Mechanical Vibrations and Shocks: v.1-5

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How to go to your page This eBook contains five volumes. Each volume has its own page numbering scheme, consisting of a volume number and a page number, separated by a colon. For example, to go to page 5 of Volume 1, type 1:5 in the “page #” box at the top of the screen and click “Go.” To go to page 5 of Volume 2, type 2:5… and so forth.

sinusoidal Vibration

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Mechanical Vibration & Shock

Sinusoidal Vibration Volume I

Christian Lalanne

HPS

HERMES PENTIN 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 Nl 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 9603 1

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

Contents

Foreword to series

xi

Introduction

xv

List of symbols 1 Basic mechanics 1.1. Static effects/dynamic effects 1.2. Behaviour under dynamic load (impact) 1.2.1. Tension 1.3. Elements of a mechanical system 1.3.1. Mass 1.3.2. Stiffiiess 1.3.2.1. Definition 1.3.2.2. Equivalent spring constant 1.3.2.3. Stiffness of various parts 1.3.2.4. Non-linear stiffness 1.3.3. Damping 1.3.3.1. Definition 1.3.3.2. Hysteresis 1.3.3.3. Origins of damping 1.3.3.4. Specific damping energy 1.3.3.5. Viscous damping constant 1.3.3.6. Rheology 1.3.3.7. Damper combinations 1.3.3.8. Non-linear damping 1.3.4. Static modulus of elasticity 1.3.5. Dynamic modulus of elasticity 1.4. Mathematical models 1.4.1. Mechanical systems 1.4.2. Lumped parameter systems

xvii 1 1 3 5 6 6 7 7 8 11 14 15 15 17 18 23 24 25 26 27 29 30 32 32 33

vi

Sinusoidal vibration

1.4.3 Degrees of freedom 1.4.4. Mode 1.4.5. Linear systems 1.4.6. Linear one-degree-of-freedom mechanical systems 1.4.7. Setting an equation for n-degrees-of-freedom lumped parameter mechanical system 2 Response of a linear single-degree-of-freedom mechanical system to an arbitrary excitation 2.1. Definitions and notation 2.2. Excitation defined by force versus time 2.3. Excitation defined by acceleration 2.4. Reduced form 2.4.1. Excitation defined by a force on a mass or by an acceleration of support 2.4.2. Excitation defined by velocity or displacement imposed on support 2.5. Solution of the differential equation of movement 2.5.1. Methods 2.5.2. Relative response 2.5.2.1. General expression for response 2.5.2.2. Subcritical damping 2.5.2.3. Critical damping 2.5.2.4. Supercritical damping 2.5.3. Absolute response 2.5.3.1. General expression for response 2.5.3.2. Subcritical damping 2.5.3.3. Critical damping 2.5.3.4. Supercritical damping 2.5.4. Summary of main results 2.6. Natural oscillations of a linear single-degree-of-freedom system 2.6.1. Damped aperiodic mode 2.6.2. Critical aperiodic mode 2.6.3. Damped oscillatory mode 2.6.3.1. Free response 2.6.3.2. Points of contact of the response with its envelope 2.6.3.3. Reduction of amplitude: logarithmic decrement 2.6.3.4. Number of cycles for a given reduction in amplitude 2.6.3.5. Influence of damping on period 2.6.3.6. Particular case of zero damping 2.6.3.7. Quality factor

34

35 37 37 38

41 41 43 47 49 49 50 53 53 53 53 55 56 56 57 57 58 59 60 62 63 64 68 71 71 75 75 83 85 86 88

Contents

vii

3 Impulse and step responses 3.1. Response of a mass-spring system to a unit step function (step or indicial response) 3.1.1. Response defined by relative displacement 3.1.1.1. Expression for response 3.1.1.2. Extremum for response 3.1.1.3. First excursion of response to unit value 3.1.2. Response defined by absolute displacement, velocity or acceleration 3.1.2.1. Expression for response 3.1.2.2. Extremum for response 3.1.2.3. First passage of the response by the unit value 3.2. Response of a mass-spring system to a unit impulse excitation 3.2.1. Response defined by relative displacement 3.2.1.1. Expression forresponse 3.2.1.2. Extremum for response 3.2.2. Response defined by absolute parameter 3.2.2.1. Expression for response 3.2.2.2. Peaks of response 3.3. Use of step and impulse responses 3.4. Transfer function of a linear one-degree-of-freedom system 3.4.1. Definition 3.4.2. Calculation of H(h) for relative response 3.4.3. Calculation of H(h) for absolute response 3.4.4. Other definitions of transfer function 3.4.4.1. Notation 3.4.4.2. Relative response 3.4.4.3. Absolute response 3.4.4.4. Summary tables 3.5. Measurement of transfer function

89

97 97 98 101 102 102 102 106 108 108 110 112 119 119 121 122 125 125 125 126 126 130

4 Sinusoidal vibration 4.1. Definitions 4.1.1. Sinusoidal vibration 4.1.2. Mean value 4.1.3. Mean square value -rms value 4.1.4. Periodic excitation 4.1.5. Quasi periodic signals 4.2. Periodic and sinusoidal vibrations in the real environment 4.3. Sinusoidal vibration tests

131 131 131 132 133 135 138 139 139

5 Response of a linear single-degree-of-freedom mechanical system to a sinusoidal excitation 5.1. General equations of motion

143 144

89 89 89 93 95

viii


5.1.1. 5.1.2. 5.1.3. 5.1.4. 5.1.5. 5.1.6.

5.2. 5.3.

5.4.

Relative response Absolute response Summary Discussion Response to periodic excitation Application to calculation for vehicle suspension response Transient response 5.2.1. Relative response 5.2.2. Absolute response Steady state response 5.3.1. Relative response 5.3.2. Absolute response Responses

O)Q z x

,

Q0 z x

m

and

m

144 147 149 150 152 153 155 155 159 159 159 160

vkm z

161

''m

5.4.1.

5.5.

Variations of velocity amplitude 5.4.1.1. Quality factor 5.4.1.2. Hysteresis loop 5.4.1.3. Energy dissipated during a cycle 5.4.1.4. Half-power points 5.4.1.5. Bandwidth 5.4.2. Variations in velocity phase k z o)0 z Responses and rF A xm m 5.5.1. Variation in response amplitude 5.5.1.1. Dynamic amplification factor

162 162 164 165 168 170 174 175 175 175

TT

5.5.1.2. Width of H(h) for HRD = 5.5.2.

Variations in phase V

V

5.6. Responses —, x

5.7.

m

x

m

V

, x

m

^^ v2

179 180

FT

and —

189

Mn

5.6.1. Movement transmissibility 5.6.2. Variations in amplitude 5.6.3. Variations in phase Graphical representation of transfer functions

6 Non-viscous damping 6.1. Damping observed in real structures 6.2. Linearization of non-linear hysteresis loops - equivalent viscous damping 6.3. Main types of damping

189 190 192 194 197 197 198 202

Contents

6.3.1.

6.4.

6.5.

Damping force proportional to the power b of the relative velocity 6.3.2. Constant damping force 6.3.3. Damping force proportional to the square of velocity 6.3.4. Damping force proportional to the square of displacement 6.3.5. Structural or hysteretic damping 6.3.6. Combination of several types of damping 6.3.7. Validity of simplification by equivalent viscous damping Measurement of damping of a system 6.4.1. Measurement of amplification factor at resonance 6.4.2. Bandwidth or V2 method 6.4.3. Decreased rate method (logarithmic decrement) 6.4.4. Evaluation of energy dissipation under permanent sinusoidal vibration 6.4.5. Other methods Non-linear stiffness

ix

7 Swept sine 7.1. Definitions 7.1.1. Swept sine 7.1.2. Octave - number of octaves in a frequency interval (f l s f2) 7.1.3. Decade 7.2. 'Swept sine' vibration in the real environment 7.3. 'Swept sine' vibration in tests 7.4. Origin and properties of main types of sweepings 7.4.1. The problem 7.4.2. Case n°l: sweep where time At spent in each interval Af is constant for all values of f0 7.4.3. Casen°2: sweep with constant rate 7.4.4. Case n°3: sweep ensuring a number of identical cycles AN in all intervals Af (delimited by the half-power points) for all values of f0 8 Response of a one-degree-of-freedom linear system to a swept sine vibration 8.1. Influence of sweep rate 8.2. Response of a linear one-degree-of-freedom system to a swept sine excitation 8.2.1. Methods used for obtaining response 8.2.2. Convolution integral (or DuhamePs integral) 8.2.3. Response of a linear one-degree-of-freedom system to a linear swept sine excitation

202 203 205 206 207 208 209 210 210 212 213 220 224 224 227 227 227 228 228 229 229 231 231 234 245

246

251 251 252 252 253 255

x


8.2.4.

8.3. 8.4. 8.5.

Response of a linear one-degree-of-freedom system to a logarithmic swept sine Choice of duration of swept sine test Choice of amplitude Choice of sweep mode

Appendix. Laplace transformations A.I. Definition A.2. Properties A.2.1. Linearity A.2.2. Shifting theorem (or time displacement theorem) A.2.3. Complex translation A.2.4. Laplace transform of the derivative of f(t) with respect to time A.2.5. Derivative in the/? domain A.2.6. Laplace transform of the integral of a function f(t) with respect to time A.2.7. Integral of the transform F(p) A.2.8. Scaling theorem A.2.9. Theorem of damping or rule of attenuation A.3. Application of Laplace transformation to the resolution of linear differential equations A.4. Calculation of inverse transform: Mellin-Fourier integral or Bromwich transform A.5. Laplace transforms A.6. Generalized impedance - the transfer function

265 269 270 271 281 281 282 282 282 283 283 283 284 284 284 285 285 287 289 292

Vibration tests: a brief historical background

295

Bibliography

297

Index

309

Foreword to series

In the course of their lifetime simple items in everyday use such as mobile telephones, wristwatches, electronic components in cars or more specific items such as satellite equipment or flight systems in aircraft, can be subjected to various conditions of temperature and humidity and more especially to mechanical shock and vibrations, which form the subject of this work. They must therefore be designed in such a way that they can withstand the effect of the environment conditions they are exposed to without being damaged. Their design must be verified using a prototype or by calculations and/or significant laboratory testing. Currently, sizing and testing are performed on the basis of specifications taken from national or international standards. The initial standards, drawn up in the 1940s, were blanket specifications, often extremely stringent, consisting of a sinusoidal vibration the frequency of which was set to the resonance of the equipment. They were essentially designed to demonstrate a certain standard resistance of the equipment, with the implicit hypothesis that if the equipment survived the particular environment it would withstand undamaged the vibrations to which it would be subjected in service. The evolution of those standards, occasionally with some delay related to a certain conservatism, followed that of the testing facilities: the possibility of producing swept sine tests, the production of narrow-band random vibrations swept over a wide range, and finally the generation of wide-band random vibrations. At the end of the 1970s, it was felt that there was a basic need to reduce the weight and cost of on-board equipment, and to produce specifications closer to the real conditions of use. This evolution was taken into account between 1980 and 1985 concerning American standards (MIL-STD 810), French standards (GAM EG 13) or international standards (NATO) which all recommended the tailoring of tests. Current preference is to talk of the tailoring of the product to its environment in order to assert more clearly that the environment must be taken into account as from the very start of the project, rather than to check the behaviour of the material a posteriori. These concepts, originating with the military, are currently being increasingly echoed in the civil field.

xii


Tailoring is based on an analysis of the life profile of the equipment, on the measurement of the environmental conditions associated with each condition of use and on the synthesis of all the data into a simple specification, which should be of the same severity as the actual environment. This approach presupposes a proper understanding of the mechanical systems subjected to dynamic loads, and knowledge of the most frequent failure modes. Generally speaking, a good assessment of the stresses in a system subjected to vibration is possible only on the basis of a finite elements model and relatively complex calculations. Such calculations can only be undertaken at a relatively advanced stage of the project once the structure has been sufficiently defined for such a model to be established. Considerable work on the environment must be performed, independently of the equipment concerned, either at the very beginning of the project at a time where there are no drawings available or at the qualification stage, in order to define the test conditions. In the absence of a precise and validated model of the structure, the simplest possible mechanical system is frequently used consisting of mass, stiffness and damping (a linear system with one degree of freedom), especially for: -the comparison of the severity of several shocks (shock response spectrum) or of several vibrations (extreme response and fatigue damage spectra); - the drafting of specifications: determining a vibration which produces the same effects on the model as the real environment, with the underlying hypothesis that the equivalent value will remain valid on the real, more complex structure; - the calculations for pre-sizing at the start of the project; - the establishment of rules for analysis of the vibrations (choice of the number of calculation points of a power spectral density) or for the definition of the tests (choice of the sweep rate of a swept sine test). This explains the importance given to this simple model in this work of five volumes on "Vibration and Mechanical Shock": Volume 1 is devoted to sinusoidal vibration. The responses, relative and absolute, of a mechanical one-degree-of-freedom system to an arbitrary excitation are considered, and its transfer function in various forms defined. By placing the properties of sinusoidal vibrations in the contexts of the environment and of laboratory tests, the transitory and steady state response of a single-degree-offreedom system with viscous and then with non-linear damping is evolved. The

Foreword to series

xiii

various sinusoidal modes of sweeping with their properties are described, and then, starting from the response of a one-degree-of-freedom system, the consequences of an unsuitable choice of the sweep rate are shown and a rule for choice of this rate deduced from it. Volume 2 deals with mechanical shock. This volume presents the shock response spectrum (SRS) with its different definitions, its properties and the precautions to be taken in calculating it. The shock shapes most widely used with the usual test facilities are presented with their characteristics, with indications how to establish test specifications of the same severity as the real, measured environment. A demonstration is then given on how these specifications can be made with classic laboratory equipment: shock machines, electrodynamic exciters driven by a time signal or by a response spectrum, indicating the limits, advantages and disadvantages of each solution. Volume 3 examines the analysis of random vibration, which encompass the vast majority of the vibrations encountered in the real environment. This volume describes the properties of the process enabling simplification of the analysis, before presenting the analysis of the signal in the frequency domain. The definition of the power spectral density is reviewed as well as the precautions to be taken in calculating it, together with the processes used to improve results (windowing, overlapping). A complementary third approach consists of analyzing the statistical properties of the time signal. In particular, this study makes it possible to determine the distribution law of the maxima of a random Gaussian signal and to simplify the calculations of fatigue damage by avoiding direct counting of the peaks (Volumes 4 and 5). Having established the relationships which provide the response of a linear system with one degree of freedom to a random vibration, Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behaviour of a material subjected to fatigue, the laws of damage accumulation, together with the methods for counting the peaks of the response, used to establish a histogram when it is impossible to use the probability density of the peaks obtained with a Gaussian signal. The expressions of mean damage and of its standard deviation are established. A few cases are then examined using other hypotheses (mean not equal to zero, taking account of the fatigue limit, non linear accumulation law, etc.). Volume 5 is more especially dedicated to presenting the method of specification development according to the principle of tailoring. The extreme response and fatigue damage spectra are defined for each type of stress (sinusoidal vibrations, swept sine, shocks, random vibrations, etc.). The process for establishing a specification as from the life cycle profile of the equipment is then detailed, taking account of an uncertainty factor, designed to cover the uncertainties related to the dispersion of the real environment and of the mechanical strength, and of another

xiv


coefficient, the test factor, which takes into account the number of tests performed to demonstrate the resistance of the equipment. This work is intended first and foremost for engineers and technicians working in design teams, which are responsible for sizing equipment, for project teams given the task of writing the various sizing and testing specifications (validation, qualification, certification, etc.) and for laboratories in charge of defining the tests and their performance, following the choice of the most suitable simulation means.

Introduction

Sinusoidal vibrations were the first to be used in laboratory tests to verify the ability of equipment to withstand their future vibratory environment in service without damage. Following the evolution of standards and testing facilities, these vibrations, generally speaking, are currently studied only to simulate vibratory conditions of the same nature, as encountered, for example, in equipment situated close to revolving machinery (motors, transmission shafts, etc). Nevertheless, their value lies in their simplicity, enabling the behaviour of a mechanical system subjected to dynamic stress to be demonstrated and the introduction of basic definitions. Given that generally speaking the real environment is more or less random in nature, with a continuous frequency spectrum in a relatively wide range, in order to overcome the inadequacies of the initial testing facilities testing rapidly moved to the "swept sine" type. Here the vibration applied is a sinusoid the frequency of which varies over time according to a sinusoidal or exponential law. Despite the relatively rapid evolution of electrodynamic exciters and electrohydraulic vibration exciters, capable of generating wide-band random vibrations, these swept sine standards have lasted, and are in fact still used, for example in aerospace applications. They are also widely used for measuring the dynamic characteristics of structures. Following a few brief reminders of basic mechanics (Chapter 1), Chapter 2 of this volume on "Sinusoidal Vibration" examines the relative and absolute response of a mechanical system with one degree of freedom subjected to a given excitation, and defines the transfer function in different forms. Chapter 3 is devoted more particularly to the response of such a system to a unit impulse or to a unit step. The properties of sinusoidal vibrations are then presented hi the context of the environment and in laboratory tests (Chapter 4). The transitory and steady state

xvi


response of a system with one degree of freedom to viscous damping (Chapter 5) and to non-linear damping (Chapter 6) is then examined. Chapter 7 defines the various sinusoidal sweeping modes, with their properties and eventual justification. Chapter 7 is also devoted to the response of a system with one degree of freedom subjected to linear and exponential sweeping vibrations, to illustrate the consequences of an unsuitable choice of sweep rate, resulting in the presentation of a rule for the choice of rate. In the appendix are reviewed the major properties of the Laplace transform, which provides a powerful tool for the analytical calculation of the response of a system with one degree of freedom to a given excitation. Inverse transforms more particularly suitable for this application are given in a table.

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 confusion. A(t) A(p) c Cgq C( 0) D e E Ed E( ) f fm f f0 F(t) Ft,

Indicial admittance or step response Laplace transform of A(t) Viscous damping constant Equivalent viscous damping constant Part of the response relating to non-zero initial conditions Damping capacity Neper number Young's modulus Dynamic modulus of elasticity Function characteristic of sweep mode Frequency of excitation Expected frequency Sweep rate Natural frequency External force applied to a system Damping force

Fc Ff Fm g G G(TI) h H^ HRD HRV h(t) H()

iI

J k ^ rms tm

Peak factor (or crest factor) Form factor Maximum value of F(t) Acceleration due to gravity Coulomb modulus Attenuation related to sweep rate Interval (f/f 0 ) Transmissibility Dynamic amplification factor Relative transmissibility Impulse response Transfer function

v=i

Moment of inertia Damping constant Stiffness or uncertainty coefficient Tms value of t( t) Maximum value of i( t)

xviii

^(t)


^(t)

Generalized excitation (displacement) First derivative of t(t)

"(.(t) L( p)

Second derivative of ^(t) Laplace transform of i( t)

L(Q) m n

Fourier transform of t( t) Mass Number of cycles

nd N Ns

Number of decades Normal force Number of cycles performed during swept sine test Laplace variable Reduced pseudo-pulsation Maximum value of q(6) Value of q(0) for 6 = 0 Reduced response Value of q(0) for 0=0 First derivative of q (0) Second derivative of q(0) Q factor (quality factor) Laplace transform of q(0) Ultimate tensile strength Number of octaves per minute Number of octaves per second Time Sweep duration Duration of application of vibration Natural period Time-constant of logarithmic swept sine Generalized response Maximum elastic strain energy stored during one cycle

p P qm q0 q(0) q0 q (9) q(0) Q Q(p) Rm ^•om Ros t ts T TQ Tj u(t) Us

Uts U(p) U(Q) xm x(t) x(t) xm x(t) X(Q) y(t)

y(t) y(t)

zm zs z(t)

z(t) z(t) z(p)

Elastic strain energy per unit volume Laplace transform of u(t) Fourier transform of u(t) Maximum value of x(t) Absolute displacement of the base of a one-degreeof-freedom system Absolute velocity of the base of a one-degree-offreedom system Maximum value of x(t) Absolute acceleration of the base of a one-degreeof-freedom system Fourier transform of x(t) Absolute displacement response of the mass of a one-degree-of-freedom system Absolute velocity response of the mass of a onedegree-of-freedom system Absolute acceleration response of the mass of a one-degree-of-freedom system Maximum value of z(t) Maximum static relative displacement Relative displacement response of the mass of a one-degree-of-freedom system with respect to its base Relative velocity response Relative acceleration response Generalized impedance

List of symbols

8 5g( ) A AEd Af AN e e r|

Logarithmic decrement Dirac delta function Energy dissipated per unit time Energy dissipated by damping in one cycle Interval of frequency between half-power points Number of cycles between half-power points Relative deformation Velocity of relative deformation Coefficient of dissipation (or of loss) or reduced sweep rate

cp X,(9) A (p) |ii 7i 0 9jj © a am co0 Q £ H,eq \j/

xix

Phase Reduced excitation Laplace transform of >.( 6) Coefficient of friction 3.14159265... Reduced time q (6) Reduced sweep rate Reduced pseudo-period Stress Mean stress Natural pulsation (2 TT f0) Pulsation of excitation (2*f) Damping factor Equivalent viscous damping factor Phase


Chapter 1

Basic mechanics

1.1. Static effects/dynamic effects In order to evaluate the mechanical characteristics of materials, it is important to be aware of the nature of stresses [HAU 65], There are two main load types that need to be considered when doing this: - those which can be considered as applied statically; - those which are the subject of a dynamic analysis of signal versus time. Materials exhibit different behaviours under static and dynamic loads. Dynamic loads can be evaluated according to the following two criteria: - the load varies quickly if it communicates notable velocities to the particles of the body in deformation, so that the total kinetic energy of the moving masses constitutes a large part of the total work of the external forces; this first criterion is that used during the analysis of the oscillations of the elastic bodies; - the speed of variation of the load can be related to the velocity of evolution of the plastic deformation process occurring at a tune of fast deformation whilst studying the mechanical properties of the material. According to this last criterion, plastic deformation does not have time to be completed when the loading is fast. The material becomes more fragile as the deformation velocity grows; elongation at rupture decreases and the ultimate load increases (Figure 1.1).

2


Figure 1.1. Tension diagram for static and dynamic loads

Thus, a material can sometimes sustain an important dynamic load without damage, whereas the same load, statically, would lead to plastic deformation or to failure. Many materials subjected to short duration loads have ultimate strengths higher than those observed when they are static [BLA 56], [CLA 49], [CLA 54] and [TAY 46]. The important parameter is in fact the relative deformation velocity, defined by:

where kt is the deformation observed in time At on a test-bar of initial length 10 subjected to stress. If a test-bar of initial length 10 cm undergoes in 1 s a lengthening of 0.5 cm, the relative deformation velocity is equal to 0.05 s"1. The observed phenomena vary according to the value of this parameter. Table 1.1 shows the principal fields of study and the usable test facilities. This book will focus on the values in the region 10"1 to 101 s-1 (these ranges being very approximate). Certain dynamic characteristics require the data of the dynamic loads to be specified (the order of application being particularly important). Dynamic fatigue strength at any time t depends, for example, on the properties inherent hi the material concerning the characteristics of the load over time, and on the previous use of the part (which can reflect a certain combination of residual stresses or corrosion).

Basic mechanics

3

Table 1.1. Fields of deformation velocity

lo-1

5 10~ 10

0

Evolution of Constant Phenomenon creep velocity velocity of in time deformation Type of test

Creep

Statics

Test facility

Machines with constant load

Hydraulic machine

Negligible inertia forces

101

Response of structure, resonance

105

Propagation of Propagation of elastoplastic shock waves waves

Very fast Fast dynamics dynamics (impact) (hypervelocity) Hydraulic Impact Explosives vibration metal - metal Pyrotechnic Gas guns machine Shakers shocks Important inertia forces Slow dynamics

1.2. Behaviour under dynamic load (impact) Hopkinson [HOP 04] noted that copper and steel wire can withstand stresses that are higher than their static elastic limit and are well beyond the static ultimate limit without separating proportionality between the stresses and the strains. This is provided that the length of time during which the stress exceeds the yield stress is of the order of 10

seconds or less.

From tests carried out on steel (annealed steel with a low percentage of carbon) it was noted that the initiation of plastic deformation requires a definite time when stresses greater than the yield stress are applied [CLA 49]. It was observed that this time can vary between 5 ms (under a stress of approximately 352 MPa) and 6 s (with approximately 255 MPa; static yield stress being equal to 214 MPa). Other tests carried out on five other materials showed that this delay exists only for materials for which the curve of static stress deformation presents a definite yield stress, and the plastic deformation then occurs for the load period. Under dynamic loading an elastic strain is propagated in a material with a velocity corresponding to the sound velocity c0 in this material [CLA 54]. This velocity is a function of the modulus of elasticity, E, and of the density, p, of material. For a long, narrow part, we have:

4


The longitudinal deflection produced in the part is given by:

where Vj= velocity of the particles of material. In the case of plastic deformation, we have [KAR 50]:

a l):

Arbitrary excitation of simple mechanical system

where

since

yielding

2.5.3. Absolute response 2.5.3.1. General expression for response The solution of a differential second order equation of the form:

has as a Laplace transform Q(p) [LAL 75]:

57

58


where

As previously:

q(0) is obtained by searching the original of Q(p)

(a = variable of integration). Particular case

2.5.3.2. Subcritical damping The roots of p2 + 2 £ p +1 are complex (0 < £ < l)


59

While replacing p1 and p2 by their expressions in q(0), it becomes:

If moreover £ = 0

2.5.3.3. Critical damping The equation p2 +2 £ p + l = 0 has a double root (p = -l). In this case £ = 1

and

60


2.5.3.4. Supercritical damping The equation p + 2 4 p +1 = 0 has two real roots. This condition is carried out when £ > 1. Let us replace expressions in [2.56] [KIM 26]

yielding

where

by their

Arbitrary excitation of simple mechanical system Another form

with

If

If moreover £ = 0:

61

62


2.5.4. Summary of main results Zero initial conditions: Relative response

Absolute response

If the initial conditions are not zero, we have to add to these expressions, according to the nature of the response.


63

For Relative response

Absolute response

For Relative response Absolute response

For Relative response

L

Absolute response

In all these relations, the only difference between the cases 0 < £ < 1 and £, > 1 resides in the nature of the sine and cosine functions (hyperbolic for £ > 1).

2.6. Natural oscillations of a linear single-degree-of-freedom system We have just shown that the response q(0) can be written for non-zero initial conditions:

The response qic(0) is equal to the sum of the response q(9) obtained for zero initial conditions and the term C(0) corresponds to a damped oscillatory response

64


produced by non-zero initial conditions. So A,(0) = 0 is set in the differential equation of the movement F2.211:

it then becomes, after a Laplace transformation,

The various cases related to the nature of the roots of:

are considered here. 2.6.1. Damped aperiodic mode In this case £, > 1. The two roots of p2 + 2 £ p + l = 0 are real. Suppose that the response is defined by an absolute movement. If this were not the case, it would be enough to make X0 = 0 in the relations of this paragraph. The response q(0) of the system around its equilibrium position is written:

q(0) can also be written in the form:

where

and


65

It is noticed that the roots -£ + V£ -1 and -£ - v£ -1 of the equation p + 2 £ , p + l = 0are both negative, their sum negative and their product positive. So the two exponential terms are decreasing functions of time, like q(9). dq The velocity —, which is equal to:

ae

is also, for the same reason, a decreasing function of time. Therefore, the movement cannot be oscillatory. It is a damped exponential motion. q(0) can also be written:

-2^~l

When 9 tends towards infinity, e ^^^ * tends towards zero (%> l). After a certain time, the second term thus becomes negligible hi comparison with the unit and q(6) behaves then like:

As I -£ + -v£ -1 I is always negative, q(0) decreases constantly according to tune. If the system is moved away from its equilibrium position and released with a zero velocity qn with an elongation qn at tune t = 0, coefficient a becomes, for

66


q0 being supposed positive, a being positive and q(0) always remaining positive: the system returns towards its equilibrium position without crossing it. The velocity can also be written:

Figure 2.8. Damped aperiodic response

We have:

when 6 is sufficiently large. The velocity is then always negative.

Variations of the roots pj and P2 according to % The characteristic equation [2.86] p + 2 £ p + l = 0is that of a hyperbole (in the axes p, £) whose asymptotic directions are:


67

i.e.

The tangent parallel with the axis Op is given by 2 p + 2 £ = 0, i.e. p = -£ .

Figure 2.9. Characteristic equation This yields £, = 1 while using [2.86] (since £ is positive or zero), i.e.

Any parallel with the axis Op such as £ > 1 crosses the curve at two points corresponding to the values p1 and p2 of p. The system returns all the more quickly to its equilibrium position, so that q(0) decreases quickly, therefore p1 is larger (the time-constant, in the expression is of value

, i.e. the relative damping £

is still smaller (or the coefficient of energy dissipation c). 2 P! has the greatest possible value when the equation p + 2 £ p +1 = 0 has a double root, i.e. when £ = 1.

68


NOTE: If the system is released out of its equilibrium position with a zero initial velocity, the resulting movement is characterized by a velocity which changes only once in sign. The system tends to its equilibrium position without reaching it. It is said that the motion is 'damped aperiodic' (t, > \). Damping is supercritical.

Figure 2.10. Aperiodic damped response

Figure 2.11. Aperiodic damped response

2.6.2. Critical aperiodic mode On the assumption that £ = 1, the two roots ofp -1. By definition

2

+ 2£p + l = 0 a r e equal to

yielding c = cc = 2Vkm. The parameter cc, the critical damping coefficient, is the smallest value of c for which the damped movement is non-oscillatory. It is the reason why £ is also defined as infraction of critical damping or critical damping ratio. According to whether the response is relative or absolute, the response q(0) is equal to:


69

or

As an example, Figures 2.12-2.14 show q(9), respectively, for A,0 = 0, q0 =q0 = 1, q 0 = 0 and q0 = l,thenq 0 =1 and q0 = 0.

Figure 2.12. Critical aperiodic response

Figure 2.13. Critical aperiodic response

q(9) can be written q(0) = — + (q0 + q0) 0 e . For rather large 0, — becomes _ 9 J 9 negligible and q(9) behaves like (q0 +q 0 ) 9e~ : q(9) thus tends towards zero when 9 tends towards infinity. This mode, known as critical, is not oscillatory. It corresponds to the fastest possible return of the system towards the equilibrium position fastest possible among all the damped exponential movements. If we write qc(9) is written as the expression of q(9) corresponding to the critical mode, this proposal can be verified while calculating:

70


Consider the expression of q(0) [2.88] given for £ > 1 in the form of a sum, the exponential terms eventually become:

Figure 2.14. Critical aperiodic response ( 1, this means

that the exponential term tends towards zero when 0 tends towards infinity and consequently:

This return towards zero is thus performed more quickly in critical mode than in damped exponential mode.

2.6.3. Damped oscillatory mode It is assumed that 0 < £, < 1. 2.6.3.1. Free response The equation p + 2 £ p + l = 0 has two complex roots. Let us suppose that the response is defined by an absolute movement (A,0 = 0 for a relative movement). The response

can be also written:

with

72


The response is of the damped oscillatory type with a pulsation equal to I

2n

2

P = VI - £ » which corresponds to a period 0 = —. It is said that the movement is P damped sinusoidal or pseudo-sinusoidal. The pseudo-pulsation P is always lower than 1. For the usual values of £, the pulsation is equal to 1 at first approximation

fe L the points of contact of the curve can be determined with its envelope by seeking 9 solutions o f s i n i - \ / l - ^ 9 + <J> I = 1.

e

These points are separated by time intervals equal to —. 2 The points of intersection of the curve with the time axis are such that s i n ( p 0 + <J>) = 0. The maximum response is located a little before the point of contact of the curve with its envelope.

0 The system needs a little more than — to pass from a maximum to the next 4 0 position of zero displacement and little less than — to pass from this equilibrium 4 position to the position of maximum displacement.

Figure 2.21. Points of contact with the envelope 2.6.3.3. Reduction of amplitude: logarithmic decrement Considering two successive maximum displacements qlM and q2M

76


Figure 2.22. Successive maxima of the response

where the times 01 and 02 are such that

i.e. if


77

However, ±sin(p 9j + <j>) = ±sin(P 62 + <j>) (± according to whether they are two maxima or two minima, but the two signs are taken to be identical), yielding

The difference 92 - 6j is the pseudo-period®.

while 8 = £ ® is called the logarithmic decrement.

This is a quantity accessible experimentally. In practice, if damping is weak, the measurement of 5 is inprecise when carried out from two successive positive peaks. It is better to consider n pseudo-periods and 5 is then given by [HAB 68]:

i*i

Here n is the number of positive peaks. Indeed, the ratio of the amplitude of any two consecutive peaks is [HAL 78], [LAZ 68]:

yielding

78


Figure 2.23. Successive peaks of the response

NOTES: I. It is useful to be able to calculate the logarithmic decrement 8 starting from a maximum and a successive minimum. In this case, we can 'write:

yielding


79

I f n i s even (positive or negative peaks), which corresponds to a first positive peak and the last negative peak (or the reverse), we have:

2. The decrement 6 can also be expressed according to the difference successive peaks:

(indices 1 and 2 or more generally n and n + \).

Figure 2.24. Different peaks If damping is weak, qlM and q2M are not very different and if we set:

Aq can be considered as infinitely small:

From

of two

80


yielding

In the case of several peaks, we have:

and

Since

we can connect the relative damping, £, and the logarithmic

decrement, 8, by:

Knowing that [HAB

or


81

Example | LAZ 501 Table 2.3. Examples of decrement and damping values Material

8

t

Concrete

0.06

0.010

Bolted steel

0.05

0.008

Welded steel

0.03

0.005

NOTE: If^ is very small, in practice less than 0.10, £ can, at first approximation, be neglected. Then:

Figure 2.25 represents the variations in the decrement 8 with damping £ and shows how, in the vicinity of the origin, one can at first approximation confuse the curve with its tangent [THO 65a].

Figure 2.25. Variations in decrement with damping

82


We defined 5 starting from [2.127]:

yielding, by replacing 5 with the expression [2.137],

i.e.

9

Figure 2.26. Reduction of amplitude with damping

The curves in Figure 2.26 give the ratio n. For very small £, we have at first approximation:

versus £, for various values of


2.6.3.4. Number of cycles for a given reduction in amplitude Amplitude reduction of 50% On the assumption that

relation [2.125] becomes

If ^ is small:

83

84


Figure 2.27. Number of cycles for amplitude reduction of 50%

The curve in Figure 2.27 shows the variations of n versus £, in the domain where the approximation 5 » 2rc £ is correct [THO 65a].

Amplitude reduction of 90% In the same way, we have:

and for small values of £:


85

Figure 2.28. Number of cycles for an amplitude reduction of 90%

Reduction ofo% More generally, the number of cycles described for an amplitude reduction of a% will be, for small £:

2.6.3.5. Influence of damping on period Except if it is very large, damping in general has little influence over the period; we have

for

and, for small £,

For most current calculations, it is possible to confuse ® with 00. For the first order, the pulsation and the period are not modified by damping. For the second

86


order, the pulsation is modified by a corrective term that is always negative and the period is increased:

and

(P0 = l) or

NOTE: The logarithmic decrement also represents the energy variation during a cycle of decrease. For sufficiently small o we have [LAZ 50]:

|Energy(n -1) = Energy to the (n - if1 cycle] In practical cases where £, lies between 0 and 1, the energy initially provided to the system dissipates itself little by little in the external medium in various forms (friction between solid bodies, with air or another fluid, internal slips in the metal during elastic strain, radiation, energy dissipation in electromagnetic form). Consequently, the amplitude of the oscillations decreases constantly with time. If we wanted to keep a constant amplitude, it would be necessary to restore the system with the energy which it loses every time. The system is then no longer free: the oscillations are maintained or forced. We will study this case in Chapter 5.

2.6.3.6. Particular case of zero damping In this case, q(0) becomes:


87

which can be also written as:

where

If it is supposed that the mechanical system has moved away from its equilibrium position and then is released in the absence of any external forces at time t = 0, the response is then of the non-damped oscillatory type for £, = 0. In this (theoretical) case, the movement of natural pulsation co0 should last indefinitely since the characteristic equation does not contain a first order term. This is the consequence of the absence of a damping element. The potential energy of the spring decreases by increasing the kinetic energy of the mass and vice versa: the system is known as conservative. In summary, when the relative damping ^ varies continuously, the mode passes without discontinuity to one of the following: £=0

Undamped oscillatory mode.

0 < % < 1 Damped oscillatory mode. The system moves away from its equilibrium position, oscillating around the equilibrium point before stabilizing. £=1

Critical aperiodic mode, corresponding to the fastest possible return of the system without crossing the equilibrium position.

2, > 1

Damped aperiodic mode. The system returns to its equilibrium position without any oscillation, all the more quickly because £ is closer to the unit.

In the following chapters, we will focus more specifically on the case 0 < £ < 1, which corresponds to the values observed in the majority of real structures.

88


Figure 2.29. Various modes for q0 = 1, q0 = 1

Figure 2.30. Various modes for q0 = 1, q0 = 0 2.6.3.7. Quality factor The quality factor or Q factor of the oscillator is the number Q defined by:

The properties of this factor will be considered in more detail in the following chapters.

Chapter 3

Impulse and step responses

3.1. Response of a mass-spring system to a unit step function (step or indicial response) 3.1.1. Response defined by relative displacement 3.1.1.1. Expression for response Let us consider a damped mass-spring system. Before the initial time t = 0 the mass is assumed to be at rest. At time t = 0, a constant excitation of unit amplitude continuously acts for all t > 0 [BRO 53], [KAR 40]. We have seen that, for zero initial conditions the Laplace transform of the response of a one-degree-of-freedom system is given by [2.29]:

Here

(unit step transform); yielding the response:

90


with tm = I [HAB 68], [KAR 40]. NOTE: This calculation is identical to that carried out to obtain the primary response spectrum to a rectangular shock [LAL 75].

Figure 3.1. Example of relative displacement response to a unit step excitation

Figure 3.2. Response to a unit step excitation for £ = 1

Impulse and step response

91

Specific cases

l.If| = l

and, for £m =1,

2. Zero damping In the reduced form, the equation of movement can be written using the notation of the previous paragraphs:

or

with\the initial conditions being constant, namely, for 9 = 0

or, according to the case, t = 0 and

After integration, this becomes as before:

92


and

the expression in which, by definition of the excitation, im = 1:

Example If the excitation is a force, the equation of the movement is, for t > 0, with, for initial conditions at t = 0, yielding, after integration,

NOTE: The dimensions of [3.15] do not seem correct. It should be remembered that 1

the excitation used is a force of amplitude unit equal to — which is thus k homogeneous -with a displacement.

Figure 3.3. Step response for £ = 0

The function z(t), response with the step unit function, is often termed indicial admittance or step response and is written A(t).


93

It can be seen in this example that if we set, according to our notation, the ratio of the maximum elongation zm to the static deflection zs which the mass would take if the force were statically applied reached a value of 2. The spring, in dynamics, performs twice more often than in statics, although it is unlikely to undergo stresses that are twice as large. Often, however, the materials resist the transient stresses better than static stresses (Chapter 1). This remark relates to the first instants of time during which F(t) is transitory and is raised from 0 to 1. For this example where F(t) remains equal to one for all t positive values and where the system is undamped, the effect of shock would be followed by a fatigue effect.

3.1.1.2. Extremumfor response The expression for the response

has a zero derivative

for

such that

The first maximum (which would correspond to the point of the positive primary shock response spectrum at the natural frequency f0 of the resonator) occurs for at time

94


From this the value q(0) is deduced:

(always a positive quantity). The first maximum amplitude qm tends towards 1 when J; tends towards 1.

Figure 3.4. First maximum amplitude versus £,

NOTES: 1. qm is independent of the natural frequency of the resonator. 2./^

=0 , q m = 2 .


Figure 3.5. Amplitude of the first minimum versus £

Fork = 2,

and

qm is negative for all

3.1.1.3. First excursion of response to unit value 6j is searched such that:

95

96


i.e.

This yields, since tan simultaneously:

and

must be present

Arc tan

Figure 3.6. Resolution of [3.23]

Figure 3.7. Time of first excursion by the unit step response


Ifq(e) = l

If£ = l

The only positive root exists for infinite 0.

3.1.2. Response defined by absolute displacement, velocity or acceleration 3.1.2.1. Expression for response In this case, for any £, and zero initial conditions,

with

If£=0

97

98


If£=l

Cm-D.

3.1.2.2. Extremum for response The extremum of the response q(0) = A(0) occurs for 6 = 6m such that which leads to

i.e. to


Figure 3.8. Example of absolute response

For 2 ^ -1 > 0 (and since 0m is positive):

and if

99

100


Figure 3.9. Amplitude of the absolute response for % = 1

For

i.e.

If

Then

if

or if

It yields

or

Figure 3.10. Time ofextremum versus £


If

and

Figure 3.11. Amplitude of absolute response versus E,

3.1.2.3. First passage of the response by the unit value The first excursion of the unit value happens at time Qj such that

101

102


If

If

yielding 6

infinity.

Figure 3.12. Time of first transit of the unit value versus £

3.2. Response of a mass-spring system to a unit impulse excitation 3.2.1. Response defined by relative displacement 3.2.1.1. Expression for response Let us consider a Dirac delta function 6g(6) obeying


such that, with

and

103

is the impulse

The quantity

transmitted to the mass m by a force acting for a small interval of time A6 [KAR 40]. The contribution of the restoring force of the spring to the impulse is negligible during the very short time interval A9. Depending on whether the impulse is defined by a force or an acceleration, then

or

If

the generalized delta function is equal, according to the case, to

or

then the generalized equation is obtained as follows:

to

Then

(I = 1). To make the differential equation dimensionless each member is divided by the quantity becomes:

homogeneous with length and set q

and

This

104


The Laplace transform of this equation is written with the notation already used,

and since the transform of a Dirac delta function is equal to the unit [LAL 75], this gives

(£ * 1) and

Figure 3.13. Impulse response

Specific cases

l.For and


105

Example If the impulse is defined by a force,

then

This relation is quite homogeneous, since the 'number' 1 corresponds to the impulse

which has the dimension of a displacement.

Figure 3.14. Impulse response

The unit impulse response is denoted by h(t) [BRO 53] [KAR 40]. It is termed impulse response, impulsive response or -weight function [GUI 63]. 2. If

106


Figure 3.15. Examples of impulse responses versus

3.2.1.2. Extremumfor response q(9) presents a peak qm when

This yields

For

for

such that


107

yielding

For

Figure 3.16. Time of the first maximum versus J;

Figure 3.17. Amplitude of the first maximum versus £

NOTE: The comparison of the Laplace transforms of the response -with the unit step function

and the unit impulse

108


1 shows that these two transforms differ by a factor — and that, consequently [BRO P 537, [KAR 40],

3.2.2. Response defined by absolute parameter 3.2.2.1. Expression for response

Figure 3.18. Absolute response


NOTE: For

the preceding case is found

In non-reduced coordinates the impulse response is written [BRO 62]:

Figure 3.19. Absolute response for £ = 1

If

109

110


3.2.2.2. Peaks of response The response h(6) presents a peak when

i.e., after simplification, if

and if

this yields

i.e.

i.e. for 6 = 0m such that


111

Figure 3.20. Time of the first maximum of the Figure 3.21. Amplitude of the first maximum absolute response versus £ of the absolute response versus £,

If,

Since

we obtain

If

and

NOTE: 1. The equation [2.37], which can be written

is none other than a convolution integral applied to the functions ^(t) and

112


(h(t) = impulse response or weight function). 2. The Fourier transform of a convolution product of two functions t and H is equal to the product of their Fourier transforms [LAL 75]. Ifu = t*h

The function H(Q), Fourier transform of the impulse response, is the transfer function of the system [LAL 75]. 3. In addition, the Laplace transformation applied to a linear one-degree-of-freedom differential equation leads to a similar relation:

A(P) is termed operational admittance and

generalized impedance of

the system. 4. In the same way, the relation [2.60] can be considered as the convolution product of the two functions f(i) and

3.3. Use of step and impulse responses The preceding results can be used to calculate the response of the linear onedegree-of-freedom system (k, m) to an arbitrary excitation ^(t). This response can be considered in two ways [BRO 53]: - either as the sum of the responses of the system to a succession of impulses of very short duration (the envelope of these impulses corresponding to the excitation) (Figure 3.22); - or as the sum of the responses of the system to a series of step functions (Figure 3.23). The application of the superposition principle supposes that the system is linear, i.e. described by linear differential equations [KAR 40] [MUS 68].


113

Figure 3.22. Arbitrary pulse as a series of impulses

Figure 3.23. Arbitrary pulse as a series of step functions

Let us initially regard the excitation ^(t) as a succession of very short duration Aa impulses and let l(a) be the impulse amplitude at time a. By hypothesis, ^(a) = 0 f o r a < 0 . Set h(t - a) as the response of the system at time t, resulting from the impulse at the time a pertaining to the time interval (0, t) (paragraph 3.2.1). The response z(t) of the system to all the impulses occurring between a = 0 and a = t is:

If the excitation is a continuous function, the intervals Aa can tend towards zero; it then becomes (Duhamel's formula):

114


The calculation of this integral requires knowledge of the excitation function ^(t) and of the response h(t - a) to the unit impulse at time a. The integral [3.90] is none other than a convolution integral [LAL 75]; this can then be written as:

According to properties of the convolution [LAL 75]:

Figure 3.24. Summation of impulse responses

NOTE: It is supposed in this calculation that at time t the response to an impulse applied at time a is observed, so that this response is only one function of the time interval t - a, but not oft or of a separately. This is the case if the coefficients of the differential equation of the system are constant. This assumption is in general not justified if these coefficients are Junctions of time [KAR 40].


115

Consider the excitation as a sum of step functions separated by equal time intervals Aa (Figure 3.25). Atfa) , , The amplitude of each step function is A^(a), i.e. Aa. Set A ( t - a ) as Aa the step response at time t, resulting from the application of a unit step function at time a (with 0 < a < t). Set ^(0) as the value of the excitation at time a = 0 and A(t) as the response of the system at time t corresponding to the application of the unit step at the instant a = 0. The response of the system to a single unit step function is equal to

Figure 3.25. Summation of step responses

The response of the linear system to all the step functions applied between the times a = 0 and a = t and separated by Aa is thus:

116


If the excitation function is continuous, the response tends, when Act tends towards zero, towards the limit

where

This is the superposition integral or Rocard integral. In the majority of practical cases, and according to our assumptions, ^(0) = 0 and

NOTES: 1. The expression [3.95] is sometimes called Duhamel's integral and sometimes [3.90] Rocard's integral [RID 69]. 2. The integral [3.90] can be obtained by the integration by parts of [3.94] while setting U = A(t - a) and dV = i?(a) da using integration by parts, by noting that A(O) is often zero in most current practical problems (knowing moreover that

If u is a continuous and derivable function in (0, t), integration by parts gives

yielding, if *0 = 40),


117

and, since A(o) = 0, by Duhamel's formula:

The functions h(t) and A(t) were calculated directly in the preceding paragraphs. Their expression could be obtained by starting from the general equation of movement in its reduced form. The next step will be to find, for example, h(t). The unit impulse can be defined, in generalized form, by the integral:

a being a variable of integration ( a < 9). This relation defines an excitation where the duration is infinitely small and whose integral in time domain is equal to the unit. Since it corresponds to an excitation of duration tending towards zero, it can be regarded as an initial condition to the solution of the equation of motion

(while supposing E, = 0) i.e.

The initial value of the response q(0) is equal to Cj and, for a system initially at rest (Cj = 0), the initial velocity is C2. The amplitude of the response being zero for 0 = 0, the initial velocity change is obtained by setting q = 0 in the equation of movement [3.101], while integrating

over time and taking the limit when 9

tends towards zero [SUT 68]:

yielding

This then gives the expression of the response to the generalized unit impulse of an undamped simple system:

118


- for zero damping, the indicial admittance and the impulse response to the generalized excitation are written, respectively:

and

This yields

for arbitrary £ damping,

and

yielding

Figure 3.26. Decline in impulses


119

The response of the simple system of natural pulsation o>0 can therefore be calculated after a decline of the excitation l(t) in a series of impulses of duration Aa. For a signal of given form, the displacement u(t) is a function oft, (00 and £.

3.4. Transfer function of a linear one-degree-of-freedom system 3.4.1. Definition It was shown in [3.90] that the behaviour of a linear system can be characterized by its weight function (response of the system to an unit impulse function)

where, if the response is relative,

and, if it is absolute:

The function h( ) can be expressed versus time. We have, for example, for the relative response:

The Fourier transform of h(t) is the transfer function H(Q) of the system [BEN 63]:

120


Let us set h

The variable h is defined as the interval. In reduced

coordinates:

NOTE: Rigorously, H(h) is the response function in the frequency domain, whereas the transfer function is the Laplace transform o/h(0) [KIM 24]. Commonly, H(h) is also known as the transfer function. The function H(h)' is complex and can be put in the form [BEN 63]

Sometimes the module JH(h)j is called the gain factor [KIM 24] or gain, or amplification factor when h(9) is the relative response function; or transmissibility when h(6) is the absolute response function and (h) is the associated phase (phase factor). Taking into account the characteristics of real physical systems, H(h) satisfies the following properties:

where H is the complex conjugate of H

4. If two mechanical systems having transfer functions Hj(h) and H 2 (h) are put in series and if there is no coupling between the two systems thus associated, the transfer function of the unit is equal to [BEN 63]:

i.e. . The dimensionless term 'h' is used throughout this and succeeding chapters. This is equivalent to the frequency ratios f / f 0 or Q) 16>G .

Impulse and step response 121

This can be found in references [LAL 75] [LAL 82] [LAL 95a] and examples of use of this transfer function for the calculation of the response of a structure in a given point when it is subjected to a sinusoidal, random or shock excitation are given in the following chapters. In a more general way, the transfer function can be defined as the ratio of the response of a structure (with several degrees of freedom) to the excitation, according to the frequency. The stated properties of H(h) remain valid with this definition. The function H(h) depends only on the structural characteristics.

3.4.2. Calculation of H(h) for relative response By definition,

Knowing that

it becomes

122


3.4.3. Calculation of H(h) for absolute response In this case,


For

and for

123

124


The complex transfer function can also be studied through its real and imaginary parts (Nyquist diagram):

Figure 3.27. Real and imaginary parts of H(h)

Figure 3.28. Nyquist diagram


125

3.4.4. Other definitions of transfer function 3 A A.I. Notation According to choice of the parameters for excitation and response, the transfer function can be defined differently. In order to avoid any confusion, the two letters are placed in as subscripts after the letter H; the first letter specifies the nature of the input, the second that of the response. The letter H will be used without indeces only in the case of reduced coordinates. We will use the same rule for the impedance

3.4.4.2. Relative response

The function |H is equal to the function

To distinguish it from the

transfer function giving the absolute responses we will denote it by HR.

Calculation of

Calculation of

126


Response defined by the relative velocity

It is supposed here that the excitation, and consequently the response, are sinusoidal and of frequency Q, or that the excitation is resoluble into a Fourier series, with each component being a sinusoid. This yields

Thus, if

3.4.4.3. Absolute response In the same way, starting from [3.137],

noting HA, it is possible to calculate all the expressions of the usual transfer functions of this nature.

3.4.4.4. Summary tables Table 3.1 states the values of HA and HR for each parameter input and each parameter response.


127

Example Suppose that the excitation and the response are, respectively, the velocities x and z. Table 3.1 indicates that the transfer function can be obtained from the relation

Yielding

Table 3.2 gives this relation more directly. To continue to use reduced parameters, and in particular of reduced transfer functions (which is not the case for the transfer functions in Table 3.2), these functions can be defined as follows.

Table 3.1. Transfer function corresponding to excitation and response

128


These results are also be presented as in Table 3.2. For a given excitation, we obtain the acceleration and velocity and acceleration transfer function while multiplying respectively by h and h the displacement transfer function (relative or absolute response). This is used to draw the transfer function in a four coordinate representation from which can be read (starting from only one curve plotted against the reduced frequency h) the transfer function for the displacement, velocity and acceleration (paragraph 5.7).

NOTE: The transfer functions are sometimes expressed in decibels

where H(h) is the amplitude of the transfer /unction as defined in the preceding tables. A variation of H(h) by a factor of10 corresponds to an amplification of 20 dB. Table 3.2. Transfer function corresponding to excitation and response Response Excitation

Displacement

Velocity

Acceleration

(m)

(m/s)

(m/s2)

Absolute Relative Absolute Relative Absolute Relative y(t)

Displacement x(t)(m) e 4>

•**

E

v Velocity x(t) o (m/s) E n we can still represent l(t) graphically by a line spectrum. 4.2. Periodic and sinusoidal vibrations in the real environment Perfectly sinusoidal vibrations are seldom met in the real environment. In certain cases, however, the signal can be treated similarly to a sinusoid in order to facilitate the analyses. Such vibrations are observed, for example, in rotating machines, and in badly balanced parts in rotation (inbalances of the trees, defects of coaxiality of the reducer shafts with the driving shafts, electric motor, gears) [RUB 64]. The more frequent case of periodic vibrations decomposable in Fourier series is reduced to a sinusoidal vibrations problem, by studying the effect of each harmonic component and by applying the superposition theorem (if the necessary assumptions are respected, in particular that of linearity). They can be observed on machines generating periodic impacts (presses), in internal combustion engines with several cylinders and so on [BEN 71], [BRO], [KLE 71b] and [TUS 72]. Quasi-periodic vibrations can be studied in the same manner, component by component, insofar as each component can be characterized. They are measured, for example, in plane structures propelled by several badly synchronized engines [BEN 71]. 4.3. Sinusoidal vibration tests The sinusoidal vibration tests carried out using electrodynamic shakers or hydraulic vibration machines can have several aims: - the simulation of an environment of the same nature; - the search for resonance frequencies (identification of the dynamic behaviour of a structure). This research can be carried out by measuring the response of the structure at various points when it is subjected to random excitation, shocks or swept frequency sinusoidal vibrations. In this last case, the frequency of the sinusoid varies over time according to a law which is in general exponential, sometimes linear. When the swept sine test is controlled by an analogical control system, the

140


frequency varies in a continuous way with time. When numerical control means are used, the frequency remains constant at a certain time with each selected value, and varies between two successive values by increments that may or may not be constant depending on the type of sweeping selected; Example The amplitude is supposed to be xm =10 cm at a frequency of 0.5 Hz. Maximum velocity: xm = 2 f x

m

=0.314 m / s

Maximum acceleration: x m = ( 2 7 i f ) 2 x m =0.987 m/s2

At 3 Hz, xm =10 cm Velocity:

xm = 1.885 m/s

Figure 4.6. Acceleration, velocity and displacement versus frequency


141

Acceleration:

At 10 Hz, if xm = 5 m/s2, the velocity is equal to

p/s and

the displacement is xm

- fatigue tests either on test-bars or directly on structures, the frequency of the sinusoid often being fixed on the resonance frequency of the structure. In this last case, the test is often intended to simulate the fatigue effects of a more complex real environment, generally random and making the assumption that induced fatigue is dominant around resonance [GAM 92]. The problems to be solved are then the following [CUR 71]: -the determination of an equivalence between random and sinusoidal vibration. There are rules to choose the severity and the duration of an equivalent sine test [GAM 92]; -it is necessary to know the frequencies of resonance of the material (determined by a preliminary test); -for these frequencies, it is necessary to choose the number of test frequencies, in general lower than the number of resonances (in order for a sufficient fraction of the total time of test to remain at each frequency), and then to define the severity, and the duration of each sinusoid at each frequency of resonance selected. The choice of the frequencies is very important. As far as possible, those for which rupture by fatigue is most probable are chosen i.e. those for which the Q factor is higher than a given value (2generally). This choice can be questioned since, being based on previously measured transfer functions, it is a function of the position of the sensors and can thus lead to errors; - thefrequentcontrol of the resonancefrequency,which varies appreciably at the end of the material's lifetime. For the sine tests, the specifications specify the frequency of the sinusoid, its duration of application and its amplitude. The amplitude of the excitation is in general defined by a zero to peak acceleration (sometimes peak-to-peak); for very low frequencies (less than a few Hertz), it is often preferred to describe the excitation by a displacement because the acceleration is in general very weak. With intermediate frequencies, velocity is sometimes chosen.


Chapter 5

Response of a linear single-degree-offreedom mechanical system to a sinusoidal excitation

In Chapter 2 simple harmonic movements, damped and undamped, were considered, where the mechanical system, displaced from its equilibrium position and released at the initial moment, was simply subjected to a restoring force and, possibly, to a damping force. In this chapter the movement of a system subjected to steady state excitation, whose amplitude varies sinusoidally with time, with its restoring force in the same direction will be studied. The two possibilities of an excitation defined by a force applied to the mass of the system or by a movement of the support of the system, this movement itself being defined by an displacement, a velocity or an acceleration varying with time will also be examined. The two types of excitation focussed on will be: - the case close to reality where there are damping forces; - the ideal case where damping is zero.

144


Figure 5.1. Excitation by a force

Figure 5.2. Excitation by an acceleration

5.1. General equations of motion 5.1.1. Relative response In Chapter 2 the differential equation of the movement was established. The Laplace transform is written, for the relative response:

q0 and q0 being initial conditions. To simplify the calculations, and by taking account of the remarks of this chapter, it is supposed that q0 = q0 = 0. If this were not the case, it would be enough to add to the final expression of q(9) the term C(9) previously calculated. The transform of a sinusoid

is given by

where h

(Q being the pulsation of the sinusoid and con the natural pulsation of V

the undamped one-degree-of-freedom mechanical system), yielding

Response of a linear single-degree-of-freedom mechanical system 145 Case 1: 0 < £, < 1 (underdamped system)

For non-zero initial conditions, this must be added to q(0)

Case 2:^ = 1 (critical damping] For zero initial conditions,

For non-zero initial conditions, we have to add to q(0):

146


Case 3: £ > 1 (overdamped system)

The denominator can be written p2 + 2 £ p +1, for £, > 1,

yielding

with, for non-zero initial conditions,

Response of a linear single-degree-of-freedom mechanical system

5.1.2. Absolute response Case 1: 0 < £, < 1 Zero initial conditions

If the initial conditions are not zero, it must be added to q(0)

Case 2: £ = 1

147

148


Non-zero initial conditions

Case3:£>l Zero initial conditions

i.e.

Non-zero initial conditions

This must be added to q(0).

Response of a linear single degree of freedom mechanical system

149

5.1.3. Summary The principal relations obtained for zero initial conditions are brought together below. Relative response

Absolute response

150


5.1.4. Discussion Whatever the value of £, the response q(0) is made up of three terms: -the first, C(0), related to conditions initially non-zero, which disappears when 0 increases, because of the presence of the term e~E0 ; -the second, which corresponds to the transient movement at the reduced frequency -\jl-Z,2

resulting from the sinusoid application at time 9 = 0. This

oscillation attenuates and disappears after a while from £ because of the factor e~ E 0 . In the case of the relative response, for example, for 0 < £, < 1, this term is equal to

- the third term corresponds to an oscillation of reduced pulsation h, which is that of the sinusoid applied to the system. The vibration of the system is forced, the frequency of the response being imposed on the system by the excitation. The sinusoid applied theoretically having one unlimited duration, it is said that the response, described by this third term, is steady state.

Table 5.1. Expressions for reduced response

Response

Displacement

Velocity

Acceleration

Reaction force

on Excitation

Absolute

y(t) Displacement x(t) •w o> 4> O

B

Velocity x(t)

CO

Acceleration x(t) Force on the mass m (here, z & y)

Relative z(t)

Absolute

Relative

Absolute

y(t)

z(t)

y(t)

Relative z(t)

base F T (t)

152


The steady state response for 0 < £ < 1 will be considered in detail in the following paragraphs. The reduced parameter q(0) is used to calculate the response of the mechanical system. This is particularly interesting because of the possibility of easily deducing expressions for relative or absolute response q(0), irrespective of the way the excitation (force, acceleration, velocity or displacement of the support) is defined, as in Table 5.1.

5.1.5. Response to periodic excitation The response to a periodic excitation can be calculated by development of a Fourier series for the excitation [HAB 68].

The response of a one-degree-of-freedom system obeys the differential equation

This equation being linear, the solutions of the equation calculated successively for each term in sine and cosine can be superimposed. This yields


153

with

5.1.6. Application to calculation for vehicle suspension response Considering a vehicle rolling at velocity v on a sinusoidal road as shown in Figure 5.3.

Figure 5.3. Example of a vehicle on road

[5.36]

s = distance between a maximum of sinusoid and the vehicle L = sinusoid.period It is supposed that [VOL 65]: - the wheels are small, so that the hub of each wheel is at a constant distance from the road; - the tires have negligible deformation. We have with the notation already used in the preceding paragraphs:

154


The distance s is related to time by s = v t yielding

with

where

• The displacement y must be smallest possible to make the suspension effective. It is necessary therefore that h or the velocity be large. If £ tends towards zero, y tends towards infinity when h tends towards 1, with critical velocity

When £ is different from zero, the value of y for h = 1 is


155

5.2. Transient response 5.2.1. Relative response For 0 < £ < 1

The response

can be also written

where

and

A pseudo-sinusoidal movement occurs. The total response q(0) is zero for 0 = 0 since the term representing the transient response is then equal to

This response qT never takes place alone. It is superimposed on the steady state response q p (0) studied in the following paragraph.

156


The amplitude A(h) is maximum when

dA(h)

= 0, i.e. when

dh

when h = 1 (h >0). In this case,

The movement has a logarithmic decrement equal to [KIM 29]:

and for the reduced pseudo-period

The transient response qT has an amplitude equal to cycle number n such that i.e.

For £ small, it becomes

i.e.

of the first peak after


157

Figure 5.4. Cycle number for attenuation N of the transient relative response

If N « 23, we have n » Q. When a system is subjected to a sine wave excitation, the amplitude of the response is established gradually during the transitional stage up to a level proportional to that of the excitation and which corresponds to the I

2~

steady state response. In the following paragraph it is seen that, if h = \ 1 ~ £ > the response tends in steady state mode towards

The number of cycles necessary to reach this steady state response is independent of h. For £ small, this number is roughly proportional to the Q factor of the system.

158


Figure 5.5. Establishment of the relative response for £ = 0

Figure 5.6. Establishment of the relative response for £, = 0.1

Figure 5.7. Ratio of transient response/steady state response

For the particular case where


5.2.2. Absolute response

For

or

with

and

If E = l

5.3. Steady state response 5.3.1. Relative response For 0 < £, < 1, the steady state response is written

159

160


This expression can be also put in the form

In the amplitude of this response HRJ-, it is noted that, the first index recalls that the response is relative and the second is about a displacement. Therefore

The phase is such that

5.3.2. Absolute response For 0 < £, < 1, the steady state response is expressed

As previously, this response can be written

where

and

HAD is

terme

d the transmissibility factor or transmissibility or transmittance.


5.4. Responses

161

and

Starting with the study of the responses

and

some

important definitions are introduced. They are equal to

where

More interesting is the case where the input is an acceleration xm, and the reduced response q(0) gives the relative displacement z(t) yielding

with

and

162


5.4.1. Variations of velocity amplitude 5.4.1.1. Quality factor The amplitude HRV of the velocity passes through a mazimum when the derivative

is zero.

This function is equal to zero when h = 1 (h > 0). The response thus is at a maximum (whatever be £) for h = 1. There is then velocity resonance, and

At resonance, the amplitude of the forced vibration q(0) is Q times that of the excitation (here the physical significance of the Q factor is seen). It should be noted that this resonance takes place for a frequency equal to the natural frequency of the undamped system, and not for a frequency equal to that of the free oscillation of the damped system. It tends towards the unit when h tends towards zero. The curve thus starts from the origin with a slope equal to 1 (whatever £,). For h = 0, HRV = 0. The slope tends towards zero when h —> <x>, like H RV . The expression of HRV 1 does not change when h is replaced by —; thus taking a logarithmic scale for the h abscissae, the curves H RV (h) are symmetrical with respect to the line h = 1.

For

HRV ->• oo when h -> 1.

Since tan


HRV can be written in the form

Setting y = 2 £ HRV and x = tan y, the curves

valid for all the systems m, k, and c are known as universal. In the case of an excitation by force, the quantity 2 £ HRV is equal to

Figure 5.8. Amplitude of velocity response

163

164


5.4.1.2. Hysteresis loop It has been supposed up to now that damping was viscous, with the damping force being proportional to the relative velocity between the ends of the damping device and of the form Fd = -c z, where c is the damping constant, and in a direction opposed to that of the movement. This damping, which leads to a linear differential equation of the movement, is itself known as linear. If the relative displacement response z(t) has the form

the damping force is equal to

where

The curve Fd(z) (hysteresis loop) has the equations, in parametric coordinates,

i.e., after elimination of time t:

Figure 5.9. Viscous damping force


165

Figure 5.10 Hysteresis loop for visc

The hysteresis loop is thus presented in the form of an ellipse of half the smaller axis Fdm = c Q zm and half the larger axis zm.

5.4.1.3. Energy dissipated during a cycle The energy dissipated during a cycle can be written:

Knowing that

i.e., since i

or

we have

166


For a viscously damped system, in which the dDAMPING CONSTANT C IS INDEPENDENT OF THE FREQUENCY, THE RELATIVE DAMPING E IS INVERSELY PROPORTIONAL TO THE FREQUENCY:

We can deduce from this the energy A consumed per time unit. If T is the period of the excitation

Since (Chapter 3)

Consumed energy is at a maximum when the function equal to H2RV(h), is at a maximum, i.e. for h = 1, yielding

and


167

The dissipated energy is thus inversely proportional to £. When £ decreases, the resonance curve A ( h ) presents a larger and narrower peak [LAN]. It can however be shown that the surface under the curve A (h) remains unchanged. This surface S is described by:

2 . 2

Figure 5.11. Energy dissipated by damping

168


Figure 5.12. Reduced energy dissipated by damping

71

The integral is equal to — (Volume 4), yielding

df

The surface S is thus quite independent of £. Therefore

5.4.1.4. Half-power points The half-power

points are defined by the values of h such as the energy

dissipated per unit time is equal to

yielding


169

i.e., since h and £ are positive,

A logarithmic scale is sometimes used to represent the transmissibility and a unit, the Bel, is introduced, or generally in practice, a subunit, the decibel. It is said that a power Pt is higher by n decibels (dB) than a power P0 if

If P1 > P0, the system has a gain of n dB. If P1 < P0, the system produces an attenuation of n dB [GUI 63]. If instead of the powers, forces or velocities are considered here, the definition of the gain (or attenuation which is none other than a negative gain) remains identical on the condition of replacing the constant 10 by factor of 20 (log P = 2 logVe + Constant), since the power is proportional to the square of the rms velocity [LAL 95 a]. The curve 2 £ HRV or HRV is close to a horizontal line for £ small, passes through a maximum for h = 1, then decreases and tends towards zero when h becomes large compared to 1. By analogy with an electric resonant system, the mechanical system can be characterized by the interval (bandwidth) between two 1 frequencies h1 and h2 selected in such a way that 2 £, HRV is either equal to ——, V2 or for h such that

The values h1 and h2 correspond to the abscissae of two points N1 and N2 named half-power points because the power which can be dissipated by the shock absorber during a simple harmonic movement at a given frequency is proportional to the square of the reduced amplitude HRV [MEI 67].

170


5A.1.5. Bandwidth yielding

The quantity Q

is the dissonance. It is zero with resonance and

equivalent to Q ( h - l ) in its vicinity [GUI 63]. The condition tanvj/=±l, is (modulo TI) which shows that \j/ undergoes, when h varies from h1 to h2, a variation of

i.e. of

h1,and h2 are calculated from [5.100]

This becomes

The bandwidth of the system Ah = h2 - h1 can be also written


171

This is all the narrower when Q is larger.

Selectivity In the general case of a system having its largest response for a pulsation co m . the selectivity a is defined by

where AQ is the previously definite bandwidth. a characterizes the function of the filter of the system, by its ability, by eliminating near-frequencies (of A£2) to allow through a single frequency. For a resonant system, com = co0 and <s = Q. In electricity, the interval Q2 ~ î characterizes the selectivity of the system. The resonant system is a simple model of a filter where the selective transmissibility can make it possible to choose signals in the useful band (Q1? Cl^j among other signals external with this band and which are undesirable. The selectivity is improved as the peak becomes more acute. In mechanics, this property is used for protection against vibrations (filtering by choosing the frequency of resonance smaller than the frequency of the vibration).

Figure 5.13. Domains of the transfer function HRV

172


It can also be shown [LAL 95a] [LAL 95b] that the response of a one-degree-offreedom system is primarily produced by the contents of the excitation in this frequency band. From these relations, the expressions of h1 and h2 can be deduced:

The bandwidth Ah = h2 - h1 can be also written

yielding, since h

NOTE: The ratio

- h — 1 is also sometimes considered. For the abscissae ~u QI and Q2 of the half-power points, and for £ small, this ratio is equal, respectively,

to

and

The Q factor of the mechanical systems does not exceed a few tens of units and those of the electric circuits do not exceed a few hundreds. In [2.139] it was seen that

yielding [GUR 59]:


173

Figure 5.14. Bandwidth

The bandwidth can thus be also defined as the field of the frequencies transmitted with an attenuation of 10 log 2 « 3.03 dB below the maximum level

Q (attenuation between the levels Q and —) [DEN 56], [THU 71]. V2

Figure 5.15. Representation in dB of the transfer function HRV

Figure 5.15 represents some resonance curves, plotted versus the variable h and for various values of the Q factor with the ordinate being in dB [GUI 63].

174 sINUSOIDAL VIBRATION

5.4.2. Variations in velocity phase In [5.77] it was seen that q(9) = HRV sin(he-\|/) where [5.791

yielding

To obtain the curves \j/(h), it is therefore enough to shift by — the already 2 71

71

plotted curves (p(h), while keeping £ the same. The phase v|/ varies from -— to + — 2

2

since 9 varies from 0 to n. It is zero for h = 1, i.e. when the frequency of the system is equal to that of the excitation (whatever £).

Figure 5.16. Velocity phase versus h The velocity of the mass is thus always in phase with the excitation in this case.


175

When h is lower than 1, the velocity of the mass is in advance of the excitation (\\i < 0, i.e. -\|/ > 0). When h is larger than 1, the velocity of the mass has a phase lag with respect to excitation. In passing through resonance, the curve \\i(h) presents a point of inflection. Around this point there is then a roughly linear variation of the phase varying with h (in an interval that is larger as £ becomes smaller).

5.5. Responses

k z

2

and

F

m

O>Q z *m

In these cases,

The response

q(0)

is at a maximum when

sin(h 6 - (p) = 1,

i.e. for

h 0-cp = (4k + l)-. 2

5.5.1. Variation in response amplitude 5.5.1.1. Dynamic amplification factor Given that the excitation is a force applied to the mass or an acceleration communicated to the support, the reduced response makes it possible to calculate the relative displacement z. The ratio Hnr, of the amplitude of the response relative to displacement to the equivalent static displacement

I is often called the

dynamic amplification factor. NOTE: Some authors [RUZ 71] call the amplification factor of the quantities

or

(amplification factor of the displacement, of the velocity and of

the acceleration respectively) and relative transmissibility (acceleration, velocity or displacement).

176


The function HRD(h) depends on the parameter £,. It is always a positive function which passes through a maximum when the denominator passes through a /

2\2

22

minimum. The derivative of 11 - h ) + 4 £, h2 is cancelled when

( h > 0 ) , provided that 1 - 2 £ 2 > 0 , i.e.

When h tends towards zero,

HRjj>(h) tends towards 1 whatever the value of £. There is resonance for h = hm, the function H R D (h) is maximum and is then equal to

When h -» co, H RD (h) -» 0. In addition, Hm -> oo when £ -> 0. In this case, hm = 1. Resonance is all the more acute since the relative damping £ is smaller; the damping has two effects: it lowers the maximum and makes the peak less acute.

Figure 5.17. Frequency of the maximum of HRD versus £

Figure 5.18. Maximum o/Hj^ versus %

It can be interesting to chart Hm versus h; it can be seen that the calculation of £, versus hm from [5.113] gives £ =


177

This yields:

where hm can only be positive. Here interest will focus just on the branch of the curve belonging to the interval 0 < h < 1. There can be a maximum only for h < 1 (i.e. for a frequency of the excitation 1 lower than that of the resonator a> 0 ), the condition £, < —, is assumed to be V2 realized. If h = 1 is not a condition of resonance, then there is resonance only if at the same time £ = 0. Otherwise, resonance takes place when h < 1.

Figure 5.19. Maximum of HRD versus the peak frequency

It can be seen that the condition £, = 1 corresponds to the critical modes.

178


Figure 5.20. Dynamic amplification factor around the critical mode Like all the curves of H(h), the one corresponding to £, = —, which separates V2 the areas from the curves with or without a maximum, has a horizontal level in the vicinity of the ordinate axis (h = 0).

1 L, = — gives optimum damping. It is for this value that Hm varies less versus h

-n

(an interesting property in electro-acoustics).

Figure 5.21. Dynamic amplification factor for various values of t,


It can also be seen that when £ =

1

, the three first derivatives from H

179

are

V2

zero for h = 0. Finally it should be noted that this £ =

1

value is lower than that for critical V2 damping (£, = 1). It could be thought that the existence of the transient state (£, < 1) does not disturb the response but in practice it has little influence. Setting 5, the logarithmic decrement, it was shown that

For £ =

i

, thus 8 = 2 TI . This is an enormous damping: the ratio of two

V2

successive maximum displacements is then equal to e = e * « 560. The transient state disappears very quickly and is negligible as of the second oscillation.

5.5.1.2. Width of H(h) for HRD =

RDmax V2

By analogy with the definition of the half-power points given for HRV in paragraph 5.4.1.4, we can calculate the width Ah of the peak of H^ for the H

ordinate HRd =

and

RD pr^. V2

Q It has been seen that Hpj-^^ = . y1 -£

yielding

180

h

sINUSOIDAL VIBRATION

must be positive, which requires for the first root that 1 + 2^12

I

£, > 2£2 .

2

The other root leads to 2 e2 +2£,\l-£i < 1. Let us set h1 and h2 the two roots. This gives

If E is small,

and

Particular case If E is small with respect to 1, we have, at first approximation,

In the particular case where E is small, the abscissa of the points for whicl TT

HRD =

J^ is approximately equal to the abscissa of the half-power point; V2 (definite from HRV). The bandwidth can be calculated from

5.5.2. Variations in phase The phase is given by


181

It is noted that: 1 - tan cp is unchanged when h is replaced by —; h

- tan -> 00 when h —»1, therefore 9 —» —: the response is in quadrature 2 phase lead with respect to the excitation; . - tan 9 = 0, i.e. 9 = 0 when h = 0 (in the interval considered); - the derivatives below do not cancel;

- tan< -> 0, i.e. 9 -> , when h -> oo (9 cannot tend towards zero since there is no maximum. The function which is cancelled already when h = 0 cannot cancel one second time): the response and the excitation are in opposite phase;

Figure 5.22. Phase of response

n - for all values of £, q> is equal to — when h = 1; all the curves thus pass through 2 71

the point h = 1, 9 = —; 2 71

-for £ < 1, all the curves have a point of inflection in h = 1, 9 = —. The slope at 2 this point is greater as £ gets smaller.

182


Particular cases

- When h is small, the mass m practically has a movement in phase with the excitation (~ 0). In this case, qmax being closer to 1 as h is smaller, the mass follows the movement of the support. Values of angle
1 (the choice of the sign is uninportatnt). If the value - is taken in (1, 00), then for example, for 0 < h < 1:

and, for


187

Particular case where Here

or

with

and

NOTE: The resonance frequency, defined as the frequency for which the response is at a maximum, has the following values: Table 5.2. Resonance frequency and maximum of the transfer function Response Displacement

Velocity

Acceleration

Resonance frequency

Amplitude of the relative response

188


(the natural frequency of the system being equal to h = -\/l - £ ). For the majority of real physical systems, % is small and the difference between these frequencies is negligible.

Figure 5.28. Resonance frequency versus £

Figure 5.29. Error made by always considering h = 1

Figure 5.30. Peak amplitude of the transfer function

Figure 5.31. Error made by always taking l/2£


189

5.6. Responses

5.6.1. Movement transmissibility Here

The maximum amplitude of q(0) obtained for sin(h 6 - q>) = 1, occurring for is equal to

If the excitation is an absolute displacement of the support, the response is the absolute displacement of the mass m. The movement transmissibility is defined as the ratio of the amplitude of these two displacements:

For certain applications, in particular in the case of calculations of vibration isolators or package cushioning, it is more useful to know the fraction of the force amplitude applied to m which is transmitted to the support through the system [BLA 61] [HAB 68]. Then a force transmission coefficient or force transmissibility Tf is defined by

Tf = Tm = HAD is then obtained according to Table 5.1.

190


5.6.2. Variations in amplitude The amplitude H AD (h) is at a maximum when that

Figure 5.32. Transmissibility

This derivative is zero if h = 0 or if

i.e. for

or, since

. for h such


Figure 5.33. Frequency of the maximum of transmissibility versus E,

191

Figure 5.34. Maximum oftransmissibility versus £,

yielding HAD max

When h tends towards zero, the amplitude H^ tends towards 1 (whatever £). When h -» oo, HAJQ -> 0. From the relation [5.147] is drawn

yielding h < 1. The locus of the maxima thus has as an equation H

AD

This gives the same law as that obtained for relative displacement.

192


Figure 5.35. Locus of maximum oftransmissibility versus h

Case oft, = 0 With this assumption, H^ pass

through

for

Hnr). For all values of^, all the curves HAE

and

for

all the curves are above

Indeed,

H^j)

HAD

Indeed, the condition

is carried out only if In the same way, for h

5.6.3. Variations in phase

If

all the curves are below the straight line

H^


193

Figure 5.36. Phase variations

for

The denominator is zero if or, since

In this case, tan

and

All the curves have, for £, < 1, a point of inflection at h = 1. The slope at this point gets larger as £ gets smaller. when

194


<j> then gets smaller as £, gets larger.

Figure 5.37. Phase versus E, for h = 1

For

5.7. Graphical representation of transfer functions The transfer functions can be plotted in a traditional way on linear or logarithmic axes, but also on a four coordinate nomographic grid, which makes it possible to directly reach the transfer functions of the displacements, the velocities and the accelerations. In this plane diagram at four inputs, the frequency is always carried on the abscissa. Knowing that HRV = Q HRD and that Hj^ = H RV , from the ordinate, along the vertical axis the following can be read:

Response of a linear single-degree-of-fireedom mechanical system

195

- either the velocity (Figure 5.38). Accelerations are then located on an axis of negative slope (-45°) with respect to the axis of the velocities while the amplitude of the displacements are on an axis at 45° with respect to the same vertical axis. Indeed (Figure 5.39):

However, a line at 45° with respect to the vertical axis,

O'K is thus proportional to log

Figure 5.38. Four coordinate diagram

- the amplitude of the displacements. A similar calculation shows that the axis of the velocities forms an angle of + 45° with respect to the horizontal line and that of the accelerations an angle of 90° with respect to the axis of the velocities.

196


Figure 5.39. Construction of the four input diagram

Chapter 6

Non-viscous damping

6.1. Damping observed in real structures In real structures, damping, which is not perfectly viscous, is actually a combination of several forms of damping. The equation of movement is in c consequence more complex, but the definition of damping ratio £ remains —, c c where cc is the critical damping of the mode of vibration considered. The exact calculation of £, is impossible for several reasons [LEV 60]: probably insufficient knowledge of the exact mode of vibration, and of the effective mass of the system, the stiffnesses, the friction of the connections, the constant c and so on. It is therefore important to measure these parameters when possible. In practice, non-linear damping can often be compared to one of the following categories, which will be considered in the following paragraphs: - damping force proportional to the power b of the relative velocity z; - constant damping force (Coulomb or dry damping), which corresponds to the case where b = 0; - damping force proportional to the square of the velocity (b = 2); - damping force proportional to the square of the relative displacement; -hysteretic damping, with force proportional to the relative velocity and inversely proportional to the excitation frequency.

198


Such damping produces a force which is opposed to the direction or the velocity of the movement. 6.2. Linearization of non-linear hysteresis loops - equivalent viscous damping Generally, the differential equation of the movement can be written [DEN 56]:

with, for viscous damping, f(z, z) = c z. Because of the presence of this term, the movement is no longer harmonic in the general case and the equation of the movement is no longer linear, such damping leads to nonlinear equations which make calculations complex in a way seldom justified by the result obtained. Except in some particular cases, such as the Coulomb damping, there is no exact solution. The solution of the differential equation must be carried out numerically. The problem can sometimes be solved by using a development of the Fourier series of the damping force [LEV 60]. Very often in practice damping is fortunately rather weak so that the response can be approached using a sinusoid. This makes it possible to go back to a linear problem, which is easier to treat analytically, by replacing the term f(z, z) by a force of viscous damping equivalent c^ z; by supposing that the movement response is sinusoidal, the equivalent damping constant ceq of a system with viscous damping is calculated which would dissipate the same energy per cycle as nonlinear damping. The practice therefore consists of determining the nature and the amplitude of the dissipation of energy of the real damping device, rather than substituting in the mathematical models with a viscous damping device having a dissipation of equivalent energy [CRE 65]. It is equivalent of saying that the hysteresis loop is modified. In distinction from structures with viscous damping, nonlinear structures have non-elliptic hysteresis loops Fd(z) whose form approaches, for example, those shown in Figures 6.1 and 6.2 (dotted curve).

Non-viscous damping

199

Figure 6.1. Hysteresis loops of non-linear systems

Linearization results in the transformation of the real hysteresis loop into an equivalent ellipse (Figure 6.2) [CAU 59], [CRE 65], [KAY 77] and [LAZ 68].

Figure 6.2. Linearization of hysteresis loop

Equivalence thus consists in seeking the characteristics of a viscous damping which include: - the surface delimited by me cycle Fd(z) (same energy dissipation); - the amplitude of the displacement zm. The curve obtained is equivalent only for the selected criteria. For example, the remanent deformation and the coercive force are not exactly the same. Equivalence leads to results which are much better when the non-linearity of the system is lower.

200


This method, developed in 1930 by L. S. Jacobsen [JAC 30], is general in application and its author was able to show good correlation with the computed results carried out in an exact way when such calculations are possible (Coulomb damping [DEN 30aJ) and with experimental results. This can, in addition, be extended to the case of systems with several degrees of freedom.

Figure 6.3. Linearization of hysteresis loop

If the response can be written in the form z(t) = zm sin(Qt - cp), the energy dissipated by cycle can be calculated using

Energy AEd is equal to that dissipated by an equivalent viscous damping c^, if

[HAB 68]:

i.e. if [BYE 67], [DEN 56], [LAZ 68] and [THO 65a]:

Non-viscous damping

The transfer function of a one-degree-of-freedom system general way

In addition,

yielding

(or in a more

can be written while replacing ceq by this value in the

relation established, for viscous damping:

(since

201

\ and for the phase

202

Sinusoidal vibration is the energy dissipated by the cycle, the amplitude of the equivalent force

applied is

6.3. Main types of damping 6.3.1. Damping force proportional to the power b of the relative velocity Table 6.1. Expressions for damping proportional to power b of relative velocity

Damping force

Equation of the hysteresis loop

Energy dissipated by damping during a cycle

Equivalent viscous damping Equivalent damping ratio

Amplitude of the response

Non-viscous damping

203

Phase of the response

References in Table 6.1: [DEN 30b], [GAM 92], [HAB 68], [JAC 30], [JAC 58], [MOR 63a], [PLU 59], [VAN 57] and [VAN 58]. Relation between b and the parameter J the B. J. Lazan expression It has been shown [JAC 30] [LAL 96] that if the stress is proportional to the relative displacement zm (a = K zm), the coefficient J of the relation of B.J. Lazan (D = J a ) is related to the parameter b by

J depends on parameters related to the dynamic behaviour of the structure being considered (K and co0).

6.3.2. Constant damping force If the damping force opposed to the movement is independent of displacement and velocity, the damping is known as Coulomb or dry damping. This damping is observed during friction between two surfaces (dry friction) applied one against the other with a normal force N (mechanical assemblies). It is [BAN 77], [BYE 67], [NEL 80] and [VOL 65]: - a function of the materials in contact and of their surface quality; - proportional to the force normal to the interface; - mainly independent of the relative velocity of slipping between two surfaces; -larger before the beginning of the relative movement than during the movement in steady state mode.

204


Figure 6.4. One-degree-of-freedom system with dry friction

The difference between the coefficients of static and dynamic friction is in general neglected and force N is supposed to be constant and independent of the frequency and of the displacement. A one-degree-of-freedom system damped by dry friction is represented in Figure 6.4. Table 6.2. Express ions for a constant damping force

Damping force Equation of the hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping Equivalent damping ratio


Non-viscous damping

205

Phase of the response

[BEA 80], [CRE 61], [CRE 65], [DEN 29], [DEN 56], [EAR 72], [HAB 68], [JAC 30], [JAC 58], [LEV 60], [MOR 63b], [PAI 59], [PLU 59], [ROO 82], [RUZ 57], [RUZ 71], [UNO 73] and [VAN 58].

6.3.3. Damping force proportional to the square of velocity A damping of this type is observed in the case of a body moving in a fluid .2 Z

(applications in fluid dynamics, the force of damping being of the form Cx p A —) 2

or during the turbulent flow of a fluid through an orifice (with high velocities of the fluid, from 2 to 200 m/s, resistance to the movement ceases to be linear with the velocity). When the movement becomes fast [BAN 77], the flow becomes turbulent and the resistance non-linear. Resistance varies with the square of the velocity [BAN 77], [BYE 67] and [VOL 65]. Table 6.3. Expressions for quadratic damping

Damping force Equation of the hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping Equivalent damping ratio

206



Phase of response

[CRE 65], [HAB 68], [JAC 30], [RUZ 71], [SNO 68] and [UNG 73]. The constant (3 is termed the quadratic damping coefficient. It is characteristic of the geometry of the damping device and of the properties of the fluid [VOL 65]. 6.3.4. Damping force proportional to the square of displacement Table 6.4. Expressions for damping force proportional to the square of the displacement

Damping force

Equation of hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping

Equivalent damping ratio

Non-viscous damping

207

Amplitude of response

Phase of response

Such damping is representative of the internal damping of materials, of the structural connections, and cases where the specific energy of damping can be expressed as a function of the level of stress, independent of the form and distribution of the stresses and volume of the material [BAN 77], [BYE 67], [KIM 26] and [KIM 27]. 6.3.5. Structural or hysteretic damping Table 6.5. Expressions for structural damping Damping coefficient function of Q

Damping force

Equation of the hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping

Damping force proportional to the displacement

Complex stiffness

208


Equivalent damping ratio

Amplitude of response

Phase of response

This kind of damping is observed when the elastic material is imperfect when in a system the dissipation of energy is mainly obtained by deformation of material and slip, or friction in the connections. Under a cyclic load, the curve Q, s of the material forms a closed hysteresis loop rather than only one line [BAN 77]. The dissipation of energy per cycle is proportional to the surface enclosed by the hysteresis loop. This type of mechanism is observable when repeated stresses are applied to an elastic body causing a rise in temperature of the material. This is called internal friction, hysteretic damping, structural damping or displacement damping. Various formulations are used [BER 76], [BER 73], [BIR77], [BIS 55], [CLO 80], [GAN 85], [GUR 59], [HAY 72], [HOB 76], [JEN 59], [KIM 27], [LAL 75], [LAL 80], [LAZ 50], [LAZ 53], [LAZ 68], [MEI 67], [MOR 63a], [MYK 52], [PLU 59], [REI 56], [RUZ 71], [SCA 63], [SOR 49] and [WEG 35]. 6.3.6 Combination of several types of damping . If several types of damping, as is often the case, are simultaneously present combined with a linear stiffness [BEN 62] [DEN 30a], equivalent viscous damping can be obtained by calculating the energy AEdi dissipated by each damping device and by computing ceq [JAC 30] [JAC 58]:

Non-viscous damping

209

Example Viscous damping and Coulomb damping [JAC 30], [JAC 58], [LEV 60] and [RUZ71]

Fm F c Q

= maximum F(t) (excitation) = frictional force = viscous damping ratio = pulsation of the excitation

6.3.7. Validity of simplification by equivalent viscous damping The cases considered above do not cover all the possibilities, but are representative of many situations. The viscous approach supposes that although non-linear mechanisms of damping are present their effect is relatively small. It is thus applicable if the term for viscous damping is selected to dissipate the same energy by cycle as the system with nonlinear damping [BAN 77]. Equivalent viscous damping tends to underestimate the energy dissipated in the cycle and the amplitude of a steady state forced vibration: the real response can larger than envisaged with this simplification. The decrease of the transient vibration calculated for equivalent viscous damping takes a form different from that observed with Coulomb damping, with a damping force proportional to the square of the displacement or with structural damping. This difference should not be neglected if the duration of the decrease of the response is an important parameter in the problem being considered.

210


The damped natural frequency is itself different in the case of equivalent viscous damping and in the non-linear case. But this difference is in general so small that it can be neglected. When damping is sufficiently small (10%), the method of equivalent viscous damping is a precise technique for the approximate solution of non-linear damping problems.

6.4. Measurement of damping of a system All moving mechanical systems dissipate energy. This dissipation is often undesirable (in an engine, for example), but can be required in certain cases (vehicle suspension, isolation of a material to the shocks and vibrations and so on). Generally, mass and stiffness parameters can be calculated quite easily. It is much more difficult to evaluate damping by calculation because of ignorance of the concerned phenomena and difficulties in their modelling. It is thus desirable to define this parameter experimentally. The methods of measurement of damping in general require the object under test to be subjected to vibration and to measure dissipated vibratory energy or a parameter directly related to this energy. Damping is generally studied through the properties of the response of a one-degree-of-freedom mass-spring-damping system [BIR 77] [CLO 80] [PLU 59]. There are several possible methods for evaluating the damping of a system: - amplitude of the response or amplification factor; - quality factor; - logarithmic decrement; - equivalent viscous damping; - complex module; Af -bandwidth —. f

6.4.1. Measurement of amplification factor at resonance The damping of the one-degree-of-freedom system tends to reduce the amplitude of the response to a sine wave excitation. If the system were subjected to no external forces, the oscillations response created by a short excitation would attenuate and disappear in some cycles. So that the response preserves a constant amplitude, the

Non-viscous damping

211

excitation must bring a quantity of energy equal to the energy dissipated by damping in the system. The amplitude of the velocity response z is at a maximum when the frequency of the sinewave excitation is equal to the resonance frequency f0 of the system. Since the response depends on the damping of the system, this damping can be deduced from measurement of the amplitude of the response, the one degree of linear freedom system supposedly being linear:

or

For sufficiently small £, it has been seen that with a weak error, the amplification 2

factor, defined by HRD =

co0 zm

, was equal to Q. The experimental determination

*m of E, can thus consist of plotting the curve HRD or HRV and of calculating £, from the peak value of this function. If the amplitude of the excitation is constant, the sum of potential and kinetic energies is constant. The stored energy is thus equal to the

maximum of one or the other; it will be, for example Us = — k z^ . The energy 2 dissipated during a cycle is equal to [5.87] AE<j = TccQz^, yielding, since it is supposed that £} = co0:

i.e.

212


NOTE: The measurement of the response/excitation ratio depends on the configuration of the structure as much as the material. The system is therefore characterized by this rather than the basic properties of the material. This method is not applicable to non-linear systems, since the result is a function of the level of excitation.

6.4.2. Bandwidth or >/2 method Another evaluation method (known as Kennedy-Pancu [KEN 47]) consists of measuring the bandwidth Af between the half-power points relating to one peak of the transfer function [AER 62], with the height equal to the maximum of the curve HRD (or H RV ) divided by V2 (Figure 6.5). From the curve H RV (h), we will have if hj and h2 are the abscissae of the halfpower points:

where (f0 = peak frequency, h}

and

If T0 is the natural period and T1] and T2 are the periods corresponding to an attenuation of

•ft , damping c is given by 2

Non-viscous damping

213

Figure 6.5. Bandwidth associated with resonance

since

and

and

i.e., with the approximation

From the curve HRD, these relations are valid only if ^ is small. The curve could also be used for small £.

H^

6.4.3. Decreased rate method (logarithmic decrement) The precision of the method of the bandwidth is often limited by the non-linear behaviour of the material or the reading of the curves. Sometimes it is better to use the traditional relation of logarithmic decrement, defined from the free response of the system after cessation of the exciting force (Figure 6.6).

214


Figure 6.6. Measurement of logarithmic decrement [BUR 59]

The amplitude ratio of two successive peaks allows the calculation of the logarithmic decrement 8 from

In addition, the existence of the following relation between this decrement and damping ratio is also shown

The measurement of the response of a one-degree-of-freedom system to an impulse load thus makes it possible to calculate 5 or £, from the peaks of the curve [FOR 37] and [MAC 58]:

Non-viscous damping

215

Figure 6.7. Calculation of damping from §

The curve of Figure 6.7 can be use to determine £, from 5. In order to improve the precision of the estimate of 8, it is preferable to consider two non-consecutive peaks. The definition then used is:

where zmi and zmn are, respectively, the first and the n* peak of the response (Figure 6.8). In the particular case where £, is much lower than 1, from [6.29] is obtained:

yielding

and

216

S inusoidal vibration

with

with 4 being small

yielding the approximate value £a

The error caused by using this approximate relation can be evaluated by plotting £a~£ the curve according to £ (Figure 6.8) or that giving the exact value of £ £ according to the approximate value £a (Figure 6.9). This gives

Figure 6.8. Error related to the approximate relation £(5)

Figure 6.9. Exact value of£> versus approximate value ^

Non-viscous damping

217

yielding

and

The specific damping capacity p, the ratio of the specific energy dissipated by damping to the elastic deformation energy per unit of volume, is thus equal to

In a more precise way, p can be also written

while assuming that U^ is proportional to the square of the amplitude of the response. For a cylindrical test-bar,

(potential energy + kinetic energy)

i.e., since constant

218

Sinusoidal vibration 2

Ute is thus proportional to zm yielding, from [6.31] and [6.41] for two successive peaks:

Figure 6.10. Specific damping capacity versus £,

Figure 6.11. Specific damping capacity versus 6

The use of the decrement to calculate p from the experimental results supposes that 8 is constant during n cycles. This is not always the case. It was seen that damping increases as a power of the stress, i.e. of the deformation and it is thus desirable to use this method only for very low levels of stress. For 8 small, we can write [6.45] in the form of a series:

If

we find

The method of logarithmic decrement takes no account of the non-linear effects. The logarithmic decrement 8 can be also expressed according to the resonance peak amplitude Hmax and its width Af at an arbitrary height H [BIR 77], [PLU 59]. F. Forester [FOR 37] showed that

Non-viscous damping

219

Figure 6.12. Bandwidth at amplitude H

Setting as ne the number of cycles such that the amplitude decreases by a factor e (number of Neper), it becomes

where te= time to reach the amplitude

If the envelope Z(t) of the response

(which is roughly a damped sinusoid) is considered this gives,

and if the amplitude in decibels is expressed as

220


For a value of H such that

yielding

a relation already obtained. The calculation

of the Q factor from this result and from the curve H(f) can lead to errors if the damping is not viscous. In addition, it was supposed that the damping was viscous. If this assumption is not checked different values of £are obtained depending on the peaks chosen, particularly for peaks chosen at the beginning or end of the response [MAC 58]. Another difficulty can arise in the case of a several degrees-of-freedom system for which it can be difficult to excite only one mode. If several modes are excited, the response of a combination of several sinusoids to various frequencies will be presented.

6.4.4. Evaluation of energy dissipation under permanent sinusoidal vibration An alternative method can consist of subjecting the mechanical system to harmonic excitation and to evaluate, during a cycle, the energy dissipated in the damping device [CAP 82], where the quantity is largely accepted as a measure of the damping ratio. This method can be applied to an oscillator whose spring is not perfectly elastic, this then leads to the constant k and c of an equivalent simple oscillator.

Non-viscous damping

221

Figure 6.13. Force in a spring It has been seen that, if a one-degree-of-freedom mechanical system is subjected to a sinusoidal force F(t) = Fm sinQ t such that the pulsation is equal to the natural pulsation of the system (co0), the displacement response is given by

where

The force Fs in the spring is equal to Fs = k z(t) and the force F^ in the damping device to Fd = cz = 2 m £ Q z = 2 k £ z m s i n Q t , yielding Fd according toz:

Figure 6.14. Damping force versus displacement

222


This function is represented by an ellipse. During a complete cycle the potential energy stored in the spring is entirely restored. On the other hand, there is energy AE 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 wtaâ 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

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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).

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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;

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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]:

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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

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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

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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.

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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

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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.

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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

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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].

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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

( t 2 J =

lun

and joint moments:

, J"J

1.8.2. Central moments The central moment of order n (with regard to the mean) is the quantity: E {[4t,)-m]

^

n

}= Urn -IP4)-ml" J

[1.16]

N->co N •_,

in the case of a discrete ensemble and, for p(^) continuous: [1.17]

1.8.3. Variance The vor/ance is the central moment of order 2

10

Random vibration

[U8] By definition:

S.f

-2 m

p[4t,)] d^t,) + m2 } "p[4t,)] d/

1.8.4. Standard deviation The quantity s^ \ is called the standard deviation. If the mean is zero, *

When the mean m is known, an absolutely unbiased estimator of s

2

is

2

. When m is unknown, the estimator of s is ^

N m' = — N

where

N-l

Statistical properties of a random process

11

Example Let us consider 5 samples of a random vibration given time t = tj (Figure 1.4).

and the values of i at a

Figure 1.4. Example of stochastic process

If the exact mean m is known (m = 4.2 m/s2 for example), the variance is estimated from: S2 =

(2 - 4.2)* + (5 - 4.2?

S2 =

+

(2 - 4.2? + (4 - 4.2? + (7 - 4.2f

= 3.64 (m/s2)2

If the mean m is unknown, it can be evaluated from

m =—

2+5+2+4+7

s2 = — = 4.50 (m/s2)2 4

5

20 2 = — = 4 m/s 5

2\2

12

Random vibration

1.8.5. Autocorrelation function Given a random process '^(t), the autocorrelation function is the function defined, in the discrete case, by:

R( t l ,t 1 + t)= lim -2/4',) I 4 t i + ^)

[L22]

[1-23] or, for a continuous process, by: [1-24]

1 .8.6. Cross-correlation function Given the two processes {t(t)} and |u(t)} (for example, the excitation and the response of a mechanical system), the cross-correlation function is the function: [1.25] or

R(T)= lim — Z / A ' i J - X t i 4 N-»°oN

j

The correlation is a number measuring the degree of resemblance or similarity between two functions of the same parameter (time generally) [BOD 72]. 1.8.7. Autocovariance

Autocovariance is the quantity:

C(t,, tl +T) = R(t,,t, -f t)- 3

Presence of peaks of high value (more than in the Gaussian case).

Figure 1.9. Kurtosis influence on probability density


21

Examples 1. Let us consider an acceleration signal sampled by a step of At = 0.01 s at 10 points (to facilitate calculation), each point representing the value of the signal for time interval At x

_ x 2 rms ~

+X+--.-H ~

T = N At = 10 . 0.01 s

0.1 3 8

2 2

*ms = - (m/s ) and

Xrms

=1 95

-

This signal has as a mean

m=

l + 3 + 2 + - - - + (-2)+0 n ^—'0.01 = -0.4 m/s2 0.1

22

Random vibration

And for standard deviation s such that 1

\2 At|

^("

T ^-l*i-mJ

s2 = -U(l + 0.4)2 + (3 + 0.4)2 + (2 + 0.4)2 + (0 + 0.4)2 + (-1 + 0.4)2 + (- 3 + 0.4)2

S 2 =3.64 (m/s2)2

s = 6.03 m/s2 2. Let us consider a sinusoid x(t) = x m sin(Q t + cp)

xm

|x(t)|=

2 .. x

=

(for a Gaussian distribution, |x(t)| » 0.798 s ).

1.9.7. Temporal autocorrelation function We define in the time domain the autocorrelation function R^(T) of the calculated signal, for a given T delay, of the product t(t) l(t + T) [BEA 72] [BEN 58] [BEN 63] [BEN 80] [BOD 72] [JAM 47] [MAX 65] [RAC 69] [SVE 80].

Figure 1.10. Sample of random signal


23

[1.48] [1-49]

The result is independent of the selected signal sample i. The delay i being given, one thus creates, for each value of t, the product £(t) and ^(t + T) and one calculates the mean of all the products thus obtained. The function R^(t) indicates the influence of the value of t at time t on the value of the function i at time t + T . Indeed let us consider the mean square of the variation between ^(t) and ^(t + t),

One notes that the weaker the autocorrelation R^(T), the greater the mean square of the difference [^(t)-^(t + t)] and, consequently, the less ^(t) and 4t + t) resemble each other.

Figure 1.11. Examples of autocorrelation functions

The autocorrelation function measures the correlation between two values of considered at different times t. If R/ tends towards zero quickly when T

24

Random vibration

becomes large, the random signal probably fluctuates quickly and contains high frequency components. If R^ tends slowly towards zero, the changes in the random function are probably very slow [BEN 63] [BEN 80] [RAC 69]. R^ is thus a measurement of the degree of random fluctuation of a signal. Discrete form The autocorrelation function calculated for a sample of signal digitized with N points separated by At is equal, for T = m At, to [BEA 72]:

[1.51]

Catalogues of correlograms exist allowing typological study and facilitating the identification of the parameters characteristic of a vibratory phenomenon [VTN 72]. Their use makes it possible to analyse, with some care, the composition of a signal (white noise, narrow band noise, sinusoids etc).

Figure 1.12. Examples of autocorrelation functions


25

Calculation of the autocorrelation function of a sinusoid 40 = tm sin(Q t) T) = — J *

sin Ot sin Q t + i dt

[1-52] The correlation function of a sinusoid of amplitude lm and angular frequency Q

4

is a cosine of amplitude — and pulsation H. The amplitude of the sinusoid thus 2 can, conversely, be deduced from the autocorrelation function: max

[1.53]

1.9.8. Properties of the autocorrelation function 1. Re(o)=Ep(t)]=£ 2 (t) = quadratic mean [1.54] For a centered signal (g = 0], the ordinate at the origin of the autocorrelation function is equal to the variance of the signal. 2. The autocorrelation function is even [BEN 63] [BEN 80] [RAC 69]: R,(T) =* R,(-r)

[1.55]

3.

R,M|• ~t when T -> QO.

26

Random vibration 4. It is shown that: [1.57]

[1.58]

Figure 1.13. Correlation coefficient

NOTES. 1. The autocorrelation function is sometimes expressed in the reduced form:

MO)

[1.59]

or the normalized autocorrelation function [BOD 72] or the correlation coefficient p^(i) varies between -1 and+l p^ = 1 if the signals are identical (superimposable) p^ = -1 if the signals are identical in absolute value and of opposite sign. 2. If the mean m is not zero, the correlation coefficient is given by


27

1.9.9. Correlation duration Correlation duration is the term given to a signal the value T O oft for which the function of reduced autocorrelation p^ is always lower, in absolute value, than a certain value pfc . o

Correlation duration of: - a wide-band noise is weak, - a narrow band noise is large; in extreme cases, a sinusoidal signal, which is thus deterministic, has an infinite correlation duration. This last remark is sometimes used to detect in a signal ^(t) a sinusoidal wave s(t) = S sin Q t embedded in a random noise b(t) : [1.60] The autocorrelation is written: R,(T) = R s M + R b (T)

[1.61]

If the signal is centered, for T sufficiently large, R b (t) becomes negligible so that S2 R^(T) = R S (T) = — cos Q T

t1-62!

2

This calculation makes it possible to detect a sinusoidal wave of low amplitude embedded in a very significant noise [SHI 70a]. Examples of application of the correlation method [MAX 69]: - determination of the dynamic characteristics of a systems, - extraction of a periodic signal embedded in a noise, - detection of periodic vibrations of a vibratory phenomenon, - study of transmission of vibrations (cross-correlation between two points of a structure), - study of turbubences, - calculation of power spectral densities [FAU 69], - more generally, applications in the field of signal processing, in particular in medicine, astrophysics, geophysics etc [JEN 68].

28

Random vibration

1.9.10. Cross-correlation Let us consider two random functions ^(t) and u(t); the cross-correlation function is defined by: = lim

1 2T

fT I A t ) u ( t + T)dt -T

[1-63]

The cross-correlation function makes it possible to establish the degree of resemblance between two functions of the same variable (time in general). Discrete form [BEA 72] If N is the number of sampled points and T a delay such that T = m At , where At is the temporal step between two successive points, the cross-correlation between two signals t and u is given by

,

N-m

N-m 1.9.11. Cross-correlation coefficient Cross-correlation coefficient P^ U (T) or normalized cross-correlation function or normalized covariance is the quantity [JEN 68]

^/R,(0) R u (0) It is shown that:

If t(\) is a random signal input of a system and u(t) the signal response at a point of this system, p^ u (t) is characteristic of the degree of linear dependence of the signal u with respect to I. At the limit, if ^(t) and u(t) are independent,

If the joint probability density of the random variables /(t) and u(t) is equal to p(4 u), one can show that the cross-correlation coefficient p^ u can be written in the form:


29

[1.66] where m^ m u s^ and s u are respectively the mean values and the standard deviations of ^(t) and u(t) .

NOTE. For a digitized signal, the cross-correlation function is calculated using the relation:

1

N

R, u (m At) = - £>(p At) u[(p-m) At]

[1-67]

1.9.12. Ergodicity A process is known as ergodic if all the temporal averages exist and have the same value as the corresponding ensemble averages calculated at an arbitrary given moment [BEN 58] [CRA 67] [JAM 47] [SVE 80]. A ergodic process is thus necessarily stationary. One disposes in general only of a very restricted number of records not permitting experimental evaluation of the ensemble averages. In practice, one simply calculates the temporal averages by making the assumption that the process is stationary and ergodic [ELD 61]. The concept of ergodicity is thus particularly important. Each particular realization of the random function makes it possible to consider the statistical properties of the whole ensemble of the particular realizations.

NOTE. A condition necessary and sufficient such that a stationary random vibration ^(t) is ergodic is that its correlation function satisfies the condition [SVE 80]. liml

frf,_llR(T)dt

T^«T M T;

=

o

[1-68]

-where R^(T) is the autocorrelation function calculated from the centered variable

30

Random vibration

1.10. Significance of the statistical analysis (ensemble or temporal) Checking of stationarity and ergodicity should in theory be carried out before any analysis of a vibratory mechanical environment, in order to be sure that consideration of only one sample is representative of the whole process. Very often, for lack of experimental data and to save time, one makes these assumptions without checking (which is regrettable) [MIX 69] [RAC 69] [SVE 80].

1.11. Stationary and pseudo-stationary signals We saw that the signal is known as stationary if the rms value as well as the other statistical properties remain constant over long periods of time. In the real environment, this is not the case. The rms value of the load varies in a continuous or discrete way and gives the shape of signal known as random pseudostationary. For a road vehicle for example, variations are due to the changes in road roughness, to changes of velocity of the vehicle, to mass transfers during turns, to wind effect etc. The temporal random function ^(t) is known as quasi-stationary if it can be divided into intervals of duration T sufficiently long compared with the characteristic correlation time, but sufficiently short to allow treatment in each interval as if the signal were stationary. Thus, the quasi-stationary random function is a function having characteristics which vary sufficiently slowly [BOL 84]. The study of the stationarity and ergodicity is an important stage in the analysis of vibration, but it is not in general sufficient; it in fact by itself alone does not make it possible to answer the most frequently encountered problems, for example the estimate of the severity of a vibration or the comparison of several stresses of this nature. 1.12 Summary chart of main definitions (Table 1.2) to be found on the next page.

Table 1.2

Main definitions

Through the process (ensemble averages)

Along the process (temporal averages)

Moment of order n

Central moment of order n

Variance

Autocorrelat ion

Crosscorrelation

Stationarity if all the averages of order n are Ergodicity if the temporal averages are equal to the ensemble averages. independent of the selected time tj.


Chapter 2

Properties of random vibration in the frequency domain

The frequency content of the random signal must produce useful information by comparison with the natural frequencies of the mechanical system which undergoes the vibration. This chapter is concerned with power spectral density, with its properties, an estimate of statistical error necessarily introduced by its calculation and means of reducing it. Following chapters will show that this spectrum provides a powerful tool to enable description of random vibrations. It also provides basic data for many other analyses of signal properties.

2.1. Fourier transform The Fourier transform of a non-periodic ^(t) signal, having a finite total energy, is given by the relationship: (t) e-lQt dt

L(Q) =

[2-1]

-00

This expression is complex; it is therefore necessary in order to represent it graphically to plot: - either the real and the imaginary part versus the angular frequency Q, - or the amplitude and the phase, versus Q. Very often, one limits oneself to amplitude data. The curve thus obtained is called the Fourier spectrum [BEN 58].

34

Random vibration

The random signals are not of finite energy. One can thus calculate only the Fourier transform of a sample of signal of duration T by supposing this sample representative of the whole phenomenon. It is in addition possible, starting from the expression of L(n), to return to the temporal signal by calculation of the inverse transform.

[2.2]

One could envisage the comparison of two random vibrations (assumed to be ergodic) from their Fourier spectra calculated using samples of duration T. This work is difficult, for it supposes the comparison of four curves two by two, each transform being made up of a real part and an imaginary part (or amplitude and phase). One could however limit oneself to a comparison of the amplitudes of the transforms, by neglecting the phases. We will see in the following paragraphs that, for reasons related to the randomness of the signal and the miscalculation which results from it, it is preferable to proceed with an average of the modules of Fourier transforms calculated for several signal samples (more exactly, an average of the squares of the amplitudes). This is the idea behind power spectral density.

Figure 2.1. Example of Fourier transform

In an indirect way, the Fourier transform is thus very much used in the analysis of random vibrations.


35

2.2. Power spectral density 2.2.1. Need The search for a criterion for estimating the severity of a vibration naturally results in examination of the following characteristics: - The maximum acceleration of the signal: this parameter neglects the smaller amplitudes which can excite the system for a prolonged length of time, - The mean value of the signal: this parameter has only a little sense as a criterion of severity, because negative accelerations are subtractive and the mean value is in general zero. If that is not the case, it does not produce information sufficient to characterize the severity of the vibration, - The rms value: for a long time this was used to characterize the voltages in electrical circuits, the rms value is much more interesting data [MOR 55]: - if the mean is zero, the rms value is in fact the standard deviation of instantaneous acceleration and is thus one of the characteristics of the statistical distribution, -even if two or several signal samples have very different frequency contents, their rms values can be combined by using the square root of the sum of their squares. This quantity is thus often used as a relative instantaneous severity criterion of the vibrations [MAR 58]. It however has the disadvantage of being global data and of not revealing the distribution of levels according to frequency, nevertheless very important. For this purpose, a solution can be provided by [WIE 30]: - filtering the signal ^(t) using a series of rectangular filters of central frequency f and bandwidth Af (Figure 2.2), - calculating the rms value L^g of the signal collected at the output of each filter. The curve which would give Lrms with respect to f would be indeed a description of the spectrum of signal t(t), but the result would be different depending on the width Af derived from the filters chosen for the analysis. So, for a stationary noise, one filters the supposed broad band signal using a rectangular filter of filter width Af, centered around a central frequency f c , the obtained response having the aspect of a stable, permanent signal. Its rms value is more or less constant with time. If, by preserving its central frequency, one reduces the filter width Af, maintaining its gain, the output signal will seem unstable, fluctuating greatly with time (as well as its rms value), and more especially so if Af is weaker.

36

Random vibration

Figure 2.2. Filtering of the random signal

To obtain a characteristic value of the signal, it is thus necessary to calculate the mean over a much longer length of time, or to calculate the mean of several rms values for various samples of the signal. One in addition notes that the smaller Af is, the more the signal response at the filter output has a low rms value [TIP 77]. L2 To be liberated from the width Af, one considers rather the variations of — Af with f. The rms value is squared by analogy with electrical power. 2.2.2. Definition If one considers a tension u(t) applied to the terminals of a resistance R = 1 Q, passing current i(t), the energy dissipated (Joule effect) in the resistance during time dt is equal to: dE=Ri2(t)dt=i2(t)

[2.3]


37

Figure 2.3. Electrical circuit with source of tension and resistance

The instantaneous power of the signal is thus:

[2.4] and the energy dissipated during time T, between t and t + T, is written:

[2.5] P(t) depends on time t (if i varies with t). It is possible to calculate a mean power in the interval T using:

[2-6l The total energy of the signal is therefore:

[2.7] and total mean power:

[2.8]

By analogy with these calculations, one defines [BEN 58] [TUS 72] in vibration mechanics the mean power of an excitation ^(t) between -T/2 and +T/2 by:

[2.9]

38

Random vibration

where J.*T=*(t)

for|tT/2

Let us suppose that the function ^ T (t) has as a Fourier transform L T (f). According to Parseval's equality, p.10]

yielding, since [JAM 47] *

Pm = lim - (""VrMl2 df = lim - J°° L T (f)P df T->«>T -°°

[2.11]

[2.12]

T-»ooT °

This relation gives the mean power contained in f(t) when all the frequencies are considered. Let us find the mean power contained in a frequency band Af . For that, let us suppose that the excitation ^(t) is applied to a linear system with constant parameters whose weighting function is h(t) and the transfer function H(f) . The response r T (t) is given by the convolution integral r1T ( t ) =J| h ( X ) M t-X)dX 1 Q

[2.13]

where X is a constant of integration. The mean power of the response is written: Pmresponse =^1(^(0^

[2.14]

i.e., according to Parseval's theorem: 2 fT,

, tf

Pmresponse = lim — 'p JUI I R-rlf 1 v /II df T

f) 11^1 L^-'J

If one takes the Fourier transform of the two members of [2.13], one can show that: R T (f) = H(f) L T (f) yielding

[2.16]


^response '^f f N'f

L

l(ff

df

39

P-171

Examples 1 . If H(f ) = 1 for any value of f, rP

mresponse

=lim

_

r 2 l L TOf

df = pr

. mmput

[2.18]

a result which a priori is obvious. Af

Af

2

2

2. IfH(f) = l f o r O < f - — < f < f + — H(f) = 0 elsewhere

In this last case, let us set: |2

GT(f)="LT(f)^

[2.20]

The mean power corresponding to the record ^j(t), finite length T, in the band Af centered on f, is written: T (f)df

[2.21]

and total mean power in all the record P(f,Af) = lim T 4 " 2 G T (f) df

[2.22]

One terms power spectral density the quantity: lim ^

T-»oo

Af

[2.23]

40

Random vibration

In what follows, we will call this function PSD.

Figure 2.4. Example of PSD (aircraft)

NOTE. By using the angular frequency Q, we would obtain:

[2.24] with

[2.25]

Taking into account the above relations, and [2.10] in particular, the PSD G(f) can be written [BEA 72] [BEN 63] [BEN 80]: [2.26]

where ^ T (t, Af) is the part of the signal ranging between the frequencies f - Af/2 and f + Af/2. This relation shows that the PSD can be obtained by filtering the signal using a narrow band filter of given width, by squaring the response and by taking the mean of the results for a given time interval [BEA 72]. This method is used for analog computations.


41

The expression [2.26] defines theoretically the PSD. In practice, this relation cannot be respected exactly since the calculation of G(f) would require an infinite integration time and an infinitely narrow bandwidth.

NOTES. - The function G(f) is positive or zero whatever the value off. - The PSD was defined above for f ranging between 0 and infinity, which corresponds to the practical case. In a more general way, one could define S(f) mathematically between - oo and + <x>, in such a way that S(-f) = S(f)

[2.27]

- The pulsation Q = 2 n f is sometimes used as variable instead off. 7/"Gn(Q) is the corresponding PSD, we have G(f) = 2 TC GQ(Q)

[2.28]

The relations between these various definitions of the PSD can be easily obtained starting from the expression of the rms value: = fG(f)df = «u

One then deduces: G(f) = 2 S(f)

[2.30]

'j — /"• ô^

r^? ^ 11

G(f) = 4 n Sn(Q)

[2.32]

f(f\

NOTE. A sample of duration Tqfa stationary random signal can be represented by a Fourier series, the term aj of the development in an exponential Fourier series being equal to: sin

27ikt T

d; — 1

J T

-T/2

cos

2nkt

dt

T The signal t(t) can be written in complex form

42

Random vibration

P.34] 1 / v where ck = — ^ctj - Pj ij 2

77ze power spectral density can also be defined from this development in a Fourier series. It is shown that [PRE 54] [RAC 69] [SKO 59] [SVE 80]

T-»oo 2 Af The power spectral density is a curve very much used in the analysis of vibrations: - either in a direct way, to compare the frequency contents of several vibrations, to calculate, in a given frequency band, the rms value of the signal, to transfer a vibration from one point in a structure to another, ... - or as intermediate data, to evaluate certain statistical properties of the vibration (frequency expected, probability density of the peaks of the signal, number of peaks expected per unit time etc). NOTE. The function G(f), although termed power, does not have the dimension of it. This term is often used because the square of the fluctuating quantity appears often in the expression for the power, but it is unsuitable here [LAL 95]. So it is often preferred to name it 'acceleration spectral density' or 'acceleration density' [BOO 56] or 'power spectral density of acceleration' or 'intensity spectrum' [MAR 58].

2.3. Cross-power spectral density From two samples of random signal records ^j(t) and ^(t)' one defines the cross-power spectrum by

T-»oo

if the limit exists, Lj and L2 being respectively the Fourier transforms of ^(t) and ^ ( t ) calculated between 0 and T.


43

2.4. Power spectral density of a random process The PSD was defined above for only one function of time ^(t). Let us consider the case here where the function of time belongs to a random process, where each function will be noted J t(t). A sample of this signal of duration T will be denoted by 1 0 T (t), and its Fourier transform ^(f). Its PSD is

i

'L T (f) G T (f) =

[2.37]

By definition, the PSD of the random process is, over time T, equal to:

n being the number of functions ! i(\) and, for T infinite, G(f) = lim G T (f)

[2.39]

T->oo

If the process is stationary and ergodic, the PSD of the process can be calculated starting from several samples of one recording only.

2.5. Cross-power spectral density of two processes As previously, one defines the cross-power spectrum between two records of duration T each one taken in one of the processes by: [

GT(f) =

9 'T*l 'T 2 T

The cross-power spectrum of the two processes is, over T,

Z'G T (f) n

and, for T infinite,

[2-40]

44

Random vibration

G(f) = lim G T (f)

[2.42]

2.6. Relation between PSD and correlation function of a process It is shown that, for a stationary process [BEN 58] [BEN 80] [JAM 47] [LEY 65] [NEW 75]:

G(f)=2 r°R(T)

[2.43]

J-OO

R(T) being an even function of T, we have: G(f) = 4 J°° R(T) cos(2 it f t) dr

[2.44]

If we take the inverse transform of G(f) given in [2.43], it becomes: [2-45] i.e., since G(f) is an even function of f [LEY 65]: R(T)= J°°G(f)cos(27tf T)

[2.46]

R(Q) = r (t) = ff G(f) df = (rms value)^

[2.47]

and

NOTE. These relations, named Wiener-Khinchine relations', can be expressed in terms of the angular frequency Q in the form [BEN 58] [KOW 69] [MIX 69]: G(Q) = 71

R(T) cos(Q T) di

[2.48]

°

R(t)= j°°G(Q)cos(QT)dT

[2.49]


45

2.7. Quadspectrum - cospectrura The cross-power spectral density G^ u (f) can be written in the form [BEN 80]:

where the function C^ u (f) = 2 J R j u ( t ) cos(2 it f T) (h

[2-51]

—GO

is the cospectrum or coincident spectral density, and where Q^f) = 2 J^R^r) sin(2 w f T) dt

f2-52]

—CO

is the quadspectrum or quadrature spectral density function. We have:

R

fti W = Jr[ C û( f ) cos(2 * f T ) + Q*u( f ) sin(2 * f T)] df

f2>54]

[2.55]

[2.57]

2.8. Definitions 2.8.1. Broad-band process A broad-band process is a random stationary process whose power spectral density G(Q) has significant values hi a frequency band or a frequency domain which is rigorously of the same order of magnitude as the central frequency of the band[PRE56a].

46

Random vibration

Figure 2.5. Wide-band process Such processes appear in pressure fluctuations on the skin of a missile rocket (jet noise and turbulence of supersonic boundary layer).

2.8.2. White noise When carrying out analytical studies, it is now usual to idealize the wide-band process by considering a uniform spectral density G(f) = G 0 .

Figure 2.6. White noise A process having such a spectrum is named white noise by analogy with white light which contains the visible spectrum. An ideal white noise, which is supposed to have a uniform density at all frequencies, is a theoretical concept, physically unrealizable, since the area under the curve would be infinite (and therefore also the rms value). Nevertheless, model ideal white noise is often used to simplify calculations and to obtain suitable orders of magnitude of the solution, in particular for the evaluation of the response of a onedegree-of-freedom system to wide-band noise. This response is indeed primarily produced by the values of the PSD in the frequency band ranging between the halfpower points. If the PSD does not vary too much in this interval, one can compare it at a first approximation to that of a white noise of the same amplitude. It should however be ensured that the results of this simplified analysis do indeed provide a


47

correct approximation to that which would be obtained with physically attainable excitation.

2.8.3. Band-limited white noise One also uses in the calculations the spectra of band-limited white noises, such as that in Figure 2.7, which are correct approximations to many realizable random processes on exciters.

Figure 2.7. Band-limited white noise

2.8.4. Narrow-band process A narrow-band process is a random stationary process whose PSD has significant values in one frequency band only or a frequency domain whose width is small compared with the value of the central frequency of the band [FUL 62].

Figure 2.8. PSD of narrow-band noise

The signal as a function of time i(i) looks like a sinusoid of angular frequency Q0, with amplitude and phase varying randomly. There is only one peak between two zero crossings.

48

Random vibration

Figure 2.9. Narrow-band noise It is of interest to consider individual cycles and envelopes, whose significance we will note later on. If the process is Gaussian, it is possible to calculate from G(Q) the expected frequency of the cycles and the probability distribution of the points on the envelope These processes relate in particular to the response of low damped mechanical systems, when the excitation is a broad-band noise. 2.8.5. Pink noise A pink noise is a vibration of which the power spectral density amplitude is of inverse proportion to the frequency.

2.9. Autocorrelation function of white noise The relation [2.45] can be also written, since G(f) = 4 re S(n) [BEN 58] [CRA 63]: [2.58] If S(Q) is constant equal to S0 when Q varies, this expression becomes:

where the integral is the Dirac delta function S(T) , such as:


6(t) -> oo

when T -» 0

6(r) = 0

when T = 0

49

[2.60]

f 5(t) dt = 1

yielding R(T) = 2 TI S0 5(i)

[2.61]

NOTE. If the PSD is defined by G(Q) in (0, <x>), this expression becomes

R(T) = * ^- 8(1) 2

[2-62]

[2-63]

T = 0, R -> oo. Knowing that R(0) w e.vf_

[3.65]

and Xrmc

3.5. Case: periodic signals It is known that any periodic signal can be represented by a Fourier series in accordance with: = L0 +

Ln sin(2 7i n f,

Power spectral density [2,26]: G(f)= lira - j 4 ( t , Af)dt T-»ooTAf ° Af->0

is zero for f * f n (with f n = n f t ) and infinite for f = fn since the spectrum of is a discrete spectrum, in which each component Ln has zero width Af . If one wishes to standardize the representations and to be able to define the PSD f 00

of a periodic function, so that the integral J G(f) G(f) df is equal to the mean square value of ^(t), one must consider that each component is related to Dirac delta function, the area under the curve of this function being equal to the mean square value of the component. With this definition,

n=0

Rms value of random vibration

83

where ^Sn is the mean square value of the n* harmonic ^ n (t) defined by ,2 -rmsn

J(t)dt 1

n

[3.69]

[3.70]

~nf,

(n = 1,2, 3,...). ln(t) is the value of the n* component and

where T is arbitrary and i is the mean value of the signal l(t). The Dirac delta function §(f - n fj] at the frequency f n is such that: [3.71]

and [3.72]

S(f-f n ) =

for f *• f n (E = positive constant different from zero, arbitrarily small). The definition of the PSD in this particular case of a periodic signal does not require taking the limit for infinite T, since the mean square value of a periodic signal can be calculated over only one period or a whole number of periods.

Figure 3.12. PSD of a periodic signal The chart of the PSD of a periodic signal is that of a discrete spectrum, the amplitude of each component being proportional to the area representing its mean square value (and not its amplitude).

84

Random vibration

We have, with the preceding notations, relationships of the same form as those obtained for a random signal: [3.73] li

^

\

/

n=0

'o+ "

n=l

" [3.75]

and, between two frequencies fj and f; (fj = i fj - 8, f; = j fj + e, i and j integers,

[3.76]

Lastly, if for a random signal, we had: J f f _ + G(f)df = 0

[3-77]

we have here: fo(f)df = f n * [0

forf = n f i for f *n ^ et f * 0

[378]

The area under the PSD at a given frequency is either zero, or equal to the mean square value of the component if f = n fj (whereas, for a random signal, this area is always zero). 3.6. Case: periodic signal superimposed onto random noise Let us suppose that: 4t) = a(t) + P(t)

[3.79]

a(t) = random signal, of PSD G a (f) defined in [2.26] p(t)= periodic signal, of PSD G p (f) defined in the preceding paragraph.

Rms value of random vibration

85

The PSD of *(t) is equal to: G,(f) = G a (f) + G p (f)

[3.80]

G,(f) = G a ( f ) + Z 4 6 ( f - f n )

P.81]

n=0

where fn = « f,

n = integer e (0, °o) fj= fundamental frequency of the periodic signal in = mean square value of the n* component ^ n (t) of ^(t) The rms value of this composite signal is, as previously, equal to the square root of the area under G «(f).


Chapter 4

Practical calculation of power spectral density

Analysis of random vibration is carried out most of the time by supposing that it is stationary and ergodic. This assumption makes it possible to replace a study based on the statistical properties of a great number of signals by that of only one sample of finite duration T. Several approaches are possible for the calculation of the PSD of such a sample.

4.1. Sampling of the signal Sampling consists in transforming a vibratory signal continuous at the outset by a succession of sample points regularly distributed in time. If 5t is the time interval separating two successive points, the sampling frequency is equal to fsamp = 1 / 8t. So that the digitized signal is correctly represented, it is necessary that the sampling frequency is sufficiently high compared to the largest frequency of the signal to be analysed. A too low sampling frequency can thus lead to an aliasing phenomenon, characterized by the appearance of frequency components having no physical reality.

88

Random vibration

Example Figure 4.1 thus shows a component of frequency 70 Hz artificially created by the sampling of 200 points/s of a sinusoidal signal of frequency 350 Hz.

Figure 4.1. Highlighting of the aliasing phenomenon due to under-sampling

Shannon's theorem indicates that if a function contains no frequencies higher than fmax Hz, it is completely determined by its ordinates at a series of points spaced 1/2 fmax seconds apart [SHA 49]. Given a signal which one wishes to analyse up to the frequency fmax , it is thus appropriate, to avoid aliasing - to filter it using a low-pass filter in order to eliminate frequencies higher than fmax (the high frequency part of the spectrum which can have a physical reality or noise), -to sample it with a frequency at least equal to 2f max [CUR87] [GIL88] [PRE 90] [ROT 70].

NOTE. f

Nyquist = fsamp. / 2 » called Nyquist frequency.

In practice however, the low-pass filters are imperfect and filter incompletely the frequencies higher than the wanted value. Let us consider a low-pass filter having a decrease of 120 dB per octave beyond the desired cut-out frequency (f max ). It is


89

considered that the signal is sufficiently attenuated at - 40 dB. It thus should be considered that the true contents of the filtered signal extend to the frequency corresponding to this attenuation (f_ 40 ), calculated as follows.

Figure 4.2. Taking account of the real characteristics of the low-pass filter for the determination of the sampling frequency

An attenuation of 120 dB per octave means that

-120 =

where AQ and Aj are respectively the amplitudes of the signal not attenuated (frequency fmax ) and attenuated at -40 dB (frequency f_ 4 Q ). Yielding

1*2

-120 =

and log 2 f

-40 = 10 3

«L26

90

Random vibration

The frequency f_40 being the greatest frequency of the signal requires that, according to Shannon's theorem, fsamp = 2 f_4o , yielding f

samp. _

The frequency f_4Q is also the Nyquist frequency This result often resulted in the belief that Shannon's theorem imposes a sampling frequency at least equal to 2.6 times the highest frequency of the signal to be analysed. We will use this value in the followings paragraphs.

4.2. Calculation of PSD from rms value of filtered signal The theoretical relation [2.26] which would assume one infinite duration T and a zero analysis bandwidth Af is replaced by the approximate relation [KEL 67]:

where '(.^ is the mean square value of the sample of finite duration T, calculated at the output of a filter of central frequency f and non zero width Af [MOR 56]. NOTE. Given a random vibration l(t) of white noise type and a perfect rectangular filter, the result of filtering is a signal having a constant spectrum over the width of the filter, zero elsewhere [CUR 64]. The result can be obtained by multiplying the PSD G0 of the input l(t) by the square of the transmission characteristic of the filter (frequency-response characteristic) at each frequency (transfer function, defined as the ratio of the amplitude of the filter response to the amplitude of the sinewave excitation as a function of the frequency. If this ratio is independent of the excitation amplitude, the filter is said to be linear). In practice, the filters are not perfectly rectangular. The mean square value of the response is equal to G0 multiplied by the area squared under the transfer function of the filter. This surface is defined as the 'rms bandwidth of the filter '.


91

If the PSD of the signal to be analysed varies with the frequency, the mean square response of a perfect filter divided by the -width Af of the filter gives a point on the PSD (mean value of the PSD over the width of the filter). With a real filter, this approximate value of the PSD is obtained by considering the ratio of the mean square value of the response to the rms bandwidth of the filter Af, defined by [BEN 62], [GOL 53] and [PIE 64]:

Af =

H(f)

[4.2]

dt

H,

•where H(f) is the frequency response function of the (narrow) band-pass filter used and Hmax its maximum value.

4.3. Calculation of PSD starting from Fourier transform The most used method consists in considering expression [2.39]: ^

r~

~i

[4.3]

G M (f) = lim - d |L(f, T)P L

-

J

NOTES. 1. Knowing that the discrete Fourier transform can be written [KAY81] L

( m T ) - T y i c::f t 2 * j m 1X1 ^ J v1 N N N

[4.4]

j=o

r/ze expression of the PSD can be expressed for calculation in the form [BEN 71] [ROT 70]: N-l

G(mAf) = N

m

i exp -1

j=o

[4.5]

N

where 0 < m < M and Cj = j 5t. 2. The calculation of the PSD can also be carried out by using relation [2.26], by evaluating the correlation in the time domain and by carrying out a Fourier transformation (Wiener-Khintchine method) (correlation analysers) [MAX 86].

92

Random vibration

The calculation data are in general the following: - the maximumfrequencyof the spectrum, - the number of points of the PSD (or the frequency step Af ), - the maximum statistical error tolerated.

4.3.1. Maximum frequency Given an already sampled signal (frequency fsamp. ) and taking into account the elements of paragraph 4.1, the PSD will be correct only for frequencies lower than ^ max = * ' 2-

4.3.2. Extraction of sample of duration T Two approaches are possible for the calculation of the PSD: - to suppose that the signal is periodic and composed of the repetition of the sample of duration T, - to suppose that the signal has zero values at all the points outside the time time corresponding to the sample. These two approaches are equivalent [BEN 75]. In both cases, one is led to isolate by truncation a part of the signal, which amounts to applying to it a rectangular temporal window r(t) of amplitude 1 for 0 < t < T and zero elsewhere. If l(i) is the signal to be analysed, the Fourier transform is thus calculated in practice with f(t) = ^(t) r(t).


93

Figure 4.3. Application of a temporal window

In the frequency domain, the transform of a product is equal to the convolution of the Fourier transforms L(Q) and R(£i) of each term:

F(Q) = J L(o>) R(Q - G>) dco (CQ is a variable of integration).

Figure 4.4. Fourier transform of a rectangular waveform

[4.6]

94

Random vibration

The Fourier transform of a rectangular temporal window appears as a principal central lobe surrounded by small lobes of decreasing amplitude (cf. Volume 2, Chapter 1). The transform cancels out regularly for Q a multiple of

(i.e. a T

1 frequency f multiple of —). The effect of the convolution is to widen the peaks of T the spectrum, the resolution, consequence of the width of the central lobe, not being 1

able to better Af = —. T The expression [4.6] shows that, for each point of the spectrum of frequency Q 2n (multiple of ), the side lobes have a parasitic influence on the calculated value of T the transform (leakage). To reduce this influence and to improve the precision of calculation, their amplitude needs to be reduced.

t

Figure 4.5. The Manning window

This result can be obtained by considering a modified window which removes discontinuities of the beginning and end of the rectangular window in the time domain. Many shapes of temporal windows are used [BLA 91] [DAS 89] [JEN 68] [NUT 81]. One of best known and the most used is the Manning window, which is represented by a versed sine function (Figure 4.5): / x

if

r(t) = - 1-cos

27C

0

[4.7]


95

Figure 4.6. The Bingham -window [BIN 67]

This shape is only sometimes used to constitute the rising and decaying parts of the window (Bingham window, Figure 4.6). Weighting coefficient of the window is the term given to the percentage of rise time (equal to the decay time) of the total length T of the window. This ratio cannot naturally exceed 0.5, corresponding to the case of the previously defined Harming window. Examples of windows The advantages of the various have been discussed in the literature [BIN 67] [NUT 81]. These advantages are related to the nature of the signal to be analysed. Actually, the most important point in the analysis is not the type of window, but rather the choice of the bandwidth [JEN 68]. The Harming window is nevertheless recommended. The replacement of the rectangular window by a more smoothed shape modifies the signal actually treated through attenuation of its ends, which results in a reduction of the rms duration of the sample and in consequence in a reduction of the resolution, depending on the width of the central lobe. One should not forget to correct the result of the calculation of the PSD to compensate for the difference in area related to the shape of the new window. Given a temporal window defined by r(t), having R(f) for Fourier transform, the area intervening in the calculation of the PSD is equal to: Q = J4* |R(f)|2 df

[4.8]

96

Random vibration

From Parse val's theorem, this expression can be written in a form utilizing the N points of the digitized signal:

[4.9]


97

Table 4.1. The principal windows Window type

Compensation factor

Definition

/x i!

[ io

7t(t-9T/lo)~|{

ft f 1 _L J~/-»C T\l) — ..' , .' 4*\L1 + COS

Bingham (Figure 4.6)

j

1/0.875

1 T JJ T 9T for 0 < t < — and <x>

s= lim s

[4.17]

N-»°o

s being the true standard deviation for a measurement G(f), is a description of uncertainty of this measure. In practice, one will make only one calculation of G(f) at the frequency f and one will try to estimate the error carried out according to the conditions of the analysis.

4.6.2. Definition

Statistical error or normalized rms error is the quantity defined by the ratio: £_

S

_4

[4.18]

(variation coefficient) where t ^ is the mean square value of the signal filtered in the filter of width Af (quantity proportional to G(f)) and s 2 is the standard *

deviation of the measurement of finite duration T.

related to the error introduced by taking a

NOTE. We are interested here in the statistical error related to calculation of the PSD. One makes also an error of comparable nature during the calculation of other quantities such as coherence, transfer function etc (cf. paragraph 4. 12).

104

Random vibration

4.7. Statistical error calculation 4.7.1. Distribution of measured PSD

V

If the ratio s = •=£=• is small, one can ensure with a high confidence level that a

4

measurement of the PSD is close to the true average [NEW 75]. If on the contrary s is large, the confidence level is small. We propose below to calculate the confidence level which can be associated with a measurement of the PSD when s is known. The analysis is based on an assumption concerning the distribution of the measured values of the PSD. 2

The measured value of the mean square z of the response of a filter Af to a random vibration is itself a random variable. It is supposed in what follows that z can be expressed as the sum of the squares of a certain number of Gaussian random variables statistically independent, zero average and of the same variance:

p /n x 2 (t)d t+ /

J

J

'2T/n ..2/ 'T/n X ( t ' d t + ""' JT(l-l/n)

[4.19]

One can indeed think that z satisfies this assumption, but one cannot prove that these terms have an equal weight or that they are statistically independent. One notes however in experiments [KOR 66] that the measured values of z roughly have the distribution which would be obtained if these assumptions were checked, namely a chi-square law, of the form: X2 = X2 + X2 + X32 + -+Xn

[4.20]

If it can be considered that the random signal follows a Gaussian law, it can be shown ([BEN 71] [BLA 58] [DEN 62] [GOL 53] [JEN 68] [NEW 75]) that 2

measurements G(f) of the true PSD G(f) are distributed as G(f) — where x n *s n the chi-square law with n degrees of freedom, mean n and variance 2 n (if the mean value of each independent variable is zero and their variance equal to 1 [BLA 58] [PIE 64]). Figure 4.10 shows some curves of the probability density of this law for various values of n. One notices that, when n grows, the density approaches that of a normal law (consequence of the central limit theorem).


105

Figure 4.10. Probability density: the chi-square law

NOTE. Some authors [OSG 69] consider that measurements G(f) are distributed more Y

2

like G(f)

, basing themselves on the following reasoning. From the values n-1

Xj, X 2 , X3, • • • , X n of a normally distributed population, of mean m (unknown value) and standard deviation s, one can calculate —.\2 (x 1 -x) +(x 2 -x) +(x 3 -x) + ... + (x n -x)

[4.21]

-where [4.22] (mean of the various values taken by variable X by each of the n elements). Let us consider the reduced variable X: -X

[4.23]

106

Random vibration

The variables Uj are no longer independent, since there is a relationship between them: according to a property of the arithmetic mean, the algebraic sum of the deviations with respect to the mean is zero, therefore ^(Xj - X] = 0, and = 0 yielding:

consequently,

In the sample of size n, only n -1 data are really independent, for if n — 1 variations are known, the last results from this. If there is n -1 independent data, there are also n -1 degrees of freedom. The majority of authors however agree to consider that it is necessary to use a law with n degrees of freedom. This dissension has little incidence in practice, the number of degrees of freedom to be taken into account being necessarily higher than 90 so that the statistical error remains, according to the rules of the art, lower than 15% approximately.

4.7.2. Variance of measured PSD The variance of G(f) is given by: G(O Xn S

G(f) ~

S

G(f) ~

[4.24]

var . n.

However the variance of a chi-square law is equal to twice the number of degrees of freedom: [4.25]

Var

yielding 2

o1" S

G(f)~

G 2 (f)

*) ______ n 2 n

The mean of this law is equal to n.

_2

n

~

O 2

G 2 (f)

n

[4.26]


107

4.7.3. Statistical error

Figure 4.11. Statistical error as function of the number ofdof

G(f) =

G(f)

G(f) = G(f)

[4.27]

The statistical error is thus such as:

£

2

S

G(f)

— .. _- .

G(f) [4.28] s is also termed 'standarderror'.

108

Random vibration

4.7.4. Relationship between number of degrees of freedom, duration and bandwidth of analysis This relation can be obtained,

either using

a series expansion

of

E^ [G(f ) - G( f ) j r , or starting from the autocorrelation function.

From a series expansion: It is shown that [BEN 61b] [BEN 62]:

TAf

[4.29]

576 bias

variability

Except when the slope of the PSD varies greatly with Af, the bias is in general negligible. Then 2 8 =

E{[6(f)-G(f)f

G 2 (f)

1 TAf

This relation is a good approximation as long as e is lower than approximately 0.2 (i.e. for T Af > 25). [4.30]

The error is thus only a function of the duration T of the sample and of the width Af of the analysis filter (always assumed ideal [BEA 72] [BEN 63] [NEW 75]).

Figure 4.12. Statistical error


109

Figure 4.12 shows the variations of this quantity with the product TAf. The number of events n represented by a record of white noise type signal, duration T, filtered by a filter of width A f , is thus, starting from [4.28]: n = 2AfT

[4.31]

Definition The quantity n = 2 Af T is called number of degrees of freedom (dof). From the autocorrelation function Let us consider ^(t) a vibratory signal response collected at the output of a filter of width Af. The mean square value of i(i) is given by [COO 65]:

2

2

Setting t^f the measured value of \^, we have, by definition: ft/2 ,2

,2

[4.32] e=

"Af

.4 A Af ~

8 = :

Af

However, we can write: 9

dv

i.e., while setting t = u and t = v-u = v - t ,

110

Random vibration

yielding

dt where D{T) is the autocorrelation coefficient. Given a narrow band random signal, we saw that the coefficient p is symmetrical with regard to the axis i = 0 and that p decrease when T| becomes large. If T is sufficiently large, as well as the majority of the values oft: 2 fT f+oo 2 / x — j dt J P2(l) dT

2 S =

T^

0

-00

•Af

yielding the standardized variance 8 [BEN 62]: 2

_ 2 f+oo

[4.33]

T ' .

[4-34]

*

Particular cases 1. Rectangular band-pass filter We saw [2.70] that in this case [MOR 58]:

cos 2 7t fft T sin 7i Af T P(T) = TC T Af

yielding 2

2

2 4 foo cos 2 7i f 0 T sin rc Af T s » —J T—T rn ô T u

2 2 Ar.2 71 T Af


e

2

111

1 *

TAf and [BEN 62] [KOR 66] [MOR 63]:

[4.35]

Example For e to be lower than 0.1, it is necessary that the product T Af be greater than 100, which can be achieved, for example, either with T = 1 s and Af = 100 Hz, or with T = 100 s and Af = 1 Hz. We will see, later on, the incidence of these choices on the calculation of the PSD.

2. Resonant circuit For a resonant circuit: /

\

^

r-

P(T) = cos 2 7t f 0 T e

-ft t Af

yielding 2

4

f°°

2 ~

-,

-2 it T Af ,

E « — J cos 2 7i f 0 T e 00

dt

-27iiAf

[4.36]

4.7.5. Confidence interval Uncertainty concerning G(f) can also be expressed in term of confidence interval. If the signal ^(t) has.a roughly Gaussian probability density function, the distribution of

X2 -/ v , for any f, is the same as —. Given an estimate G(f) obtained G(f) n

G(f)

112

Random vibration

from a signal sample, for n = 2 Af T events, the confidence interval in which the true PSD G(f) is located is, on the confidence level (l - a):

n G(f)

[4.37]

2 Cn, l-oc/2

2

, a/2 2

where x n a/2 and % n i_ ot /2 âve n degrees of freedom. Table 4.3 gives some values 2 of % n a according to the number of degrees of freedom n for various values of a.

•/ Figure 4.13. Values of%n

a with

respect to the number of degrees of freedom and ofo.

Figure 4.13 represents graphically the function parameterized by the probability a.

xa with respect to n,

Example 99% of the values lie between 0.995 and 0.005. One reads from Figure 4.13, for n = 10, that the limits are %2 = 25.2 and 2.16.


113

Figure 4.14. Example of use of the curves X n a (n)

Example Figure 4.14 shows how in a particular case these curves can be used to evaluate numerically the limits of the confidence interval defined by the relation [4.37]. Let us set n = 10. One notes from this Figure that 80% of the values are within the interval 4.87 and 15.99 with mean value m = 10. If the true value of the mean of the calculated PSD S0 is m, it cannot be determined exactly, nevertheless it is known that 4.87

10

S

15.99 10

2.05 S0 > m > 0.625 S0

roi

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z [PL IdS]foPfo

N. Jaqutnu 3t/;/o

uoijounfsv^'^fo

20 21 22 23 24

40.0 41.4 42.8 44.2 45.6

37.6 38.9 40.3 41.6 43.0

34.2 35.5 36.8 38.1 39.4

31.4 32.7 33.9 35.2 36.4

28.4 29.6 30.8 32.0 33.2

23.8 24.9 26.0 27.1 28.2

19.3 20.3 21.3 22.3 23.3

15.5 16.3 17.2 18.1 19.0

12.4 13.2 14.0 14.8 15.7

10.9 11.6 12.3 13.1 13.8

9.59 10.3 11.0 11.7 12.4

8.26 8.90 9.54 10.2 10.9

7.43 8.03 8.64 9.26 9.89

25 26 27 28 29

46.9 48.3 49.6 51.0 52.3

44.3 45.6 47.0 48.3 49.6

40.6 41.9 43.2 44.5 45.7

37.7 38.9 40.1 41.3 42.6

34.4 35.6 36.7 37.9 39.1

29.3 30.4 31.5 32.6 33.7

24.3 25.3 26.3 27.3 28.3

19.9 20.8 21.7 22.7 23.6

16.5 17.3 18.1 18.9 19.8

14.6 15.4 16.2 16.9 17.7

13.1 13.8 14.6 15.3 16.0

11.5 12.2 12.9 13.6 14.3

10.5 11.2 11.8 12.5 13.1

30 40 50 60

53.7 66.8 79.5 92.0

50.9 63.7 76.2 88.4

47.0 59.3 71.4 83.3

43.8 55.8 67.5 79.1

40.3 51.8 63.2 74.4

34.8 45.6 56.3 67.0

29.3 39.3 49.3 59.3

24.5 33.7 42.9 52.3

20.6 29.1 37.7 46.5

18.5 26.5 34.8 43.2

16.8 24.4 32.4 40.5

15.0 22.2 29.7 37.5

13.8 20.7 28.0 35.5

70 80 90 100

104.2 116.3 128.3 140.2

100.4 112.3 124.1 135.8

95.0 106.6 118.1 129.6

90.5 101.9 113.1 124.3

85.5 96.6 107.6 118.5

77.6 88.1 98.6 109.1

69.3 79.3 89.3 99.3

61.7 71.1 80.6 90.1

55.3 64.3 73.3 82.4

51.7 60.4 69.1 77.9

48.8 57.2 65.6 74.2

45.4 53.5 61.8 70.1

43.3 51.2 59.2 67.3

116

Random vibration

More specific tables or curves were published to provide directly the value of the limits [DAR 72] [MOO 61] [PIE 64]. For example, Table 4.4 gives the confidence interval defined in [4.37] for three values of 1 - a [PIE 64], Table 4.4. Confidence limits for the calculation of a PSD [PIE 64] Degrees of freedom

Confidence interval limits relating to a measured power spectral density G(f) = l

(l- a) = 0.90

(l - a) = 0.95

(l - a) = 0.99

n

Lower limit

Higher limit

Lower limit

Higher limit

Lower limit

Higher limit

10 15 20

0.546 0.599 0.637

2.54

0.483 0.546 0.585

3.03 2.39 2.08

0.397 0.457 0.500

4.63 3.26 2.69

25

0.662

0.615

1.90

0.532

2.38

30

0.685

1.71 1.62

0.637

0.559

2.17

40 50

0.719 0.741

1.51 1.44

0.676 0.699

1.78 1.64

0.599 0.629

1.93

75

0.781 0.806

1.34 1.28

0.743

1.42

0.769

1.35

0.680 0.714

1.59

100 150 200

0.833 0.855

1.22 1.19

0.806 0.826

1.27 1.23

0.758 0.781

1.49 1.37 1.31

250

0.870 0.877

1.16 1.15

0.847 0.855

1.20

300

1.18

0.800 0.820

1.25

400

0.893

0.877

1.15

0.840

1.21

500

0.901

1.13 1.11

0.885

1.14

0.855

1.18

750 1000 5000

0.917

1.09 1.08 1.03

0.909 0.917 0.962

1.11 1.09 1.04

0.877

1.15 1.12

0.934 0.971

2.07 1.84

1.54

0.893 0.952

Multiply the lower and higher limits in the table by the measured value G(f ) to obtain the limits of the confidence interval of the true value G(f).

1.78

1.27

1.05


117

NOTE. When n > 30, -^2 x,n follows a law close to a Gaussian law of mean ^2 n -1 and standard deviation 1 (Fisher's law). Let \ be a normal reduced variable and a a value of the probability such that

Probjx 120], i.e. for E small, it is shown that the chi-square law tends towards the normal law and that the distribution of the values o/G(f) can itself be approximated by a normal law of mean n and standard deviation ^2 n (law of large numbers). In this case,


119

Figures 4.15 to 4.18 provide, for a confidence level of 99%, and then 90%: — variations in the confidence interval limits depending to the number of degrees of freedom n, obtained using an exact calculation (chi-square law), by considering the Fisher and Gauss assumptions, - the error made using each one of these simplifying assumptions. These curves show that the Fisher assumption constitutes an approximation acceptable for n greater than 30 approximately (according to the confidence level), •with relatively simple analytical expressions for the limits.

Figure 4.19. Confidence limits (G/G)

Figure 4.20. Confidence limits (G/G) [MOO 61]

120

Random vibration

G G The ratio — (or —, depending on the case) is plotted in Figures 4.19 and 4.20 G G with respect to n, for various values of the confidence level.

Example Let us suppose that a PSD level G = 2 has been measured with a filter of width Af = 2.5 Hz and from a signal sample of duration T= 10 s. The number of degrees of freedom is n = 2 T Af = 50 (yielding s = ..•

= 0.2). Table 4.4

gives, for 1 - a = 0.90 : 0.741G 50.

126

Random vibration

Example If it is required that s = ±0.5 dB , i.e. that s = ± 12.2 % , it is necessary, at confidence level 84%, that TAf = 67.17, or that the number of degrees of freedom is equal to n = 2 T Af « 135. If A f = 24 Hz, T1

i1Un4

- ° ^..ij5 s. At confidence level 90%, the (0.122fAf variatioiis of the PSI) are, in the interval [BAN 78]:

n 50 100 250

-

Lower limit (dB)

Upper limit (dB)

-1.570 -1.077 -0.665

1.329 0.958 0.617

4.7.7. Statistical error calculation from digitized signal Let N be the number of sampling points of the signal x(t) of duration T, M the number of points in frequency of the PSD fsamp the sampling frequency of the signal f fmax the maximum frequency of the PSD, lower or equal to -samp 2.6 (modified Shannon's theorem, paragraph 4.1)

5t the time interval between two points. We have: T=N5t

[4.53] [4.54]

2M


127

NOTE. M points separated by an interval Af lead to a maximum frequency f = M Af = -samp' . To fulfill the condition of paragraph 4.3.1, it is necessary l

to limit in practice the useful field of the PSD to f m a x *

samp. 2.6

If we need a PSD calculated based on M points, we need at least AN = 2 M points per block. Since the signal is composed of N points, we will cut up it into K=

N

T

blocks of duration AT = —. 2M K

Knowing that fsanm = —: St

Af = 2M8t yielding

i.e. [4.55]

128

Random vibration

Example N = 32 768 points

M = 512 points

T = 64 s

yielding 2M = 1 024 points per sample N

K=

= 32 samples (of 2 s)

2M

K 32 Af = — = — = 0.5 Hz T 64

N 32768 fsamp = — = = 512 points/s T 64

32768 Even if M Af = 512 0.5 Hz = 256 Hz, we must have, in practice, 2.6

2.6

4.8. Overlapping 4.8.1. Utility One can carry out an overlapping of blocks for three reasons: - to limit the loss of information related to the use of a window on sequential blocks, which results in ignorance of a significant part of the signal because of the low values of the window at its ends [GAD 87]; - to reduce the length of analysis (interesting for real time analyses) [CON 95]; - to reduce the statistical error when the duration T of the signal sample cannot be increased. We saw that this error is related to the number of blocks taken in the sample of duration T. If all the blocks are sequential, the maximum number K of blocks of fixed duration AT (arising from the frequency resolution desired) is equal to the integer part of T/AT [WEL 67]. An overlapping makes it possible to increase this number of blocks whilst preserving their size AT.


129

Overlapping rate The overlapping rate R is the ratio of the duration of the block overlapped by the following block over the total duration of the block. This rate is in general limited to the interval between 0 and 0.75.

Figure 4.26. Overlapping of blocks

Overlapping in addition makes it possible to minimize the influence of the side lobes of the windows [CAR 80] [NUT 71] [NUT 76].

4.8.2. Influence on number of degrees of freedom Let N be the number of points of the signal sample, N' (> N) the number of points necessary to respect the desired statistical error with K blocks of size A N ( N ' = K A N ) . The difference N'-N must be distributed over K - l possible overlappings [NUT 71]:

N'-N = ( K - l ) R A N yielding R=

N'-N AN ( K - l )

=

N'-N

.. ... [4.56]

N'-AN

For R to be equal to 0.5 for example, it is necessary that N'= 2 N - AN. Overlapping modifies the number of degrees of freedom of the analysis since the blocks cannot be regarded any more as independent and noncorrelated. The

130

Random vibration

estimated value of the PSD no longer obeys a one chi-square law. The variance of the PSD measured from an overlapping is less than that calculated from contiguous blocks [WEL 67]. R. Potter and J. Lortscher [POT 78] showed however that, when K is sufficiently large, the calculation could still be carried out on the assumption of non overlapping, on the condition the result could still be corrected by a reduction factor depending on the type of window and the selected overlapping rate. The correlation as a function of overlapping can be estimated using the coefficient: r(t)r[t + ( l - R ) A T ] d t [4.57]

Table 4.5. Reduction factor Window

Correlation coefficient C

Coefficient \i

R = 25%

R = 50%

R = 75%

R = 50%

R = 75%

Rectangle

0.25000

0.50000

0.75000

0.66667

0.36364

Bingham

0.17143

0.45714

0.74286

0.70524

0.38754

Hamming

0.02685

0.23377

0.70692

0.90147

0.47389

Hanning

0.00751

0.16667

0.65915

0.94737

0.51958

Parzen

0.00041

0.04967

0.49296

0.999509

0.67071

Flat signal

0.00051

-0.01539

0.04553

0.99953

0.99540

Kaiser-Bessel

0.00121

0.07255

0.53823

0.98958

0.62896

4.8.3. Influence on statistical error When the blocks are statistically independent, the number of degrees of freedom is equal t o n = 2K = 2 T A f whatever the window. With overlappings of K blocks, the effective number of blocks to consider in order to calculate the statistical error is given [HAR 78] [WEL 67]: - f o r R = 50%by: K

1 50 =

2

2

CSQ%

K

K 2

2 c50%

K

'50%

[4.58]


131

-for R = 75% by: K

75%

* ~ ~,1 + T~2 2C

T~2 ^ ~ 2 _ c2 o + c2 T~2 75% + 2 c50o/0 + 2 c2s% 75 /0 50% + 3 C25o/0 K K2

1

+

2 75% +2 C50% + 2 C25%

= u« K

[4.59]

2C

(the approximation being acceptable for K > 10). Under these conditions, the statistical error is no longer equal to 1/vK , but to: e

[4.60]

=

The coefficient |i being less than 1, the statistical error is, for a given K, all the larger as overlapping is greater. But with an overlapping, the total duration of the treated signal is smaller, which makes it possible to carry out more quickly the analyses in real time (control of the test facilities). The time saving can be calculated from [4.56]: n _

N'-N

T-T

N'-AN

T-AT

(AT =duration of a block). To avoid a confusion of notations, we will let To be the duration of the signal to be treated with an overlapping and T the duration without overlapping. We then have:

N'-N

T-TO

N'-AN

T-AT

yielding

TO = T ( I - R ) + R A T

[4.6i]

Since R < 1 and AT « T, we have in general To « T (l - R) . The time saving T is thus approximately equal to — « (l - R) .

132

Random vibration

Example T = 25 s, Af = 4 Hz (i.e. K = T Af = 100), s0 = 0.1 (without overlapping). With R = 0.75 and a Hanning window, p,«0.52; yielding E = 1/^/0.52 x 25 x 4 » 0.139. But this result is obtained for a signal of duration T0 « (l - 0.75) 25 « 6.25 s.

If we consider now a sample of given duration T, overlapping makes it possible to define a greater number of blocks. This K1 number can be deducted from [4.56]: N'=

N-RAN 1-R

yielding, if N ' = K ' A N K-R

[4.62]

K' = 1-R

The increase in the number of blocks makes it possible to reduce the statistical error which becomes equal to:

1-R

1 8 =

= S0

K-R

[4.63]

1-R

Example With the data of the above example, the statistical error would be equal to 1 - 0.75 S«80

* 0.693 80 = 0.0693.

4.8.4. Choice of overlapping rate The calculation of the PSD uses the square of the signal values to be analysed. In this calculation the square of the function describing the window for each block thus intervenes in an indirect way, by taking account of the selected overlapping rate R. For a linear average, this leads to an effective weighting function r^^t) such as [GAD 87]:


133

[4.64] where T is the duration of the window used (duration of the block), i is the number of the window in the sum, K is the number of windows at time t

Figure 4.27. Ripple on the Manning -window (R = 0.58;

Figure 4.28. Manning \vindo\vfor R = 0.75

Figure 4.29. Ripple amplitude versus 1 - R

With the Harming window, one of the most used, it can be observed (Figure 4.27) that there is a ripple on r^^t), except when 1 - R is of the form 1 / p where p is an integer equal to or higher than 3 (Figure 4.28). The ripple has a negligible amplitude when 1-R is small (lower than 1/3) [CON 95] [GAD 87]. This property can be observed in Figure 4.29, which represents the variations of the ratio of the maximum and minimum amplitudes of the ripple (in dB) with respect to 1-R.

134

Random vibration

This remark makes it possible to justify the use, in practice, of an overlapping equal to 0.75 which guarantees a constant weighting on a broad part of the window (the other possible values, 2/3, 3/4, 4/5, etc..., are less used, because they do not lead as 3/4 to a integer number of points when the block size is a power of two).

4.9. Calculation of PSD for given statistical error 4.9.1. Case: digitalization of signal is to be carried out Given a vibration ^(t), one sets out to calculate its power spectral density between 0 and fmax with M points (M must be a power of 2), for a statistical error not exceeding a selected value 8. The procedure is summarized in Table 4.6 [BEA 72] [LEL 73] [NUT 80]. Table 4.6. Computing process of a PSD starting from a non-digitized signal The signal of total duration T (to be defined) will be cut out in K blocks of unit duration AT, under the following conditions: f samp. — 2-6 tmax _ *• Nyquist

*samp.

Condition to avoid the aliasing phenomenon (modified Shannon's theorem). Nyquist frequency [PRE 90].

£r

1f

Af

Nyquist M

st

* *samp.

Interval between two points of the PSD (this interval limits the possible precision of the analysis starting from the PSD). Temporal step (time interval between two points of the signal), if the preceding condition is observed.

AN=2M

Number of points per block.

N -2M N "s2

Minimum number of signal points to analyse hi order to respect the statistical error.

T = N8t

Minimum total duration of the sample to be treated.

Y

N

2M

AT = — ( = 2 M 6 t = — ) K Af

Number of blocks. Duration of one block.


135

Calculation of — |L/f )| for

AT '

^

Calculation from the FFT of each block.

each point of the PSD, where f = mAf ( 0 < m < M )

-t-^f fcf * K

AT

1V

Averaging of the spectra obtained for each of the K blocks (stationary and ergodic process)

With these conditions, the maximum frequency of the PSD computed is equal to f'max = ^Nyquist • But it is preferable to consider the PSD only in the interval (0, fmax).

NOTE. It is supposed here that the signal has frequency components greater than fmax and that it was thus filtered by a low-pass filter to avoid aliasing. If it is known that the signal has no frequency beyond f max , this filtering is not necessary and 1 f' max = fLmax -

4.9.2. Case: only one sample of an already digitized signal is available If the signal sample of duration T has already been digitized with N points, one can use the value of the statistical error to calculate the number of points M of the PSD (i.e. the frequency interval Af), which is thus no longer to be freely selected (but it is nevertheless possible to increase the number of points of the PSD by overlapping and/or addition of zeros). Table 4.7. Computing process of a PSD starting from an already digitized signal

Data: The digitized signal, fmax and 8. p 1

Af

max

samp.

Lf

rifnaX

5t-

Theoretical maximum frequency of the PSD (see preceding note).

samp. 2.6

Practical maximum frequency.

l

Temporal step (time interval between two points of the signal).

*samp.

136

Random vibration

T N =— 8t

Number of signal points of duration T. Number of points of the PSD necessary to respect the statistical error (one will take the number immediately beneath that equal to the power of 2).

M~Ns2 2 ,, * Nyquist Af

*samp.

Nyquist frequency.

Nyquist

Interval between two points of the PSD.

M AN = 2 M K-

Number of points per block.

N

Number of blocks.

2M

Etc

If the number of points M of the PSD to be plotted is itself imposed, it would be necessary to have a signal defined by N' points instead of N given points (N < N'). One can avoid this difficulty in two complementary ways: - either by using an overlapping of the blocks (of 2 M points). One will set the overlapping rate R equal to 0.5 and 0.75 while taking smallest of these two values (for a Harming window) which satisfies the inequality: -R 2M b

[4.98]

If a random variable is uniformly distributed about [0, 1], the variable a+b y = a + ( b - a ) r is uniformly distributed about [a, b], having a mean of and 2 b-a standard deviation s =2 V3

4.13.2.2. Other laws

We want here to create a signal whose instantaneous values obey a given distribution law F(x). This function being nondecreasing, the probability that x < X is equal to [DAH 74]: P(x < X) = P[F(x) < F(X)]

[4.99]

156

Random vibration

Let us set F(x) = r where r is a random variable uniformly distributed about [0,1]. It then becomes: p[F(x)

Mechanical Vibrations and Shocks: v.1-5

Mechanical Vibrations and Shocks: Volume 1, Sinusoidal Vibration

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