rao2004-briefhistoryofvibration

8/13/2019 Rao2004-BriefHistoryOfVibration

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CHAPTER FI.JNDA}IENTALSFVIBRATIONFrom about 3000 n.c., stringed instruments such as harps appearedon the walls ofEgyptian ombs I I .2]. In fact, the British Museum exhibits a harp with bull-headed sound-box found on an nlaid panelof a royal tomb at Ur from 2600 s.c. Stringed musical instru-ments probably originated with the hunter's bow, a weapon favored by the armies ofancientEgypt. One of the most primitive stringed nstruments,called the nanga, datingback o 1500n.c., can be seen n the British Museum. Our presentsystemof music is basedin ancientGreekcivilization. Sinceancient imes, both musiciansandphilosopherssoughout the rules and laws of soundproduction, used hem in improving musical instrumentsandpassedhem on from generation o generation.

The Greekphilosopherand mathematicianPythagoras 582-507 n.c.) is considered obe the first person o investigatemusicalsoundson a scientific basis Fig. 1.1).Amongother things, Pythagorasconducted experimentson a vibrating string by using a simpleapparatus alled a monochord. In the monochord shown in Fig. 1.2 the wooden bridgeslabeledI and3 are fixed. Bridge 2 is made movable while the tension n the string is heldconstantby the hanging weight. Pythagorasobserved hat if two like strings of differentlengths are subject o the same ension, he shorterone emits a higher note; in addition, ifthe length of the shorter string is half the longer one, the shorterone will emit a note anoctaveabove he other. However,Pythagoras eft no written accountof his work (Fig. 1.3)Although the concept of pitch wasdevelopedby the time of Pythagoras, ts relation withthe frequencyof vibration of the soundingbody was not understood. n fact the relationbetween he pitch and the frequencywas not understooduntil the time of Galileo in thesixteenthcentury.

Around 350 n.c., Aristotle wrote treatiseson music and sound, making observationsuch as the voice is sweeter han the sound of instruments, and the sound of the flute issweeter han that of the lyre. In 320 s.c., Aristoxenus,a pupil of Aristotle and a musicianwrote a three-volume work entitled Elements of Harmony. These books are perhaps he

FIGURE 1.1 Pythagoras.Reprintedwith permission rom L. E. Navia,fo thagoras An Annotated B b iography,Garland Publishing, Inc., New York, 1990).


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1.2 BRIEFHISTORY FVIBRATION

FIGURE.2 Monochord.

FIGURE l'3 Pythagoras-as .musician.(Reprinted ith permissionromD. E. smith,Flsroryf Mathematics,ol. , Doverpublications,fn.., N , yort, fSSS).oldest ones available on the subject of music written directly by the author. Euclid, in about300s'c'' wrotebrieflyaboutmusicwitlout any eference itr ptryti alnature f soundin a treatisecalled'ntroductiono Harmonics.No furtherscientincadditionso soundweremadeby theGreeks.t appearshat heRomans erived heirknowledge f musiccompletelyrom the Greeks, xcept hatvitruvius,a famousRomanarcnitect,wrote nabout20 B'c' on theacoustic ropirtiesof theaters. is treatise, ntitledDeArchitecturaLibri Decem'was ost for severai enturies,o berediscou r o niyin thefifteenthcen-tury'There ppearso havebeennodevelopmentn the heorie,or rounaandvibrationornearly16centuriesollowing heworksof Vitruvius.chinaexperienced anyearthquakesn ancientimes.ZhangHeng,whoserved sahistorian ndastronomern the-secondentury, erceived needo aevltopan nstrumentto measurearthquakesrecisely.n a.o. r32-heinventedheworld,s irstseismographomeasurearthquakes1.3,1.41. hisseismographasmade f finecastbronze, adadiam_eterof eightchi (a chi is-equalo 0.237 t r), andwasshapedike a wine ar (Fig. r.a).Inside he ar wasa mechanismonsisting f pendulumsurroundedy a groupof eightlevermechanismsointing n eightairecti-ons.ight dragon igures,with a bronze all in

PITACOTTAS

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r.2.2From Galileo oRayleigh

CHAPTER FUNDAI{ENTALSOFVIBRATION

FIGURE 1.4 Theworld's irstseismo-graph,nventedn China n e.o. 132. ReprintedwithpermissionromR.TatonEd.),History fScience,asic ooks,nc., ew ork, 957).

the mouth of each,were arrangedon the outsideof the seismograph.There were toadswitmouths open upward underneatheachdragon.A strong earthquake n any direction woultilt the pendulum n that direction, triggering the lever n the dragon head.This opened hmouth of the dragon, hereby releasing ts bronzeball, which fell in the mouth of the toawith a clanging sound. Thus the seismographenabled he monitoring personnel o knowboth the time and directionof occunence of the earthquake.Galileo Galilei (1564-1642) s considered o be the founderof modernexperimental cence. n fact, the seventeenth entury is often considered he century of genius since thfoundations of modernphilosophy and sciencewere laid during that period. Galileo wainspired o study he behaviorof a simplependulumby observing he pendulummovements of a lamp in a church in Pisa. One day, while feeling bored during a sermoGalileo wasstaringat theceiling of the church.A swing ing amp caughthis attention.Hstarted measuring he period of the pendulummovementsof the lamp with his pulse anfound to his amazement hat the time period was ndependentof the amplitude of swingsThis led him to conduct more experiments on the simple pendulum. ln DiscourseConcerningTwoNew Sciences, ublished n 1638,Galileo discussed ibrating bodies.Hdescribed he dependenceof the frequencyof vibration on the length of a simple pendulum, alongwith the phenomenon f sympathetic ibrations (resonance).Galileo's writings also indicate that he had a clear understanding of the relationship between thfrequency, ength, tension,and density of a vibrating stretched string [1.5]. However, thfirst correct publishedaccount of the vibration of strings was given by the French mathematicianandtheologian,Marin Mersenne 1588-1@8) in his book Harmonicorum Libepublished n 1636.Mersenne also measured, or the first time, the frequency of vibratioof a long string and from that predicted he frequencyof a shorter string having the samdensity and tension.Mersenne s consideredby many the father of acoustics.He is ofte


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1.2 BRIEFHISTORYOFITBRATIcredited with the discoveryof the laws of vibrating strings because e publisheesults n 1636' wo yearsbeforeGalileo.However, he creditbelongs o Galileo sincaws werewritten manyyearsearlierbut their publication wasprohibitedby the ordehe Inquisitorof Romeunril 163g.Inspiredby the work of Galileo, theAcademiadel cimento was fbunded n Floren 1657; his was followed by the formationsor tn Royal Societyof London in 1662he ParisAcademiedes Sciencesn 1666.Larer,Robert Hooke (1635_1703)arsocuctedexperiments o frnd a relation uetween he pitch and frequencyof vibration otring' However, t wasJoseph auveur 1653-1716)*il;;;A; .,t r experimhoroughlyand coined he word u ourilrl'for the scienceoii o [1.6]. sauveuranceandJohnwallis (1616-^1703)in Enslandobserved,independentry,he phenoon of mode shapes,and they found that a iibrating ,o r ir a ,-, rinf un t uu no mott certain points and violent motion at intermediatepoints. suuuJu. called the formointsnodesand the latter ones oops. rt was found tt ut ,u t -uii.uiton, had higher fuenciescompared o the one associatedwith the ,i*pr; ;i;;Jn or rn string withodes. n fact, the higher frequencieswere6i*., be integrar,nuirifr , of the frequef simple vibration' and Sauveurcalled the higher frequencieshannonics and the fuencyof simplevibration the fundamental requency.Sauveuralsofound thata stringcibratewith severalof its-harmonicspresentat thesame ime. ln aJaiiion, he observedhenomenonof beatswhen. wo organ pip , of slightly different pitches are soundogether' n 1700Sauveurcalculated]by a somewhataubiou, methoi, the frequencyotretchedstring from the measured ug of it, middle point.Sir Isaac Newto-n (1642-1727; pubrished his monumenrar work, ph,osophiaNaturalis Principia Mathematica. n't686, describing the law oiuniu rrut gravitationell as the threerawsof motion and otherdiscoveries.Newton,s , ono law of motionoutinely used in modern books on vibrations to derive the equationsof motion ofibrating body'The theoretical(dynamical) sotutionor tn p.oul=m or thevibrating srrinas found by rheEnglish mathematician'Brookrhyror (16g5_1731)in 1713,who alsresented he famousThylor's theoremon infinite series.The natural'frequencyof vibraion obtained rom the equationof motion derived bv Thylor ug. J*irt the experimeal values observedby Garileo and Mersenne.The procedure adopted by Thyror waerfectedhrought l1tr9gy9tir ;; ;;;;i derivativesn ,rr q i.i . of motionbanielBernoulli1700-1782).ean^o'ei ru .t (1717-17g3),ndLeonard ul1707-1783) \r ' r '-r ' oJ)' ''tlThe possibilityof a string vibrating with severalof its harmonicspresentat the samime (with dispracement f any point ut'uny nrtunt beingequal to the algebraicsum of dislacements for each harmonic) *u, p.o*a through the dynamic equations of Danieernoulli in his memoir,publishedoy tit n rrin Academy n 1755 l.7l. This characreristwasreferred'toas he principleof thecoexistenceofsmalloscillations,-*t i.r,, in present-daterminology, s the principle of superposition.This principle ; ;;;;;;; to be mosr valuble in the developmentof the theoryof vibrationsand ed to ttrepossiuility of expressinany arbitrary function (t: :,.ury initial shapeof the string) using an infinire seriesof sinend cosines.Becauseof this implication,'D'Alembert and Eul-erdoubtedthe validity ofhisprinciple.However,jh: {tii t ortrris type orexpansionwasprovedby J. B. J. Fourie1768-1830)inhis AnatyticatThiory of Uioii \SZZ.


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CH,APTER FT]NDAMENTALSF VIBMTIONThe analytical solution of the vibrating string was presentedby Joseph Lagra(1736-1813) in his memoir published by the Turin Academy in 1759. In his stu

Lagrange assumed hat the string was made up of a finite number of equally spaced dtical massparticles,and he established he existenceof a numberof independent requcies equal to the number of massparticles.When the number of particles was allowedbe infinite, the resulting frequencieswere found to be the sameas the harmonic frequcies of the stretched tring. The method of setting up the differential equationof the motof a string (called the wave equation),presentedn most modern books on vibration tory, was first developedby D'Alembert in his memoir publishedby the Berlin Acadein 1750.The vibration of thin beams supported and clamped in different ways was fstudied by Euler in 17 4 and Daniel Bernoulli in 175 . Their approachhasbecomeknoas the Euler-Bernoulli or thin beam theory.

CharlesCoulomb did both theoreticaland experimentalstudies n 1784 on the torsiooscillationsof a metalcylinder suspended y a wire (Fig. 1.5). By assuming hat the resing torque of the twisted wire is proportionalto the angleof twist, he derived the equaof motion for the torsional vibration of the suspended ylinder.By integratingthe equaof motion, he found that the period of oscillation is independentof the angle of twist.

There is an interestingstory related to the developmentof the theory of vibrationplates 1.8]. In 1802 he German scientist,E. F. F. Chladni (1756-1824) developedmethod of placing sand on a vibrating plate to find its mode shapesand observed

FIGURE 1.5 Coulomb's device for tor-sional vibration tests. Reprinted ith permis-sion rom S. P.Timoshenko, istoryof Strength fMaterials,McGraw-HillBookCompany,nc.,NewYork,1953).


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teJ 'n-n-tou-)ne-ly,St/nalit-)n)nof1ele

1.2 BRIEFHISTORYOFWBMTIONbeauty and intricary of the modal patternsof the vibrating plates. In lg0g the FreAcademy invited chladni to give a demonstration of his experiments. NapoldBonaparte,who attended he meeting,was very impressed nd presented sum of 3,0francsto the academy, o be awardeJ o the first peison to give u ,uiirru ,o.y mathemcal theory of the vibrationof plates. By the cloiing date Jf the comperition in octobl8ll, only one candidate, ophieGermain,had entered he contest.But Lagrange,wwas one of the judges, noticed an elror in the derivation of her differential equationmotion' The academy pened he competitionagain,with a new closing dateof octobl8 13' SophieGermainagainentered he contest,presenting he correct form of the diffeential equation.However, he academydid not award the prize to her because he udgwantedphysical ustification of the assumptionsmade in her derivation.The competitiwasopenedoncemore. n her third attempt,SophieGermainwas inally awarded he priin 1815,although hejudgeswerenot completelysatisfiedwith her theory. n fact, t wlater found that her differential equation was correct but the boundary conditions weelroneous'The correctboundaryconditions for the vibration orput s were given in lg5by G. R. Kirchhoff(1824-1887).In the meantime, he problemof vibration of a rectangurar lexible membrane,whicis important for the understanding f the soundemitted by drums, was solved for the firtime by simeon Poisson 1781-1840).The vibrationof a circularun rurun wasstudiby R' F A' clebsch (1833-1872) n 1862.After this, vibration studieswere done onnumberof practicalmechanicaland structuralsystems. n rgiT Lord Baron Rayleigh publishedhis book on the theoryofsound [1.9]; it is considered classicon the subjectosoundand vibration even oday.Notableu*ong the many contributionsof Rayleigh is thmethod of finding the fundamental frequency of vibration of a conservativesystem bmaking useof the principle of conservationof n rgy-now known asRayleigh,smethodThis methodproved to.be a helpful technique or the solution of difficult vibration problems'An extension f themethod,which canbe used o find multiple natural requencieis known as the Rayleigh-Ritzmethod.

In 1902Frahm nvestigatedhe mportanceoftorsional vibration study in thedesignofthepropeller shaftsof steamships. he dynamic vibration absorber,which involves the addition of a secondaryspring-mass ystem o eliminate the vibrationsoi u ,,'uin system,waalso proposedby Frahm in 1909.Among the modern contributers o the theory of vibrations, the names of Stodola, De Lavall Timoshenko, and Mindlin are notable. AureStodola 1859-1943) contributed o the srudyof vibration of beams,plates,and membranes'He developeda method or analyzingvibrating beams hat is atsoappricable o tur-bineblades.Noting thatev_ery ajo. typ of prime movergivesrise to vibration problemsc' G' P' De Laval (1s45-1913) preseniea practicalsoluion to the proulem of.-vibrarioof an unbalanced otaring disk. After noticing failures of steershafl in high_speedur_bines,he useda bamboofishing rod as a shalt to mount the rotor. He observed hat thissystemnot only eliminated the vibration of the unbalanced otor but also survived up tospeeds shigh as 100,000 pm tl.l0l.stephenTimoshenko rg7g-rg7z) presented n improved heoryof vibrationof beamswhichhasbecomeknownas heTimoshenkoor thick beam heory,by considering heeffectsof rotary inertiaandsheardeformation.A similar theorywasp*r [a C/ n. D. Mindlin for

t.2.3RecentContributions


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CT1APTER FUNDAMENTATSFVIBRATIONthe vibration analysis of thick platesby including the effects of rotary inertia and shedeformation.

It has ong been recognized hat many basic problemsof mechanics, ncluding thoof vibrations, are nonlinear.Although the linear treatmentscommonly adoptedare quitsatisfactory or most purposes, hey are not adequate n all cases. n nonlinear systemphenonmenamay occur that are theoretically impossible in linear systems. The mathmatical theory of nonlinear vibrations began to develop in the works of Poincard anLyapunov at the end of the nineteenth century. Poincard developed the perturbatiomethod in 1892 in connection with the approximate solution of nonlinear celestimechanicsproblems.Lyapunov laid the foundationsof modem stability theory in 189which is applicable o all types of dynamical systems. After L920, he studies undertakeby Duffing and van der Pol brought the fust definite solutions nto the theory of nonlinevibrationsanddrew attention o its importance n engineering. n the last 30 years,autholike Minorsky and Stoker haveendeavoredo collect in monographs he main results concerning nonlinear vibrations. Most practical applicationsof nonlinear vibration involvethe use of some ype of a perturbation heory approach.The modernmethodsof perturbtion theory were surveyedby Nayfeh t . I 11.Random characteristics represent n diverse phenomena uchas earthquakes,windtransportationof goodson wheeledvehicles,and rocket and et engine noise. It becamnecessaryo deviseconceptsand methodsof vibration analysis or these andom effectAlthough EinsteinconsideredBrownianmovement,a particular typeof random vibrationas ong ago as 1905,no applicationswere investigateduntil 1930.The introduction oftheconelation function by Taylor in 1920and of the spectraldensityby Wiener and Khinchiin the early 1930sopenednew prospects or progress n the theory of random vibrationPapersby Lin and Rice, publishedbetween 1943 and 1945,pavedrhe way for the applcation of random vibrations to practical engineering problems. The monographs oCrandall and Mark, and Robsonsystematizedhe existing knowledge n the theory of random vibrations1.12,1.13].Until about30 yearsago, vibration studies,even hosedealing with complex engneering systems,were done by using gross models, with only a few degreesof freedom. However, the advent of high-speeddigital computers in the 1950s made possible o treat moderatelycomplex systemsand to generateapproximate solutionin semidefinite orm, relying on classicalsolution methodsbut using numerical evauation of certain terms that cannot be expressed n closed form. The simultaneoudevelopmentof the finite elementmethod enabledengineers o use digital computeto conduct numerically detailed vibration analysisof complex mechanical,vehiculaand structural systemsdisplaying thousandsof degreesof freedom [1.14]. Althoughthe finite elementmethod was not so nameduntil recently, he concept was usedseveral centuriesback. For.example,ancient mathematicians ound the circumferenceoa circle by approximating t as a polygon, where each side of the polygon, in presenday notation,canbe called a finite element.The finite elementmethod asknown todawas presented y Turner,Clough, Martin, and Topp in connectionwith the analysis oaircraft structures 1.151.Figure 1.6shows he finite element dealizationof the bod

o f a bus 1 . 161 .

rao2004-briefhistoryofvibration

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