This page intentionally left blank Vibration of Mechanical Systems This is a textbook for a first course in mechanical vibrations There are many books in this area that try to include everything, thus they have become exhaustive compendiums that are overwhelming for an undergraduate In this book, all the basic concepts in mechanical vibrations are clearly identified and presented in a concise and simple manner with illustrative and practical examples Vibration concepts include a review of selected topics in mechanics; a description of single-degree-of-freedom (SDOF) systems in terms of equivalent mass, equivalent stiffness, and equivalent damping; a unified treatment of various forced response problems (base excitation and rotating balance); an introduction to systems thinking, highlighting the fact that SDOF analysis is a building block for multi-degree-of-freedom (MDOF) and continuous system analyses via modal analysis; and a simple introduction to finite element analysis to connect continuous system and MDOF analyses There are more than 60 exercise problems and a complete solutions manual The use of MATLAB R software is emphasized Alok Sinha is a Professor of Mechanical Engineering at The Pennsylvania State University (PSU), University Park He received his PhD degree in mechanical engineering from Carnegie Mellon University He has been a PSU faculty member since August 1983 His areas of teaching and research are vibration, control systems, jet engines, robotics, neural networks, and nanotechnology He is the author of Linear Systems: Optimal and Robust Control He has served as a Visiting Associate Professor of Aeronautics and Astronautics at MIT, Cambridge, MA, and as a researcher at Pratt & Whitney, East Hartford, CT He has also been an associate editor of ASME Journal of Dynamic Systems, Measurement, and Control At present, he serves as an associate editor of ASME Journal of Turbomachinery and AIAA Journal Alok Sinha is a Fellow of ASME He has received the NASA certificate of recognition for significant contributions to the Space Shuttle Microgravity Mission VIBRATION OF MECHANICAL SYSTEMS Alok Sinha The Pennsylvania State University CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521518734 © Alok Sinha 2010 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2010 ISBN-13 978-0-511-77527-7 eBook (EBL) ISBN-13 978-0-521-51873-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate To My Wife Hansa and My Daughters Divya and Swarna CONTENTS Preface page xiii Equivalent Single-Degree-of-Freedom System and Free Vibration 1.1 Degrees of Freedom 1.2 Elements of a Vibratory System 1.2.1 Mass and/or Mass-Moment of Inertia Pure Translational Motion Pure Rotational Motion Planar Motion (Combined Rotation and Translation) of a Rigid Body Special Case: Pure Rotation about a Fixed Point 1.2.2 Spring Pure Translational Motion Pure Rotational Motion 1.2.3 Damper Pure Translational Motion Pure Rotational Motion 1.3 Equivalent Mass, Equivalent Stiffness, and Equivalent Damping Constant for an SDOF System 1.3.1 A Rotor–Shaft System 1.3.2 Equivalent Mass of a Spring 1.3.3 Springs in Series and Parallel Springs in Series Springs in Parallel 1.3.4 An SDOF System with Two Springs and Combined Rotational and Translational Motion 1.3.5 Viscous Dampers in Series and Parallel vii 5 6 8 10 10 11 12 13 14 16 16 17 19 22 viii Contents Dampers in Series Dampers in Parallel 1.4 Free Vibration of an Undamped SDOF System 1.4.1 Differential Equation of Motion Energy Approach 1.4.2 Solution of the Differential Equation of Motion Governing Free Vibration of an Undamped Spring–Mass System 1.5 Free Vibration of a Viscously Damped SDOF System 1.5.1 Differential Equation of Motion 1.5.2 Solution of the Differential Equation of Motion Governing Free Vibration of a Damped Spring–Mass System Case I: Underdamped (0 < ξ < or < ceq < cc ) Case II: Critically Damped (ξ = or ceq = cc ) Case III: Overdamped (ξ > or ceq > cc ) 1.5.3 Logarithmic Decrement: Identification of Damping Ratio from Free Response of an Underdamped System (0 < ξ < 1) Solution 1.6 Stability of an SDOF Spring–Mass–Damper System Exercise Problems Vibration of a Single-Degree-of-Freedom System Under Constant and Purely Harmonic Excitation 2.1 Responses of Undamped and Damped SDOF Systems to a Constant Force Case I: Undamped (ξ = 0) and Underdamped (0 < ξ < 1) Case II: Critically Damped (ξ = or ceq = cc ) Case III: Overdamped (ξ > or ceq > cc ) 2.2 Response of an Undamped SDOF System to a Harmonic Excitation Case I: ω = ωn Case II: ω = ωn (Resonance) Case I: ω = ωn Case II: ω = ωn 2.3 Response of a Damped SDOF System to a Harmonic Excitation Particular Solution Case I: Underdamped (0 < ξ < or < ceq < cc ) 22 23 25 25 27 34 40 40 41 42 45 46 51 55 58 63 72 72 74 75 76 82 83 84 87 87 88 89 92 298 Vibration of Mechanical Systems P5.7 Find the natural frequencies and the mode shapes of a cantilever beam attached to a spring at its end (Figure P5.7) E, Ia , ρ, A, l k Figure P5.7 A cantilever beam attached to a spring P5.8 A sinusoidal force is applied at the midpoint of a fixed–fixed elastic beam Determine the response of the system using zero initial conditions f sin ωt E, Ia , ρ, A, l Figure P5.8 A fixed–fixed beam excited by a sinusoidal force P5.9 Consider a fixed–fixed steel bar with the length = 0.1 m and the cross-sectional area = × 10−4 m2 Using the finite element method, determine the first three natural frequencies and compare them to the theoretical values P5.10 Consider a fixed–fixed steel beam with the length = 0.1 m, the cross-sectional area = × 10−4 m2 and the area moment of inertia = 0.5 × 10−8 m4 Using the finite element method, determine the first three natural frequencies and compare them to the theoretical values APPENDIX A EQUIVALENT STIFFNESSES (SPRING C ONSTANTS) OF BEAMS, TORSIONAL SHAFT, A ND LONGITUDINAL B AR In this Appendix, equivalent stiffnesses of beams, a torsional shaft, and a longitudinal bar are presented Derivations of these stiffnesses are based on static deflection of a structure A.1 FIXED–FIXED BEAM Force F a b x Deflection, δ(x) ⎧ ⎪ ⎪ ⎪ ⎨ F = keq (x) = δ(x) ⎪ ⎪ ⎪ ⎩ E, I a a+b=l 6EIa ; − x(3a + b)) 0≤x≤a 6EIa ; − ( − x)(3b + a)) a≤x≤ x b2 (3a ( − x)2 a (3b (A.1) keq (x) : Equivalent stiffness at a distance x from left end E : Young’s modulus of elasticity Ia : Area moment of inertia 299 300 Appendix A A.2 SIM PLY SUPPORTED BEAM Force F a b E, I a x Deflection, δ(x) ⎧ ⎪ ⎪ ⎨ F = keq (x) = δ(x) ⎪ ⎪ ⎩ a+b=l 6EIa ; − x − b2 ) 6EIa ; ( − x)a(2 x − x − a ) xb( 0≤x≤a (A.2) a≤x≤ A.3 CANTILEVER BEAM Force F a b x Deflection, δ(x) E, I a a+b=l ⎧ 6EIa ⎪ ⎪ ⎨ x (3a − x) ; F = keq (x) = δ(x) ⎪ ⎪ ⎩ 6EIa ; a (3x − a) 0≤x≤a (A.3) a≤x≤ A.4 SHAFT UNDER TORSION l Outer dia = d2 Inner dia = d1 keq = 4 π G(d2 − d1 ) ; 32 G: Shear modulus of elasticity (A.4) Appendix A 301 A.5 ELASTIC BAR UNDER AXIAL LOAD l keq = EA ; A: Cross-sectional area (A.5) APPENDIX B SOME M AT HE MA TICA L FORMULAE B.1 TRIGONOM ETRIC IDENTITY sin(−x) = − sin x cos(−x) = cos x sin(x ± y) = sin x cos y ± cos x sin y cos(x ± y) = cos x cos y ∓ sin x sin y sin x + sin y = sin x+y x−y cos 2 sin x − sin y = cos x−y x+y sin 2 cos x + cos y = cos x−y x+y cos 2 x−y x+y sin 2 cos x − cos y = −2 sin tan(x ± y) = tan x ± tan y ∓ tan x tan y e j x = cos x + j sin x; j= sinh γ x = eγ x − e−γ x cosh γ x = eγ x + e−γ x 302 −1 303 Appendix B sin γ x = e jγ x − e− jγ x ; 2j j= −1 cos γ x = e jγ x + e− jγ x ; j= −1 cosh2 x − sinh2 x = B.2 POWER SERIES EXPANSION ex = + x + x3 x2 + + ··· 2! 3! sin x = x − x3 x5 x7 + − + ···; 3! 5! 7! x in radians cos x = − x2 x4 x6 + − + ···; 2! 4! 6! x in radians B.3 BINOM IAL EXPANSION (1 − x)− p = + px + p( p + 1)x + · · · ; |x| < APPENDIX C LAPLACE T RANSFORM TABLE f (t) f (s) δ(t); unit impulse function us (t); unit step function s tn−1 us (t); n = 1, 2, 3, , (n − 1)! sn e−at us (t) s+a tn−1 e−at us (t); n = 1, 2, 3, , (n − 1)! (s + a)n a s2 + a s s2 + a sin(at)us (t) cos(at)us (t) −ξ ωn t e sin ωd t us (t) ωd = ωn − ξ ωd ωn − e−ξ ωn t sin(ωd t − φ1 )us (t); ωd = ωn − ξ ωd 1− ωn −ξ ωn t e sin(ωd t + φ1 ) us (t) ωd ωd = ωn − ξ ; φ1 cos−1 ξ ; s2 + 2ξ ωn s + ωn s s2 + 2ξ ωn s + ωn s(s2 ξ