8434_Harris_fm_b.qxd 09/20/2001 11:40 AM Page iii
`
`HARRIS’
`SHOCK AND
`VIBRATION
`HANDBOOK
`
`Cyril M. Harris Editor
`Charles Batchelor Professor Emeritus
`of Electrical Engineering
`Columbia University
`New York, New York
`
`Allan G. Piersol
`Consultant
`Piersol Engineering Company
`Woodland Hills, California
`
`Editor
`
`Fifth Edition
`
`McGRAW-HILL
`New York Chicago San Francisco Lisbon London Madrid
`Mexico City Milan New Delhi San Juan Seoul
`Singapore Sydney Toronto
`
`Invensys Ex. 2020,
`Micro Motion v. Invensys IPR 2014-00393, page 1
`
`

`
`8434_Harris_fm_b.qxd 09/20/2001 11:40 AM Page iv
`
`Library of Congress Cataloging-in-Publication Data
`
`Harris’ shock and vibration handbook / Cyril M. Harris, editor, Allan G.
`Piersol, editor.—5th ed.
`p.
`cm.
`ISBN 0-07-137081-1
`1. Vibration—Handbooks, manuals, etc. 2. Shock (Mechanics)—
`Handbooks, manuals, etc.
`I. Harris, Cyril M., date.
`II. Piersol, Allan G.
`TA355.H35
`2002
`620.3—dc21
`
`2001044228
`
`Copyright © 2002, 1996, 1988, 1976, 1961 by The McGraw-Hill Companies,
`Inc. All rights reserved. Printed in the United States of America. Except as
`permitted under the United States Copyright Act of 1976, no part of this pub-
`lication may be reproduced or distributed in any form or by any means, or
`stored in a data base or retrieval system, without the prior written permission
`of the publisher.
`
`1 2 3 4 5 6 7 8 9 0 DOC/DOC 0 7 6 5 4 3 2 1
`
`ISBN 0-07-137081-1
`
`The sponsoring editor for this book was Kenneth P. McCombs, the editing
`supervisor was Stephen M. Smith, and the production supervisor was Sherri
`Souffrance. It was set in Times Roman by North Market Street Graphics.
`
`Printed and bound by R. R. Donnelley & Sons Company.
`
`McGraw-Hill books are available at special quantity discounts to use as pre-
`miums and sales promotions, or for use in corporate training programs. For
`more information, please write to the Director of Special Sales, McGraw-Hill
`Professional, Two Penn Plaza, New York, NY 10121-2298. Or contact your
`local bookstore.
`
`This book is printed on acid-free paper.
`
`Information contained in this work has been obtained by The
`McGraw-Hill Companies, Inc. (“McGraw-Hill”) from sources
`believed to be reliable. However, neither McGraw-Hill nor its
`authors guarantee the accuracy or completeness of any infor-
`mation published herein and neither McGraw-Hill nor its
`authors shall be responsible for any errors, omissions, or dam-
`ages arising out of use of this information. This work is pub-
`lished with the understanding that McGraw-Hill and its authors
`are supplying information but are not attempting to render
`engineering or other professional services. If such services are
`required, the assistance of an appropriate professional should
`be sought.
`
`Invensys Ex. 2020,
`Micro Motion v. Invensys IPR 2014-00393, page 2
`
`

`
`8434_Harris_23_b.qxd 09/20/2001 12:02 PM Page 23.25
`
`CONCEPTS IN SHOCK DATA ANALYSIS
`
`23.25
`
`(δr)max =
`
`(Aeq)r =
`
`F(ωn)
`
`1ᎏωn
`ωn
`2(δr)maxᎏᎏ
`g
`
`=
`
`ωnᎏ
`g
`
`F(ωn)
`
`(23.54)
`
`This result is clearly evident for the Fourier spectrum and undamped shock response
`spectrum of the acceleration impulse. The Fourier spectrum is the horizontal line
`(independent of frequency) shown in Fig. 23.3A and the shock response spectrum is
`the inclined straight line (increasing linearly with frequency) shown in Fig. 23.7A.
`Since the impulse exists only at t = 0, the entire response is residual. The undamped
`shock spectra in the impulsive region of the half-sine pulse and the decaying sinu-
`soidal acceleration, Fig. 23.7C and D, respectively, also are related to the Fourier spec-
`tra of these shocks, Fig. 23.3C and D, in a similar manner. This results from the fact
`that the maximum response occurs in the residual motion for systems with small nat-
`ural frequencies. Another example is the entire negative shock response spectrum
`with no damping for the half-sine pulse in Fig. 23.7C, whose values are ωn/g times the
`values of the Fourier spectrum in Fig. 23.3C.
`
`METHODS OF DATA REDUCTION
`
`Even though preceding sections of this chapter include several analytic functions as
`examples of typical shocks, data reduction in general is applied to measurements of
`shock that are not definable by analytic functions. The following sections outline
`data reduction methods that are adapted for use with any general type of function,
`obtained in digital form in practice. Standard forms for presenting the analysis
`results are given in Ref. 8.
`
`FOURIER SPECTRUM
`
`The Fourier spectrum is computed using the discrete Fourier transform (DFT)
`defined in Eq. (14.6). The DFT is commonly computed using a fast Fourier trans-
`form (FFT) algorithm, as discussed in Chap. 14 (see Ref. 9 for details on FFT com-
`putations). Fourier spectra can be computed as a function of either radial frequency
`ω in radians/sec or cyclical frequency f in Hz, that is,
`F1(f) =冕∞
`F2(ω) =冕∞
`where the two functions are related by F2(ω) = 2πF1(f).
`
`x(t)e−j2πftdt
`
`or
`
`−∞
`
`x(t)e−jωtdt
`
`−∞
`
`(23.55)
`
`SHOCK RESPONSE SPECTRUM
`
`The shock response spectrum can be computed by the following techniques: (a)
`direct numerical or recursive integration of the Duhamel integral in Eq. (23.33), or
`(b) convolution or recursive filtering procedures. One of the most widely used pro-
`grams for computing the shock response spectrum is the “ramp invariant method”
`detailed in Ref. 10.Any of these computational procedures can be modified to count
`
`Invensys Ex. 2020,
`Micro Motion v. Invensys IPR 2014-00393, page 3
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`

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`8434_Harris_22_b.qxd 09/20/2001 12:06 PM Page 22.4
`
`22.4
`
`CHAPTER TWENTY-TWO
`
`FINITE FOURIER TRANSFORMS
`
`Since frequency domain descriptions of vibrations are generally of the greatest engi-
`neering value, the Fourier transform plays a major role in both the theoretical defi-
`nitions of properties and the analysis algorithms for vibration data. The finite
`Fourier transform of a sample record x(t) is defined as
`
`x(t) sin (2πft)dtx(t) cos (2πft)dt − j冕TX(f,T) =冕Tx(t)e−j2πftdt =冕T
`
`
`where j = 兹−1苶. Three properties of the definition in Eq. (22.3) should be noted, as
`follows:
`
`0
`
`0
`
`
`
`0
`
`(22.3)
`
`1. The finite Fourier transform is generally a complex number that is defined for
`both positive and negative frequencies, that is, X(f,T); −∞ < f < ∞. However,
`X(−f,T) = X*(f,T), where the asterisk denotes the complex conjugate, meaning
`that values at mathematically negative frequencies are redundant and provide no
`information beyond that provided by the values at positive frequencies. Since
`engineers typically think of frequency as a positive value, it is common to present
`finite Fourier transforms as 2X(f,T); 0 < f < ∞.
`2. Fourier transforms are often defined as a function of radial frequency ω in radi-
`ans/sec, as opposed to cyclical frequency f in Hz, particularly for analytical appli-
`cations. However, data analysis is usually accomplished in terms of cyclical
`frequency f, as defined in Eq. (22.3). The two definitions are interrelated by
`X(f,T) = 2π X(ω,T).
`3. The finite Fourier transform X(f,T) is equivalent to the Fourier series of x(t)
`assumed to have a period T.
`
`STATIONARY DETERMINISTIC VIBRATIONS
`
`Stationary deterministic vibration environments generally fall into one of two cate-
`gories, namely, periodic vibrations or almost-periodic vibrations.
`
`Periodic Vibrations. Periodic vibrations are those with time-histories that exactly
`repeat themselves after a time interval TP, that is, x(t) = x(t + iTP); i = 1, 2, 3, . . . ,
`where TP is called the period of the vibration. All periodic vibrations can be decom-
`posed into a Fourier series, which consists of a collection of commensurately related
`sine waves,1,2 that is,
`x(t) = a0 + 冱
`
`ak sin (2πkf1t + θk)
`
`k = 1, 2, 3, . . .
`
`(22.4)
`
`k
`
`where a0 is the mean value, kf1 is the kth frequency component (harmonic), and ak
`and θk are the amplitude and phase angle associated with the kth frequency compo-
`nent of the periodic vibration. The k = 1 component is called the fundamental fre-
`quency of the periodic vibration, and is given by f1 = 1/TP. The magnitude of the
`frequency components in Eq. (22.4) are given by
`2|X(f,TP)|ᎏᎏ
`TP
`
`Lx(f) =
`
`0 < f
`
`(22.5)
`
`Invensys Ex. 2020,
`Micro Motion v. Invensys IPR 2014-00393, page 4
`
`

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`8434_Harris_11_b.qxd 09/20/2001 11:17 AM Page 11.8
`
`11.8
`
`CHAPTER ELEVEN
`
`If f1 = 0, the first zero crossing of the autocorrelation function occurs at a delay τ = 1/f2.
`The average relation between two variables x(t) and y(t) is represented by the
`cross-correlation Rxy (τ,t1) defined by
`
`Rxy(τ,t1) = x苶(苶t苶1苶)苶y苶(苶t苶1苶苶+苶苶τ苶)苶
`
`(11.18)
`
`For variables of a stationary process, the cross-correlation is a function only of the
`delay τ. However, the maximum value does not necessarily occur at τ = 0. The cross-
`correlation function can be approximated by the time average:
`Rxy(τ) ⯝ 冕T
`1ᎏ
`x(t)y(t + τ) dt
`T
`
`(11.19)
`
`0
`
`POWER SPECTRAL DENSITY
`
`The frequency content of a random variable x(t) is represented by the power spec-
`tral density Wx(f), defined as the mean-square response of an ideal narrow-band fil-
`ter to x(t), divided by the bandwidth ∆f of the filter in the limit as ∆f → 0 at frequency
`f (Hz):
`
`x苶 2苶⌬苶f苶
`ᎏ∆f
`This is illustrated in Fig. 22.5. By this definition the sum of the power spectral com-
`ponents over the entire frequency range must equal the total mean-square value
`of x:
`
`(11.20)
`
`(11.21)
`
`Wx( f ) = lim∆f→0
`
`x苶2苶 =冕∞
`
`Wx( f ) df
`
`0
`
`The term power is used because the dynamical power in a vibrating system is pro-
`portional to the square of the vibration amplitude.
`An alternate approach to the power spectral density of stationary variables uses
`the Fourier series representation of x(t) over a finite time period 0 ≤ t ≤ T, defined in
`Eq. (22.4) as
`
`x(t) = x苶 + 冱∞
`
`An cos(2πfnt) + 冱∞
`
`Bn sin(2πfnt)
`n = 1
`n = 1
`where fn = n/T. The coefficients of the Fourier series are found by
`冕T
`x(t)cos(2πfnt) dt
`冕T
`x(t)sin(2πfnt) dt
`
`An =
`
`Bn =
`
`2ᎏ
`T
`
`2ᎏ
`T
`
`0
`
`0
`
`(11.22)
`
`(11.23)
`
`Comparing this to Eq. (11.19), it follows that the coefficients of the Fourier series are
`a measure of the correlation of x(t) with the cosine and sine waves at a particular
`frequency.
`
`Invensys Ex. 2020,
`Micro Motion v. Invensys IPR 2014-00393, page 5