- Covers vibration measurement, finite element analysis, and eigenvalue determination - learn how to use Mathcad, Maple, Matlab, and Mathematica to solve vibrations problems - I ncludes 313 solved problems completely explained - ~ ,- - Use with your textbook or for independent study The world's leading course outline series Us, with 1I"s, C1JUrm : !E! r ';r: : . ' trrr~YiWa [if ' d'=~IIiSb -Hllt l" 1Ef ' .lIrtf l '" [il" .,.~ ulliMbtrsu- lt r.lblllrlC' s,.:w CopylOjIllPd M_' SCHAUM'S OUT LI NE OF THEORY AND PROBLEMS of MECHANICAL VIBRATIONS S. GRAH AM KELLY, Ph. D. Associate Professor of Me chanical En gineering alld Assistant Provost Th e University of Akr oll : SCHAUM'S OUTLINE SERIES McGRAW- HI LL New York St. Louis San Francisco Auckland Bogota C ar acas Lisbon Londoll Madrid Mexico City Milan Montreal New Delhi San iuan Singapore Sydney Tokyo Toronto \ S. GRAHAM KELLY is Associate Professor of M ec hanical Engineering and Assistant Provost at Th e U niv ers it y of Akron. He has been on the facuity at Akron s in ce 19 82, serv in g before at the University of Notre Dam e. He ho ld s a B.S. in Eng in eering Science and Mechanics and an M.S. a nd a Ph.D. in Engin eer ing Mechanics from Virginia Tech. He is also th e author of Fundamenrals of Mechanical Vibrations and the accompany in g software VIBES , published by McGraw·Hill. Cred its Selected mater ial reprinted from S. Gra ham Kelly. Fun da me nt als of Mechanical Vibrations, © 1993 McGraw-Hili. Jn c. Reprinted by permission of McGr aw-Hi li , In c. Maple-based ma terial produced by per mi ss ion of Water loo M aple In c. Mapl e an d Maple V are registered trademarks of Waterl oo Maple Inc.: Mapl e sys tem itself copyrighted by Wate rl oo Maple Inc. Mathcad mate ri al produced by permission of MathSoft, Inc. Schaum's Outline of T he ory and Problems of MEC H ANICAL VI BRAT I ONS Copyright © 1996 by The McGraw-Hi li Compan ies, Inc. All rights reserved. Prin ted in the United States of America. Except as p er mitted under th e Co pyright Act of 1976. no part of this publication may be r eproduced or distributed in any form or by any means. or st ored in a data base or retrieval syst em. with out the prior written permission of the puhlisher. I 2345 6789 JO t I 12 13 14 15 16 17 18 1 920 PRS PRS 9 8 7 6 ISBN 0-07-034041-2 Sponsoring Edito r: Jo hn A li ano Production Supe rvi sor: Suzanne Rapcavage Project Supervision: The T ota l Book Library or Congress Cataloging.in.Publication Data Ke ll y. S. Graham. Schaum's ou tline of theory and problems of mechanical vibrations I S. Graham Kelly. p. cm. - (Schaum 's outl ine series) Includes in dex. ISBN 0-07-03 4041 ·2 1. Vibration-Proble ms. exercises. etc. 2. Vibration-Out lines. syllabi. elc. I. Title. OA935.K383 t996 620.3·076-<1c20 95-4 75 54 CIP McGraw-Hill· r;z A Dili s iOJ1 o{T'hcMcG rawB iUCompanies sc h. Gl.,p , "\35 . K3. '23 i"l."\.b c ~ 1 Preface A student of mechanical vibrations must draw upon knowledge of many ar eas of engineering science (statics, dynamics, mechanics of materials, and even fluid mechanics) as well as mathematics (calculus, differential equations, and lin ear algebra). Th e student must then sy nthes iz e this knowledge to formulate the solution of a mechanical vibrations problem. Many mechanical systems require modeling before their vi brations can be analyzed. After appropriate assumptions are made , including the numb er of degrees of free dom necessar y, bas ic conservation laws are applied to derive governing differential eq uations. Appropriate mathematical methods are applied to sol ve the differential equations. Often the modeling results in a differential equation whose solution is well known, in which case the existing solution is used. If this is the case the solution must be studied and written in a form which can be used in anal ys is and design applications. A student of mechanical vibrations must learn how to use existing knowledge to do all of the above. The purpose of this book is to provide a supplement f or a student studying mechanical vibrations that will guide the stude nt through a ll aspects of vibration analysis. Each chapter has a short introduc ti on of the th eo ry used in the chapter, followed by a large number of so lv ed problems. The so lv ed proble ms mostly show how the theory is used in design and analysis applications. A few problems in each chapter examine the th eo ry in m ore d eta il. Th e coverage of the book is quite broad and includes free and forced vibrations of I-degree-of-freedom, multi-degr ee -of-freedom, and continuous sys- tems. Undamped systems and systems with viscous damping are considered. Systems with Coulomb damping and hy steretic damping are considered for I-degree-of-free dom systems. There are several chapters of special note. Chapter 8 focuses on design of vibratIon control devices such as vibration isolators and vibration absorbers. Chapter 9 introduces the finite element method from an - analytical viewpoint. Th e problems in Chapter 9 us e the finite element method using only a few elements to analyze the vibrations of bars and beams. Chapter 10 focuses on nonlinear vibration s, mainly discussing the differences between line ar and nonlinear systems including self-excited vibrations and chaotic motion. Chapter 11 shows how applications software can be used in vibration analysis and design . The book can be used to supplement a course using any of the popular vibrations textbooks, or can be used as a textb oo k in a course where theoretical development is limited. In any case the book is a good source for studying the solutions of vibrations problems. The author would like to thank the staff at McGr aw-Hill, especially John Aliano, for making this book po ss ible. He would also like to thank his wife and son, Seala and Graham , for patie nc e during preparation of the manuscript and Gara Alderman and Peggy Du ckworth for clerical help. S. GRAHAM K E LLY v Contents PROBLEMS AND EXAMPLES ALSO FO U1\" D IN TH E COMPANION SCHAUM'S IlVTER4.CTIVE OuTLI NE . . ix Chapter 1 Chapter 2 Chapter 3 C hapter 4 C hapt er 5 Chapter 6 MECHANICAL SYSTEM ANALySiS . . . . • .• . . .• . . 1.1 Degrees of Freedom and Generalized Coordinates. 1.2 Mecha ni cal System Components. 1.3 Equival ent Systems Ana lysis. 1.4 Torsional Sy stems . 1.5 Static Eq uili brium P os ition. FR EE VIBRATIO:-IS OF I·DEGREE·OF·FREEDOM SYSTEMS 2.1 Derivation of Differential Equat ions. 2.2 Standard Form of Diff erentia l Equations. 2.3 Undamped Respons.!. 2.4 Dam ped Re sponse. 2.5 Fr ee Vibration Rt: spon st! for Sy ste ms Subject to Coulomb Damping. HAR:VIONIC EXCITATION OF l ·DEG R EE ·OF·FREEDOM 36 SYSTEMS . . . • . . .•• . •. • 64 3. 1 Dcriva:ion of Differenti a! Equa tion s. 3.2 -H armo nic Excitation. 3.3 L :ndamped SY S[~ :-:1 Response. 3 Dnmpt:d System Response. 3.5 Frequt!ncy Squared Exc i~J.[ions. 3.6 Harmonic Support Excitation. 3. 7 Multifrequency Excit ations. 3.8 General Periodic Excitations: Fouri!!:- Series. 3.9 Coulomb Dampi:1g. 3.:0 Hyste retic Damping. GE NERAL FORCED RESPONSE OF I-DEGREE·OF -F REEDOM SYSTEMS . . . • . . • • 109 4.1 Gc! neral Differentia l E quation. t2 Convo l utio n Integra l. 4.3 LaP b.: "! Transform Solutions. 4A Cnit I m!=,ulse Function and L'nit Step Funct ic:1 4.5 ;\u mer ical Methods. 4.6 Response Sp~ctrum. FREE VIBRATIONS OF MULTI·DEGREE·OF ·F REEDOM SY STEMS • . . •• . •. • • 13 6 5.1 LaGr ange's EqL:J;ions. 5.2 M:::.trix Formul atio n of Di ffere ntial Eq u:l:!ons for Li ne ar Systems. 5.3 Stiffness Influence Cod ficients. 5.4 F1e:~ibili[v ~latrix . 5.5 Normal Mode Solu ti on . 5.6 Mode S hap e Orthogonality. " 5.7 ~Iatrix Iterati on. 5.8 Damped Systems. F ORCED VIBRA TIO NS OF o\IULTI-DEGREE·OF·FREEDOi\l SYSTEMS . .• . . . • • •• . . .•. . . . . . 180 6.1 General System. 6.2 HJrmon ic Excitation. 6 3 LaPlace Transform Solutions. 6.4 M oda l Analysis for Systc!ms with Proportio nal D amp ing 6.5 ~Iodal Analy s is fo r Sysr~ms with General Damping. vi i v iii Ch ap ter 7 Ch apter 8 Chapter 9 Ch apte r 10 , Chapter 11 Append ix CO NTEN TS V IBR ATIONS OF CONTINUOUS SySTEMS . . . . . . . . 201 7 .1 Wave Equa ti on. 7.2 Wave Solution. 7.3 Normal Mode So lution. 7.4 Beam Equat io n. 7.5 Modal Superp os ition. 7. 6 Ray leigh's Quo ti ent. 7 .7 Rayleigh-Ritz Method. VIBRATION CONTROL 235 8.1 Vibrat io n I so la ti o n. 8.2 I so la ti on f ro m Ha rm o ni c E xci tatio n. 8.3 Sh oc k Is ol a ti o n. 8 .4 Impulse Isolation. 8.5 Vibra ti on Ab so rbers. 8.6 Damped Absorber s. 8.7 Houda ill e Damper s. 8.8 Wh irlin g. FINITE ELEMENT METHOD . . . . . . . 264 9. 1 Gen eral Method. 9. 2 Forced Vibra ti on s. 9 .3 Bar Element. 9 .4 Beam Elemen t. NONLI NEA R SYSTEMS . . . . . . 285 10.1 Di ffe rences from Lin ear System s. 10.2 Qual it ati ve Analysis. 10 .3 Du ffi ng's Equat io n. 10 .4 Sel f- E xc it ed Vibrat io n s. COMPUTER APPLICATIONS . . . . . 301 11.1 Software Speci fic to Vibra ti ons Appli ca ti ons. 11 .2 Spreadsheet Programs. 11.3 Electro ni c Notepad s. 11 .4 S ym bo li c Processors. SAMPLE SCREENS FROM THE COMPANION INTERACTIVE OUTLINE . . . . . . . . . . . . . . . . . . . . 333 Index • . . ••. • • • • . . . . • . . . . 349 Problems and Examples Also Found in the Companion SCHAUM'S ELECTRONIC TUTOR Some of the problems and examples in this book have software components in the companion Schaum's Electronic' Tutor. The Mathcad Engine, which "drives" t he Electronic Tutor, allows every number, formula, and graph chosen to be completely l ive and interactive. To identify those items that are avai lable in the Electronic Tutor software, please look for the Mathcad icons, ~, placed under the problem number or adjacent to a numbered item. A complete list of these Mathcad entries follows below. For more information about the software, including the sample screens, see Appendix on page 333. Problem 1.4 Problem 3.10 Problem 4.26 Problem 7.1 Problem 8.24 Problem 1.5 Problem 3.12 Problem 5.19 Problem 7.4 Problem 8.25 Problem 1.7 Problem 3. 15 Problem 5.20 Problem 7.5 Problem 8.26 Problem 1.12 Problem 3.18 Problem 5.25 Problem 7.6 Problem 8.27 Problem 1.14 Problem 3. 19 Problem 5.26 Problem 7. [3 Problem 8.28 Problem 1.19 Problem 3.20 Problem 5.27 Problem 7.[6 Problem 8.32 Problem 2 .8 Problem 3.23 Problem 5.28 Problem 7.22 Prob[em 8.34 Problem 2.9 Problem 3.24 Problem 5.30 Problem 7.23 Problem 8.35 Problem 2.14 Problem 3.25 Problem 5.3 [ Problem 7.25 Problem 8.37 Problem 2. [5 Problem 3.26 Problem 5.32 Problem 8.3 Problem 9.5 Problem 2. [6 Problem 3.27 Problem 5.35 Problem 8.4 Problem 9.6 Problem 2. [7 Problem 3.28 Problem 5.38 Problem 8.5 Problem 9.7 Problem 2.[8 Problem 3.34 Problem 5.40 Problem 8.6 Problem 9.13 Problem 2.19 Problem 3.35 Problem 5.41 Problem 8. [0 Problem 9.[4 Probl em 2.20 Problem 3.36 Problem 5.42 Problem 8. [ [ Problem 10.6 Problem 2.2 [ Problem 3.38 Problem 5.44 Problem 8. 12 Problem 10.8 Problem 2.22 Problem 3.40 Problem 5.45 Problem 8. 13 Problem 10. 11 Problem 2.23 Problem 4.3 Problem 6.3 Problem 8.14 Problem 11.4 Proble",! 2.25 Problem 4.5 Problem 6.9 Problem 8. 15 Problem 11.5 Problem 2.29 Prob[em 4.6 Problem 6.10 Problem 8.16 Problem 11.6 Probl em 3.4 Problem 4.13 Problem 6.1 [ Problem 8.17 Problem 11.7 Problem 3.5 Problem 4.18 Problem 6.12 Problem 8.18 Problem 1l .8 Problem 3.7 Problem 4.19 Problem 6.15 Problem 8.19 Problem 11. 9 Pro bl em 3.8 Problem 4.24 Problem 6.16 ix Chapter 1 Mechanical System Analysis 1.1 DEGREES OF FREEDOM AND GENERALIZED COORDINATES The number of degrees of freedom used in the analysis of a mechanical system is the number of ki nematica ll y independent coordinates necessary to completely describe the motion of every particle in the system. Any such set of coordinates is called a set of generaliz ed coordinat es . The choice of a set of ge nera li zed coordinates is not unique. Kinemat ic qu antities such as di splacements, ve loci ti es, and accelera ti ons are written as functions of the generalized coordinates and their tim e derivatives. A system with a finite number of degrees of fr eedom is ca ll ed a discrete system, while a system with an infinite number of degrees of freedom is called a continuous system or · a distributed parameter system. 1.2 MECHANICAL SYSTEM COMPONENTS A mecha ni cal system comprises in ertia components, sti ff ness component s, and damping components. Th e in ertia compon en ts have kine ti c energy when the system is in motion. The kinetic energy of a rigid body un de rgo in g pl anar motion is _ T =! mv 2+U w 2 ~ (I. I) where v is the velocity of the body's mass center, w is its angul ar velocity about an axis perpendi cu l ar to the plane of motion, m is the body's mass, and I is it s mass moment of in ertia about an aXIS parallel to the axis of rot ation through the mass cente r. A linear stiffness component (a linear spring) has a fo rc e displacement rela ti on of the form F=kx (1 .2 ) where F is app lied force and x is the component's change in length from its unstretched length. The stiffness k has dimensions of force p er length. A dashpot is a mechanical device that adds vi scous damping to a mechanical system. A lin ear viscous damping co mponent has a force-velocity relation of the form F=cv<ft- (1.3) wh ere c::, is the damping coefficient of dimensions mass per time. 1.3 EQUIVALENT SYSTEMS ANALYSIS All lin ear I-degree -o f- fre edom systems with vi scous damping can be modeled by the simple mass-spr in g- dashpot sys tem of Fi g. 1-1. Let x be the chosen generalized coordinate. The kinetic energy of a linear system can be written in the form (1.4) 2 MECHANICAL SYSTEM ANALYSIS [CHAP. 1 The potential energy of a lin ear system can be written in the form (1.5) The work done by the viscous damping force in a lin ear system between two arbitrary locations X, and X, can be written as W = - f co.x dx (1.6) x, Co. Fig. 1·1 1.4 TORSIONAL SYSTEMS When an ang ul ar coordinate is used as a generalized coordinate for a lin ea r system, the system can be modeled by the equivalent torsional system of Fig. 1·2. The moment applied to a linear torsional spring is proportional to its ang ul ar rotation while the moment applied to a linear torsional viscous damper is proportiona l to its ang ul ar velocity. The equivalent system coefficients for a tor sional system are determined by calculating the total kinetic energy, potential energy, and work done by viscous damping forces for the original syst em in terms of the chosen generalized coordinate and sett in g them equal to v = ! k,,,e ' 8, W = - f c tJde '" 8, ' 1' 'Fig. 1·2 (1.7) (1.8) (1 .9) CHAP.1J MECHANICAL SYSTEM ANALYSIS 3 1.5 STATIC EQUILIBRIUM POSITION Systems, such as the one in Fig. 1·3, have e l ~J.tic elements that are subject to force when the system is in equilibrium. The resulting deflecllon in the elastic el ement is ca ll ed its static deflection, usually den oted by 11. ". The static deflec ti on of an elastic elem,ent in a lin ear sys tem has no effect on the system's eq ui va lent s ti ff ness. Fig. 1·3 Solved Problems 1.1 Determine the number of degrees of freedom to be used in the vibra ti on analysis of the rigid b ar of F ig . 1-4, and specify a set of generalized coor dinates that can be used in its vibration anal ys i s. f- 30em -t 70em 1: k=2 000N/m (11U ' \ m -2.3 kg Fig. 1·4 Since the bar is r igid, the sys tem h as only 1 degree of freedom. One po ssib le c hoice for th e ge nera li zed coordinate is 8, th e angular displacement of th e bar me as ur ed positiv e clockwi se fr om the syste m's eq uil ibri um pos it ion. 1.2 I?etermine the number of degrees of freedom needed f or the analysis of the mechanical system of Fi g. 1·5, and specify a set of generalized coordinates that can be used in its vibration analysi s. Fig. 1·5 [...]... parameters for an equivalent system model of the system of Fig, 1-30 when 20 M~CHANICAL [CHAP 1 SYSTEM ANALYSIS e, the clockwise angular displace ment of the bar, measured from the system 's equilibrium position, is used as the generalized coordinate Include approximations for the inertia effects of the springs and assume small e Slender bar of mass m -+ 2L ~ 3 Fig 1-30 The inertia of the springs is... numbe r of degrees of freedom necessary for the analysis of the system of Fig 1-38 mass m l moment of inertia I Fig 1-38 Ans 3 28 1.35 MECHANICAL SYSTEM ANALYSIS [CHAP I Determine the numbe r of degrees of freedom necessary for the analysis of the syste m of Fi g 1-39 rmm Fig 1-39 Ans 1.36 D e te rmine the longitudinal stiffness of a rectangul ar, 30 x 50 mm steel bar (£ = 210 x 10" N/ m' ) of length... ~(O.OI S m)'(sox 10' ;,) O .S m 1.65 ~ (0.025 m)'( 40 X 10' ~) 12 m N-m m X 10' ~ N-m 2.05 X 10' ~ CHAP 11 MECHANICAL SYSTEM ANALYSIS 11 The angle of twist of the end of aluminum core of shaft AB is the same as the angle of twist of the end of the steel shell of shaft AB Also, the total torque on the end of shaft AB is the sum of the resisting torque in the aluminum core and the res isting torque in the steel... valent systems model is the to rsional system T hus from the above k '., = k,' CHAP 1.30 II MECHANICAL SYSTEM ANALYS IS 25 D etermine the parameters in an equivale nt system model of the system of F ig 1·35 when e, the clockwise ang ular displaceme nt of the bar fro m the system 's equilibrium position , is used as the genera lized coord ina te Assume small e 1 r- 3L +k1 Fig 1·35 Ass uming small... torsional viscous damper of Fig 1·22 consists of a thin disk attached to a rotating shaft The face of the disk has a radius R and rotates in a dish of fluid of depth hand viscosity 1-' Determine the torsional viscous damping coefficient for this damper OJ Fig 1·22 14 [C HAP 1 MECHANICAL SYSTEM ANALYSIS Let r be the shear stress acting on the differentia l area dA = r dr d8 on the surface of the disk,... equilibrium position, is used as the generalized coordinate Assume small e, e Let q, be the counterclockwise displaeement of bar CD, measured from th e system 's equili brium position Since the ends of bars AB and CD are connected by a rigid link, their displacements must be the same Thus ass uming small e and q" I Le = Lq, -> q, = 1e MECHANICAL SYSTEM ANALYSIS CHAP II 21 The kinetic energy of th e system at... e l the sys tem sho wn in Fig 1-1 5 by a block a ttac h e d to a sing le s pring o f a n equivale nt stiffness Fig 1-15 The first step is to replace the paralle l combinati o ns by springs of equ ivale nt sti lfnesses using the res ults of Pro blem 1.9 The result is s ho wn in Fig 1-16a The springs on the left of th e block a re in series with o ne ano the r The res ult of Proble m 1.10 is used to... in its equilibrium position If 8 is sm all, then the displaceme nt of the right end of the bar is x + (L/2)8 Thus the system has 2 degrees of freedom , and x and 8 are a possible set of generalized coordinates, as illustrated in Fig 1-6 Fig 1-6 1.3 Detennine the numbe r of d egrees of freedom used in the a nalysis of the m ech a nica l system of Fig 1-7 Specify 'a set of gene ra lized coordinates that... block is th e sum of th e fo rces developed in the springs Thus th ese springs behave as if they are in parallel and ca n be repl aced by a spring of stilfness k 2k 7k 2 3 6 - + - =as illustrated in Fig 1-16c 10 MECHAN ICAL SYSTEM ANA LYSIS [CHAP I (a) k ~ ~ 2 3 m (b ) (e) Fig 1-16 1.12 sa rf+ Model the torsional system of Fig 1-17 by a disk attached to a torsional spring of an equivalent stiffness Mathcad... the same as that of the linear spring assuming a linear displacement function ~l ~ f x (0) x u(t)=x u(O)~ (b) Fig 1-24 1.18 What is the mass of a particle that should be added to the block of the system of Fig 1-25 to approximate the inertia effects of the series combination of springs? Fig 1-25 16 [CHAP I MECHANICAL SYSTEM ANALYSIS Using the results of Problem 1.17, the inerti a effects of the . M_' SCHAUM& apos ;S OUT LI NE OF THEORY AND PROBLEMS of MECHANICAL VIBRATIONS S. GRAH AM KELLY, Ph. D. Associate Professor of Me chanical En gineering alld Assistant Provost. angle of twi st of the end of aluminum core of shaft AB is the same as the angle of twi st of the end of the steel shell of shaft AB . Also, the total torque on the end of shaft AB is. single s prin g o f a n equival e nt s ti ff n ess . Fig. 1-15 Th e first st ep is to replace the para ll el combina ti ons by springs of equ iva lent s ti lfnesses using the results of