferable, with the next shooting, either to change the position of the pots, or to increase the number by using weaker charges. The advantages of this method are the following: - the equipment can be tested in its actual assembly configuration; - high intensity shocks can be obtained simultaneously along the three principal axes of the equipment. There are also some drawbacks: -no analytical method of determination a priori of the charge necessary to obtain a given shock exists; - the use of explosive requires testing under specific conditions to ensure safety; -the shocks obtained are not very reproducible, with many influential parameters; - the tests can be expensive if, each time, the structure is deformed [AER 66].

230

Mechanical shock

4. A test facility made up of a basic rectangular steel plate (Figure 8.3) suspended horizontally. This plate receives on its lower part, directly or by the intermediary of an 'expendable' item, an explosive load (chalk line, explosive in plate or bread).

Figure 8.3. Plate with resonant system subjected to detonation

A second plate supporting the test item rests on the base plate via four elastic supports. Tests carried out by this means showed that the shock spectrum generated at the input of the test item depends on: - the explosive charge; - the nature and thickness of the plate carrying the test item; - the nature of the elastic supports and their prestressing; - the nature of material of the base plate and its dimensions; constituting -the mass of the test item [THO 73]. The reproducibility of the shocks is better if the load is not in direct contact with the base plate.

8.2. Simulation using metal to metal impact The shock obtained by a metal to rnetal impact has similar characteristics to those of a pyrotechnical shock in an intermediate field: great amplitude; short duration; high frequency content; shock response spectrum comparable with a low frequency slope of 12 dB per octave etc. Simulation is in general satisfactory up to approximately 10 kHz.


231

Figure 8.4. Simulation by metal to metal impact (Hopkinson bar)

The shock can be created by the impact of a hammer on the structure itself, a Hopkinson bar or a resonant plate [BAI 79], [DAY 85], [DAV 92] and [LUH 81].

Figure 8.5. Simulation by the impact of a ball on a steel beam

With all these devices, the amplitude of the shock is controlled while acting on the velocity of impact. The frequency components are adjusted by modifying the resonant geometry of system (length of the bar between two points of fixing, the addition or removal of runners, etc) or by the addition of a deformable material between the hammer and the anvil. To generate shocks of great amplitude, the hammer can be replaced by a ball or a projectile with a plane front face made out of steel or aluminium, and launched by a pneumatic gun (air or nitrogen) [DAV 92]. The impact can be carried out directly on the resonant beam or to a surmounted plate of a resonant mechanical system composed of a plate supporting the test item connecting it to the impact plate. 8.3. Simulation using electrodynamic shakers The possibilities of creating shocks using an electrodynamic shaker are limited by the maximum stroke of the table and more especially by the acceptable maximum force. The limitation relating to the stroke is not very constraining for the pyrotechnical shocks, since they are at high frequencies. There remains a limitation on the maximum acceleration of the shock [CAR 77], [CON 76], [LUH 76] and

232

Mechanical shock

[POW 76]. If, with the reservations of Section 4.3.6, one agrees to cover only part of the spectrum, then when one makes a possible simulation on the shaker; this gives a better approach to matching the real spectrum. Exciters have the advantage of allowing the realization of any signal shape such as shocks of simple shapes [DIN 64], [GAL 66], but also random noise or a combination of simple elementary signals with the characteristics to reproduce a specified response spectrum (direct control from a shock spectrum, of. Chapter 9). The problem of the over-testing at low frequencies as previously discussed is eliminated and it is possible, in certain cases, to reproduce the real spectrum up to 1000 Hz. If one is sufficiently far away from the source of the shock, the transient has a lower level of acceleration and the only limitation is the bandwidth of the shaker, which is about 2000 Hz. Certain facilities of this type were modified in the USA to make it possible to simulate the effects of pyrotechnical shocks up to 4000 Hz. One can thus manage to simulate shocks whose spectrum can reach 7000 g [MOE 86]. We will see, however, in Chapter 9 the limits and disadvantages of this method.

8.4. Simulation using conventional shock machines We saw that, generally, the method of development of a specification of a shock consists of replacing the transient of the real environment, whose shape is in general complex, by a simple shape shock, such as half-sine, triangle, trapezoid etc, starting from the 'shock response spectrum' equivalence criterion (with the application of a given or calculated uncertainty factor^ to the shock amplitude) [LUH 76]. With the examination of the shapes of the response spectra of standard simple shocks, it seems that the signal best adapted is the terminal peak saw tooth pulse, whose spectra are also appreciably symmetrical. The research of the characteristics of such a triangular impulse (amplitude, duration) having a spectrum envelope of that of a pyrotechnical shock led often to a duration of about 1ms and to an amplitude being able to reach several tens of thousands of ms"1. Except in the case of very small test items, it is in general not possible to carry out such shocks on the usual drop tables: - limitation in amplitude (acceptable maximum force on the table); - duration limit: the pneumatic programmers do not allow it to go below 3 to 4 ms. Even with the lead programmers, it is difficult to obtain a duration of less than 2 ms. However spectra of the pyrotechnical shocks with, in general, averages close

1 cf. Volume 5.


233

to zero have a very weak slope at low frequencies, which leads to a very small duration of simple shock, of about one millisecond (or less); - the spectra of the pyrotechnical shocks are much more sensitive to the choice of damping than simple shocks carried out on shock machines. To escape the first limitation, one accepts, in certain cases, simulation of the effects of the shock only at low frequencies, as indicated in Figure 8.6. The 'equivalent' shock has in this case a larger amplitude since fa, the last covered frequency is higher.

Figure 8.6. Need for a TPS shock pulse of very short duration

Figure 8.7. Realizable durations lead to an over-test

With this approximation, the shape of the shock has little importance, all the shocks of simple shape having in the zone which interests us (impulse zone) symmetrical spectra. One however often chooses the terminal peak saw tooth to be able to reach, with lead programmers, levels of acceleration difficult to obtain with other types of programmers. This procedure, one of the first used, is open to criticism for several reasons: -if the tested item has only one frequency fa, simulation can be regarded as correct (insofar as the test facilities are able to carry out the specified shock perfectly). But very often, in addition to a fundamental frequency ff of rather low resonance such as one can realize easily for fa > ff, the specimen has other resonances at higher frequencies with substantial Q factors. In this case, all resonances are excited by shock and because of the frequency content particular to this kind of shock, the responses of the modes at high

234

Mechanical shock

frequencies can be dominating. This process can thus lead to important undertesting; -by covering only the low frequencies, one can define an 'equivalent' shock of sufficiently low amplitude to be realizable on the drop testers. However, nothing is solved from the point of view of shock duration. The limitation of 2ms on the crusher programmers or 4ms approximately on the pneumatic programmers will not make it possible to carry out a sufficiently short shock. Its spectrum will in general envelop much too much of the pyrotechnical shock at low frequencies (Figure 8.7). Except for the intersection point of the spectra (f = f a ), simulation will then be incorrect over all the frequency band. Over-testing issometimes acceptable for f < fa, and under-test beyond. We tried to show in this chapter how mechanical shocks could be simulated on materials in the laboratory. The facilities described are the most current, but the list is far from being exhaustive. Many other processes were or are still used to satisfy particular needs [CON 76], [NEL 74], [POW 74] and [POW 76].

Chapter 9

Control of a shaker using a shock response spectrum

9.1. Principle of control by a shock response spectrum 9.1.1. Problems The response spectra of shocks measured in the real environment often have a complicated shape which is impossible to envelop by the spectrum of a shock of simple shape realizable with the usual test facilities of the drop table type. This problem arises in particular when the spectrum presents an important peak [SMA 73]. The spectrum of a shock of simple shape will be: - either an envelope of the peak, which will lead to significant over-testing compared with the other Frequencies; - or envelope of the spectrum except the peak with, consequently, under-testing at the frequencies close to the peak. The simulation of shocks of pyrotechnic origin leads to this kind of situation. Shock pulses of simple shape (half-sine, terminal peak saw tooth) have, in logarithmic scales, a slope of 6 dB/octave (i.e. 45°) at low frequencies incompatible with those larger ones, of spectra of pyrotechnic shocks (> 9 dB/octave). When the levels of acceleration do not exceed the possibilities of the shakers, simulation with control using spectra are of interest.

236

Mechanical shock

Figure 9.1. Examples of SRS which are difficult to envelop with the SRS of a simple shock

The exciters are actually always controlled by a signal which is a function of time. The calculation of a shock spectrum is an unambiguous operation. There is an infinity of acceleration-time signals with a given spectrum. The general principle thus consists in searching out one of the signals x(t) having the specified spectrum. Historically, the simulation of shocks with spectrum control was first carried out using analogue and then digital methods [SMA 74a] [SMA 75].

9.1.2. Method of parallel filters The analogue method, suggested in 1964 by G.W. Painter and H.J. Parry ([PAI 64], [ROB 67], [SMA 74a], [SMA 75] and [VAN 72]) consists of using the responses of a series of filters placed simultaneously at the output of a generator of (rectangular) impulses. The filters, distributed into the third octave, are selected to cover the range of frequency of interest. Each filter output is a response impulse. If the filters are of narrow bands, each response resembles a narrow band signal which becomes established and then attenuates. If the filters are equivalent to one-degreeof-freedom systems, the response is of the decaying sinusoidal type and the reconstituted signal is oscillatory [USH 72]. Each filter is followed by an amplifier allowing regulation of the intensity of the response. All the responses are then added together and sent to the input of the amplifier which controls the shaker. One approaches the spectrum specified by modifying the gain of the amplifiers at the output of each filter. It is admitted that the output of a given filter affects only the point of the shock spectrum whose frequency is equal to the central frequency of the filter and to which the shock spectrum is insensitive with the dephasing caused by the filters or the shaker. The complete signal


237

corresponding to a flat spectrum resembles a swept sine of initial frequency equal to the central frequency of the highest filter, whose frequency decrease logarithmically to the central frequency of the lower filter [BAR 74], [HUG] and [MET 67]. The disadvantage of this process is that one does not have practically any check on the characteristics of the total control signal (shape, amplitude and duration). According to the velocity of convergence towards the specified spectrum, the adjustment of the overall signals can be in addition be extensive and result in applying several shocks to the test item to develop the control signal [MET 67]. This method also was used digitally [SMA 75], the essential difference being a greater number of possible shapes of shocks. Thereafter, one benefited from the development of data processing tools to make numerical control systems which are easier to use and use elementary signals of various shapes (according to the manufacturer) to constitute the control signal [BAR 74].

9.1.3. Current numerical methods From the data of selected points on the shock spectrum to be simulated, the calculator of the control system uses an acceleration signal with a very tight spectrum. For that, the calculation software proceeds as follows: - At each frequency f0 of the reference shock spectrum, the software generates an elementary acceleration signal, for example a decaying sinusoid. Such a signal has the property of having a shock response spectrum presenting a peak of the frequency of the sinusoid whose amplitude is a function of the damping of the sinusoid. With an identical shock spectrum, this property makes it possible to realize on the shaker shocks which would be unrealizable with a control carried out by a temporal signal of simple shape (cf. Figure 9.2). For high frequencies, the spectrum of the sinusoid tends roughly towards the amplitude of the signal. - All the elementary signals are added by possibly introducing a given delay (and variable) between each one of them, in order to control to a certain extent the total duration of the shock (which is primarily due to the lower frequency components). - The total signal being thus made up, the software proceeds to processes correcting the amplitudes of each elementary signal so that the spectrum of the total signal converges towards the reference spectrum after some interations.

238

Mechanical shock

Figure 9.2. Elementary shock (a) and its SRS (b)

Figure 9.3. SRS of the components of the required shock

The operator must provide to the software, at each frequency of the reference spectrum: -the frequency of the spectrum; - its amplitude; - a delay; - the damping of sinusoids or other parameters characterizing the number of oscillations of the signal. When a satisfactory spectrum time signal has been obtained, it remains to be checked that the maximum velocity and displacement during the shock are within


239

the authorized limits of the test facility (by integration of the acceleration signal). Lastly, after measurement of the transfer function of the facility, one calculates the electric excitation which will make it possible to reproduce on the table the acceleration pulse with the desired spectrum (as in the case of control from a signal according to time) [FAV 74]. We propose to examine below the principal shapes of elementary signals used or usable.

9.2. Decaying sinusoid 9.2.1. Definition The shocks measured in the field environment are very often responses of structures to an excitation applied upstream and are thus composed of a damped sine type of the superposition of several modal responses of [BOI 81], [CRI 78], [SMA 75] and [SMA 85]. Electrodynamic shakers are completely adapted to the reproduction of this type of signals. According to this, one should be able to reconstitute a given SRS from such signals, of the form:

where: O=2nf f = frequency of the sinusoid n = damping factor NOTE: The constant A is not the amplitude of the sinusoid, which is actually equal to [CAR 74], [NEL 74], [SMA 73], [SMA 74a], [SMA 74b] and [SMA 75]:

9.2.2. Response spectrum This elementary signal a(t) has a shock spectrum which presents a more or less significant peak to the frequency f0 = f according to the value of n. This peak increases when n decreases. It can, for very weak n (about 10-3), reach an amplitude

240

Mechanical shock

exceeding by a factor 10 the amplitude of shock according to time [SMA 73]. It is an interesting property, since it allows, for equal SRS, reduction in the amplitude of the acceleration signal by an important factor and thus the ability to carry out shocks on a shaker which could not be carried out with simple shapes.

Figure 9.4. SRS of a decaying sinusoid for various values of n

Figure 9.5. SRS of a decaying sinusoid for various values of the Q factor

When n - 0.5, the SRS tends towards that of a half-sine pulse. One should not confuse the damping factor n, which characterizes the exponential decay of the


241

acceleration signal a(t), and the damping factor E chosen for the plotting of the SRS. For given n, the SRS of the decaying sinusoid presents also a peak whose amplitude varies according to E or Q = 1/2 E (Figure 9.5). The ratio R of the peak of the spectrum to the value of the spectrum at the very high frequency is given in Figure 9.6 for various values of the damping factors n (sinusoid) and E (SRS) [SMA 75].

Figure 9.6. Amplitude of the peak of the SRS of a decaying sinusoid versus n and E

for approximated using the relation [GAL 73]:

the value of this ration can be

242

Mechanical shock

Particular case where n = E

Let us set E = n + e. It becomes 2

yielding

If e is small, we have In

and if

9.2.3. Velocity and displacement With this type of signal, the velocity and the displacement are not zero at the end of the shock. The velocity, calculated by integration of a(t) = A e"11 equal to

If t - o

The displacement is given by:

If t - o, x (t) -

o (Figure 9.7).

sin Q t, is

Control of a shaker using a shock response spectrum 243

Figure 9.7. The velocity and the displacement are not zero at the end of the damped sine

These zero values of the velocity and the displacement at the end of the shock are very awkward for a test on shaker.

9.2.4. Constitution of the total signal The total control signal is made up initially of the sum (with or without delay) of elementary signals defined separately at frequencies at each point of the SRS, added to a compensation signal of the velocity and displacement. The first stage consists of determining the constants Ai and ni of the elementary decaying sinusoids. The procedure can be as follows [SMA 74b]: - Choice of a certain number of points of the spectrum of specified shock, sufficient for correctly describing the curve (couples frequency fi, value of the spectrum Si). - Choice of damping constant ni of the sinusoids, if possible close to actual values in the real environment. This choice can be guided by examination of the shock spectra of a decaying sinusoid in reduced coordinates (plotted with the same Q factor as that of the specified spectrum), for various values of n (Figure 9.4). These curves underline the influence of n on the magnitude of the peak of the spectrum and over its width. One can also rely on the curves of Figure 9.6. But in practice, one prefers to have a rule easier to introduce into the software. The value ni ~ 0.1 gives good results [CRI 78]. It is, however, preferable to choose a variable damping factor according to the frequency of the sinusoid, strong at the low

244

Mechanical shock

frequencies and weak at the high frequencies. It can, for example, decrease in a linear way from 0.3 to 0.01 between the two ends of the spectrum. NOTE: If we have acceleration signals which lead to the specified spectrum, we could use the Prony method to estimate the frequencies and damping factors [GAR 86]. 1 -n being chosen, we can calculate, using the relation [9.3], for Q = — given

(damping chosen for plotting of the shock spectrum of reference), the ratio R of the peak of the spectrum to the amplitude of the decaying sinusoid. This value of R makes it possible to determine the amplitude amax of the decaying sinusoid at the particular frequency. Knowing that the amplitude amax of the first peak of the decaying sinusoid is related to the constant A by the relation [9.2]:

we determine the value of A for each elementary sinusoid. For n small ( < 0.08 ), we have

9.2.5. Methods of signal compensation Compensation can be carried out in several ways. 1. By truncating the total signal until it is realizable on the shaker. This correction can, however, lead to an important degradation of the corresponding spectrum [SMA 73]. 2. By adding to the total signal (sum of all the elementary signals) a highly damped decaying sinusoid, shifted in time, defined to compensate for the velocity and the displacement [SMA 74b] [SMA 75] [SMA 85]. 3. By adding to each component two exponential compensation functions, with a phase in the sinusoid [NEL 74] [SMA 75]


245

Compensation using a decaying sinusoid In order to calculate the characteristics of the compensating pulse, the complete acceleration signal used to simulate the specified spectrum can be written in the form:

where

0i is the delay applied to the ith elementary signal. Ac, wc, nc and 9 are the characteristics of the compensating signal (decaying sinusoid). The calculation of these constants is carried out by cancelling the expressions of the velocity x and of the displacement x obtained by integration of x.

246

Mechanical shock The cancellation of the velocity and displacement for t equal to infinity leads to:

yielding

Figure 9.8. Acceleration pulse compensated by a decaying sinusoid

Figure 9.9. Velocity associated with the signal compensated by a decaying sinusoid


247

Figure 9.10. Displacement associated with the signal compensated by a decaying sinusoid

The constants Ac and 0 characterizing the compensating sinusoid are thus a function of the other parameters (Qc, nc). The frequency of the compensating

(

Mechanical Vibrations and Shocks: Mechanical Shock v. 2 (Mechanical vibration & shock)

Mechanical Vibration and Shock, Mechanical Shock

Mechanical Vibrations and Shocks: Random Vibrations v. 3 (Mechanical vibration & shock)

Mechanical Vibration and Shock: Mechanical Shock (Volume 2)

Mechanical Vibrations and Shocks: Fatigue Damage v. 4 (Mechanical vibration & shock)

Mechanical Vibrations and Shocks: Specification Development v. 5 (Mechanical vibration & shock)

Mechanical Vibrations and Shocks: v.1-5

Vibration and Shock Handbook (Mechanical Engineering)

Mechanical Vibrations and Shocks: Volume 1, Sinusoidal Vibration