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Reza N. Jazar Advanced Vibrations Theory and Application Second Edition Advanced Vibrations Reza N. Jazar Advanced Vibrations Theory and Application Second Edition RezaN.Jazar SchoolofEngineering RMITUniversity Melbourne,VIC,Australia SchoolofCivilEngineering andArchitecture XiamenUniversityofTechnology Xiamen,Fuzhou,China ISBN978-3-031-16355-5 ISBN978-3-031-16356-2 (eBook) https://doi.org/10.1007/978-3-031-16356-2 1stedition:©SpringerScience+BusinessMediaNewYork2013 2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Donottrustanysciencethatjustifiesbutfails topredict. to: Kavosh Vazan Mojgan Preface For several years, my students complained that the available books in the market on the topic of vibrations are all treating this science inconsistent from different viewpoints. They were asking me to write a book on vibrations that covers all important and must-known subjects deep enough; shows applications well, has enough examples, right to the point; and prepare them to tackle real engineering problemsaswellasmakethemreadyforthenextstepinnonlinearvibrationsand vibrationsofcontinuoussystems.Thisbookispreparedtoaddressthoserequests.It isamodified,improved,andupdatedversionoftheAdvancedVibrations:AModern Approachthatispublishedin2013.Thepresentbookhasaneducationaldivisionof chapters,sections,andsubsectionsthatmakeittobemoreappropriatefortraditional classroomteaching. This book is for designers, practicing engineers, and students of engineering. It introduces the fundamental knowledge used in mechanical vibrations. This knowledgecanbeutilizedtodevelopcomputerprogramsforanalyzing,designing, and optimizing vibration problems in industrial systems.The main problem in the study of vibrations is to determine the motion of a system subjected to a given excitation. The excitation and the response can be expressed by either kinematic quantitiessuchasdisplacements,velocities,andaccelerations,orbyforces. Thesubjectofvibrations hasbeen intheengineering curriculumformorethan hundred years. I consider the 1896 Lord Rayleigh’s (John William Strutt, 3rd BaronRayleigh,aBritishscientist)"TheoryofSound"asthefirstmodernbookon the subject of sound, wave propagation, 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, theRussian-Americanengineer,StephenP.Timoshenko(1878–1972),andthenas “Mechanical Vibrations” (1934) by the Dutch-American engineer, Jacob P. Den Hartog.AlthoughTimoshenkowasthearchitectofmodernstructureofengineering educationandhisvariousbookswereusedforalongtimeineducatingmechanical and civil engineers, it was Den Hartog’s Mechanical Vibrations that was globally accepted as a classical educational book. Almost all mechanical vibration books that appeared after 1940 follow the structure of Den Hartog’s, starting from time ix x Preface responseandendingwithfrequencyresponseanalysisandoptimization.Sometimes aglanceatrandomvibrations,nonlinearvibrations,continuoussystems,vibrations control,ormodalanalysismayalsobeseeninvariousbooks.Thecurrentbookalso follows the same order as Den Hartog’s. It begins with a review of fundamental concepts of mechanical vibrations and continues with time response, and ends withfrequencyresponses.Aglanceonmusicfromvibrationsviewpoint,first-order vibrating systems, vibration absorber design, and root mean square optimization methodaresometopicsthatsolelyappearinthisbook. Level ofthe Book This book has evolved from nearly three decades of research and working on real engineering vibrating systems, and teaching courses in fundamental and advanced vibrations.Itisprimarilydesignedtoteachthelastyearofanundergraduatestudy and/or the first year of graduate study in engineering. Hence, it is an intermediate textbook. It provides the reader with both fundamental and advanced topics. The whole book can be covered in two successive courses; however, it is possible to jumpoversomesectionsandcoverthebookinonecourse.Studentsarerequiredto knowthefundamentalsofkinematicsanddynamics,aswellasabasicknowledge oflinearalgebra,differentialequations,andnumericalmethods. 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 topic which is not designed for undergraduateteachingandcanbedroppedinthefirstreading. Organizationofthe Book The text is organized such that it can be used for teaching or for self-study. It is organizedin3parts: Part I, “Vibration Fundamentals,” introduces the vibrations as a cause of trans- formationofenergy.Itcoverskinematicsofvibrationsanddevelopspracticalskills toderivetheequationsofmotionofvibratingsystems.TheconceptsoftheNewton- Euler dynamics and Lagrangian method are both used equally for the derivation ofequationsofmotion.PartIalsocoversallmathematicalanalysisthatthereader needstoabsorbtheadvancedtopicsthroughoutthebook. Part II, “Time Response,” covers the time and transient responses of vibrating systemsandnon-harmonicexcitations,aswellasfreevibrations. Preface xi Part III, “Frequency Response,” covers the methods of developing the steady- state frequency response of vibrating systems to harmonic excitations. The princi- plesofvibrationoptimizationwillbeintroducedinthispart.Therootmeansquare optimizationtechniqueforsuspensiondesignofmechanicalsystemsisintroduced andapplied.Theoutcomeoftheoptimizationtechniqueistheoptimalstiffnessand dampingforsuspendedequipment. Method ofPresentation Thestructureofpresentationisina“fact-reason-application”fashion.The“fact”is themainsubjectweintroduceineachsection.Thenthereasonisgivenasa“proof.” Finally,theapplicationofthefactisexaminedinsome“examples.”The“examples” areaveryimportantpartofthebook.Theyshowhowtoimplementtheknowledge introduced in “facts.” They also cover some other facts that are needed to expand thesubject. Prerequisites Thebookiswrittenforseniorundergraduateandfirst-yeargraduate-levelstudents inengineering.Theassumptionisthatusersarefamiliarwithmatrixalgebraaswell asbasicdynamicsanddifferentialequations.Prerequisitesarethefundamentalsof kinematics,dynamics,calculus,fundamentalsofdifferentialequations,andmatrix theory.Thesetopicsareusuallytaughtinthefirstthreeundergraduateyears. UnitSystem Thesystemofunitsadoptedinthisbookis,unlessotherwisestated,theInternational System of Units (SI). The units of degree (deg) or radian (rad) are utilized for variablesrepresentingangularquantities. Melbourne,VIC,Australia RezaN.Jazar How to Use This Book This book is suitable for the first course in vibrations and is written for a full semester of 16 weeks long in graduate level. If the level of the course is undergraduateormixed,andifthelengthofthesemesterisshorterthan16weeks, then the following pattern can be suggested to fit the contents to different classes. Teaching Chap. 3, Vibration Dynamic, is optional in all cases depending on the background of students. The main topic of this chapter is to review dynamics and remindstudentsoftheLagrangeanmethodofderivingequationsofmotion.Chapter 2mayalsobeleftforstudentstoreadbythemselves. Undergraduate level, 10 or 12 weeks semester: Skip all asterisk sections and teachthefollowingchapters. IVibrationFundamentals 1VibrationKinematics IITimeResponse 3OneDegreeofFreedom IIIFrequencyResponse 6OneDegreeofFreedomSystems 7MultiDegreesofFreedomSystems Undergraduatelevel,16weekssemester:Skipallasterisksectionsandteachthe followingchapters. IVibrationFundamentals 1VibrationKinematics 2VibrationDynamics IITimeResponse 3OneDegreeofFreedom 4MultiDegreesofFreedom 5First-OrderSystems IIIFrequencyResponse xiii