Introduction to Mechanical Vibrations

CHAPTER 1 Iroducio o Mechaical Vibraios Vibraio is he oio of a paricle or a body or syse of coeced bodies displaced fro a posiio of equilibriu. Mos vibraios are udesirable i achies ad srucures because hey produce icreased sresses, eergy losses, cause added wear, icrease bearig loads, iduce faigue, creae passeger discofor i vehicles, ad absorb eergy fro he syse. Roaig achie pars eed careful balacig i order o preve daage fro vibraios. Vibraio occurs whe a syse is displaced fro a posiio of sable equilibriu. The syse eds o reur o his equilibriu posiio uder he acio of resorig forces (such as he elasic forces, as for a ass aached o a sprig, or graviaioal forces, as for a siple pedulu). The syse keeps ovig back ad forh across is posiio of equilibriu. A syse is a cobiaio of elees ieded o ac ogeher o accoplish a objecive. For exaple, a auoobile is a syse whose elees are he wheels, suspesio, car body, ad so forh. A saic elee is oe whose oupu a ay give ie depeds oly o he ipu a ha ie while a dyaic elee is oe whose prese oupu depeds o pas ipus. I he sae way we also speak of saic ad dyaic syses. A saic syse coais all elees while a dyaic syse coais a leas oe dyaic elee. A physical syse udergoig a ie-varyig ierchage or dissipaio of eergy aog or wihi is eleeary sorage or dissipaive devices is said o be i a dyaic sae. All of he elees i geeral are called passive, i.e., hey are icapable of geeraig e eergy. A dyaic syse coposed of a fiie uber of sorage elees is said o be luped or discree, while a syse coaiig elees, which are dese i physical space, is called coiuous. The aalyical descripio of he dyaics of he discree case is a se of ordiary differeial equaios, while for he coiuous case i is a se of parial differeial equaios. The aalyical foraio of a dyaic syse depeds upo he kieaic or geoeric cosrais ad he physical laws goverig he behaviour of he syse. 1.1 CLASSIFICATION OF VIBRATIONS Vibraios ca be classified io hree caegories: free, forced, ad self-excied. Free vibraio of a syse is vibraio ha occurs i he absece of exeral force. A exeral force ha acs o he syse causes forced vibraios. I his case, he exciig force coiuously supplies eergy o he syse. Forced vibraios ay be eiher deeriisic or rado (see Fig. 1.1). Selfexcied vibraios are periodic ad deeriisic oscillaios. Uder cerai codiios, he 1

2 SOLVING VIBRATION ANALYSIS PROBLEMS USING MATLAB equilibriu sae i such a vibraio syse becoes usable, ad ay disurbace causes he perurbaios o grow uil soe effec liis ay furher growh. I coras o forced vibraios, he exciig force is idepede of he vibraios ad ca sill persis eve whe he syse is preveed fro vibraig. x x = x() Fig. 1.1(a) A deeriisic (periodic) exciaio. x Fig. 1.1(b) Rado exciaio. 1.2 ELEMENTARY PARTS OF VIBRATING SYSTEMS I geeral, a vibraig syse cosiss of a sprig (a eas for sorig poeial eergy), a ass or ieria (a eas for sorig kieic eergy), ad a daper (a eas by which eergy is gradually los) as show i Fig. 1.2. A udaped vibraig syse ivolves he rasfer of is poeial eergy o kieic eergy ad kieic eergy o poeial eergy, aleraively. I a daped vibraig syse, soe eergy is dissipaed i each cycle of vibraio ad should be replaced by a exeral source if a seady sae of vibraio is o be aiaied.

INTRODUCTION TO MECHANICAL VIBRATIONS 3 Sprig k Daper c Saic equilibriu posiio 0 Mass Exciaio force F() Displacee x Fig. 1.2 Eleeary pars of vibraig syses. 1.3 PERIODIC MOTION Whe he oio is repeaed i equal iervals of ie, i is kow as periodic oio. Siple haroic oio is he siples for of periodic oio. If x() represes he displacee of a ass i a vibraory syse, he oio ca be expressed by he equaio x = A cos ω = A cos 2π τ where A is he apliude of oscillaio easured fro he equilibriu posiio of he ass. The repeiio ie τ is called he period of he oscillaio, ad is reciprocal, f = 1, is called he τ frequecy. Ay periodic oio saisfies he relaioship x () = x ( + τ) 2π Tha is Period τ = s/cycle ω Frequecy f = 1 τ = ω cycles/s, or Hz 2π ω is called he circular frequecy easured i rad/sec. The velociy ad acceleraio of a haroic displacee are also haroic of he sae frequecy, bu lead he displacee by π/2 ad π radias, respecively. Whe he acceleraio X && of a paricle wih reciliear oio is always proporioal o is displacee fro a fixed poi o he pah ad is direced owards he fixed poi, he paricle is said o have siple haroic oio. The oio of ay vibraig syses i geeral is o haroic. I ay cases he vibraios are periodic as i he ipac force geeraed by a forgig haer. If x() is a periodic fucio wih period τ, is Fourier series represeaio is give by x() = a 0 2 + (a cos ω + b si ω) = 1 where ω = 2π/τ is he fudaeal frequecy ad a 0, a 1, a 2,, b 1, b 2, are cosa coefficies, which are give by: zτ a 0 = 2 τ 0 x() d

4 SOLVING VIBRATION ANALYSIS PROBLEMS USING MATLAB zτ a = 2 x() cos ω d τ 0 b = 2 zτ x() si ω d τ 0 The expoeial for of x() is give by: x() = iω ce = The Fourier coefficies c ca be deeried, usig c = 1 z τ (x) e iω d τ 0 The haroic fucios a cos ω or b si ω are kow as he haroics of order of he periodic fucio x(). The haroic of order has a period τ/. These haroics ca be ploed as verical lies i a diagra of apliude (a ad b ) versus frequecy (ω) ad is called frequecy specru. 1.4 DISCRETE AND CONTINUOUS SYSTEMS Mos of he echaical ad srucural syses ca be described usig a fiie uber of degrees of freedo. However, here are soe syses, especially hose iclude coiuous elasic ebers, have a ifiie uber of degree of freedo. Mos echaical ad srucural syses have elasic (deforable) elees or copoes as ebers ad hece have a ifiie uber of degrees of freedo. Syses which have a fiie uber of degrees of freedo are kow as discree or luped paraeer syses, ad hose syses wih a ifiie uber of degrees of freedo are called coiuous or disribued syses. 1.5 VIBRATION ANALYSIS The oupus of a vibraig syse, i geeral, deped upo he iiial codiios, ad exeral exciaios. The vibraio aalysis of a physical syse ay be suarised by he four seps: 1. Maheaical Modellig of a Physical Syse 2. Forulaio of Goverig Equaios 3. Maheaical Soluio of he Goverig Equaios 1. Maheaical odellig of a physical syse The purpose of he aheaical odellig is o deerie he exisece ad aure of he syse, is feaures ad aspecs, ad he physical elees or copoes ivolved i he physical syse. Necessary assupios are ade o siplify he odellig. Iplici assupios are used ha iclude: (a) A physical syse ca be reaed as a coiuous piece of aer (b) Newo s laws of oio ca be applied by assuig ha he earh is a ieral frae (c) Igore or eglec he relaivisic effecs All copoes or elees of he physical syse are liear. The resulig aheaical odel ay be liear or o-liear, depedig o he give physical syse. Geerally speakig, all physical syses exhibi o-liear behaviour. Accurae aheaical odel-

INTRODUCTION TO MECHANICAL VIBRATIONS 5 lig of ay physical syse will lead o o-liear differeial equaios goverig he behaviour of he syse. Ofe, hese o-liear differeial equaios have eiher o soluio or difficul o fid a soluio. Assupios are ade o liearise a syse, which peris quick soluios for pracical purposes. The advaages of liear odels are he followig: (1) heir respose is proporioal o ipu (2) superposiio is applicable (3) hey closely approxiae he behaviour of ay dyaic syses (4) heir respose characerisics ca be obaied fro he for of syse equaios wihou a deailed soluio (5) a closed-for soluio is ofe possible (6) uerical aalysis echiques are well developed, ad (7) hey serve as a basis for udersadig ore coplex o-liear syse behaviours. I should, however, be oed ha i os o-liear probles i is o possible o obai closed-for aalyic soluios for he equaios of oio. Therefore, a copuer siulaio is ofe used for he respose aalysis. Whe aalysig he resuls obaied fro he aheaical odel, oe should realise ha he aheaical odel is oly a approxiaio o he rue or real physical syse ad herefore he acual behaviour of he syse ay be differe. 2. Forulaio of goverig equaios Oce he aheaical odel is developed, we ca apply he basic laws of aure ad he priciples of dyaics ad obai he differeial equaios ha gover he behaviour of he syse. A basic law of aure is a physical law ha is applicable o all physical syses irrespecive of he aerial fro which he syse is cosruced. Differe aerials behave differely uder differe operaig codiios. Cosiuive equaios provide iforaio abou he aerials of which a syse is ade. Applicaio of geoeric cosrais such as he kieaic relaioship bewee displacee, velociy, ad acceleraio is ofe ecessary o coplee he aheaical odellig of he physical syse. The applicaio of geoeric cosrais is ecessary i order o forulae he required boudary ad/or iiial codiios. The resulig aheaical odel ay be liear or o-liear, depedig upo he behaviour of he elees or copoes of he dyaic syse. 3. Maheaical soluio of he goverig equaios The aheaical odellig of a physical vibraig syse resuls i he forulaio of he goverig equaios of oio. Maheaical odellig of ypical syses leads o a syse of differeial equaios of oio. The goverig equaios of oio of a syse are solved o fid he respose of he syse. There are ay echiques available for fidig he soluio, aely, he sadard ehods for he soluio of ordiary differeial equaios, Laplace rasforaio ehods, arix ehods, ad uerical ehods. I geeral, exac aalyical soluios are available for ay liear dyaic syses, bu for oly a few oliear syses. Of course, exac aalyical soluios are always preferable o uerical or approxiae soluios. 4. Physical ierpreaio of he resuls The soluio of he goverig equaios of oio for he physical syse geerally gives he perforace. To verify he validiy of he odel, he prediced perforace is copared wih he experieal resuls. The odel ay have o be refied or a ew odel is developed ad a ew predicio copared wih he experieal resuls. Physical ierprea-

6 SOLVING VIBRATION ANALYSIS PROBLEMS USING MATLAB io of he resuls is a ipora ad fial sep i he aalysis procedure. I soe siuaios, his ay ivolve (a) drawig geeral ifereces fro he aheaical soluio, (b) develope of desig curves, (c) arrive a a siple ariheic o arrive a a coclusio (for a ypical or specific proble), ad (d) recoedaios regardig he sigificace of he resuls ad ay chages (if ay) required or desirable i he syse ivolved. 1.5.1 COMPONENTS OF VIBRATING SYSTEMS (a) Siffess elees Soe ies i requires fidig ou he equivale sprig siffess values whe a coiuous syse is aached o a discree syse or whe here are a uber of sprig elees i he syse. Siffess of coiuous elasic elees such as rods, beas, ad shafs, which produce resorig elasic forces, is obaied fro deflecio cosideraios. The siffess coefficie of he rod (Fig. 1.3) is give by k = EA l The cailever bea (Fig.1.4) siffess is k = 3 EI l3 The orsioal siffess of he shaf (Fig.1.5) is K = GJ l E,A, l EA k= l u F Fig.1.3 Logiudial vibraio of rods. E,I, l F v k= 3EI l 3 Fig.1.4 Trasverse vibraio of cailever beas.

INTRODUCTION TO MECHANICAL VIBRATIONS 7 G,J, l T k= GJ l Fig. 1.5 Torsioal syse. Whe here are several sprigs arraged i parallel as show i Fig. 1.6, he equivale sprig cosa is give by algebraic su of he siffess of idividual sprigs. Maheaically, k eq = i = 1 k i k 1 k 2 k Fig. 1.6 Sprigs i parallel. Whe he sprigs are arraged i series as show i Fig. 1.7, he sae force is developed i each sprig ad is equal o he force acig o he ass. k 1 k 2 k 3 k Fig. 1.7 Sprigs i series. The equivale siffess k eq is give by: 1 1/k eq = 1 i = 1 k i Hece, whe elasic elees are i series, he reciprocal of he equivale elasic cosa is equal o he reciprocals of he elasic cosas of he elees i he origial syse. (b) Mass or ieria elees The ass or ieria elee is assued o be a rigid body. Oce he aheaical odel of he physical vibraig syse is developed, he ass or ieria elees of he syse ca be easily ideified.

8 SOLVING VIBRATION ANALYSIS PROBLEMS USING MATLAB (c) Dapig elees I real echaical syses, here is always eergy dissipaio i oe for or aoher. The process of eergy dissipaio is referred o i he sudy of vibraio as dapig. A daper is cosidered o have eiher ass or elasiciy. The hree ai fors of dapig are viscous dapig, Coulob or dry-fricio dapig, ad hyseresis dapig. The os coo ype of eergy-dissipaig elee used i vibraios sudy is he viscous daper, which is also referred o as a dashpo. I viscous dapig, he dapig force is proporioal o he velociy of he body. Coulob or dry-fricio dapig occurs whe slidig coac ha exiss bewee surfaces i coac are dry or have isufficie lubricaio. I his case, he dapig force is cosa i agiude bu opposie i direcio o ha of he oio. I dry-fricio dapig eergy is dissipaed as hea. Solid aerials are o perfecly elasic ad whe hey are defored, eergy is absorbed ad dissipaed by he aerial. The effec is due o he ieral fricio due o he relaive oio bewee he ieral plaes of he aerial durig he deforaio process. Such aerials are kow as visco-elasic solids ad he ype of dapig which hey exhibi is called as srucural or hysereic dapig, or aerial or solid dapig. I ay pracical applicaios, several dashpos are used i cobiaio. I is quie possible o replace hese cobiaios of dashpos by a sigle dashpo of a equivale dapig coefficie so ha he behaviour of he syse wih he equivale dashpo is cosidered ideical o he behaviour of he acual syse. 1.6 FREE VIBRATION OF SINGLE DEGREE OF FREEDOM SYSTEMS The os basic echaical syse is he sigle-degree-of-freedo syse, which is characerized by he fac ha is oio is described by a sigle variable or coordiaes. Such a odel is ofe used as a approxiaio for a geerally ore coplex syse. Exciaios ca be broadly divided io wo ypes, iiial exciaios ad exerally applied forces. The behavior of a syse characerized by he oio caused by hese exciaios is called as he syse respose. The oio is geerally described by displacees. 1.6.1 FREE VIBRATION OF AN UNDAMPED TRANSLATIONAL SYSTEM The siples odel of a vibraig echaical syse cosiss of a sigle ass elee which is coeced o a rigid suppor hrough a liearly elasic assless sprig as show i Fig. 1.8. The ass is cosraied o ove oly i he verical direcio. The oio of he syse is described by a sigle coordiae x() ad hece i has oe degree of freedo (DOF). k L Fig. 1.8 Sprig ass syse.

INTRODUCTION TO MECHANICAL VIBRATIONS 9 The equaio of oio for he free vibraio of a udaped sigle degree of freedo syse ca be rewrie as &&x() + kx () = 0 Dividig hrough by, he equaio ca be wrie i he for &&x() + ω 2 x () = 0 i which ω = codiios k / is a real cosa. The soluio of his equaio is obaied fro he iiial x(0) = x 0, &x(0) = v 0 where x 0 ad v 0 are he iiial displacee ad iiial velociy, respecively. The geeral soluio ca be wrie as x() = A 1 e i ω + A e i ω 2 i which A 1 ad A 2 are cosas of iegraio, boh coplex quaiies. I ca be fially siplified as: x() = X e + e = X cos (ω 2 φ) so ha ow he cosas of iegraio are X ad φ. This equaio represes haroic oscillaio, for which reaso such a syse is called a haroic oscillaor. There are hree quaiies defiig he respose, he apliude X, he phase agle φ ad he frequecy ω, he firs wo depedig o exeral facors, aely, he iiial exciaios, ad he hird depedig o ieral facors, aely, he syse paraeers. O he oher had, for a give syse, he frequecy of he respose is a characerisic of he syse ha says always he sae, idepedely of he iiial exciaios. For his reaso, ω is called he aural frequecy of he haroic oscillaor. The cosas X ad φ are obaied fro he iiial codiios of he syse as follows: X = x i ( ω φ) i ( ω φ) F v + H G I ω K J 0 2 0 v ad φ = a 1 x L NM 0 ω 0 2 O QP The ie period τ, is defied as he ie ecessary for he syse o coplee oe vibraio cycle, or as he ie bewee wo cosecuive peaks. I is relaed o he aural frequecy by τ = 2 π = 2π ω k Noe ha he aural frequecy ca also be defied as he reciprocal of he period, or f = 1 1 k = τ 2π i which case i has uis of cycles per secod (cps), where oe cycle per secod is kow as oe Herz (Hz).

10 SOLVING VIBRATION ANALYSIS PROBLEMS USING MATLAB 1.6.2 FREE VIBRATION OF AN UNDAMPED TORSIONAL SYSTEM A ass aached o he ed of he shaf is a siple orsioal syse (Fig. 1.9). The ass of he shaf is cosidered o be sall i copariso o he ass of he disk ad is herefore egleced. k l I G Fig. 1.9 Torsioal syse. The orque ha produces he wis M is give by M = GJ l where J = he polar ass oe of ieria of he shaf J = diaeer d I K G = shear odulus of he aerial of he shaf. l = legh of he shaf. The orsioal sprig cosa k is defied as k = T GJ θ = l The equaio of oio of he syse ca be wrie as: I G && θ + k θ = 0 F HG πd 4 32 The aural circular frequecy of such a orsioal syse is ω = The geeral soluio of equaio of oio is give by &θ 0 θ() = θ 0 cos ω + si ω ω 1.6.3 ENERGY METHOD for a circular shaf of F HG k I I GKJ 1/ 2 Free vibraio of syses ivolves he cyclic ierchage of kieic ad poeial eergy. I udaped free vibraig syses, o eergy is dissipaed or reoved fro he syse. The kieic eergy T is sored i he ass by virue of is velociy ad he poeial eergy U is sored i he for of srai eergy i elasic deforaio. Sice he oal eergy i he syse

INTRODUCTION TO MECHANICAL VIBRATIONS 11 is cosa, he priciple of coservaio of echaical eergy applies. Sice he echaical eergy is coserved, he su of he kieic eergy ad poeial eergy is cosa ad is rae of chage is zero. This priciple ca be expressed as T + U = cosa d or (T + U) = 0 d where T ad U deoe he kieic ad poeial eergy, respecively. The priciple of coservaio of eergy ca be resaed by T 1 + U 1 = T 2 + U 2 where he subscrips 1 ad 2 deoe wo differe isaces of ie whe he ass is passig hrough is saic equilibriu posiio ad selec U 1 = 0 as referece for he poeial eergy. Subscrip 2 idicaes he ie correspodig o he axiu displacee of he ass a his posiio, we have he T 2 = 0 ad T 1 + 0 = 0 + U 2 If he syse is udergoig haroic oio, he T 1 ad U 2 deoe he axiu values of T ad U, respecively ad herefore las equaio becoes T ax = U ax I is quie useful i calculaig he aural frequecy direcly. 1.6.4 STABILITY OF UNDAMPED LINEAR SYSTEMS The ass/ieria ad siffess paraeers have a affec o he sabiliy of a udaped sigle degree of freedo vibraory syse. The ass ad siffess coefficies eer io he characerisic equaio which defies he respose of he syse. Hece, ay chages i hese coefficie will lead o chages i he syse behavior or respose. I his secio, he effecs of he syse ieria ad siffess paraeers o he sabiliy of he oio of a udaped sigle degree of freedo syse are exaied. I ca be show ha by a proper selecio of he ieria ad siffess coefficies, he isabiliy of he oio of he syse ca be avoided. A sable syse is oe which execues bouded oscillaios abou he equilibriu posiio. 1.6.5 FREE VIBRATION WITH VISCOUS DAMPING Viscous dapig force is proporioal o he velociy &x of he ass ad acig i he direcio opposie o he velociy of he ass ad ca be expressed as F = c &x where c is he dapig cosa or coefficie of viscous dapig. The differeial equaio of oio for free vibraio of a daped sprig-ass syse (Fig. 1.10) is wrie as: x&& + c & x + k x =0

12 SOLVING VIBRATION ANALYSIS PROBLEMS USING MATLAB k c k( + x) cẋ x g (a) (b) Fig. 1.10 Daped sprig-ass syse. By assuig x() = Ce s as he soluio, he auxiliary equaio obaied is c s s k + + = which has he roos 2 0 s 1, 2 = c c 2 2 2 ± F H G I K J The soluio akes oe of hree fors, depedig o wheher he quaiy (c/2) 2 k/ is zero, posiive, or egaive. If his quaiy is zero, c = 2ω This resuls i repeaed roos s 1 = s 2 = c/2, ad he soluio is x() = (A + B)e (c/2) As he case i which repeaed roos occur has special sigificace, we shall refer o he correspodig value of he dapig cosa as he criical dapig cosa, deoed by C c = 2ω. The roos ca be wrie as: e j s 1, 2 = ζ ± ζ2 1 ω where ω = (k/) 1/2 is he circular frequecy of he correspodig udaped syse, ad ζ = c c C = 2 ω c is kow as he dapig facor. If ζ < 1, he roos are boh iagiary ad he soluio for he oio is x() = Xe ζω k si ( ω + φ) where ω d = 1 ζ2 ω is called he daped circular frequecy which is always less ha ω, ad φ is he phase agle of he daped oscillaios. The geeral for of he oio is show i Fig. 1.11. For oio of his ype, he syse is said o be uderdaped. d

INTRODUCTION TO MECHANICAL VIBRATIONS 13 x() Xe <1 Fig. 1.11 The geeral for of oio. If ζ = 1, he dapig cosa is equal o he criical dapig cosa, ad he syse is said o be criically daped. The displacee is give by x() = (A + B)e ω The soluio is he produc of a liear fucio of ie ad a decayig expoeial. Depedig o he values of A ad B, ay fors of oio are possible, bu each for is characerized by apliude which decays wihou oscillaios, such as is show i Fig. 1.12. x() =1 Fig. 1.12 Apliude decayig wihou oscillaios. I his case ζ > 1, ad he syse is said o be overdaped. The soluio is give by: x() = Ce ( ζ + ζ2 1) ω ( ζ ζ2 1) ω + Ce 1 2 The oio will be o-oscillaory ad will be siilar o ha show i Fig. 1.13. x() >1 Fig. 1.13 No-oscillaory oio. 1.6.6 LOGARITHMIC DECREMENT The logarihic decree represes he rae a which he apliude of a free daped vibraio decreases. I is defied as he aural logarih of he raio of ay wo successive apliudes.