ME3 Setio 4.1 Fored Vibratio Respose of Liear Syste Nov 4, Whe a liear ehaial syste is exited by a exteral fore, its respose will deped o the for of the exitatio fore F(t) ad the aout of dapig whih is preset. The free-body diagra is show below. The equatio of otio is: x && + x& + kx = F() t (1) k kl l x kx x& g F(t) The solutio to equatio (1) will have two parts. If Ft ( ) =, we have the hoogeeous differetial equatio whose solutio orrespods physially to daped free vibratio. This is also alled the trasiet solutio, sie it will always evetually die out i tie. (Reall fro earlier lass that the hoogeeous solutio is ultiplied by a expoetial futio with egative expoet this is the effet of dapig.) Whe Ft ( ), we obtai a partiular solutio (the fored respose) that is due to the exitatio. This is also alled the steady-state solutio ad persists as log as F(t) is preset. Review of Trasiet Solutio Proedure Whe Ft ( ) =, we get the hoogeeous equatio: x && + x& + kx = () The traditioal solutio ethod (sie, ad k are all ostats) is to assue a solutio of the for x = e rt, whih yields the harateristi equatio: r r k + + = (3) 1
The roots of this equatio are: r, r 1 k = ± (4) ad the geeral solutio to the hoogeeous equatio is give by: x() t = Ae + Be rt 1 rt ( ( / ) k / ) t ( ( / ) k / ) ( / ) t = e Ae + Be t (5) ( / ) t The first ter e is siply a expoetially deayig futio of tie. The behaviour of the ters i brakets depeds o whether the disriiat is positive, zero or egative. Whe > 4k, the expoets of the braketed ters are real ubers ad o osillatios are possible. This ase is referred to as overdaped. A overdaped syste rapidly returs to equilibriu whe perturbed. Whe < 4k, the expoets beoe iagiary ubers result a the be expressed as sies ad osies, viz.: ± i k/ ( / ) t. The ( ) ( / ) t x( t) = e Aos( k/ ( / ) t) + Bsi( k/ ( / ) t) (6) Critial Dapig. For the liitig ase betwee the osillatory ad o-osillatory otio, we defie ritial dapig as the value of whih gives a zero disriiat: k = = ω (7) or = k = ω (8) Here ω = k/ is the atural frequey of vibratio of the udaped syste. It is oveiet to defie the dapig ratio ζ, ζ = = (9) ω
so that = ζω (1) With this relatio, the roots of the harateristi equatio (4) beoe: r r = ω ζ ± ζ (11) 1, ( 1) The three ases of dapig ow deped o whether ζ > 1 (overdaped), ζ < 1 (uderdaped) or ζ = 1 (ritially daped). We are usually ost iterested i the uderdaped ase, with ζ < 1. For this ase the roots are oplex: r r 1, ω( ζ i 1 ζ ) = ± (1) ad ( ζ ω ζ ω ) ζω x( t) = e Aos( 1 t) + Bsi( 1 t) (13) t This a also be writte as, xt = e X t (14) ζωt () os( 1 ζ ω φ) where X = A + B ad ϕ = Ta A 1 B (15) N.B. Note that the agle ϕ i equatio (14) is always expressed i radias!! Equatio (14) idiates that the frequey of daped osillatio is equal to = 1 (16) µ ω ζ For sall dapig (ζ << 1), we a expad (16) i a bioial series ad take oly the lead ter: 1 µ ω(1 ζ ) = ω 1 (17) 8 k (I equatio (17) we have used the bioial series ( 1) + + + +.) (1 ε) 1 ε ε... 3
Thus we see that the frequey of osillatio of the daped syste is always less tha that of the udaped syste. The geeral behaviour of the uderdaped solutio (equatio 14) is plotted below i Figure 1. Xosϕ Figure 1. Daped osillatio behaviour for ζ < 1. Fored Osillatios The solutio (13) to the hoogeeous (ufored) equatio of otio is soeties alled the trasiet solutio, sie it will always deay to zero after suffiiet tie has passed. I ost vibratio probles of iterest there is a o-zero forig futio F(t). This is usually a haroi fore, e.g.: Ft () = Fos( ωt) (18) The fored equatio of otio beoes: x && + x& + kx = F os( ωt) (19) The trasiet (opleetary) solutio is still give by equatios (13) or (14), but ow we eed to fid a partiular solutio x () t satisfyig equatio (19), whih is the respose p to the forig futio. We a use the ethod of udeteried oeffiiets, with xp () t = Dos( ωt) + Esi( ωt) () This is also alled the steady-state solutio. Substitutig ito the O.D.E. we fid: ( ) ( ) ω D+ ωe+ kd os( ωt) + ω E ωd+ ke si( ωt) = F os( ωt) (1) 4
This yields two equatios for D ad E: ( ω ) k D+ ωe = F ωd + k E = ( ω ) () fro whih ( k ω ) ω D = F ad E = F ( k ω ) + ( ω) ( k ω ) + ( ω) (3) We a rewrite the partiular solutio () as xp () t = X os( ωt ϕ) where X = D + E = F ( k ω ) + ( ω ) (4) ad 1 ω ϕ = Ta k ω (5) Before plottig these results, it is useful to express the o-diesioally. Reallig that: k ω = = atural frequey of udaped osillatios = ω = ritial dapig ζ = = dapig fator (6) We a write the o-diesioal aplitude ad phase as: Xk F 1 ζωω ( / ) 1 ( ω/ ω) 1 = ; ϕ = ta 1 ( ω/ ω) + [ ζ( ω/ ω) ] (7) These equatios idiate that the o-diesioal aplitude Xk/F ad phase agle are futios oly of the frequey ratio ( ω / ω ) ad the dapig fator ζ. The results are plotted i Figure. 5
Figure. Plot of o-diesioal aplitude ad phase agle. The aplitude of the fored vibratio depeds o the ratio of the drivig frequey ω to the atural frequey of vibratio ω. Whe ( ω / ω ) 1, the aplitude approahes X = F / k, whih is equivalet to pullig the sprig with a ostat fore F. Whe the forig frequey is very high, ( ω / ω ) 1, the aplitude of respose goes to zero, i.e. it is ot possible to exite the ass with very high frequey exitatio i this ase. The pheoeo of resoae ours whe ω ω. This leads to high aplitude vibratios, liited oly by the dapig of the syste. To fid preisely the frequey of vibratio whih produes the axiu respose, we differetiate equatio (7) with respet to ω. Settig this derivative to zero, we fid ax = 1 = 1 (8) k ω ω ζ ω Thus for sall dapig ( ζ ), ωax ω. The pheoeo of resoae is geerally soethig you wat to avoid i a egieerig struture. If there is a possibility of a exitig fore ear a atural frequey of vibratio, the you should esure that the dapig is high eough to prevet atastrophi aplitudes of vibratio. 6