de Silva, Clarence W. FrontmatterVibration: Fundamentals and PracticeClarence W. de SilvaBoca Raton: CRC Press LLC, 2000Boca Raton London New York Washington, D.C.CRC PressClarence W. de SilvaVIBRATIONFundamentals and Practice 2000 CRC Press Library of Congress Cataloging-in-Publication Data De Silva, Clarence W.Vibration : fundamentals and practice / Clarence W. de Silva p. cm.Includes bibliographical references and index.ISBN 0-8493-1808-4 (alk. paper)1. Vibration.I. Title.TA355.D384 1999620.3dc2199-16238CIPThis book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted withpermission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publishreliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materialsor for the consequences of their use.Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical,includingphotocopying,microlming,andrecording,orbyanyinformationstorageorretrievalsystem,withoutpriorpermission in writing from the publisher.TheconsentofCRCPressLLCdoesnotextendtocopyingforgeneraldistribution,forpromotion,forcreatingnewworks, or for resale. Specic permission must be obtained in writing from CRC Press LLC for such copying.Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are only used for identication and explanation, without intent to infringe. Cover art is the U.S. Space Shuttle and the International Space Station. (Courtesy of NASA Langley ResearchCenter, Hampton, VA. With permission.) 2000 by CRC Press LLCNo claim to original U.S. Government worksInternational Standard Book Number 0-8493-1808-4Library of Congress Card Number 99-16238Printed in the United States of America1234567890Printed on acid-free paper 2000 CRC Press Preface This book provides the background and techniques that will allow successful modeling, analysis,monitoring,testing,design,modication,andcontrolofvibrationinengineeringsystems.Itissuitableasbothacoursetextbookforstudentsandinstructors,andapracticalreferencetoolforengineersandotherprofessionals. Asatextbook,itcanbeusedinasingle-semestercourseforthird-year (junior) and fourth-year (senior) undergraduate students, or for Masters level graduatestudentsinanybranchofengineeringsuchasaeronauticalandaerospace,civil,mechanical,andmanufacturing engineering. But, in view of the practical considerations, design issues, experimentaltechniques, and instrumentation that are presented throughout the book, and in view of the simpliedand snapshot-style presentation of fundamentals and advanced theory, the book will also serve asavaluablereferencetoolforengineers,technicians,andotherprofessionalsinindustryandinresearch laboratories.ThebookisanoutgrowthoftheauthorsexperienceinteachingundergraduateandgraduatecoursesinDynamics,MechanicalVibration,DynamicSystemModeling,InstrumentationandDesign,FeedbackControl,ModernControlEngineering,andModal AnalysisandTestingintheU.S.andCanada(CarnegieMellonUniversityandtheUniversityofBritishColumbia)formorethan20years.Theindustrialexperienceandtrainingthathereceivedinproducttestingandqualication,analysis,design,andvibrationinstrumentationatplaceslike WestinghouseElectricCorporation in Pittsburgh, IBM Corporation in Boca Raton, NASAs Langley and Lewis ResearchCenters,andBruelandKjaerinDenmarkenabledtheauthortoprovidearealisticandpracticaltreatment of the subject.Designforvibrationandcontrolofvibrationarecrucialinmaintainingahighperformancelevel and production efciency, and prolonging the useful life of machinery, structures, and indus-trialprocesses.Beforedesigningorcontrollinganengineeringsystemforgoodvibratoryperfor-mance, it is important to understand, represent (i.e., model), and analyze the vibratory characteristicsofthesystem.Suppressionoreliminationofundesirablevibrationsandgenerationofrequiredforms and levels of desired vibrations are general goals of vibration engineering. In recent years,researchers and practitioners have devoted considerable effort to studying and controlling vibrationin a range of applications in various branches of engineering. With this book, designers, engineers,and students can reap the benets of that study and experience, and learn the observation, instru-mentation, modeling, analysis, design, modication, and control techniques that produce mechan-icalandaeronauticalsystems,civilengineeringstructures,andmanufacturingprocessesthatareoptimized against the effects of vibration.The book provides the background and techniques that will allow successful modeling, analysis,design, modication, testing, and control of vibration in engineering systems. This knowledge willbe useful in the practice of vibration, regardless of the application area or the branch of engineering.A uniform and coherent treatment of the subject is presented, by introducing practical applicationsof vibration, through examples, in the very beginning of the book, along with experimental tech-niques and instrumentation, and then integrating these applications, design, and control consider-ations into fundamentals and analytical methods throughout the text. To maintain clarity and focusandtomaximizetheusefulnessofthebook,anattempthasbeenmadetodescribeandillustrateindustry-standardandstate-of-the-artinstrumentation,hardware,andcomputationaltechniquesrelated to the practice of vibration. As its main features, the book: 2000 CRC Press Introduces practical applications, design, and experimental techniques in the very begin-ning, and then uniformly integrates them throughout the book Provides36SummaryBoxesthatpresentkeymaterialcoveredinthebook,inpointform, within each chapter, for easy reference and recollection (these items are particularlysuitable for use by instructors in their presentations) Outlines mathematics, dynamics, modeling, fast Fourier transform (FFT) techniques, andreliability analysis in appendices Provides over 60 worked examples and case studies, and over 300 problems Will be accompanied by an Instructors Manual, for instructors, that contains completesolutions to all the end-of-chapter problems Describessensors,transducers,lters,ampliers,analyzers,andotherinstrumentationthat is useful in the practice of vibration Describesindustry-standardcomputertechniques,hardware,andtoolsforanalysis,design, and control of vibratory systems, with examples Provides a comprehensive coverage of vibration testing and qualication of products Offersanalogiesofmechanicalandstructuralvibration,tootheroscillatorybehaviorsuch as in electrical and uid systems, and contrasts these with thermal systems. A N OTE TO I NSTRUCTORS The book is suitable as the text for a standard undergraduate course in Mechanical Vibration or foraspecializedcoursefornal-yearundergraduatestudentsandMasterslevelgraduatestudents.Three typical course syllabuses are outlined below.A. A Standard Undergraduate Course As the textbook for an undergraduate (3rd year or 4th year) course in Mechanical Vibration, it maybe incorporated into the following syllabus for a 12 week course consisting of 36 hours of lecturesand 12 hours of laboratory experiments: Lectures Chapter 1 (1 hour)Sections 8.1, 8.2, 8.4, 9.1, 9.2, 9.8 (3 hours)Chapter 2 (6 hours)Chapter 3 (6 hours)Section 11.4 (2 hours)Chapter 5 (6 hours)Chapter 6 (6 hours)Sections 12.1, 12.2, 12.3, 12.4, 12.5 (6 hours) Laboratory Experiments The following four laboratory experiments, each of 3-hour duration, may be incorporated.1. Experiment on modal testing (hammer test and other transient tests) and damping mea-surement in the time domain (see Section 11.4)2. Experimentonshakertestinganddampingmeasurementinthefrequencydomain(seeSection 11.4)3. Experiment on single-plane and two-plane balancing (see Section 12.3)4. Experiment on modal testing of a distributed-parameter system (see Section 11.4) 2000 CRC Press B.A Course in Industrial Vibration Chapter 1 (1 hour)Chapter 4 (3 hours)Chapter 7 (5 hours)Chapter 8 (5 hours)Chapter 9 (4 hours)Chapter 10 (6 hours)Chapter 11 (6 hours)Chapter 12 (6 hours)A project may be included in place of a nal examination. C. A Course in Modal Analysis and Testing Chapter 1 (1 hour)Chapter 4 (3 hours)Chapter 5 (6 hours)Chapter 6 (6 hours)Chapter 7 (5 hours)Chapter 10 (6 hours)Chapter 11 (6 hours)Section 12.6 (hours)A project may be included in place of a nal examination.Clarence W. de SilvaVancouver, Canada 2000 CRC Press The Author Clarence W.deSilva, Fellow ASMEandFellowIEEE,isProfessorofMechanicalEngineeringat the University of British Columbia, Vancouver, Canada, and has occupied the NSERC Chair inIndustrial Automation since 1988. He obtained his rst Ph.D. from the Massachusetts Institute ofTechnology in 1978 and, 20 years later, another Ph.D. from the University of Cambridge, England.De Silva has served as a consultant to several companies, including IBM and Westinghouse in theU.S.,andhasledthedevelopmentofmanyindustrialmachines.HeisrecipientoftheEducationAwardoftheDynamicSystemsandControlDivisionoftheAmericanSocietyofMechanicalEngineers,theMeritoriousAchievementAwardoftheAssociationofProfessionalEngineersofBritish Columbia, the Outstanding Contribution Award of the IEEE Systems, Man, and CyberneticsSociety, the Outstanding Large Chapter Award of the IEEE Industry Applications Society, and theOutstanding Chapter Award from the IEEE Control Systems Society. He has authored 14 technicalbooks, 10 edited volumes, over 120 journal papers, and a similar number of conference papers andbook chapters. He has served on the editorial boards of 12 international journals, and is the Editor-in-Chief of the International Journal of Knowledge-Based Intelligent Engineering Systems, SeniorTechnical Editor of Measurements and Control , and Regional Editor, North America, of the Inter-national Journal of Intelligent Real-Time Automation . He has been a Lilly Fellow, Senior FulbrightFellow to Cambridge University, ASI Fellow, and a Killam Fellow. 2000 CRC Press Acknowledgment Preparation of this book would not have been possible if not for the support of many individualsand organizations, but it is not possible to list all of them here. I wish to recognize the followingspecic contributions:Financial assistance for my research and professional activities has been provided primarily by: Ministry of Advanced Education, Training and Technology, Province of British Columbia,for the Network of Centres of Excellence Program Natural Sciences and Engineering Research Council of Canada (NSERC) Network of Centres of Excellence (Institute of Robotics and Intelligent Systems) Advanced Systems Institute of British Columbia Science Council of British Columbia Ministry of Environment of British Columbia British Columbia Hydro and Power Authority National Research Council Killam Memorial Faculty Fellowship Program B.C. Packers, Ltd. Neptune Dynamics, Ltd. Gareld Weston FoundationSpecial acknowledgment should be made here of the Infrastructure Grant from the Ministry ofAdvanced Education, Training and Technology, Province of British Columbia, which made part ofthesecretarialsupportformyworkpossible.TheDepartmentofMechanicalEngineeringattheUniversityofBritishColumbiaprovidedmewithanexcellentenvironmenttocarryoutmyeducationalactivities,includingthepreparationofthisbook.Mygraduatestudents,researchassociates, teaching assistants, and ofce staff have contributed directly and indirectly to the successof the book. Particular mention should be made of the following people: Ricky Min-Fan Lee for systems assistance Hassan Bayoumi and Jay Choi for graphics assistance YuanChen,ScottGu,IwanKurnianto,FaragOmar,andRoyaRahbariforteachingassistance Marje Lewis and Laura Gawronski for secretarial assistance.I wish to thank the staff of CRC Press, particularly the Associate Editor, Felicia Shapiro and theProjectEditor,SaraSeltzer,fortheirneeffortintheproductionofthebook.Encouragementofvariousauthoritiesintheeldofengineeringparticularly,ProfessorDevendraGargofDukeUniversity, Professor Mo Jamshidi of the University of New Mexico, and Professor Arthur Murphy(DuPont Fellow Emeritus) is gratefully acknowledged. Finally, my family deserves an apologyfor the unintentional neglect that they may have suffered during the latter stages of production ofthe book. 2000 CRC Press Source Credits The sources of several photos, gures, and tables are recognized and given credit, as follows: Figure 1.1: Courtesy of Ms. Kimberly Land, NASA Langley Research Center, Hampton,Virginia. Figure1.3:CourtesyofProfessorCarlosE. Ventura,DepartmentofCivilEngineering,the University of British Columbia, Vancouver, Canada. Figure 1.4: Courtesy of Ms. Heather Conn of BC Transit, Vancouver, Canada. Photo byMark Van Manen. Figure 1.5: Courtesy of Ms. Jeana Dugger, Key Technologies, Inc., Walla Walla, Wash-ington. Figure1.8:Courtesyof MechanicalEngineering magazine,fromarticleSemiactiveCone Suspension Smooths the Ride by Bill Siuru, Vol. 116, No. 3, page 106. Copyright,American Society of Mechanical Engineers International, New York. Table 7.5: Reprinted from ASME BPVC, Section III-Division 1, Appendices, by permis-sion of The American Society of Mechanical Engineers, New York. All rights reserved. Figure 8.8: Courtesy of Bruel & Kjaer, Naerum, Denmark. Figure 9.36 and Figure 9.38: Courtesy of Ms. Beth Daniels. Copyright 1999 Tektronix,Inc. All rights reserved. Reproduced by permission. Figure 11.6, Figure 11.8, and Figure 12.15: Experimental setups used by the author forteachingafourth-yearcourseintheUndergraduate VibrationsLaboratory,Departmentof Mechanical Engineering, the University of British Columbia, Vancouver, Canada. Figure12.34,Figure12.35,andTable12.2:CourtesyofDr.GeorgeWang.ExtractedfromthereportActiveControlof Vibrationin WoodMachiningfor WoodRecoverybyG.Wang,J.Xi,Q.Zhong,S.Abayakoon,K.Krishnappa,andF.Lam,NationalResearch Council, Integrated Manufacturing Technologies Institute, Vancouver, Canada,pp. 5, 8, 25-28, May 1998. 2000 CRC Press Dedication Professor David N. Wormley.For the things we have to learn before we can do them,we learn by doing them. Aristotle (Author of Mechanics and Acoustics, 384322 B.C.) 2000 CRC Press Table of Contents Chapter 1Vibration Engineering 1.1Study of Vibration1.2Application Areas1.3History of Vibration1.4Organization of the BookProblemsReferences and Further ReadingAuthors WorkOther Useful Publications Chapter 2Time Response 2.1Undamped Oscillator2.1.1Energy Storage ElementsInertiaSpringGravitational Potential Energy2.1.2Conservation of Energy System 1 (Translatory)System 2 (Rotatory)System 3 (Flexural)System 4 (Swinging) System 5 (Liquid Slosh)System 6 (Electrical)CapacitorInductor2.1.3Free ResponseExample 2.1Solution2.2Heavy Springs2.2.1Kinetic Energy EquivalenceExample 2.2Solution2.3Oscillations in Fluid SystemsExample 2.3Solution2.4Damped Simple Oscillator2.4.1Case 1: Underdamped MotionInitial Conditions2.4.2Logarithmic Decrement Method2.4.3Case 2: Overdamped Motion2.4.4Case 3: Critically Damped Motion2.4.5Justication for the Trial Solution 2000 CRC Press First-Order SystemSecond-Order SystemRepeated Roots2.4.6Stability and Speed of Response Example 2.4Solution2.5Forced Response2.5.1Impulse Response Function2.5.2Forced Response2.5.3Response to a Support MotionImpulse ResponseThe Riddle of Zero Initial ConditionsStep ResponseLiebnitzs RuleProblems Chapter 3Frequency Response 3.1Response to Harmonic Excitations3.1.1Response CharacteristicsCase 1Case 2Case 3Particular Solution (Method 1)Particular Solution (Method 2): Complex Function MethodResonance3.1.2Measurement of Damping Ratio (Q-Factor Method)Example 3.1Solution3.2Transform Techniques3.2.1Transfer Function3.2.2Frequency-Response Function (Frequency-Transfer Function)Impulse ResponseCase 1 ( < 1)Case 2 ( > 1)Case 3 ( = 1)Step Response3.2.3Transfer Function MatrixExample 3.2Example 3.3Example 3.4Solution3.3Mechanical Impedance ApproachMass ElementSpring ElementDamper Element3.3.1Interconnection LawsExample 3.5Example 3.63.4Transmissibility Functions3.4.1Force Transmissibility 2000 CRC Press 3.4.2Motion TransmissibilitySystem Suspended on a Rigid Base (Force Transmissibility)System with Support Motion (Motion Transmissibility)3.4.3General Case Example 3.73.4.4Peak Values of Frequency-Response Functions3.5Receptance Method3.5.1Application of ReceptanceUndamped Simple OscillatorDynamic AbsorberProblems Chapter 4 Vibration Signal Analysis 4.1Frequency Spectrum4.1.1Frequency4.1.2Amplitude Spectrum4.1.3Phase Angle4.1.4Phasor Representation of Harmonic Signals4.1.5RMS Amplitude Spectrum4.1.6One-Sided and Two-Sided Spectra4.1.7Complex Spectrum4.2Signal Types4.3Fourier Analysis4.3.1Fourier Integral Transform (FIT)4.3.2Fourier Series Expansion (FSE)4.3.3Discrete Fourier Transform (DFT)4.3.4Aliasing DistortionSampling TheoremAliasing Distortion in the Time DomainAnti-Aliasing FilterExample 4.14.3.5Another Illustration of AliasingExample 4.24.4Analysis of Random Signals4.4.1Ergodic Random Signals4.4.2Correlation and Spectral Density4.4.3Frequency Response Using Digital Fourier Transform4.4.4Leakage (Truncation Error)4.4.5Coherence4.4.6Parsevals Theorem4.4.7Window Functions4.4.8Spectral Approach to Process Monitoring4.4.9Cepstrum4.5Other Topics of Signal Analysis 4.5.1Bandwidth4.5.2Transmission Level of a Bandpass Filter4.5.3Effective Noise Bandwidth 4.5.4Half-Power (or 3 dB) Bandwidth4.5.5Fourier Analysis Bandwidth 2000 CRC Press 4.6Resolution in Digital Fourier Results4.7Overlapped ProcessingExample 4.34.7.1Order AnalysisSpeed Spectral MapTime Spectral MapOrder TrackingProblems Chapter 5 Modal Analysis 5.1Degrees of Freedom and Independent Coordinates5.1.1Nonholonomic ConstraintsExample 5.1Example 5.25.2System Representation5.2.1Stiffness and Flexibility Matrices5.2.2Inertia Matrix5.2.3Direct Approach for Equations of Motion5.3Modal VibrationsExample 5.35.4Orthogonality of Natural Modes5.4.1Modal Mass and Normalized Modal Vectors5.5Static Modes and Rigid Body Modes5.5.1Static Modes 5.5.2Linear Independence of Modal Vectors5.5.3Modal Stiffness and Normalized Modal Vectors5.5.4Rigid Body ModesExample 5.4Equation of Heave MotionEquation of Pitch Motion Example 5.5First Mode (Rigid Body Mode)Second Mode5.5.5Modal Matrix5.5.6Conguration Space and State SpaceState Vector5.6Other Modal Formulations5.6.1Non-Symmetric Modal Formulation5.6.2Transformed Symmetric Modal FormulationExample 5.6Approach 2Approach 35.7Forced VibrationExample 5.7First Mode (Rigid Body Mode) Second Mode (Oscillatory Mode)5.8Damped Systems5.8.1Proportional DampingExample 5.8 2000 CRC Press 5.9State-Space Approach5.9.1Modal Analysis5.9.2Mode Shapes of Nonoscillatory Systems5.9.3Mode Shapes of Oscillatory SystemsExample 5.9Problems Chapter 6Distributed-Parameter Systems 6.1Transverse Vibration of Cables6.1.1Wave Equation6.1.2General (Modal) Solution 6.1.3Cable with Fixed Ends6.1.4Orthogonality of Natural ModesExample 6.1Solution6.1.5Application of Initial ConditionsExample 6.2Solution6.2Longitudinal Vibration of Rods6.2.1Equation of Motion6.2.2Boundary ConditionsExample 6.3Solution6.3Torsional Vibration of Shafts6.3.1Shaft with Circular Cross Section6.3.2Torsional Vibration of Noncircular ShaftsExample 6.4SolutionExample 6.5Solution6.4Flexural Vibration of Beams6.4.1Governing Equation for Thin BeamsMoment-Deection Relation Rotatory Dynamics (Equilibrium)Transverse Dynamics6.4.2Modal Analysis6.4.3Boundary Conditions6.4.4Free Vibration of a Simply Supported BeamNormalization of Mode Shape FunctionsInitial Conditions6.4.5Orthogonality of Mode ShapesCase of Variable Cross Section6.4.6Forced Bending VibrationExample 6.6SolutionExample 6.7Solution6.4.7Bending Vibration of Beams with Axial Loads6.4.8Bending Vibration of Thick Beams 2000 CRC Press 6.4.9Use of the Energy Approach6.4.10Orthogonality with Inertial Boundary ConditionsRotatory Inertia6.5Damped Continuous Systems 6.5.1Modal Analysis of Damped Beams Example 6.8Solution6.6Vibration of Membranes and Plates6.6.1Transverse Vibration of Membranes6.6.2Rectangular Membrane with Fixed Edges6.6.3Transverse Vibration of Thin Plates6.6.4Rectangular Plate with Simply Supported EdgesProblems Chapter 7Damping 7.1Types of Damping7.1.1Material (Internal) DampingViscoelastic Damping Hysteretic DampingExample 7.1Solution7.1.2Structural Damping7.1.3Fluid DampingExample 7.2Solution7.2Representation of Damping in Vibration Analysis7.2.1Equivalent Viscous Damping 7.2.2Complex StiffnessExample 7.3Solution7.2.3Loss Factor 7.3Measurement of Damping7.3.1Logarithmic Decrement Method7.3.2Step-Response Method7.3.3Hysteresis Loop MethodExample 7.4Solution7.3.4Magnication-Factor Method 7.3.5Bandwidth Method7.3.6General Remarks7.4Interface DampingExample 7.5Solution7.4.1Friction In Rotational Interfaces7.4.2InstabilityProblems 2000 CRC Press Chapter 8 Vibration Instrumentation 8.1Vibration Exciters8.1.1Shaker SelectionForce RatingPower RatingStroke RatingExample 8.1SolutionHydraulic ShakersInertial ShakersElectromagnetic Shakers8.1.2Dynamics of Electromagnetic ShakersTransient Exciters8.2Control System8.2.1Components of a Shaker ControllerCompressorEqualizer (Spectrum Shaper)Tracking Filter Excitation Controller (Amplitude Servo-Monitor)8.2.2Signal-Generating EquipmentOscillatorsRandom Signal GeneratorsTape PlayersData Processing8.3Performance Specication8.3.1Parameters for Performance SpecicationTime-Domain SpecicationsFrequency-Domain Specications8.3.2Linearity8.3.3Instrument RatingsRating Parameters8.3.4Accuracy and Precision8.4Motion Sensors and Transducers8.4.1PotentiometerPotentiometer ResolutionOptical Potentiometer8.4.2Variable-Inductance TransducersMutual-Induction TransducersLinear-Variable Differential Transformer (LVDT)Signal ConditioningExample 8.2Solution8.4.3Mutual-Induction Proximity Sensor8.4.4Self-Induction Transducers 8.4.5Permanent-Magnet Transducers8.4.6AC Permanent-Magnet Tachometer8.4.7AC Induction Tachometer8.4.8Eddy Current Transducers8.4.9Variable-Capacitance TransducersCapacitive Displacement Sensors 2000 CRC Press Capacitive Angular Velocity SensorCapacitance Bridge Circuit8.4.10Piezoelectric TransducersSensitivityExample 8.3SolutionPiezoelectric Accelerometer Charge Amplier8.5Torque, Force, and Other Sensors8.5.1Strain-Gage SensorsEquations for Strain-Gage MeasurementsBridge SensitivityThe Bridge ConstantExample 8.4SolutionThe Calibration ConstantExample 8.5SolutionData Acquisition Accuracy ConsiderationsSemiconductor Strain GagesForce and Torque SensorsStrain-Gage Torque SensorsDeection Torque SensorsVariable-Reluctance Torque SensorReaction Torque Sensors8.5.2Miscellaneous SensorsStroboscopeFiber-Optic Sensors and LasersFiber-Optic GyroscopeLaser Doppler InterferometerUltrasonic SensorsGyroscopic Sensors8.6Component Interconnection 8.6.1Impedance CharacteristicsCascade Connection of DevicesImpedance-Matching AmpliersOperational AmpliersVoltage FollowersCharge Ampliers8.6.2Instrumentation AmplierGround Loop NoiseProblems Chapter 9Signal Conditioning and Modication 9.1Ampliers9.1.1Operational AmplierExample 9.1Solution 2000 CRC Press 9.1.2Use of Feedback in Op-amps 9.1.3Voltage, Current, and Power Ampliers9.1.4Instrumentation AmpliersDifferential AmplierCommon ModeAmplier Performance RatingsExample 9.2SolutionCommon-Mode Rejection Ratio (CMRR) AC-Coupled Ampliers9.2Analog Filters9.2.1Passive Filters and Active FiltersNumber of Poles9.2.2Low-Pass FiltersExample 9.3SolutionLow-Pass Butterworth FilterExample 9.4Solution9.2.3High-Pass Filters9.2.4Bandpass Filters Resonance-Type Bandpass FiltersExample 9.5Solution9.2.5Band-Reject Filters9.3Modulators and Demodulators9.3.1Amplitude ModulationModulation Theorem Side Frequencies and Side Bands 9.3.2Application of Amplitude ModulationFault Detection and Diagnosis9.3.3Demodulation 9.4Analog/Digital Conversion9.4.1Digital-to-Analog Conversion (DAC)Weighted-Resistor DACLadder DACDAC Error Sources9.4.2Analog-to-Digital Conversion (ADC)Successive-Approximation ADCDual-Slope ADCCounter ADC9.4.3ADC Performance CharacteristicsResolution and Quantization ErrorMonotonicity, Nonlinearity, and Offset ErrorADC Conversion Rate 9.4.4Sample-and-Hold (S/H) Circuitry9.4.5Multiplexers (MUX)Analog MultiplexersDigital Multiplexers9.4.6Digital Filters 2000 CRC Press Software Implementation and Hardware Implementation9.5Bridge Circuits9.5.1Wheatstone Bridge9.5.2Constant-Current Bridge 9.5.3Bridge AmpliersHalf-Bridge Circuits 9.5.4Impedance BridgesOwen BridgeWien-Bridge Oscillator9.6Linearizing Devices9.6.1Linearization by Software9.6.2Linearization by Hardware Logic9.6.3Analog Linearizing Circuitry9.6.4Offsetting Circuitry9.6.5Proportional-Output CircuitryCurve-Shaping Circuitry9.7Miscellaneous Signal-Modication Circuitry9.7.1Phase Shifter9.7.2Voltage-to-Frequency Converter (VFC)9.7.3Frequency-to-Voltage Converter (FVC)9.7.4Voltage-to-Current Converter (VCC)9.7.5Peak-Hold Circuit9.8Signal Analyzers and Display Devices9.8.1Signal Analyzers9.8.2OscilloscopesTriggeringLissajous PatternsDigital OscilloscopesProblems Chapter 10Vibration Testing 10.1Representation of a Vibration Environment10.1.1Test Signals Stochastic versus Deterministic Signals10.1.2Deterministic Signal RepresentationSingle-Frequency SignalsSine SweepSine DwellDecaying SineSine BeatSine Beat with PausesMultifrequency SignalsActual Excitation RecordsSimulated Excitation Signals10.1.3Stochastic Signal RepresentationErgodic Random SignalsStationary Random SignalsIndependent and Uncorrelated SignalsTransmission of Random Excitations 2000 CRC Press 10.1.4Frequency-Domain RepresentationsFourier Spectrum MethodPower Spectral Density Method10.1.5Response SpectrumDisplacement, Velocity, and Acceleration SpectraResponse-Spectra Plotting PaperZero-Period AccelerationUses of Response Spectra10.1.6Comparison of Various Representations10.2Pretest Procedures10.2.1Purpose of Testing10.2.2Service FunctionsActive EquipmentPassive EquipmentFunctional Testing10.2.3Information AcquisitionInterface DetailsEffect of Neglecting Interface DynamicsEffects of DampingEffects of Inertia Effect of Natural FrequencyEffect of Excitation FrequencyOther Effects of Interface10.2.4Test-Program PlanningTesting of Cabinet-Mounted Equipment10.2.5Pretest Inspection10.3Testing Procedures10.3.1Resonance Search10.3.2Methods of Determining Frequency-Response FunctionsFourier Transform Method Spectral Density MethodHarmonic Excitation Method10.3.3Resonance-Search Test MethodsHammer (Bump) Test and Drop TestPluck TestShaker Tests10.3.4Mechanical AgingEquivalence for Mechanical AgingExcitation-Intensity EquivalenceDynamic-Excitation Equivalence Cumulative Damage Theory10.3.5TRS Generation10.3.6Instrument Calibration10.3.7Test-Object Mounting10.3.8Test-Input Considerations Test NomenclatureTesting with Uncorrelated ExcitationsSymmetrical Rectilinear TestingGeometry versus DynamicsSome LimitationsTesting of Black Boxes 2000 CRC Press Phasing of Excitations Testing a Gray or White Box Overtesting in Multitest Sequences10.4Product Qualication Testing10.4.1Distribution QualicationDrive-Signal GenerationDistribution SpectraTest Procedures10.4.2Seismic QualicationStages of Seismic Qualication10.4.3Test PreliminariesSingle-Frequency TestingMultifrequency Testing10.4.4Generation of RRS SpecicationsProblems Chapter 11Experimental Modal Analysis 11.1Frequency-Domain Formulation 11.1.1Transfer Function Matrix11.1.2Principle of ReciprocityExample 11.111.2Experimental Model Development11.2.1Extraction of the Time-Domain Model11.3Curve-Fitting of Transfer Functions 11.3.1Problem Identication11.3.2Single-Degree-of-Freedom and Multi-Degree-of-Freedom Techniques11.3.3Single-Degree-of-Freedom Parameter Extraction in the Frequency DomainCircle-Fit MethodPeak Picking Method11.3.4Multi-Degree-of-Freedom Curve Fitting Formulation of the Method11.3.5A Comment on Static Modes and Rigid Body Modes11.3.6Residue Extraction11.4Laboratory Experiments11.4.1Lumped-Parameter SystemFrequency-Domain TestTime-Domain Test11.4.2Distributed-Parameter System11.5Commercial EMA Systems11.5.1System CongurationFFT Analysis OptionsModal Analysis ComponentsProblems Chapter 12Vibration Design and Control Shock and Vibration12.1Specication of Vibration Limits12.1.1Peak Level Specication12.1.2RMS Value Specication 2000 CRC Press 12.1.3Frequency-Domain Specication12.2Vibration IsolationExample 12.1Solution12.2.1Design ConsiderationsExample 12.2Solution12.2.2Vibration Isolation of Flexible Systems12.3Balancing of Rotating Machinery12.3.1Static BalancingBalancing Approach12.3.2Complex Number/Vector ApproachExample 12.3Solution12.3.3Dynamic (Two-Plane) BalancingExample 12.4Solution12.3.4Experimental Procedure of Balancing12.4Balancing of Reciprocating Machines12.4.1Single-Cylinder Engine12.4.2Balancing the Inertia Load of the Piston12.4.3Multicylinder EnginesTwo-Cylinder EngineSix-Cylinder EngineExample 12.5Solution12.4.4Combustion/Pressure Load12.5Whirling of Shafts12.5.1Equations of Motion 12.5.2Steady-State Whirling Example 12.6Solution12.5.3Self-Excited Vibrations12.6Design Through Modal Testing 12.6.1Component ModicationExample 12.7Solution12.6.2Substructuring12.7Passive Control of Vibration12.7.1Undamped Vibration AbsorberExample 12.8Solution12.7.2Damped Vibration AbsorberOptimal Absorber DesignExample 12.9Solution12.7.3Vibration Dampers12.8Active Control of Vibration12.8.1Active Control System 12.8.2Control TechniquesState-Space Models 2000 CRC Press Example 12.10SolutionPosition and Velocity FeedbackLinear Quadratic Regulator (LQR) ControlModal Control12.8.3Active Control of Saw Blade Vibration12.9Control of Beam Vibrations12.9.1State-Space Model of Beam Dynamics12.9.2Control Problem12.9.3Use of Linear DampersDesign ExampleProblems Appendix ADynamic Models and Analogies A.1Model Development A.2AnalogiesA.3Mechanical ElementsA.3.1Mass (Inertia) ElementA.3.2Spring (Stiffness) ElementA.4Electrical ElementsA.4.1Capacitor ElementA.4.2Inductor ElementA.5Thermal ElementsA.5.1Thermal CapacitorA.5.2Thermal ResistanceA.6Fluid ElementsA.6.1Fluid CapacitorA.6.2Fluid InertorA.6.3Fluid ResistanceA.6.4Natural OscillationsA.7State-Space ModelsA.7.1LinearizationA.7.2Time ResponseA.7.3Some Formal DenitionsA.7.4Illustrative ExampleA.7.5Causality and Physical Realizability Appendix BNewtonian and Lagrangian Mechanics B.1Vector KinematicsB.1.1Eulers TheoremImportant CorollaryProofB.1.2Angular Velocity and Velocity at a Point of a Rigid BodyTheorem ProofB.1.3Rates of Unit Vectors Along Axes of Rotating FramesGeneral ResultCartesian CoordinatesPolar Coordinates (2-D) 2000 CRC Press Spherical Polar CoordinatesTangential-Normal (Intrinsive) Coordinates (2-D)B.1.4Acceleration Expressed in Rotating FramesSpherical Polar CoordinatesTangential-Normal Coordinates (2-D)B.2Newtonian (Vector) MechanicsB.2.1Frames of Reference Rotating at Angular Velocity B.2.2Newtons Second Law for a Particle of Mass m B.2.3Second Law for a System of Particles Rigidly or Flexibly ConnectedB.2.4Rigid Body Dynamics Inertia Matrix and Angular MomentumB.2.5Manipulation of Inertia MatrixParallel Axis Theorem Translational Transformation of [ I ] Rotational Transformation of [ I ] Principal Directions (Eigenvalue Problem)Mohrs CircleB.2.6Eulers Equations (for a Rigid Body Rotating at ) B.2.7Eulers AnglesB.3Lagrangian MechanicsB.3.1Kinetic Energy and Kinetic CoenergyB.3.2Work and Potential EnergyExamplesB.3.3Holonomic Systems, Generalized Coordinates, and Degrees of FreedomB.3.4Hamiltons PrincipleB.3.5Lagranges EquationsExampleGeneralized CoordinatesGeneralized Nonconservative ForcesLagrangian Lagranges Equations Appendix CReview of Linear Algebra C.1Vectors and MatricesC.2Vector-Matrix AlgebraC.2.1Matrix Addition and Subtraction C.2.2Null MatrixC.2.3Matrix MultiplicationC.2.4Identity MatrixC.3Matrix InverseC.3.1Matrix TransposeC.3.2Trace of a MatrixC.3.3Determinant of a Matrix C.3.4Adjoint of a MatrixC.3.5Inverse of a MatrixC.4Vector SpacesC.4.1Field ( ) C.4.2Vector Space ( )Properties Special CaseC.4.3Subspace of 2000 CRC Press C.4.4Linear DependenceC.4.5Basis and Dimension of a Vector SpaceC.4.6Inner ProductC.4.7NormPropertiesC.4.8Gram-Schmidt OrthogonalizationC.4.9Modied Gram-Schmidt ProcedureC.5DeterminantsC.5.1Properties of Determinant of a MatrixC.5.2Rank of a MatrixC.6System of Linear EquationsReferences Appendix DDigital Fourier Analysis and FFT D.1Unication of the Three Fourier Transform TypesD.1.1Relationship Between DFT and FITD.1.2Relationship Between DFT and FSED.2Fast Fourier Transform (FFT)D.2.1Development of the Radix-Two FFT AlgorithmD.2.2The Radix-Two FFT Procedure D.2.3Illustrative Example D.3Discrete Correlation and Convolution D.3.1Discrete CorrelationDiscrete Correlation Theorem Discrete Convolution TheoremD.4Digital Fourier Analysis ProceduresD.4.1Fourier Transform Using DFTD.4.2Inverse DFT Using DFTD.4.3Simultaneous DFT of Two Real Data RecordsD.4.4Reduction of Computation Time for a Real Data RecordD.4.5Convolution of Finite Duration Signals Using DFTWraparound ErrorData-Record Sectioning in Convolution Appendix EReliability Considerations for Multicomponent Units E.1Failure AnalysisE.1.1ReliabilityE.1.2UnreliabilityE.1.3InclusionExclusion FormulaExampleE.2Bayes TheoremE.2.1Product Rule for Independent EventsE.2.2Failure RateE.2.3Product Rule for ReliabilityAnswers to Numerical Problemsde Silva, Clarence W. Vibration EngineeringVibration: Fundamentals and PracticeClarence W. de SilvaBoca Raton: CRC Press LLC, 2000 1 2000 CRC Press Vibration Engineering Vibration is a repetitive, periodic, or oscillatory response of a mechanical system. The rate of thevibrationcyclesistermedfrequency.Repetitivemotionsthataresomewhatcleanandregular,and that occur at relatively low frequencies, are commonly called oscillations, while any repetitivemotion, even at high frequencies, with low amplitudes, and having irregular and random behaviorfallsintothegeneralclassofvibration.Nevertheless,thetermsvibrationandoscillationareoften used interchangeably, as is done in this book.Vibrations can naturally occur in an engineering system and may be representative of its freeandnaturaldynamicbehavior. Also,vibrationsmaybeforcedontoasystemthroughsomeformof excitation. The excitation forces may be either generated internally within the dynamic system,or transmitted to the system through an external source. When the frequency of the forcing excitationcoincides with that of the natural motion, the system will respond more vigorously with increasedamplitude. This condition is known as resonance, and the associated frequency is called the resonantfrequency. There are good vibrations, which serve a useful purpose. Also, there are bad vibra-tions, which can be unpleasant or harmful. For many engineering systems, operation at resonancewouldbeundesirableandcouldbedestructive.Suppressionoreliminationofbadvibrationsandgeneration of desired forms and levels of good vibration are general goals of vibration engineering.This book deals with1. Analysis2. Observation3. Modicationofvibrationinengineeringsystems.Applicationsofvibrationarefoundinmanybranchesofengineeringsuchasaeronauticalandaerospace,civil,manufacturing,mechanical,andevenelec-trical. Usually, an analytical or computer model is needed to analyze the vibration in an engineeringsystem. Models are also useful in the process of design and development of an engineering systemfor good performance with respect to vibrations. Vibration monitoring, testing, and experimentationare important as well in the design, implementation, maintenance, and repair of engineering systems.All these are important topics of study in the eld of vibration engineering, and the book will coverpertinent1. Theory and modeling2. Analysis3. Design4. Experimentation5. ControlIn particular, practical applications and design considerations related to modifying the vibrationalbehavior of mechanical devices and structures will be studied. This knowledge will be useful in thepractice of vibration regardless of the application area or the branch of engineering; for example, intheanalysis,design,construction,operation,andmaintenanceofcomplexstructuressuchastheSpace Shuttle and the International Space Station. Note in Figure 1.1 that long and exible compo-nents,whichwouldbepronetocomplexmodesofvibration,arepresent.Thestructuraldesignshould take this into consideration. Also, functional and servicing devices such as robotic manipu- 2000 CRC Press lators (e.g., Canadarm) can give rise to vibration interactions that need to be controlled for accurateperformance.Theapproachusedinthebookistointroducepracticalapplicationsofvibrationintheverybeginning,alongwithexperimentaltechniques,andthenintegratetheseapplicationsanddesign considerations into fundamentals and analytical methods throughout the text. 1.1 STUDY OF VIBRATION Natural,freevibrationisamanifestationoftheoscillatorybehaviorinmechanicalsystems,asaresult of repetitive interchange of kinetic and potential energies among components in the system.Such natural oscillatory response is not limited, however, to purely mechanical systems, and is foundinelectricalanduidsystemsaswell,againduetoarepetitiveexchangeoftwotypesofenergyamongsystemcomponents.But,purelythermalsystemsdonotundergofree,naturaloscillations,primarilybecauseoftheabsenceoftwoformsofreversibleenergy.Evenasystemthatcanholdtwo reversible forms of energy may not necessarily display free, natural oscillations. The reason forthis would be the strong presence of an energy dissipation mechanism that could use up the initialenergy of the system before completing a single oscillation cycle (energy interchange). Such dissi-pation is provided by damping or friction in mechanical systems, and resistance in electrical systems.Any engineering system (even a purely thermal one) is able to undergo forced oscillations, regardlessof the degree of energy dissipation. In this case, the energy necessary to sustain the oscillations willcome from the excitation source, and will be continuously replenished.Proper design and control are crucial in maintaining a high performance level and productionefciency, and prolonging the useful life of machinery, structures, and industrial processes. Beforedesigning or controlling an engineering system for good vibratory performance, it is important tounderstand, represent (model), and analyze the vibratory characteristics of the system. This can be FIGURE 1.1 The U.S. Space Shuttle and the International Space Station with the Canadarm. (Courtesy ofNASA Langley Research Center, Hampton, VA. With permission.) 2000 CRC Press accomplished through purely analytical means, computer analysis of analytical models, testing andanalysisoftestdata,oracombinationoftheseapproaches. Asanexample,aschematicdiagramof an innovative elevated guideway transit system is shown in Figure 1.2(a). This is an automatedtransitsystemthatisoperatedwithoutdrivers.Theridequality,whichdependsonthevibratorymotion of the vehicle, can be analyzed using an appropriate model. Usually, the dynamics (inertia,exibility,andenergydissipation)oftheguideway,aswellasthevehicle,mustbeincorporatedinto such a model. A simplied model is shown in Figure 1.2(b). It follows that modeling, analysis,testing, design, and control are all important aspects of study in mechanical vibration.The analysis of a vibrating system can be done either in the time domain or in the frequencydomain.Inthetimedomain,theindependentvariableofavibrationsignalistime.Inthiscase,thesystemitselfcanbemodeledasasetofdifferentialequationswithrespecttotime. Amodelof a vibrating system can be formulated by applying either force-momentum rate relations (New-tons second law) or the concepts of kinetic and potential energies. Both Newtonian (force-motion)and Lagrangian (energy) approaches will be utilized in this book.Inthefrequencydomain,theindependentvariableofavibrationsignalisfrequency.Inthiscase, the system can be modeled by input-output transfer functions which are algebraic, rather thandifferential,models.Transferfunctionrepresentationssuchasmechanicalimpedance,mobility,receptance, and transmissibility can be conveniently analyzed in the frequency domain, and effec-tively used in vibration design and evaluation. Modeling and vibration-signal analysis in both timeand frequency domains will be studied in this book. The two domains are connected by the Fouriertransformation,whichcanbetreatedasaspecialcaseoftheLaplacetransformation.Thesetransform techniques will be studied, rst in the purely analytical and analog measurement situationof continuous time. In practice, however, digital electronics and computers are commonly used insignalanalysis,sensing,andcontrol.Inthissituation,oneneedstoemployconceptsofdiscretetime, sampled data, and digital signal analysis in the time domain. Correspondingly, then, conceptsof discrete or digital Fourier transformation and techniques of fast Fourier transform (FFT) will beapplicable in the frequency domain. These concepts and techniques are also studied in this book.An engineering system, when given an initial disturbance and allowed to execute free vibrationswithoutasubsequentforcingexcitation,willtendtodosoataparticularpreferredfrequencyandmaintainingaparticularpreferredgeometricshape.Thisfrequencyistermedanaturalfrequencyofthesystem,andthecorrespondingshape(ormotionratio)ofthemovingpartsofthe system is termed a mode shape. Any arbitrary motion of a vibrating system can be representedintermsofitsnaturalfrequenciesandmodeshapes.Thesubjectofmodalanalysisprimarilyconcernsdeterminationofnaturalfrequenciesandmodeshapesofadynamicsystem.Oncethe FIGURE 1.2(a) An elevated guideway transit system. 2000 CRC Press modes are determined, they can be used in understanding the dynamic nature of the systems, andalsoindesignandcontrol.Modalanalysisisextremelyimportantinvibrationengineering,andwillbestudiedinthisbook.Naturalfrequenciesandmodeshapesofavibratingsystemcanbedeterminedexperimentallythroughproceduresofmodaltesting.Infact,adynamicmodel(an experimental model)ofthesystemcanbedeterminedinthismanner.Thesubjectofmodaltesting,experimentalmodeling(ormodelidentication),andassociatedanalysisanddesignisknown as experimental modal analysis . This subject will also be treated in this book.Energydissipation(ordamping)ispresentinanymechanicalsystem.Italtersthedynamicresponseofthesystem,andhasdesirableeffectssuchasstability,vibrationsuppression,powertransmission (e.g., in friction drives), and control. Also, it has obvious undesirable effects such asenergy wastage, reduction of the process efciency, wear and tear, noise, and heat generation. For FIGURE 1.2(b) A model for determining the ride quality of the elevated guideway transit system. 2000 CRC Press these reasons, damping is an important topic of study in the area of vibration, and will be coveredin this book. In general, energy dissipation is a nonlinear phenomenon. But, in view of well-knowndifculties of analyzing nonlinear behavior, and because an equivalent representation of the overallenergydissipationisoftenadequateinvibrationanalysis,linearmodelsareprimarilyusedtorepresent damping in the analyses herein. However, nonlinear representations are discussed as well;and how equivalent linear models can be determined for nonlinear damping are described.Properties such as mass (inertia), exibility (spring-like effect), and damping (energy dissipa-tion) are continuously distributed throughout practical mechanical devices and structures to a largeextent.Thisisthecasewithdistributedcomponentssuchascables,shafts,beams,membranes,plates,shells,andvarioussolids,aswellasstructuresmadeofsuchcomponents.Representation(i.e., modeling) of these distributed-parameter (or continuous) vibrating systems will require inde-pendentvariablesinspace(spatialcoordinates)inadditiontotime;thesemodelsarepartialdifferential equations in time and space. The analysis of distributed-parameter models will requirecomplexproceduresandspecialtools.Thisbookstudiesvibrationanalysis,particularlymodalanalysis, of several types of continuous components, as well as how approximate lumped-parametermodelscanbedevelopedforcontinuoussystems,usingproceduressuchasmodalanalysisandenergy equivalence.Vibration testing is useful in a variety of stages in the development and utilization of a product.In the design and development stage, vibration testing can be used to design, develop, and verifythe performance of individual components of a complex system before the overall system is built(assembled) and evaluated. In the production stage, vibration testing can be used for screening ofselectedbatchesofproductsforqualitycontrol.Anotheruseofvibrationtestingisinproductqualication.Here,aproductofgoodqualityistestedtoseewhetheritcanwithstandvariousdynamicenvironmentsthatitmayencounterinaspecializedapplication. Anexampleofalarge-scale shaker used for vibration testing of civil engineering structures is shown in Figure 1.3. Thesubject of vibration testing is addressed in some detail in this book.Design is a subject of paramount signicance in the practice of vibration. In particular, mechan-icalandstructuraldesignforacceptablevibrationcharacteristicswillbeimportant.Modicationof existing components and integration of new components and devices, such as vibration dampers,isolators, inertia blocks, and dynamic absorbers, can be incorporated into these practices. Further-more,eliminationofsourcesofvibrationforexample,throughcomponentalignmentandbalancing of rotating devices is a common practice. Both passive and active techniques are usedinvibrationcontrol.Inpassivecontrol,actuatorsthatrequireexternalpowersourcesarenotemployed.Inactivecontrol,vibrationiscontrolledbymeansofactuators(whichneedpower)tocounteract vibration forces. Monitoring, testing, and control of vibration will require devices suchas sensors and transducers, signal conditioning and modication hardware (e.g., lters, ampliers,modulators, demodulators, analog-digital conversion means), and actuators (e.g., vibration excitersorshakers).Theunderlyingsubjectofvibrationinstrumentationwillbecoveredinthisbook.Particularly,withinthetopicofsignalconditioning,bothhardwareandsoftware(numerical)techniques will be presented. 1.2 APPLICATION AREAS The science and engineering of vibration involve two broad categories of applications:1. Elimination or suppression of undesirable vibrations2. Generation of the necessary forms and quantities of useful vibrationsUndesirable and harmful types of vibration include structural motions generated due to earthquakes,dynamic interactions between vehicles and bridges or guideways, noise generated by constructionequipment, vibration transmitted from machinery to its supporting structures or environment, and 2000 CRC Press damage, malfunction, and failure due to dynamic loading, unacceptable motions, and fatigue causedbyvibration.Asanexample,dynamicinteractionsbetweenanautomatedtransitvehicleandabridge (see Figure 1.4) can cause structural problems as well as degradation in ride quality. Rigorousanalysis and design are needed, particularly with regard to vibration, in the development of theseground transit systems. Lowering the levels of vibration will result in reduced noise and improvedworkenvironment,maintenanceofahighperformancelevelandproductionefciency,reductionin user/operator discomfort, and prolonging the useful life of industrial machinery. Desirable typesof vibration include those generated by musical instruments, devices used in physical therapy andmedicalapplications,vibratorsusedinindustrialmixers,partfeedersandsorters,andvibratorymaterialremoverssuchasdrillsandpolishers(nishers).Forexample,productalignmentfor FIGURE1.3 Amulti-degree-of-freedomhydraulicshakerusedintestingcivilengineeringstructures.(CourtesyofProf.C.E. Ventura,UniversityofBritishColumbia. Withpermission.) 2000 CRC Press FIGURE 1.4 The SkyTrain in Vancouver, Canada, a modern automated transit system. (Photo by Mark VanManen, courtesy of BC Transit. With permission.) FIGURE 1.5 An alignment shaker. (Key Technology, Inc., of Walla Walla, WA. With permission.) 2000 CRC Press industrial processing or grading can be carried out by means of vibratory conveyors or shakers, asshown in Figure 1.5.Conceptsofvibrationhavebeenusedformanycenturiesinpracticalapplications.Recentadvances of vibration are quite signicant, and the corresponding applications are numerous. Manyof the recent developments in the eld of vibration were motivated perhaps for two primary reasons:1. The speeds of operation of machinery have doubled over the past 50 years and, conse-quently, the vibration loads generated due to rotational excitations and unbalances wouldhave quadrupled if proper actions of design and control were not taken.2. Mass, energy, and efciency considerations have resulted in lightweight, optimal designsof machinery and structures consisting of thin members with high strength. Associatedstructuralexibilityhasmadetherigid-structureassumptionunsatisfactory,andgivenrise to the need for sophisticated procedures of analysis and design that govern distrib-uted-parameter exible structures.Onecanthenvisualizeseveralpracticalapplicationswheremodeling,analysis,design,control,monitoring, and testing, related to vibration are important.Arangeofapplicationsofvibrationcanbefoundinvariousbranchesofengineering:partic-ularlycivil,mechanical,aeronauticalandaerospace,andproductionandmanufacturing.Modalanalysis and design of exible civil engineering structures such as bridges, guideways, tall buildings,and chimneys directly incorporate theory and practice of vibration. A ne example of an elongatedbuilding where vibration analysis and design are crucial is the Jefferson Memorial Arch, shown inFigure 1.6.In the area of ground transportation, vehicles are designed by incorporating vibration engineer-ing, not only to ensure structural integrity and functional operability, but also to achieve requiredlevels of ride quality and comfort. Specications such as the one shown in Figure 1.7, where limitson root-mean-square (rms) levels of vibration (expressed in units of acceleration due to gravity, g)for different frequencies of excitation (expressed in cycles per second, or hertz, or Hz) and differenttripdurations,areusedtospecifyridequalityrequirementsinthedesignoftransitsystems.Inparticular,thedesignofsuspensionsystems,bothactiveandpassive,fallswithintheeldofvibrationengineering.Figure1.8showsatestsetupusedinthedevelopmentofanautomotivesuspensionsystem.Intheareaofairtransportation,mechanicalandstructuralcomponentsofaircraftaredesignedforgoodvibrationperformance.Forexample,properdesignandbalancingcanreducehelicoptervibrationscausedbyimbalanceintheirrotors.Vibrationsinshipscanbesuppressed through structural design, propeller and rudder design, and control. Balancing of internalcombustion engines is carried out using principles of design for vibration suppression.Oscillation of transmission lines of electric power and communication signals (e.g., overheadtelephone lines) can result in faults, service interruptions, and sometimes major structural damage.Stabilization of transmission lines involves direct application of the principles of vibration in cablesand the design of vibration dampers and absorbers.Intheareaofproductionandmanufacturingengineering,mechanicalvibrationhasdirectimplicationsofproductqualityandprocessefciency.Machinetoolvibrationsareknowntonotonlydegradethedimensionalaccuracyandthenishofaproduct,butalsowillcausefastwearandtearandbreakageoftools.Millingmachines,lathes,drills,forgingmachines,andextruders,for example, should be designed for achieving low vibration levels. In addition to reducing the toollife, vibration will result in other mechanical problems in production machinery, and will requiremore frequent maintenance. Associated downtime (production loss) and cost can be quite signicant.Also,asnotedbefore,vibrationsinproductionmachinerywillgeneratenoiseproblemsandalsowillbetransmittedtootheroperationsthroughsupportstructures,therebyinterferingwiththeirperformanceaswell.Ingeneral,vibrationcandegradeperformanceandproductionefciencyof 2000 CRC Press FIGURE 1.6 Jefferson Memorial Arch in St. Louis, MO. FIGURE 1.7 A typical specification of vehicle ride quality for a specified trip duration. 2000 CRC Press manufacturing processes. Proper vibration isolation (e.g., mountings) will be needed to reduce thesetransmissibility problems.Heavy machinery in the construction industry (e.g., cranes, excavators, pile drivers, impactingand compacting machinery, and bulldozers) rely on structural integrity, reliability, and safety. Theirdesign must be based on sound principles of engineering. Although the dynamic loading in thesemachines is generally random, it is also quite repetitive from the point of view of both the excitationgeneratedbytheengineandthefunctionaloperationofthetasksperformed.Designbasedonvibration and fatigue is an important requirement for these machines: for maintaining satisfactoryperformance, prolonging the useful life, and reducing the cost and frequency of maintenance. 1.3 HISTORY OF VIBRATION The origins of the theory of vibration can be traced back to the design and development of musicalinstruments(goodvibration).Itisknownthatdrums,utes,andstringedinstrumentsexistedinChina and India for several millennia B.C. Also, ancient Egyptians and Greeks explored sound andvibration from both practical and analytical points of view. For example, while Egyptians had knownof a harp since at least 3000 B.C., the Greek philosopher, mathematician, and musician Pythagoras(ofthePythagorastheoremfame)wholivedduring582to502B.C.,experimentedonsoundsgenerated by blacksmiths and related them to music and physics. The Chinese developed a mechan-ical seismograph (an instrument to detect and record earthquake vibrations) in the 2nd century A.D.Thefoundationofthemodern-daytheoryofvibrationwasprobablylaidbyscientistsandmathematicianssuchasRobertHooke(16351703)oftheHookeslawfame,whoexperimentedon the vibration of strings; Sir Isaac Newton (16421727), who gave us calculus and the laws ofmotionforanalyzingvibrations;DanielBernoulli(17001782)andLeonardEuler(17071783),who studied beam vibrations (Bernoulli-Euler beam) and also explored dynamics and uid mechan-ics; Joseph Lagrange (17361813), who studied vibration of strings and also explored the energyapproachtoformulatingequationsofdynamics;CharlesCoulomb(17361806),whostudied FIGURE 1.8 Cone suspension system installed on a Volvo 480ES automobile for testing. (Copyright Mechan-ical Engineering magazine; the American Society of Mechanical Engineers International. With permission.) 2000 CRC Press torsionalvibrationsandfriction;JosephFourier(17681830),whodevelopedthetheoryoffre-quencyanalysisofsignals;andSimeon-DennisPoisson(17811840),whoanalyzedvibrationofmembranesandalsoanalyzedelasticity(Poissonsratio). Asaresultoftheindustrialrevolutionand associated developments of steam turbines and other rotating machinery, an urgent need wasfeltfordevelopmentsintheanalysis,design,measurement,andcontrolofvibration.Motivationformanyaspectsoftheexistingtechniquesofvibrationcanbetracedbacktorelatedactivitiessince the industrial revolution.Much credit should go to scientists and engineers of more recent history, as well. Among thenotablecontributorsareRankine(18201872),whostudiedcriticalspeedsofshafts;Kirchhoff(18241887), who analyzed vibration of plates; Rayleigh (18421919), who made contributions tothe theory of sound and vibration and developed computational techniques for determining naturalvibrations;deLaval(18451913),whostudiedthebalancingproblemofrotatingdisks;Poincar(18541912), who analyzed nonlinear vibrations; and Stodola (18591943), who studied vibrationsof rotors, bearings, and continuous systems. Distinguished engineers who made signicant contri-butionstothepublishedliteratureandalsotothepracticeofvibrationinclude Timoshenko,DenHartog, Clough, and Crandall. 1.4 ORGANIZATION OF THE BOOK This book provides the background and techniques for modeling, analysis, design, instrumentationand monitoring, modication, and control of vibration in engineering systems. This knowledge willbe useful in the practice of vibration, regardless of the application area or the branch of engineering.A uniform and coherent treatment of the subject is given by introducing practical applications ofvibration in the very beginning of the book, along with experimental techniques and instrumentation,and then integrating these applications, design and experimental techniques, and control consider-ations into fundamentals and analytical methods throughout the text.The book consists of 12 chapters and 5 appendices. The chapters have summary boxes for easyreferenceandrecollection.Manyworkedexamplesandproblems(over300)areincluded.Somebackground material is presented in the appendices, rather than in the main text, in order to avoidinterference with the continuity of the subject matter.The present introductory chapter provides some background material on the subject of vibrationengineering,andsetsthecourseforthestudy.Itgivestheobjectivesandmotivationofthestudyand indicates key application areas. A brief history of the eld of vibration is given as well.Chapter 2 provides the basics of time response analysis of vibrating systems. Both undampedanddampedsystemsarestudied.Also,analysisofbothfree(unforced)andforcedresponseisgiven.Theconceptofastatevariableisintroduced.Someanalogiesofpurelymechanicalandstructuralvibratingsystemsspecically,translatory,exural,andtorsional;toelectricalanduidoscillatorysystemsareintroduced.Anenergy-basedapproximationofadistributed-parametersystem(aheavyspring)toalumped-parametersystemisdevelopedindetail.Thelogarithmicdecrementmethodofdampingmeasurementisdeveloped. Althoughthechapterpri-marily considers single-degree-of-freedom systems, the underlying concepts can be easily extendedto multi-degree-of-freedom systems.Chapter3concernsfrequencyresponseanalysisofvibratingsystems.First,theresponseofavibratingsystemtoharmonic(sinusoidal)excitationforces(inputs)isanalyzed,primarilyusingthetime-domain concepts developed in Chapter 2. Then, its interpretation in the frequency domain is given.The link between the time domain and the frequency domain, through Fourier transform, is highlighted.Inparticular,FouriertransformisinterpretedasaspecialcaseofLaplacetransform.Theresponseanalysis using transform techniques is presented, along with the associated basic ideas of convolutionintegral,andtheimpulseresponsefunctionwhoseLaplacetransformisthetransferfunction,andFourier transform is the frequency response function. The half-power bandwidth approach of measuringdamping is given. Special types of frequency transfer functions specically, force transmissibility, 2000 CRC Press motiontransmissibility,andreceptancearestudiedandtheircomplementaryrelationshipsarehighlighted. Their use in the practice of vibration, particularly in vibration isolation, is discussed.Chapter 4 presents the fundamentals of analyzing vibration signals. First, the idea of frequencyspectrumofatimesignalisgiven.Varioustypesandclassicationsofsignalsencounteredinvibration engineering are discussed. The technique of Fourier analysis is formally introduced andlinked to the concepts presented in Chapter 3. The idea of random signals is introduced, and usefulanalytical techniques for these signals are presented. Practical issues pertaining to vibration signalanalysisareraised.Computationaltechniquesofsignalanalysisaregivenandvarioussourcesoferror, such as aliasing and truncation, are indicated; and ways of improving the accuracy of digitalsignal analysis are given.Chapter5dealswiththemodalanalysisoflumped-parametervibratingsystems.Thebasicassumptionmadeisthatdistributedeffectsofinertiaandexibilityinavibratingsystemcanberepresentedbyaninterconnectedsetoflumpedinertiaandspringelements. Thetotalnumberofpossibleindependent,incrementalmotionsoftheseinertiaelementsisthenumberofdegreesoffreedom of the system. For holonomic systems, this is also equal to the total number of independentcoordinatesneededtorepresentanarbitrarycongurationofthesystem;butfornon-holonomicsystems,therequirednumberofcoordinateswillbelarger.Forthisreason,theconceptsofholonomicandnon-holonomicsystemsandthecorrespondingtypesofconstraintsarediscussed.The representation of a general lumped-parameter vibrating system by a differential equation modelisgiven,andmethodsofobtainingsuchamodelarediscussed.ApartfromtheNewtonianandLagrangianapproaches,theinuencecoefcientapproachisgivenfordeterminingthemassandstiffnessmatrices.Theconceptsofnaturalfrequenciesandmodeshapesarediscussed,andtheprocedurefordeterminingthesecharacteristicquantities,throughmodalanalysis,isdeveloped.The orthogonality property of natural modes is derived. The ideas of static modes and rigid bodymodes are explored, and the causes of these conditions will be indicated. In addition to the standardformulationofthemodalanalysisproblem,twoothermodalformulationsaredeveloped.Theanalysisoftheproblemofforcedvibration,usingmodalanalysis,isgiven.Dampedlumped-parameter vibrating systems are studied from the point of view of modal analysis. The conditionsof existence of real modes for damped systems are explored, with specic reference to proportionaldamping. The state-space approach of representing and analyzing a vibrating system is presented.Practical problems of modal analysis are presented.Chapter 6 studies distributed-parameter vibrating systems such as cables, rods, shafts, beams,membranes, and plates. Practical examples of associated vibration problems are indicated. Vibrationofcontinuoussystemsistreatedasageneralizationoflumped-parametersystems,discussedinChapter 5. In particular, the modal analysis of continuous systems is addressed in detail. The issueoforthogonalityofmodesisstudied. Theinuenceofsystemboundaryconditionsonthemodalproblemingeneralandtheorthogonalityinparticularisdiscussed,withspecialemphasisoninertialboundaryconditions(e.g.,continuoussystemswithlumpedmassesattheboundaries).The inuence of damping on the modal analysis problem is discussed. The analysis of response toa forcing excitation is performed.Chapter7exclusivelydealswiththeproblemofenergydissipationordampinginvibratingsystems. Various types of damping present in mechanical and structural systems are discussed, withpracticalexamples,andparticularemphasisoninterfacedamping.Methodsofrepresentationormodeling of damping in the analysis of vibrating systems are indicated. Techniques and principlesof measurement of damping are given, with examples.Chapter 8 studies instrumentation issues in the practice of vibration. Applications range frommonitoringandfaultdiagnosisofindustrialprocesses,toproducttestingforqualityassessmentand qualication, experimental modal analysis for developing experimental models and for design-ingofvibratingsystems,andcontrolofvibration.Instrumentationtypes,basicsofoperation,industrial practices pertaining to vibration exciters, control systems, motion sensors and transducers,torque and force sensors, and other types of transducers are addressed. Performance specication 2000 CRC Press ofaninstrumentedsystemisdiscussed.Issuesandimplicationsofcomponentinterconnectioninthe practical use of instrumentation are addressed.Chapter 9 addresses signal conditioning and modication for practical vibration systems. TheseconsiderationsarecloselyrelatedtothesubjectofinstrumentationdiscussedinChapter8andsignal analysis discussed in Chapter 4. Particular emphasis is given to commercial instruments andhardwarethatareusefulinmonitoring,analyzing,andcontrolofvibration.Specicdevicesconsidered include ampliers, analog lters, modulators and demodulators, analog-to-digital con-verters, digital-to-analog converters, bridge circuits, linearizing devices, and other types of signalmodicationcircuitry.Commercialspectrumanalyzersanddigitaloscilloscopescommonlyemployed in the practice of vibration are discussed as well.Chapter10dealswithvibrationtesting. Thisisapracticaltopicthatisdirectlyapplicabletoproductdesignanddevelopment,experimentalmodeling,qualityassessmentandcontrol,andproductqualication. Variousmethodsofrepresentingavibrationenvironmentinatestprogramare discussed. Procedures that need to be followed prior to testing an object (i.e., pre-test procedures)are given. Available testing procedures are presented, with a discussion of appropriateness, advan-tages,anddisadvantagesofvarioustestprocedures.Thetopicofproductqualicationtestingisaddressed in some length.Chapter11studiesexperimentalmodalanalysis,whichisdirectlyrelatedtovibrationtesting(Chapter 10), experimental modeling, and design. It draws from the analytical procedures presentedin previous chapters, particularly Chapters 5 and 6. Frequency domain formulation of the problemisgiven.Theprocedureofdevelopingacompleteexperimentalmodelofavibratingsystemispresented. Procedures of curve tting of frequency transfer functions, which are essential in modelparameter extraction, are discussed. Several laboratory experiments in the area of vibration testing(modal testing) are described, giving details of the applicable instrumentation. Features and capa-bilities of several commercially available experimental modal analysis systems are described, anda comparative evaluation is given.Chapter12addressespracticalandanalyticalissuesofvibrationdesignandcontrol.Theemphasis here is in the ways of designing, modifying, or controlling a system for good performancewithregardtovibration.Waysofspecicationofvibrationlimitsforproperperformanceofanengineeringsystemarediscussed.Techniquesandpracticalconsiderationsofvibrationisolationaredescribed,withanemphasisontheuseoftransmissibilityconceptsdevelopedinChapter3.Staticanddynamicbalancingofrotatingmachineryisstudiedbypresentingbothanalyticalandpracticalprocedures.Therelatedtopicofbalancingmulti-cylinderreciprocatingmachinesisaddressed in some detail. The topic of whirling of rotating components and shafts is studied. Thesubjectofdesignthroughmodaltesting,whichisdirectlyrelatedtothematerialinchapters10and11,isdiscussed.Bothpassivecontrolandactivecontrolofvibrationarestudied,givingprocedures and practical examples.Thebackgroundmaterialthatisnotgiveninthemainbodyofthetext,butisusefulincomprehendingtheunderlyingprocedures,isgivenintheappendices.Referenceismadeinthemaintexttotheseappendices,forfurtherreading. Appendix Adealswithdynamicmodelsandanalogies. Main steps of developing analytical models for dynamic systems are indicated. Analogiesbetween mechanical, electrical, uid, and thermal systems are presented, with particular emphasisonthecauseoffreenaturaloscillations.Developmentprocedureofstate-spacemodelsforthesesystemsisindicated. AppendixBsummarizesNewtonianandLagrangianapproachestowritingequations of motion for dynamic systems. Appendix C reviews the basics of linear algebra. Vector-matrixtechniquesthatareusefulinvibrationanalysisandpracticearesummarized. AppendixDfurther explores the topic of digital Fourier analysis, with a special emphasis on the computationalprocedure of fast Fourier transform (FFT). As the background theory, the concepts of Fourier series,Fourier integral transform, and discrete Fourier transform are discussed and integrated, which leadsthedigitalcomputationofthesequantitiesusingFFT.PracticalproceduresandapplicationsofdigitalFourieranalysisaregiven. AppendixEaddressesreliabilityconsiderationsformulticom- 2000 CRC Press ponent devices. These considerations have a direct relationship to vibration monitoring and testing,failure diagnosis, product qualication, and design optimization. PROBLEMS 1.1 Explain why mechanical vibration is an important area of study for engineers. Mechanicalvibrations are known to have harmful effects as well as useful ones. Briey describe vepractical examples of good vibrations and also ve practical examples of bad vibrations. 1.2 Under some conditions it may be necessary to modify or redesign a machine with respectto its performance under vibrations. What are possible reasons for this? What are someofthemodicationsthatcanbecarriedoutonamachineinordertosuppressitsvibrations? 1.3 Ontheonehand,modernmachinesaredesignedwithsophisticatedproceduresandcomputer tools, and should perform better than the older designs, with respect to mechan-ical vibration. On the other hand, modern machines have to operate under more stringentspecicationsandrequirementsinasomewhatoptimalfashion.Ingeneral,designforsatisfactoryperformanceundervibrationtakesanincreasedimportanceformodernmachinery. Indicate some reasons for this. 1.4 Dynamic modeling both analytical and experimental (e.g., experimental modal anal-ysis) is quite important in the design and development of a product, for good perfor-mancewithregardtovibration.Indicatehowadynamicmodelcanbeutilizedinthevibration design of a device. 1.5 Outlineonepracticalapplicationofmechanicalvibrationineachofthefollowingbranches of engineering:1. Civil engineering2. Aeronautical and aerospace engineering3. Mechanical engineering4. Manufacturing engineering5. Electrical engineering REFERENCES AND FURTHER READING The book has relied on many publications, directly and indirectly, in its evolution and development.Theauthorsownworkaswellasotherexcellentbookshaveprovidedawealthofknowledge.Although it is not possible or useful to list all such material, some selected publications are listed below. A UTHOR S W ORK 1. De Silva, C.W., Dynamic Testing and Seismic Qualication Practice, D.C. Heath and Co., Lexington,MA, 1983.2. De Silva C.W. and Wormley, D.N., Automated Transit Guideways: Analysis and Design, D.C. Heathand Co., Lexington, MA, 1983.3. De Silva, C.W., Control Sensors and Actuators, Prentice-Hall, Englewood Cliffs, NJ, 1989.4. De Silva, C.W., Control System Modeling, Measurements and Data Corp., Pittsburgh, PA, 1989.5. DeSilva,C.W.,Atechniquetomodelthesimplysupportedtimoshenkobeaminthedesignofmechanical vibrating systems, International Journal of Mechanical Sciences, 17, 389-393, 1975.6. Van de Vegte, J. and de Silva, C.W., Design of passive vibration controls for internally damped beamsby modal control techniques, Journal of Sound and Vibration, 45(3), 417-425, 1976.7. De Silva, C.W., Optimal estimation of the response of internally damped beams to random loads inthe presence of measurement noise, Journal of Sound and Vibration, 47(4), 485-493, 1976. 2000 CRC Press 8. De Silva, C.W., Dynamic beam model with internal damping, rotatory inertia and shear deformation, AIAA Journal , 14(5), 676-680, 1976.9. DeSilva,C.W.andWormley,D.N.,Materialoptimizationinatorsionalguidewaytransitsystem, Journal of Advanced Transportation , 13(3), 41-60, 1979.10. De Silva, C.W., Buyukozturk, O., and Wormley, D.N., Postcracking compliance of RC beams, Journalof the Structural Division, Trans. ASCE , 105(ST1), 35-51, 1979.11. DeSilva,C.W.,Seismicqualicationofelectricalequipmentusingauniaxialtest, EarthquakeEngineering and Structural Dynamics , 8, 337-348, 1980.12. DeSilva,C.W.,Loceff,F.,andVashi,K.M.,Considerationofanoptimalprocedurefortestingtheoperability of equipment under seismic disturbances, Shock and Vibration Bulletin , 50(5), 149-158, 1980.13. DeSilva,C.W.andWormley,D.N.,Torsionalanalysisofcutoutbeams, JournaloftheStructuralDivision , Trans. ASCE, 106(ST9), 1933-1946, 1980.14. DeSilva,C.W.,Analgorithmfortheoptimaldesignofpassivevibrationcontrollersforexiblesystems, Journal of Sound and Vibration , 74(4), 495-502, 1982.15. DeSilva,C.W.,Matrixeigenvalueproblemofmultiple-shakertesting, Journalofthe EngineeringMechanics Division , Trans. ASCE, 108(EM2), 457-461, 1982.16. De Silva, C.W., Selection of shaker specications in seismic qualication tests, Journal of Sound andVibration , 91(2), 21-26, 1983.17. De Silva, C.W., Shaker test-xture design, Measurements and Control , 17(6), 152-155, 1983.18. De Silva, C.W., On the modal analysis of discrete vibratory systems, International Journal of Mechan-ical Engineering Education , 12(1), 35-44, 1984.19. DeSilva,C.W.andPalusamy,S.S.,ExperimentalmodalanalysisAmodelinganddesigntool, Mechanical Engineering , ASME, 106(6), 56-65, 1984.20. De Silva, C.W., A dynamic test procedure for improving seismic qualication guidelines, Journal ofDynamic Systems, Measurement, and Control , Trans. ASME, 106(2), 143-148, 1984.21. DeSilva,C.W.,Hardwareandsoftwareselectionforexperimentalmodalanalysis, TheShockandVibration Digest , 16(8), 3-10, 1984.22. De Silva, C.W., Computer-automated failure prediction in mechanical systems under dynamic loading, The Shock and Vibration Digest , 17(8), 3-12, 1985.23. De Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control Applicationin the distribution qualication of microcomputers, The Shock and Vibration Digest , 18(9), 3-13, 1986.24. De Silva, C.W., The digital processing of acceleration measurements for modal analysis, The Shockand Vibration Digest , 18(10), 3-10, 1986.25. DeSilva,C.W.,Price,T.E.,andKanade,T., Atorquesensorfordirect-drivemanipulators, Journalof Engineering for Industry , Trans. ASME, 109(2), 122-127, 1987.26. DeSilva,C.W.,Optimalinputdesignforthedynamictestingofmechanicalsystems, JournalofDynamic Systems, Measurement, and Control , Trans. ASME, 109(2), 111-119, 1987.27. DeSilva,C.W.,Singh,M.,andZaldonis,J.,Improvementofresponsespectrumspecicationsindynamic testing, Journal of Engineering for Industry , Trans. ASME, 112(4), 384-387, 1990.28. De Silva, C.W., Schultz, M., and Dolejsi, E., Kinematic analysis and design of a continuously-variabletransmission, Mechanism and Machine Theory , 29(1), 149-167, 1994.29. Bussani,F.anddeSilva,C.W.,Useofniteelementmethodtomodelmachineprocessingofsh, Finite Element News , 5, 36-42, 1994.30. Caron,M.,Modi,V.J.,Pradhan,S.,deSilva,C.W.,andMisra,A.K.,Planardynamicsofexiblemanipulatorswithslewingdeployablelinks, JournalofGuidance,Control,andDynamics ,21(4),572-580, 1998. O THER U SEFUL P UBLICATIONS 1. Beards, C.F., Engineering Vibration Analysis with Application to Control Systems , Halsted Press, NewYork, 1996.2. Bendat,J.S.andPiersol, A.G., RandomData:AnalysisandMeasurementProcedures ,Wiley-Inter-science, New York, 1971.3. Blevins, R.D., Flow-Induced Vibration , Van Nostrand Reinhold, New York, 1977.4. Brigham, E.O., The Fast Fourier Transform , Prentice-Hall, Englewood Cliffs, NJ, 1974. 2000 CRC Press 5. Broch,J.T., MechanicalVibrationandShockMeasurements ,BruelandKjaer,Naerum,Denmark,1980.6. Buzdugan,G.,Mihaiescu,E.,andRades,M., VibrationMeasurement ,MartinusNijhoffPublishers,Dordrecht, The Netherlands, 1986.7. Crandall, S.H., Karnopp, D.C., Kurtz, E.F., and Prodmore-Brown, D.C., Dynamics of Mechanical andElectromechanical Systems , McGraw-Hill, New York, 1968.8. Den Hartog, J.P., Mechanical Vibrations , McGraw-Hill, New York, 1956.9. Dimarogonas, A., Vibration for Engineers , 2nd edition, Prentice-Hall, Upper Saddle River, NJ, 1996.10. Ewins, D.J., Modal Testing: Theory and Practice , Research Studies Press Ltd., Letchworth, England,1984.11. Inman, D.J., Engineering Vibration , Prentice-Hall, Englewood Cliffs, NJ, 1996.12. Irwin, J.D. and Graf, E.R., Industrial Noise and Vibration Control , Prentice-Hall, Englewood Cliffs,NJ, 1979.13. McConnell, K.G., Vibration Testing , John Wiley & Sons, New York, 1995.14. Meirovitch,L., ComputationalMethodsinStructuralDynamics ,Sijthoff&Noordhoff,Rockville,MD, 1980.15. Meirovitch, L., Elements of Vibration Analysis , 2nd edition, McGraw-Hill, New York, 1986.16. Randall,R.B., ApplicationofB&KEquipmenttoFrequencyAnalysis ,BruelandKjaer,Naerum,Denmark, 1977.17. Rao, S.S., Mechanical Vibrations , 3rd edition, Addison-Wesley, Reading, MA, 1995.18. Shearer,J.L.andKulakowski,B.T., DynamicModelingandControlofEngineeringSystems ,Mac-Millan Publishing, New York, 1990.19. Shearer, J.L., Murphy, A.T., and Richardson, H.H., Introduction to System Dynamics , Addison-Wesley,Reading, MA, 1971.20. Steidel, R.F., An Introduction to Mechanical Vibrations , 2nd edition, John Wiley & Sons, New York,1979.21. Volterra, E. and Zachmanoglou, E.C., Dynamics of Vibrations , Charles E. Merrill Books, Columbus,OH, 1965.de Silva, Clarence W. Time ResponseVibration: Fundamentals and PracticeClarence W. de SilvaBoca Raton: CRC Press LLC, 2000 2 2000 CRC Press Time Response Vibrations are oscillatory responses of dynamic systems. Natural vibrations occur in these systemsdue to the presence of two modes of energy storage. Specically, when the stored energy is convertedfromoneformtotheother,repeatedlybackandforth,theresultingtimeresponseofthesystemis oscillatory in nature. In a mechanical system, natural vibrations can occur because kinetic energy,which is manifested as velocities of mass (inertia) elements, can be converted into potential energy(which has two basic types: elastic potential energy due to the deformation in spring-like elements,and gravitational potential energy due to the elevation of mass elements against the Earths grav-itational pull) and back to kinetic energy, repetitively, during motion. Similarly, natural oscillationsofelectricalsignalsoccurincircuitsduetothepresenceofelectrostaticenergy(oftheelectriccharge storage in capacitor-like elements) and electromagnetic energy (due to the magnetic eldsininductor-likeelements).Fluidsystemscanalsoexhibitnaturaloscillatoryresponsesastheypossess two forms of energy. But purely thermal systems do not produce natural oscillations becausethey, as far as anyone knows, have only one type of energy. These ideas are summarized in AppendixA. Note, however, that an oscillatory forcing function is able to make a dynamic system respondwithanoscillatorymotion(usuallyatthesamefrequencyastheforcingexcitation)evenintheabsence of two forms of energy storage. Such motions are forced responses rather than natural orfree responses. This book concerns vibrations in mechanical systems. Nevertheless, clear analogiesexist with electrical and uid systems as well as mixed systems such as electromechanical systems.Mechanicalvibrationscanoccurasbothfree(natural)responsesandforcedresponsesinnumerouspracticalsituations.Someofthesevibrationsaredesirableanduseful,andothersareundesirableandshouldbeavoidedorsuppressed.Thesoundthatisgeneratedafterastringofaguitar is plucked is a free vibration, while the sound of a violin is a mixture of both free and forcedvibrations. These sounds are generally pleasant and desirable. The response of an automobile afterit hits a road bump is an undesirable free vibration. The vibrations felt while operating a concretedrill are desirable for the drilling process itself, but are undesirable forced vibrations for the humanwho operates the drill. In the design and development of a mechanical system, regardless of whetherit is intended for generating desirable vibrations or for operating without vibrations, an analyticalmodel of the system can serve a very useful function. The model will represent the dynamic system,and can be analyzed and modied more quickly and cost effectively than one could build and testa physical prototype. Similarly, in the control or suppression of vibrations, it is possible to design,develop, and evaluate vibration isolators and control schemes through analytical means before theyarephysicallyimplemented.Itfollowsthatanalyticalmodels(see Appendix A)areusefulintheanalysis,control,andevaluationofvibrationsindynamicsystems,andalsointhedesignanddevelopment of dynamic systems for desired performance in vibration environments.An analytical model of a mechanical system is a set of equations, and can be developed eitherby the Newtonian approach where Newtons second law is explicitly applied to each inertia element,or by the Lagrangian or Hamiltonian approach, which is based on the concepts of energy (kineticand potential energies). These approaches are summarized in Appendix B. A time-domain analytical modelisasetofdifferentialequations,withrespecttotheindependentvariabletime( t ).Afrequency-domain model is a set of input-output transfer functions with respect to the independent variablefrequency( ).Thetimeresponsewilldescribehowthesystemmoves(responds)asafunction of time. Both free and forced responses are useful. The frequency response will describethe way the system moves when excited by a harmonic (sinusoidal) forcing input, and is a functionofthefrequencyofexcitation. Thischapterintroducessomebasicconceptsofvibrationanalysis 2000 CRC Press using time-domain methods. The frequency-domain analysis will be studied in subsequent chapters(Chapters 3 and 4, in particular). 2 . 1 U N D AM PED O SCI LLATO R Consider the mechanical system that is schematically shown in Figure 2.1. The inputs (or excitation)applied to the system are represented by the force f ( t ). The outputs (or response) of the system arerepresented by the displacement y . The system boundary demarcates the region of interest in thisanalysis.Thisboundarycouldbeanimaginaryone.Whatisoutsidethesystemboundaryistheenvironment in which the system operates. An analytical model of the system can be given by oneor more equations relating the outputs to the inputs. If the rates of changes of the response (outputs)arenotnegligible,thesystemisa dynamic system.Inthiscase,theanalyticalmodelinthetimedomain becomes one or more differential equations rather than algebraic equations. System param-eters (e.g., mass, stiffness, damping constant) are represented in the model, and their values shouldbe known in order to determine the response of the system to a particular excitation. State variablesareaminimumsetofvariablesthatcompletelyrepresentthedynamicstateofasystematanygiven time t . These variables are not unique (more than one choice of a valid set of state variablesispossible).Theconceptsofstatevariablesandstatemodelsareintroducedin Appendix Aandalso in this chapter. For a simple oscillator (a single-degree-of-freedom mass-spring-damper systemas in Figure 2.1), an appropriate set of state variables would be the displacement y and the velocity. An alternative set would be and the spring force.This chapter provides an introduction to the response analysis of mechanical vibrating systems inthe time domain. In this introductory chapter, single-degree-of-freedom systems that require only onecoordinate (or one independent displacement variable) in their model, are considered almost exclusively.Higher-degree-of-freedomsystemswillbeanalyzedelsewhereinthebook(e.g.,Chapter 5).Mass(inertia)andspringarethetwobasicenergystorageelementsinamechanicalvibratingsystem. Amass can store gravitational potential energy as well when located against a gravitational force. Theseelementsareanalyzedrst.Inapracticalsystem,massandstiffnesspropertiescanbedistributed(continuous)throughoutthesystem.Butinthispresentanalysis,lumped-parametermodelsareemployed where inertia, exibility, and damping effects are separately lumped into single parameters,with a single geometric coordinate used to represent the location of each lumped inertia.Thischaptersectionrstshowsthatmanytypesofoscillatorysystemscanberepresentedbythe equation of an undamped simple oscillator. In particular, mechanical, electrical, and uid systemsare considered. Please refer to Appendix A for some foundation material on this topic. The conser-vation of energy is a straightforward approach for deriving the equations of motion for undampedoscillatory systems (or conservative systems). The equations of motion for mechanical systems can FI G U RE2 . 1 A mechanical dynamic system. y y 2000 CRC Press bederivedusingthefree-bodydiagramapproachwiththedirectapplicationofNewtonssecondlaw. AnalternativeandratherconvenientapproachistheuseofLagrangeequations,asdescribedin Appendix B. The natural (free) response of an undamped simple oscillator is a simple harmonicmotion . This is a periodic, sinusoidal motion. This simple time response is also discussed. 2 . 1 . 1 E N ERG Y S TO RAG E E LEM EN TS Mass(inertia)andspringarethetwobasicenergystorageelementsinmechanicalsystems.Theconceptofstatevariablescanbeintroducedaswellthroughtheseelements(see Appendix Afordetails), and will be introduced along with associated energy and state variables. I ner t i a( m ) Consideraninertiaelementoflumpedmass m ,excitedbyforce f ,asshowninFigure2.2.Theresulting velocity is v .Newtons second law gives(2.1)Kinetic energy stored in the mass element is equal to the work done by the force f on the mass.Hence,or(2.2) Note : v is an appropriate state variable for a mass element because it can completely represent theenergy of the element. Integrate equation (2.1) from a time instant immediately before t = 0 (i.e., t = 0 ).(2.3)Hence, with t = 0 + , for a time instant immediately after t = 0, one obtains(2.4) FI G U RE2 . 2 A mass element.m dvdtf Energy E fdx fdxdt dt fvdtmdvdtvdt m vdv Kinetic energy KE mv 122v t vmfdtt( ) ( )+010v vmfdt 0 0100+ ( ) ( )++ 2000 CRC Press Since the integral of a nite quantity over an almost zero time interval is zero, these results implythataniteforcewillnotcauseaninstantaneouschangeinvelocityinaninertiaelement.Inparticular, for a mass element subjected to nite force, since the integral on the RHS of equation(2.4) is zero, one obtains(2.5) Spr i ng( k ) Consider a massless spring element of lumped stiffness k , as shown in Figure 2.3. One end of thespringisxedandtheotherendisfree. Aforce f isappliedatthefreeend,whichresultsinadisplacement (extension) x in the spring.Hookes law gives(2.6)Elasticpotentialenergystoredinthespringisequaltotheworkdonebytheforceonthespring. Hence,or(2.7) Note : f isanappropriatestatevariableforaspring,andsois x ,becausetheycancompletelyrepresent the energy in the spring.Integrate equation (2.6).(2.8)Set t = 0 + . Then,(2.9) FfI G U RE2 . 3 A spring element.v v 0 0+ ( ) ( )f kxdfdtkv orEnergy : E fdx kxdx kxfdxdt dt fvdt fkdfdt dtkfdfkf 12221 1 12Elastic potential energy PE kxfk 121222f t fkvdtt( ) ( )+010f fkvdt 0 0100+ ( ) ( )++ 2000 CRC Press Fromtheseresults,itfollowsthatatnitevelocities,therecannotbeaninstantaneouschangeinthe force of a spring. In particular, from equation (2.9) one sees that at nite velocities of a spring(2.10)Also, it follows that(2.11) G r avi t at i onal Pot ent i al Ener gy The work done in raising an object against the gravitational pull is stored as gravitational potentialenergy of the object. Consider a lumped mass m , as shown in Figure 2.4, that is raised to a height y from some reference level. The work done givesHence,(2.12) 2 . 1 . 2 C O N SERV ATI O N O F E N ERG Y There is no energy dissipation in undamped systems, which contain energy storage elements only.Inotherwords,energyisconservedinthesesystems,whichareknownasconservativesystems.For mechanical systems, conservation of energy gives(2.13)These systems tend to be oscillatory in their natural motion, as noted before. Also, as discussed inAppendix A, analogies exist with other types of systems (e.g., uid and electrical systems). Considerthe six systems sketched in Figure 2.5. Syst em1 ( Tr ansl at or y) Figure2.5(a)showsatranslatorymechanicalsystem(anundampedoscillator)thathasjustonedegree of freedom x . This can represent a simplied model of a rail car that is impacting againsta snubber. The conservation of energy (equation (2.13)) gives FI G U RE2 . 4 A mass element subjected to gravity.f f 0 0+ ( ) ( )x x 0 0+ ( ) ( )Energy :E fdy mgdy Gravitational potential energy :PE mgy KE PE const + 2000 CRC Press (2.14)Here, m is the mass and k is the spring stiffness. Differentiate equation (2.14) with respect to time t to obtainSince 0atall t ,ingeneral,onecancancelitout.Hence,bythemethodofconservationof energy, one obtains the equation of motion(2.15) Syst em2 ( Rot at or y) Figure2.5(b)showsarotationalsystemwiththesingledegreeoffreedom .Itmayrepresentasimplied model of a motor drive system. As before, the conservation energy gives(2.16)In this equation, J is the moment of inertia of the rotational element and K is the torsional stiffnessof the shaft. Then, by differentiating equation (2.16) with respect to t and canceling, one obtainsthe equation of motion(2.17) Syst em3 ( Fl exur al ) Figure2.5(c)isalateralbending(exural)system,whichisasimpliedmodelofabuildingstructure. Again, a single degree of freedom x is assumed. Conservation of energy gives(2.18)Here, m is the lumped mass at the free end of the support and k is the lateral bending stiffness ofthe support structure. Then, as before, the equation of motion becomes(2.19) Syst em4 ( Sw i ngi ng) Figure 2.5(d) shows a simple pendulum. It may represent a swinging-type building demolisher ora skilift and has a single-degree-of-freedom . Thus,12122 2mx kx const + mxx kxx + 0 x xkm x + 012122 2J K const + + KJ012122 2mx kx const + xkm x + 0KE m lPE E mglref( ) 122cos Gravitational2000 CRC PressHere, m is the pendulum mass, l is the pendulum length, g is the acceleration due to gravity, andErefis the PE at the reference point, which is a constant. Hence, conservation of energy gives(2.20)Differentiate with respect to t after canceling the common ml:Since 0 at all t, the equation of motion becomes(2.21)This system is nonlinear, in view of the term sin . For small , sin is approximately equal to .Hence, the linearized equation of motion is(2.22)FI G U RE2 . 5 SixexamplesofsingleD.O.F.oscillatorysystems:(a)translatory,(b)rotatory,(c)flexural,(d) swinging, (e) liquid slosh, and (f) electrical.122 2ml mgl constcos l gsin + 0sin + gl0 + gl02000 CRC PressSyst em5 ( Li qui dSl osh)Consider a liquid column system shown in Figure 2.5(e). It may represent two liquid tanks linkedby a pipeline. The system parameters areArea of cross section of each column = AMass density of liquid = Length of liquid mass = lThen,Note that the center of gravity of