( ) Ce, 1 System with Mass, Spring, and Viscous Damper = (2) s are unknown constants. Substituting (2) into (1), we get. Ce ms cs k. ms cs k.

Transcription

1 Syste with Mass, Sprig, a Visous Daper by We ow fro Lesso that the visous apig fore F is give F =, where is the apig ostat or oeffiiet of visous apig Vibratig systes are all subjet to apig to soe egree beause eergy is issipate by fritio a other resistaes If the apig is sall, it has very little ifluee o the atural frequeies of the syste, a hee the alulatios for the atural frequeies are geerally ae o the basis of o apig Step : Matheatial Moel 83 Mehaial Vibratios Lesso 3 Step : Goverig Equatio The syste i Figure has a equatio of otio Step 3: Solutio of the Equatio We assue a solutio of () i the for where C a + + = () st Ce, t = () s are uow ostats Substitutig () ito (), we get ( st ) ( st ) ( st ) st ( + + ) =, Cse + Cse + Ce =, Ce s s whih leas to the harateristi equatio s s + + = Δ Equilibriu Positio The roots of the harateristi equatio are s, ± = 4 = ± g Figure : A syste with ass, sprig, a aper I this setio, we will eaie losely the syste i Figure Therefore, the geeral solutio of () is give by a obiatio of the two solutios () + t t, t = Ce + Ce (3) Copyright 7 by Withit Chatlataagulhai

2 C C where a are geeral ostats that a be eterie fro iitial oitios a The solutio (3) a be rearrage to () t t t t = e Ce + Ce Step 4: Iterpretatio of the Results t e Cosier the solutio (4), the first ter is a epoetially eayig futio of tie The ters i the paretheses epes o whether the value of ters uer the square root is positive, zero, or egative Whe >, we have a overape ase I overape ase, the value of ters uer the square root is positive, therefore the powers of the epoetial futios are real ubers a there is o osillatio Whe =, we have a ritially ape ase I this ase, the value of ters uer the square root is zero, therefore there is oly oe epoetial ter left a there is o osillatio Whe <, we have a uerape ase I this ase, the value of ters uer the square root is egative, therefore the powers of the epoetial ters are ople ubers ± i / ( / ) t iθ Beause fro Euler s forula, e = osθ + isi θ, we have (4) ± i t 83 Mehaial Vibratios Lesso 3 e = os t ± isi t si The ters os a are osillatory, therefore, we will see the osillatio i this ase Critial Dapig Costat a Dapig Ratio Sie = separates the overape ase fro the uerape ase, we all the orrespoig apig ratio the ritial apig ostat, whih is give by = = Ay apig a the be epresse i ters of the ritial apig by a oiesioal uber ζ alle apig ratio, whih is give by ζ = Usig the apig ratio, the equatio of otio () a be rewritte as + + = ζ, a the roots of the harateristi equatio a be rewritte as Copyright 7 by Withit Chatlataagulhai

3 83 Mehaial Vibratios Lesso 3 s, = ± = ζ ± ζ = ζ ± ζ s, ζ ζ = ±, (5) < ζ < ζ Iagiary Ais ζ = s Figure shows a ople plae that otais loatios of s / a s / ζ = or the uape ase, the loatios of s = ± i Whe < ζ <, Startig whe, s obtaie fro (5) are s / a /, / ζ > ζ = ζ Real Ais or the uerape ase, we have loatios of / irle Whe, s /, ζ i ζ = ± The ove to the left alog the ar of the uit s a s / ζ = or the ritially ape ase, we have both s, / = Whe ζ >, or the overape ase, we have s = ζ ± ζ s / a s /, / The loatios of alog the horizotal ais a reai real ubers separate < ζ < s ζ = Figure : Loatios of / a ople plae s a s / o 3 Copyright 7 by Withit Chatlataagulhai

4 Uerape Case We wat to tae a loser loo at the solutio (4) i eah ase First, whe i uerape ase, < ζ <, we have () t t t t = e Ce + Ce = e Ce + C e ( ζ) t ( ζ) t ζt = e Ce + C e ζ t ζ t ζt i ζ t ζt e Ce = + Ce i ζ t Cos( ζ t) + icsi ( ζ t ) + Cos( ζ t) + icsi ( ζ t) os( ) si ( ) ζ ζ ' ' os( ) si ( ) ζ ζ ζt = e { } { } = + + ζt e C C t i C C t = + ζt e C t C t Usig the trigooetri ietity 83 Mehaial Vibratios Lesso 3 ζt ' ' () = os( ζ ) + si ( ζ ) ζt = Xe si ( ζ t) + φ t e C t C t ' ' The ostats ( ) ( ) (6) C, C, C, C, a ( X, φ ) a be fou after applyig iitial oitios a ( ) For the iitial oitios = a = C = a C ' a therefore the solutio beoes = + ζ ' ζ, we have t e os t si t (7) ζ ζ () t + ζ = ζ + ζ The solutio above a be plotte as Figure 3, Asiα + Bosα = A + B os α φ = A + B si α + φ, where A B φ = ta, φ = ta, B A we have Figure 3: Uerape solutio 4 Copyright 7 by Withit Chatlataagulhai

5 We a see that the otio osillates with a frequey = ζ, whih is alle the frequey of ape osillatio It a be see that < i the uerape ase, a whe ζ 3 Logarithi Dereet A oveiet way to eterie the aout of apig preset i a syste is to easure the rate of eay of free osillatios The larger the apig, the greater will be the rate of eay A ter alle logarithi ereet δ is efie as δ = l, ζ ζ πζ Whe, ζτ = l e = ζτ π = ζ πζ = ζ 83 Mehaial Vibratios Lesso 3 Eaple : [] For a vibratig syste with visous apig, w = lb, = 3 lb / i, a = lb/ i/ s Deterie the logarithi ereet a the ratio of ay two suessive aplitues where a are two suessive aplitues Fro (6), we have δ = l = l Xe = l Xe = l Xe e = l e Xe ζ t ( t τ ) ζ + ( t τ ) Xe ( t τ ) ( t τ ) ζ Xe ζ + ζ + ζ + ζ ( t) + φ ( t ) ( t) + t si si ( τ ) φ + + t si φ π si t + + φ t si ζ ( t) + φ ( t ) si + π + φ 5 Copyright 7 by Withit Chatlataagulhai

6 Solutio equal g = 386 i/ s The aeleratio of gravity is We have the ass The atural frequey is = w/ g = / 386 = 6 83 Mehaial Vibratios Lesso 3 Eaple : [] Show that the logarithi ereet is also give by the equatio δ = l, where represets the aplitue after yles have elapse The ritial apig oeffiiet is The apig ratio ζ is The logarithi ereet is 3 = = = 34 ra / s 6 = = 6 34 = 76 lb / i / s ζ = = = πζ π 68 δ = = = 49 ζ 68 The aplitue ratio for ay two oseutive yles is δ 49 = e = e = 54 6 Copyright 7 by Withit Chatlataagulhai

7 Solutio The aplitue ratio for ay two oseutive aplitues is 83 Mehaial Vibratios Lesso 3 The ratio is obtaie by = = = = = e 3 δ = = ( e ) = e δ δ Therefore, we have δ = l vt v t Eaple 3: [] Cosier a forgig haer i Figure 4 The avil together with the fouatio blo weighs 5, N The soil stiffess is 6 5 N/ The visous apig ostat is, Ns/ The tup weighs, N, is ae to fall fro a height of o to the avil If the avil is at rest before ipat by the tup, eterie the respose of the avil after the ipat Assue that the oeffiiet of restitutio betwee the avil a the tup is 4 M va v a Figure 4: A forgig haer a its atheatial oel 7 Copyright 7 by Withit Chatlataagulhai

8 83 Mehaial Vibratios Lesso 3 Solutio First, we wat to fi iitial veloity of the avil a the fouatio blo right after the ipat Fro the priiple of oservatio of oetu M v + v = Mv + v, a t a t (8) v, v where a a are veloities of the avil before a after the ipat respetively, a vt, vt are veloities of the tup before a after the ipat respetively We ow that oservatio of eergy We have v t v = To fi, we use the a = gh vt = gh = 98 = 66 / s We a obtai i ters of by usig the efiitio of the oeffiiet of restitutio vt v a va v t r =, va vt v v = 66 v = v + 5 a t 4, a t Substitute the above ito (8), we have v t 8 Copyright 7 by Withit Chatlataagulhai

9 5 ( va ) = ( 66 vt), ( va ) = ( 66 ( va 5 )), v = 46 / s a 83 Mehaial Vibratios Lesso 3 Eaple 4: [] Cosier a sho absorber syste for a otoryle of ass 4 g i Figure 5(a) The resultig isplaeet-tie urve is to be i Figure 5(b) If the ape perio of vibratio is to be s a the aplitue /4, 5 = eterie the eessary stiffess a apig ostat of the sho absorber Therefore the iitial oitios of the avil a the fouatio blo are =, = 46 / s The apig oeffiiet is equal to ζ = = = 99 M 6 5 ( 5 ) 98 The uape atural frequey is give by 6 5 = = = 9899 ra / s M 5 98 We a substitute the iforatio above ito the solutio (7) of the uerape ase Figure 5: Sho absorber of a otoryle t e t t ζ ζ () t + ζ = os ζ + si ζ to obtai the respose of the avil a the fouatio blo after the ipat 9 Copyright 7 by Withit Chatlataagulhai

10 Solutio Sie 5 5 logarithi ereet beoes 83 Mehaial Vibratios Lesso 3 = /4, = /4, therefore /6 δ = = = = πζ l l ζ The apig ratio a be fou fro the equatio above to be ζ = 44 The ape perio of vibratio is give to be s, therefore π π = τ = =, ζ = 343 ra / s The ritial apig ostat is give by = = 343 = 3735 Ns/ Thus the apig ostat is give by The stiffess is give by = ζ = = 5545 Ns/ = = 343 = 3583 N/ = The Copyright 7 by Withit Chatlataagulhai

11 4 Overape Case Whe ζ >, we have the overape ase The solutio (4) beoes t () = e Ce + Ce ζ t ζ t ζt ( + ) t ( ) ζ ζ ζ ζ t Ce = Ce + For the iitial oitios = a = C C ostats a as C C ζ + ζ + = ζ ζ ζ = ζ, we obtai the, have 83 Mehaial Vibratios Lesso 3, we After applyig the iitial oitios = a = a the solutio beoes C =, C = +, t = + ( + ) t e te t (9) We a see that this otio is ot osillatory Figure 6 shows opariso of otios with ifferet types of apig We a see that the otio is ot osillatory 5 Critially Dape Case Whe ζ =, we have the ritially ape ase Sie the roots (5) of the harateristi equatio are ouble roots s, =, the solutio of the oriary ifferetial equatio () is t = t + t Ce Cte Figure 6: Copariso of otios with ifferet types of apig Copyright 7 by Withit Chatlataagulhai

12 Eaple 5: [] Whe the ao is fire, the reatio fore pushes the barrel i the opposite iretio of the projetile The reoil ehais, osistig of a sprig a a aper, is esige to brig the barrel to rest i the shortest tie without osillatio Therefore, it is esirable to have ritially ape ase Suppose the barrel a the reoil ehais have a ass of 5 g with a reoil sprig of stiffess, N / The ao reoils 4 upo firig Fi a) The ritial apig oeffiiet of the aper b) The iitial reoil veloity of the ao ) The tie tae by the ao to retur to a positio fro its iitial positio 83 Mehaial Vibratios Lesso 3 Figure 7: Photograph of a ao Copyright 7 by Withit Chatlataagulhai

13 Solutio a) We have, = = = 447 ra / s 5 = = = 447 Ns/ Therefore, b) The respose of the ritially ape ase is give by (9) Whe we have =, Its erivative is give by t = + ( + ) t e te = te t t t = ( ) t e te t () Settig the veloity to zero to fi the tie for the aiu isplaeet t = 4 We have Substitutig ito (), we have t t ( ) e te = t 4 = te, t = 4 = t te 4 = = 486 / s ( 447 ) e ) Fro (), we have = te t = t = 86 s 447t 486 te, 83 Mehaial Vibratios Lesso 3 Syste with Mass, Sprig, a Coulob Daper I ay ehaial systes, Coulob or ry fritio apers are use beause of their ehaial sipliity a oveiee Also i vibratig strutures, wheever the opoets slie relative to eah other, ry fritio apig appears iterally Coulob apig arises whe boies slie o ry surfaes Cosier a syste i Figure 8, whih shows a ass-sprig syste with Coulob apig The Coulob fritio fore is opposite to the iretio of the ass Whe the ass oves to the right, we have a equatio of otio as The solutio of the above equatio is + = μn μn t () = Aost+ Asi t, () = is the atural frequey a where / A a A are two uow ostats to be eterie fro iitial oitios of this half yle Whe the ass oves to the left, we have a equatio of otio as The solutio of the above equatio is + = μn 3 Copyright 7 by Withit Chatlataagulhai

14 μn t () = A3ost+ A4si t+, () where = / is the atural frequey a A 3 a A 4 are two uow ostats to be eterie fro iitial oitios of this half yle 83 Mehaial Vibratios Lesso 3 This solutio is vali for half the yle oly, that is, for t π / Whe t = π /, the ass will be at its etree left positio a will the start to ove ba to the right Whe the ass oves to the right, we have that the iitial positio of the seo half of the yle is π μn t = =, π t = = By applyig the iitial oitios above to the solutio (), we have Figure 8: Sprig-ass syste with Coulob apig Let the iitial oitios be =, = The otio starts fro right to left By applyig the iitial oitios above to the solutio (), we have μn A3 =, A4 = Therefore, the solutio beoes 3μN A =, A = 3μN μn t () = os t This solutio is vali for the seo half of the yle, that is, for π / t π / Figure 9 shows the otio of the ass i this ass-sprig syste with Coulob fritio Therefore, the solutio beoes μn t t μn () = os + 4 Copyright 7 by Withit Chatlataagulhai

15 83 Mehaial Vibratios Lesso 3 stati fritio fore The otio theoretially otiues forever (with a ifiitesially sall aplitue) with visous apig 5 The aplitue reues liearly with Coulob apig, whereas it reues epoetially with visous apig 6 I eah suessive yle, the aplitue of otio is reue by 4 μ N, that is, X 4μN = X Eaple 6: [] A etal blo, plae o a rough surfae, is attahe to a sprig a is give a iitial isplaeet of fro its equilibriu positio After five yles of osillatio i s, the fial positio of the etal blo is fou to be fro its equilibriu positio Fi the oeffiiet of fritio betwee the surfae a the etal blo Figure 9: Motio of the ass with Coulob apig I suary, a syste with Coulob apig has the followig harateristis The equatio of otio is oliear with Coulob apig, while it is liear with visous apig The atural frequey of the syste oes ot hage with the aitio of Coulob apig, while it is reue with the aitio of visous apig 3 The otio is perioi with Coulob apig, while it a be operioi i a visously ape syste (overape or ritially ape ases) 4 The syste oes to rest after soe tie with Coulob apig μn Whe t (), < the syste stops sie the sprig fore is less tha the 5 Copyright 7 by Withit Chatlataagulhai

16 Solutio Sie five yles tae s, the atural frequey is give by 83 Mehaial Vibratios Lesso 3 5 = π f = π = 5 π The reutio i aplitue i five yles is Therefore, 4μN 4μg 5 = 5 = = 9 9 / 9 μ = = = Lesso 3 Hoewor Probles 84, 85, 89, 9, 93, 6,, 5 Hoewor probles are fro the require tetboo (Mehaial Vibratios, by Sigiresu S Rao, Pretie Hall, 4) Referees [] Mehaial Vibratios, by Sigiresu S Rao, Pretie Hall, 4 [] Theory of Vibratio with Appliatios, by Willia T Thoso a Marie Dillo Dahleh, Pretie Hall, Copyright 7 by Withit Chatlataagulhai