Advanced Vibrations: A Modern Approach

695 Pages·2013·12.15 MB·English

by  Reza N. Jazar (auth.)

Advanced Vibrations
Reza N. Jazar
Advanced Vibrations
A Modern Approach
RezaN.Jazar
Mechanical&AutomotiveDept.
RMITUniversity
Bundoora,Victoria,Australia
ISBN978-1-4614-4159-5 ISBN978-1-4614-4160-1(eBook)
DOI10.1007/978-1-4614-4160-1
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012941501
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Preface
Thisbookisfordesigners,partitionengineers,andstudentofengineering.Itintro-
duces the fundamental knowledge used in mechanical vibrations. This knowledge
can be utilized to develop computer programs for analyzing, designing, and opti-
mizationofvibrationproblemsinindustrialsystems.
Vibrationshavebeenintheengineeringcurriculumforaroundahundredyears.
Manygoodbooksonthesubjectareavailable.The1896LordRayleigh’s“Theory
ofSound”isthefirstmodernbookonthesubjectofsound,wavepropagation,and
vibrations. However, the first books on vibrations with a mechanical engineering
viewpoint appeared as “Vibration Problems in Engineering” (1928) by the father
of modern engineering mechanics, Stephen P. Timoshenko, and then as “Mechan-
ical Vibrations” (1934) by J.P. Den Hartog. Although Timoshenko was the archi-
tectofmodernstructureofengineeringeducationandhisvariousbookswereused
for a long time in educating mechanical and civil engineers, it was Den Hartog’s
Mechanical Vibrations that globally was accepted as a classical educational book.
Almost all mechanical vibration books that appeared after 1940 follow the struc-
ture of Den Hartog’s, starting with time response and ending with frequency re-
sponse.Sometimesaglanceatrandomvibrations,nonlinearvibrations,continuous
systems, vibrations control, or modal analysis may also be seen in various books.
Onthecontrary,thepresentbookbeginswithfrequencyresponseandcontinuesby
timeresponseandendswithoptimizationandapplication,focusingonlyondiscrete
mechanicalsystems.Startingwithfrequencyandsteady-stateresponseofvibrating
systemsismorepracticalandmorefundamentalthantimeandtransientresponse.
LeveloftheBook
This book has evolved from nearly two decades of research and optimization of
vibratingsystemsandteachingcoursesinfundamentalandadvancedvibrations.It
is addressed primarily to the last year of an undergraduate study and the first year
graduatestudyinengineering.Hence,itisanintermediatetextbook.Itprovidesthe
ix
x Preface
readerwithbothfundamentalandadvancedtopics.Thewholebookcanbecovered
in two successive courses; however, it is possible to jump over some sections and
cover the book in one course. Students are required to know the fundamentals of
kinematicsanddynamics,aswellashaveabasicknowledgeofnumericalmethods.
The contents of the book have been kept at a fairly theoretical-practical level.
Many concepts are deeply explained and their application emphasized, and most
of the related theories and formal proofs have been explained. The book places a
strong emphasis on the physical meaning and applications of the concepts. Topics
thathavebeenselectedareofhighinterestinthefield.Anattempthasbeenmade
toexposestudentstoabroadrangeoftopicsandapproaches.
An asterisk (cid:2) indicates a more advanced subject or example, which is not de-
signedforundergraduateteachingandcanbedroppedinafirstreading.
Organizationofthe Book
Thetextisorganizedsothatitcanbeusedforteachingorforself-study.
Part I, “Vibration Fundamentals,” introduces the vibrations as a cause for the
transformationofenergy.Itcoverskinematicsofvibrations,anddevelopspractical
skills to derive the equations of motion of vibrating systems. The concepts of the
Newton–EulerdynamicsandLagrangeanmethodareusedequallyforderivationof
equationsofmotion.
PartII,“FrequencyResponse,”coversthemethodsofdevelopingthesteady-state
frequencyresponseofvibratingsystemstoharmonicexcitations.
Part III, “Time Response,” covers the time and transient responses of vibrating
systemstofreeornon-harmonicexcitations.
Part IV, “Applications,” presents vibration optimization and studies the vibra-
tions of vehicles as the most observable vibrating systems. An attempt is made to
review the basic approaches and demonstrate how a vehicle can be modeled as a
vibrating multiple degree-of-freedom system. The root mean square optimization
technique for suspension design of vehicles is introduced and applied to vehicle
suspensions.Theoutcomeoftheoptimizationtechniqueistheoptimalstiffnessand
dampingforacarorsuspendedequipment.
Method ofPresentation
Thisbookusesa“fact-reason-application”structure.The“fact”isthemainsubject
weintroduceineachsection.Thenthereasonisgivenasa“proof.”Theapplication
of the fact is examinedin some “examples.” The “examples”are a very important
partofthebook.Theyshowhowtoimplementthe“facts”andalsocoversomeother
factsthatareneededtoexpandthesubject.
Preface xi
Prerequisites
Sincethebookiswrittenforseniorundergraduateandfirst-yeargraduate-levelstu-
dentsofengineering,theassumptionisthatusersarefamiliarwithmatrixalgebraas
wellasbasicdynamics.Prerequisitesarethefundamentalsofkinematics,dynamics,
and matrix theory. These topics are usually taught in the first three undergraduate
years.
Contents
PartI VibrationFundamentals
1 VibrationKinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 MechanicalVibrationElements . . . . . . . . . . . . . . . . . . . 3
1.2 KinematicsofVibrations . . . . . . . . . . . . . . . . . . . . . . 17
1.3 FourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4 KeySymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 VibrationDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1 Newton–EulerMethod . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 (cid:2)Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.3 (cid:2)RigidBodyDynamics . . . . . . . . . . . . . . . . . . . . . . 73
2.3.1 (cid:2)CoordinateFrameTransformation . . . . . . . . . . . . 73
2.3.2 (cid:2)VelocityKinematics . . . . . . . . . . . . . . . . . . . 80
2.3.3 (cid:2)AccelerationKinematics . . . . . . . . . . . . . . . . . 87
2.3.4 (cid:2)TranslationalDynamics . . . . . . . . . . . . . . . . . 100
2.3.5 (cid:2)RotationalDynamics . . . . . . . . . . . . . . . . . . . 103
2.3.6 (cid:2)MassMomentMatrix. . . . . . . . . . . . . . . . . . . 109
2.4 LagrangeMethod . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.4.1 (cid:2)LagrangeFormofNewtonEquation . . . . . . . . . . . 120
2.4.2 LagrangeanMechanics . . . . . . . . . . . . . . . . . . . 126
2.5 DissipationFunction . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.6 (cid:2)Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
2.7 (cid:2)VariationalDynamics . . . . . . . . . . . . . . . . . . . . . . . 151
2.8 KeySymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
PartII FrequencyResponse
3 OneDegreeofFreedomSystems,FrequencyResponse . . . . . . . . 175
3.1 ForcedExcitation . . . . . . . . . . . . . . . . . . . . . . . . . . 176
xiii
xiv Contents
3.2 BaseExcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.3 EccentricExcitation . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.4 (cid:2)EccentricBaseExcitation . . . . . . . . . . . . . . . . . . . . 216
3.5 ClassificationfortheFrequencyResponsesofOneDOF Forced
VibrationSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 221
3.6 KeySymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4 MultiDegreeofFreedomSystems,FrequencyResponse . . . . . . . 233
4.1 NaturalFrequencyandModeShape . . . . . . . . . . . . . . . . . 233
4.2 CouplingandDecoupling . . . . . . . . . . . . . . . . . . . . . . 273
4.3 HarmonicExcitation . . . . . . . . . . . . . . . . . . . . . . . . . 308
4.4 KeySymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
5 TwoDegreeofFreedomSystems,FrequencyResponse . . . . . . . . 343
5.1 ForcedExcitation . . . . . . . . . . . . . . . . . . . . . . . . . . 344
5.2 BaseExcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
5.3 EccentricExcitation . . . . . . . . . . . . . . . . . . . . . . . . . 373
5.4 (cid:2)EccentricBaseExcitation . . . . . . . . . . . . . . . . . . . . 386
5.5 (cid:2)ComparisonfortheFrequencyResponsesofTwoDOFForced
VibrationSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 398
5.5.1 (cid:2)PositionofExcitation. . . . . . . . . . . . . . . . . . . 398
5.5.2 (cid:2)Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
5.6 KeySymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
PartIII TimeResponse
6 First-OrderSystems,TimeResponse . . . . . . . . . . . . . . . . . . 417
6.1 NaturalMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
6.2 GeneralMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
6.3 (cid:2)CoupledSystems . . . . . . . . . . . . . . . . . . . . . . . . . 449
6.4 KeySymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
7 OneDegreeofFreedom,TimeResponse . . . . . . . . . . . . . . . . 473
7.1 FreeVibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
7.2 ForcedVibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 493
7.3 TransientVibrations . . . . . . . . . . . . . . . . . . . . . . . . . 510
7.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
7.5 KeySymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533