Structural dynamics and vibration in practice an engineering handbook douglas thorby

Structural Dynamics and Vibration in Practice This page intentionally left blank Structural Dynamics and Vibration in Practice An Engineering Handbook Douglas Thorby AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 2008 Copyright Ó 2008 Elsevier Ltd All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Catalog Number: 2007941701 ISBN: 978-0-7506-8002-8 For information on all Butterworth-Heinemann publications visit our website at http://books.elsevier.com Printed and bound in Hungary 08 09 10 11 11 10 Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org To my wife, Marjory; our children, Chris and Anne; and our grandchildren, Tom, Jenny, and Rosa This page intentionally left blank Contents xiii Preface Acknowledgements Chapter 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Statics, dynamics and structural dynamics Coordinates, displacement, velocity and acceleration Simple harmonic motion 1.3.1 Time history representation 1.3.2 Complex exponential representation Mass, stiffness and damping 1.4.1 Mass and inertia 1.4.2 Stiffness 1.4.3 Stiffness and flexibility matrices 1.4.4 Damping Energy methods in structural dynamics 1.5.1 Rayleigh’s energy method 1.5.2 The principle of virtual work 1.5.3 Lagrange’s equations Linear and non-linear systems Systems of units 1.7.1 Absolute and gravitational systems 1.7.2 Conversion between systems 1.7.3 The SI system References Chapter 2.1 Basic Concepts The Linear Single Degree of Freedom System: Classical Methods Setting up the differential equation of motion 2.1.1 Single degree of freedom system with force input 2.1.2 Single degree of freedom system with base motion input 2.2 Free response of single-DOF systems by direct solution of the equation of motion 2.3 Forced response of the system by direct solution of the equation of motion xv 1 7 10 12 14 16 17 19 21 23 23 24 26 27 28 29 29 29 33 34 38 vii viii Contents Chapter 3.1 3.2 3.3 3.4 3.5 The Linear Single Degree of Freedom System: Response in the Time Domain Exact analytical methods 3.1.1 The Laplace transform method 3.1.2 The convolution or Duhamel integral 3.1.3 Listings of standard responses ‘Semi-analytical’ methods 3.2.1 Impulse response method 3.2.2 Straight-line approximation to input function 3.2.3 Superposition of standard responses Step-by-step numerical methods using approximate derivatives 3.3.1 Euler method 3.3.2 Modified Euler method 3.3.3 Central difference method 3.3.4 The Runge–Kutta method 3.3.5 Discussion of the simpler finite difference methods Dynamic factors 3.4.1 Dynamic factor for a square step input Response spectra 3.5.1 Response spectrum for a rectangular pulse 3.5.2 Response spectrum for a sloping step References Chapter The Linear Single Degree of Freedom System: Response in the Frequency Domain 4.1 45 46 46 50 53 55 56 56 56 59 60 62 62 65 69 70 70 72 72 74 76 77 Response of a single degree of freedom system with applied force 4.1.1 Response expressed as amplitude and phase 4.1.2 Complex response functions 4.1.3 Frequency response functions 4.2 Single-DOF system excited by base motion 4.2.1 Base excitation, relative response 4.2.2 Base excitation: absolute response 4.3 Force transmissibility 4.4 Excitation by a rotating unbalance 4.4.1 Displacement response 4.4.2 Force transmitted to supports References 77 77 81 83 86 87 91 93 94 95 96 97 Chapter 99 5.1 5.2 Damping Viscous and hysteretic damping models Damping as an energy loss 5.2.1 Energy loss per cycle – viscous model 5.2.2 Energy loss per cycle – hysteretic model 5.2.3 Graphical representation of energy loss 5.2.4 Specific damping capacity 5.3 Tests on damping materials 99 103 103 104 105 106 108 Contents 5.4 5.5 5.6 5.7 5.8 Quantifying linear damping 5.4.1 Quality factor, Q 5.4.2 Logarithmic decrement 5.4.3 Number of cycles to half amplitude 5.4.4 Summary table for linear damping Heat dissipated by damping Non-linear damping 5.6.1 Coulomb damping 5.6.2 Square law damping Equivalent linear dampers 5.7.1 Viscous equivalent for coulomb damping 5.7.2 Viscous equivalent for square law damping 5.7.3 Limit cycle oscillations with square-law damping Variation of damping and natural frequency in structures with amplitude and time Chapter 6.1 6.2 6.3 6.4 6.5 6.6 Setting up the equations of motion for simple, undamped, multi-DOF systems 6.1.1 Equations of motion from Newton’s second law and d’Alembert’s principle 6.1.2 Equations of motion from the stiffness matrix 6.1.3 Equations of motion from Lagrange’s equations Matrix methods for multi-DOF systems 6.2.1 Mass and stiffness matrices: global coordinates 6.2.2 Modal coordinates 6.2.3 Transformation from global to modal coordinates Undamped normal modes 6.3.1 Introducing eigenvalues and eigenvectors Damping in multi-DOF systems 6.4.1 The damping matrix 6.4.2 Damped and undamped modes 6.4.3 Damping inserted from measurements 6.4.4 Proportional damping Response of multi-DOF systems by normal mode summation Response of multi-DOF systems by direct integration 6.6.1 Fourth-order Runge–Kutta method for multi-DOF systems Chapter 7.1 Introduction to Multi-degree-of-freedom Systems Eigenvalues and Eigenvectors The eigenvalue problem in standard form 7.1.1 The modal matrix 7.2 Some basic methods for calculating real eigenvalues and eigenvectors 7.2.1 Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination 7.2.2 Matrix iteration 7.2.3 Jacobi diagonalization ix 108 108 109 110 111 112 112 113 113 114 115 116 117 117 119 119 120 120 121 122 122 126 127 132 132 142 142 143 144 145 147 155 156 159 159 161 162 162 165 168 262 Structural dynamics and vibration in practice where k is the stiffness, the viscous damping coefficient expressed as a fraction of critical and n the excitation frequency divided by the undamped natural frequency of the single-DOF system In this case the excitation frequency is the frequency of the Fourier component dn cosðn!0 t À n Þ, which is n!0 rad/s, and the undamped natural pffiffiffiffiffiffiffiffiffi frequency is !u ¼ k=m rad/s Thus n!0 9:59ị n ẳ !u where, from Eq (9.2), !0 ¼ 2=T The phase angle, n , by which the displacement vector lags the force component dn cosðn!0 t À n Þ is given by Eq (4.10), which, in this case, becomes n ¼ tanÀ1 2 n À 2n ð9:60Þ Thus the time history, zn ðtÞ, of a single Fourier component of the displacement, for a single value of n, excluding n = 0, is zn t ị ẳ dn Á qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðn!0 t À À Á2 k 2n ỵ2 n ị2 n n ị 9:61ị The complete time history of the displacement, zðtÞ, is the summation of all such components for n = 1, 2, 3, plus the mean displacement, a0 =k, i.e., X a0 6dn zðtÞ ẳ ỵ cosn!0 t n n ị5; n ¼ 1, 2, 3, Á qÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k nẳ1 2n ỵ2 n ị2 9:62ị Sometimes the time history of the response is not required, and it is sufficient to  know  the mean square or root mean square value The mean square value, z ðtÞ , is *( !)2 +   a0 X dn z tị ẳ ỵ An cosðn!0 t À n À n Þ ð9:63Þ k k n¼1 where h i is used to indicate ‘the mean value of’ Equation (9.63) can be simplified to:   a20 X ð dn A n Þ 9:64ị z tị ẳ ỵ k 2k nẳ1 p The RMS value is hz2 tịi The mean displacement, a0 =k, affects the mean square and RMS values, and if these are being calculated primarily to give an indication of the vibration level, it is generally better to omit the mean level and deal with it separately The approach outlined here is easily adapted for other input and response quantities, for example the input could be base motion, defined as displacement, velocity or acceleration, and the response could also be in terms of any of these quantities, using any of the relationships developed in Chapter Chapter Fourier transormation and related topics 263 Example 9.3 The vertical motion of a machine tool can be represented schematically by Fig 9.10 The mass, m, of 200 kg, is carried on elastic supports, so that the natural frequency for vertical motion is 30 Hz, and the viscous damping coefficient, , is 0.1 of critical A mechanism applies a vertical, periodic force, FðtÞ, that can be approximated by a symmetrical square wave of period T = 0.1 s, and a magnitude of Ỉ 3000 N Plot the vertical displacement time history, zðtÞ; of the machine Solution The notation used is as defined in the preceding section It is first noted that since the natural frequency of the system is 30 Hz, it is likely to be excited by the third harmonic of the excitation, which is a square wave of 1/T = 10 Hz, and that the response to the 10 Hz fundamental will be essentially static From Example 9.1 it can be seen that the square wave force input, in newtons, treated as an even function, can be represented by the Fourier series:   4 4 cos !0 t cos3 !0 t ỵ cos5 !0 t cos7 !0 t ỵ Aị Ftị ẳ 3000  3 5 7 Using the fact that cos  ẳ cos ị, Eq (A) can be written as: Ftị ẳ d1 cos !0 t þ d3 cosð3!0 t À Þ þ d5 cos5 !0 t ỵ d7 cos7!0 t ị ỵ Á or in the form of Eq (9.57): X ẵdn cosn!0 t Ftị ẳ a0 ỵ n ị Bị Cị nẳ1 where !0 ẳ 2=T a0 ẳ 0; dn ẳ for even values of n: Dị Values of dn and n for odd values of n are independent of T, and are as follows: d1 ¼ ð3000 4ị= ẳ 3819 N ẳ d3 ẳ 3000 4ị=3ị ẳ 1273 N ẳ d5 ẳ 3000 4ị=5ị ẳ 763:9 N ẳ0 d7 ẳ 3000 4ị=7ị ẳ 545:7 N ẳ The following numerical values are constant throughout: !u ¼ ð2  30ị ẳ 60 rad/s = natural frequency of system = 30 Hz; m = 200 kg = mass of machine; k ẳ m!2u ẳ 2002 30ị2 ẳ 7:106 106 N/m = stiffness of supports; = 0.1 = viscous damping coefficient The displacement response is given by Eq (9.62), which can be written as: ! a0 X dn An cosn!0 t n n ị ztị ẳ ỵ k k nẳ1 Eị 264 Structural dynamics and vibration in practice where An ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á2 2n ỵ2 n ị2 Fị Table 9.3 Numerical Values for Example 9.3 Frequency (Hz) n n!0 (rad/s) dn (N) 10 30 50 70 90 110 130 11 13 20  60  100  140  180  220  260  3819 1273 763.9 545.7 424.4 347.2 293.8 4000 n (rad)    n An n (rad) 0.3333 1.0000 1.6666 2.3333 3.0000 3.6666 4.3333 1.1218 5.0000 0.5528 0.2238 0.1246 0.0802 0.0561 0.2213 1.5707 3.029 3.096 3.116 3.125 3.130 F (t ) (newtons) 3000 2000 1000 t (sec.) –1000 0.05 0.1 0.15 0.2 –2000 –3000 –4000 (a) 2.0 Z (t ) (mm) 1.5 1.0 0.5 t (sec.) 0.0 –0.5 0.05 0.1 0.15 0.2 –1.0 –1.5 –2.0 (b) Fig 9.11 (a) Input force time history, Example 9.3, (b) displacement response time history, Example 9.3 Chapter Fourier transormation and related topics n ¼ n!0 2 n n ¼ ¼ !u 2ð30ÞT 30T 265 ðGÞ and n is given by Eq (9.60): n ¼ tanÀ1 2 n À 2n ðHÞ Numerical values are shown in Table 9.3 The displacement time history, z(t), is now given by Eq (E) It can be seen from Table 9.3 that the largest contribution to the response is at 30 Hz, where the third harmonic of the force waveform coincides with the natural frequency There are also significant components at 10 and 50 Hz Figure 9.11(a) shows the input force waveform, and Figure 9.11(b) the displacement response waveform, including components up to 70 Hz References 9.1 Meirovich, L (1986) Elements of Vibration Analysis McGraw-Hill Inc 9.2 Cooley, JW and Tukey, JW (1965) An algorithm for the machine calculation of complex Fourier Coefficients Mathematics of Computation, 19, 297–301 9.3 Newland, DE (1993) Random Vibrations, Spectral and Wavelet Analysis Longman (London) and John Wiley (New York) This page intentionally left blank 10 Random Vibration Contents 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 Stationarity, ergodicity, expected and average values Amplitude probability distribution and density functions The power spectrum Response of a system to a single random input Correlation functions and cross-power spectral density functions The Response of structures to random inputs Computing power spectra and correlation functions using the discrete Fourier transform Fatigue due to random vibration References 267 270 279 286 299 305 310 320 324 Random vibration may be caused by the turbulent flow of gases or liquids, the passage of vehicles over rough surfaces, rough seas acting on ships and marine structures, and earthquakes The aerospace field also provides many examples of random vibration, and these tend to fall into three groups: flight through atmospheric turbulence, ‘separated’ airflow over wings and other surfaces and ‘mixing noise’ from rocket and jet exhaust plumes So far, we have looked at the response of systems to deterministic inputs consisting of known functions of time, and there is, potentially at least, an exact answer, provided we have all the data If the input is a random function of time, and has been recorded, then this approach is still possible, but the calculation would have to be stepwise, very lengthy, and the response would be specific to that particular input However, if the average properties of the input not change too much with time, much easier methods can be used These require the introduction of new ways of describing the average properties of time histories In this chapter, therefore, we first introduce two new concepts: amplitude probability and the power spectrum These allow two of the most common tasks in day-to-day work to be tackled: measuring random vibration and calculating the response of linear systems to it We can then fill in some of the theoretical gaps, introducing correlation functions and their relationship to power spectra and cross-power spectra; some applications of cross-power spectra; and the practical computation of basic functions, using the DFT Finally, we look at the statistical accuracy of spectral estimates; and fatigue due to random vibration 10.1 Stationarity, ergodicity, expected and average values Mathematically, the conventional analysis of random waveforms requires them to be stationary and ergodic Stationarity applies to a single waveform, and implies that its average properties are constant with time Ergodicity applies to an ensemble of a 267 268 Structural dynamics and vibration in practice t T (a) t T2 T3 T4 T1 (b) Fig 10.1 Stationary and non-stationary waveforms large number of nominally similar waveforms, recorded in similar conditions If these all have similar average properties, it suggests that it is reasonable to assume that the one or two records that we are able to analyse are typical of the process as a whole Since there are no hard and fast criteria, in practice we must rely on common sense to ensure that the records we analyse are at least reasonably stationary The concept of stationarity is illustrated by Fig 10.1, which shows sketches of two random waveforms That shown at (a) is obviously fairly stationary, and we would be justified in choosing the period T for analysis On the other hand, the record shown at (b) is clearly non-stationary, and conventional analysis over the period T1 would, at best, only give average properties over that period, and not of the periods T2, T3 and T4, which are obviously more severe, and therefore more important in practice Techniques such as wavelet analysis [10.1] have been developed, specifically for dealing with non-stationary data, but these are beyond the scope of this book However, even using conventional methods, there are steps that can be taken to mitigate the effects of non-stationarity The most obvious approach would be to analyse the periods T2, T3 and T4, in Fig 10.1(b), separately, regarding each as stationary Unfortunately, as we shall see, the use of short samples reduces the statistical accuracy of the results We can sometimes, in effect, increase the length of the samples by combining the results of several sets of measurements It may even be justifiable to modify the experimental procedure, in order to lengthen the samples As an example, an aircraft maneuver producing severe buffeting may last for less than a second, which may not be long enough to measure the random response accurately However, it is sometimes possible to prolong the maneuver artificially, enabling better measurements to be made In this way, a better understanding of the response may be obtained; for example, the modes taking part may be identified Of course, any assessment of the fatigue life of the structure should be obtained from the actual maneuvers The expected value of a random process, denoted, for example, by EðxÞ, is the value that would be obtained by averaging over a very long, theoretically infinite, time In the case of a random time history, xðtÞ say, the expectedvalue, Eẵxtị, is the mean value hxtịi Similarly, Eẵx2 tị is the mean square value, x2 ðtÞ , and so on Chapter 10 Random vibration 269 It is possible to find a few basic properties of a random waveform without considering how its amplitude and frequency components are distributed, by operating directly on a recorded time history The mean value, the mean square value, the RMS value, the variance and the standard deviation can all be found in this way Since these values can also be found, for example, from the probability density function, or in some cases, from the power spectrum, this provides a valuable cross check in practical work Assuming that a recorded time history is available, the mean value, hxðtÞi, is given by: Z T x dt 10:1ị hxi ẳ T where xtị has been abbreviated to x and the notation hxi is used to indicate the mean value of x The length of the sample, in seconds, is T In practice, the waveform will usually have been digitized at equal time intervals The mean value is then simply the average of all the recorded discrete values, xj , that is, hxi ¼ N 1X xj N ð10:2Þ where N is the total number of recorded values Clearly, N ¼ T=D where D is the sampling interval, in seconds The mean square value is Z  2 T x dt ð10:3Þ x ¼ T or for discrete, digitized values: N  2 X x2 x ¼ N j¼1 j ð10:4Þ pffiffiffiffiffiffiffiffiffi The RMS value is, of course, hx2 i The mean square and RMS values contain the effect of the mean value, if any It is usually desirable to separate out the mean or steady level before squaring, since in many practical applications, such as fatigue life assessment, the steady stress will probably be treated differently from the oscillatory stress The mean square value about the mean, i.e taking the mean as zero, is known as the variance, usually denoted by 2 , and this is given by: Z T ẵx hxi2 dt 10:5ị x ẳ T Equation (10.5) can be expressed in terms of the mean and mean square values separately, as follows First, multiplying out: Z Z Z T  T T 2 x 2xhxi ỵ hxi dt ẳ x dt 2hxi x dt ỵ hxi2 x ¼ T T 0 270 Structural dynamics and vibration in practice RT But noting that hxi2 is a constant, that from Eq (10.1), T1 x dt ¼ hxi; and from   RT Eq (10.3) that 1=T x2 dt ¼ x2 then Eq (10.5) can be written as:   10:6ị 2x ẳ x2 hxi2 where hx2 i is the mean square value and hxi2 is the mean value squared, a simpler expression than might have been expected The discrete, or digital, version is 2x ¼ N À Á2   1X xj À hxi ẳ x2 hxi2 N jẳ1 10:7ị The square root of the variance, equal to x , is known as the standard deviation In a zero-mean waveform, the mean square is equal to the variance and the RMS is equal to the standard deviation These terms are therefore often interchanged, and some care is necessary to check that this is permissible 10.2 Amplitude probability distribution and density functions Figure 10.2(a) shows a random waveform, xðtÞ, from which a sample of length, T, has been selected for analysis A small part of the waveform is shown enlarged in Fig 10.2(b) The probability, PðxÞ, that the waveform has a value less than x, a particular value of x, is given by counting up the total time that it spends below x, which will be the sum of periods such as t1 , t2 , t3 , expressed as a fraction of the total time T, thus, Pxị ẳ t1 ỵ t2 ỵ t3 ỵ ị T ð10:8Þ If this is repeated with x set to a number of different values of x, a plot of PðxÞ versus x can be obtained PðxÞ is known as the cumulative amplitude probability distribution Although many different shapes are possible, all must have the following features: (1) The value of Pxị when x ẳ 1, that is, P1ị, must always be zero, since there is no chance that the waveform lies below À1 It may, of course, be zero up to a value of x greater (less negative) than À1 (2) The value of PðxÞ can never decrease as x becomes more positive (3) The value of PðxÞ when x ¼ must always be 1, since it is certain that all of the waveform is smaller than It is, however, possible for PðxÞ to reach at a value of x less than A possible cumulative probability distribution is shown in Fig 10.2(c) The probability that the original waveform has an amplitude between x and ðx þ xÞ is Pðx þ xÞ À PðxÞ, and if x is small, this probability can be seen to be equal to dẵPxị=dxịx The quantity dẵPxị=dx, which is the slope of the graph of PðxÞ versus x, is known as the amplitude probability density function, pðxÞ, that is, p ð xị ẳ dẵPxị dx 10:9ị Chapter 10 Random vibration 271 x t T (a) t1 x t2 t3 x t (b) P (x) P (x + δx) P (x) Slope = p (x) δx x (c) p (x) Area = P (x2) – P (x1) x x1 x2 (d) Fig 10.2 Derivation of amplitude probability and density functions Figure 10.2(d) is a sketch of pðxÞ corresponding to the plot of PðxÞ shown in Fig 10.2(c) Since pðxÞ is the slope of the graph of PðxÞ versus x, it follows that PðxÞ is the integral of pxị from to x, i.e., Z Pxị ẳ x À1 pðxÞdx ð10:10Þ 272 Structural dynamics and vibration in practice The probability that any point in the waveform lies between x ¼ x1 and x ¼ x2 is therefore: Z x2 Z x1 Z x2 pxịdx pxịdx ẳ pxịdx 10:11ị Px2 ị Px1 ị ẳ 1 x1 which is simply the area under the pðxÞ curve between x1 and x2 , as shown shaded in Fig 10.2(d) The total area under the probability density curve, pðxÞ, plotted against x, must always be unity, since it is certain that any point in the waveform has a value between Æ1 (or its minimum and maximum values, if finite) Thus it is always true that Z pxị dx ẳ ð10:12Þ À1 The averages derived in Section 10.1 directly from the time history, the mean value; the mean square value; the root mean square value; the variance and the standard deviation, can also be derived from the probability density function In practice, this is now a simpler process, since the operation of counting the times spent in the various amplitude bands has already been done to produce pðxÞ The mean value, hxi, for example, can be derived from pðxÞ as follows In Fig 10.3, the probability of the waveform being in the band from x to x ỵ x is represented by the area of the strip of width x, which is pðxÞ x The contribution of this strip to the mean value is x: pðxÞ x, and the mean value, hxi, must be given by the summation of all such strips, so hxi ¼ X x Á pðxÞx ð10:13Þ À1 If pðxÞ is known (or is approximated) analytically, then Eq (10.13) can be written as an integral: Z x pxịdx 10:14ị hxi ẳ The mean square value can be found in a similar way, using either  2 X x ¼ x2 Á pðxÞx ð10:15Þ À1 p (x ) x δx x Fig 10.3 Derivation of the mean value from the probability density function Chapter 10 Random vibration 273 for discrete values or Z  2 x2 Á pðxÞdx x ¼ ð10:16Þ in the case of analytical representation pffiffiffiffiffiffiffiffiffi The RMS value is, in either case hx2 i For analytical description, the variance is given by: Z x ¼ ðx À hxiÞ2 pðxÞdx ð10:17Þ À1 À1 The discrete form is given by substituting a summation for the integral and x for dx Equation (10.17) can be simplified as follows: Z 1h Z i ÂÀ Á à 2x ẳ x hxiị2 pxị dx ẳ x 2xhxi ỵ x2 pxị dx Z1 Z 1 Z 1   ẳ x2 pxịdx 2hxi x pxịdx ỵ hxi2 pxịdx ẳ x2 2hxi2 ỵhxi2   2x ẳ x2 hxi2 1 10:18ị where Eqs (10.16), (10.14) and (10.12) have been used Thus, the variance is the mean square value minus the mean value squared, agreeing with Eq (10.6) The standard deviation, x is, of course, the square root of the variance, 2x In practical, everyday, work, the time histories to be analysed into cumulative probability distributions or probability density functions will usually have been digitized as part of the recording process, and determining the amplitude probability density function, pðxÞ, and the cumulative probability distribution function, PðxÞ become sorting and counting operations Assuming that the waveform to be analysed has been sampled at equal intervals of time, say D, it is a simple computer operation to sort the digitized amplitudes into bands, and in principle: Pxi ị ẳ ni D ni ẳ N T ð10:19Þ where Pðxi Þ is the cumulative probability distribution value corresponding to amplitude level xi ; T the time duration of the sample; ni the number of data points having an amplitude less (more negative) than xi and N ¼ T=D the total number of data points in the sample The probability density, pðxi Þ, can then be found by differencing adjacent values of Pðxi Þ Alternatively, the probability density values can be found first, by counting the number of data points between adjacent levels, and the cumulative distribution by progressively summing the results Example 10.1 The waveform shown in Fig 10.4 at (a) increases and decreases alternately at a constant rate between the two extremes ỈX, the rates being random from cycle to cycle Derive the probability density and cumulative distribution functions 274 Structural dynamics and vibration in practice x (a) X t –X p (x ) (b) 1/2X x –X X P (x ) (c) x –X X Fig 10.4 Random waveform discussed in Example 10.1 Solution The probability of any point in the waveform being outside the range ỈX is zero, and the probability density pðxÞ is also zero Within the range ỈX, the probability of any point being between x and x ỵ x is proportional to the time spent between these levels, so pðxÞ must be constant The probability density function must therefore be rectangular, as shown in Fig 10.4(b) Since the probability of the waveform having an amplitude in the range ỈX is 1, that is, Z X pxịdx ẳ X the area of the rectangle in Fig 10.4(b) is 1, and since its width is 2X, its height must be 1/2X, as shown The cumulative amplitude distribution, PðxÞ, is the integral of the density function with respect to x, and since its value must be at x ¼ ÀX and at x ¼ X, it must be as shown in Fig 10.4(c) 10.2.1 The Gaussian or Normal Distribution The majority of probability distributions occurring naturally are of this form, the shape of the probability density curve being the well-known bell shape The fact that this curve occurs almost universally in natural phenomena can be explained by the central limit theorem, which states that if a random time history is made up from the Chapter 10 Random vibration 275 sum of a large number of small, unrelated components, then it will tend to have a Gaussian amplitude distribution This tendency can be demonstrated in a simple way by the dice-throwing exercise described in the following example Example 10.2 Derive the expected probability density functions for the score when (a) a single die is thrown; (b) two dice are thrown, and the results added and (c) three dice are thrown and the results added (d) Show by plotting the probabilities that the results tend towards a Gaussian distribution as the number of dice thrown increases Solution Part (a) If a single die is thrown, the probability of any score from to is obviously 1/6; thus we have the discrete uniform distribution shown in Table 10.1 Part (b) If two dice are thrown each time, they can fall in  ¼ 36 ways, each with probability 1=36 However, there are only ten different total scores (the integers 2–12), because the same score can be obtained in a number of different ways As an example, a total score of is obtained from the three possible throws: (1+3), (3+1) and (2+2), so the probability of scoring is 3=36 The probability of each possible score is as shown in Table 10.2 Part (c) If three dice are thrown each time, and again the resulting numbers are added, there are   ¼ 216 possible outcomes, but only 16 different scores: for example, a score of or 14 can both be obtained in 15 ways, so the probability of these scores is 15=216 All the probabilities for three dice are as shown in Table 10.3 Part (d) The discrete probability functions for one, two and three dice thrown are plotted in Figs 10.5(a), (b) and (c) Figure 10.5(a), for one die, obviously shows a uniform Table 10.1 Score x Probability p(x) 1/6 1/6 1/6 1/6 1/6 1/6 Table 10.2 Score x Probability p(x) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 10 3/36 11 2/36 12 1/36 Table 10.3 Score x Probability p(x) or18 1/216 or 17 3/216 or 16 6/216 or 15 10/216 or 14 15/216 or 13 21/216 or 12 25/216 10 or 11 27/216 276 Structural dynamics and vibration in practice 0.20 0.2 0.18 0.18 0.16 0.16 0.14 0.14 0.12 0.12 p (x ) 0.10 p (x ) 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.00 0.02 7 Score x Score x (a) (b) 10 11 12 13 0.14 0.12 0.1 0.08 p (x ) 0.06 0.04 0.02 10 12 14 16 18 Score x (c) Fig 10.5 (a) One die thrown, (b) two dice thrown, (c) three dice thrown distribution However, in Fig 10.5(b), for two dice, it is apparent that the curve is already approaching the Gaussian shape For three dice, Fig 10.5(c) is remarkably close to a Gaussian probability density curve It should be noted that the probabilities plotted in Figs 10.5(a), (b) and (c) are expected values that would be realized in a very large number, theoretically an infinite number, of throws It can be shown that the basic form of the probability density function for a zeromean Gaussian process is pxị ẳ aebx 10:20ị where a and b are constants However, since these constants R are almost always chosen R1 to satisfy the two equations, pxịdx ẳ and 2 ẳ x2 pdxị, where  is the standard deviation, Eq (10.20) usually appears in the following form: x2 pxị ẳ p e22  2 ð10:21Þ If the mean value, m, say, is non-zero, then Eq (10.21) becomes xmị2 pxị ẳ p eÀ 22  2 ð10:22Þ .. .Structural Dynamics and Vibration in Practice This page intentionally left blank Structural Dynamics and Vibration in Practice An Engineering Handbook Douglas Thorby AMSTERDAM... disciplines, entering the field of structural dynamics and vibration, in industry It should also be found useful by test engineers and technicians working in this area, and by those studying the... be found useful as an introduction to structural dynamics and vibration in all branches of engineering Although the principles behind current computer software are explained, actual programs