CHAPTER 12 TWODEGREE OFFREEDOMSYSTEMS


 Judith Morris
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1 CHAPTER TWODEGREE OFFREEDOMSYSTEMS Introdution to two deree of freedo systes: The vibratin systes, whih require two oordinates to desribe its otion, are aed twodereesof freedo systes. These oordinates are aed eneraized oordinates when they are independent of eah other and equa in nuber to the derees of freedo of the syste. Unie sine deree of freedo syste, where ony one oordinate and hene one equation of otion is required to epress the vibration of the syste, in twodof systes iniu two oordinates and hene two equations of otion are required to represent the otion of the syste. For a onservative natura syste, these equations an be written by usin ass and stiffness atries. One ay find a nuber of eneraized oordinate systes to represent the otion of the sae syste. Whie usin these oordinates the ass and stiffness atries ay be ouped or unouped. When the ass atri is ouped, the syste is said to be dynaiay ouped and when the stiffness atri is ouped, the syste is nown to be statiay ouped. The set of oordinates for whih both the ass and stiffness atri are unouped, are nown as prinipa oordinates. In this ase both the syste equations are independent and individuay they an be soved as that of a sinedof syste. A twodof syste differs fro the sine dof syste in that it has two natura frequenies, and for eah of the natura frequenies there orresponds a natura state of vibration with a dispaeent onfiuration nown as the nora ode. Matheatia ters assoiated with these quantities are eienvaues and eienvetors. Nora ode vibrations are free vibrations that depend ony on the ass and stiffness of the syste and how they are distributed. A nora ode osiation is defined as one in whih eah ass of the syste underoes haroni otion of sae frequeny and passes the equiibriu position siutaneousy. The study of twodof systes is iportant beause one ay etend the sae onepts used in these ases to ore than dof systes. Aso in these ases one an easiy obtain an anaytia or osedfor soutions. But for ore derees of 09
2 freedo systes nueria anaysis usin oputer is required to find natura frequenies (eienvaues) and ode shapes (eienvetors). The above points wi be eaborated with the hep of eapes in this eture. Few eapes of twodereeoffreedo systes Fiure shows two asses and with three sprins havin sprin stiffness, and 3 free to ove on the horizonta surfae. Let and be the dispaeent of ass and respetivey. 3 Fiure As desribed in the previous etures one ay easiy derive the equation of otion by usin d Aebert prinipe or the enery prinipe (Larane prinipe or Haiton s prinipe) Usin d Aebert prinipe for ass free body diara shown in fiure (b) + ( + ) = 0, fro the () ( ) and siiary for ass + ( + ) = 0 () 3 Iportant points to reeber Inertia fore ats opposite to the diretion of aeeration, so in both the free body diaras inertia fores are shown towards eft. For sprin, assuin >, ( ) 3 Fiure (b), Free body diara 0
3 The sprin wi pu ass towards riht by ( ) and it is strethed by (towards riht) it wi eert a fore of ( ) towards eft on ass. Siiary assuin of >, the sprin et opressed by an aount and eert tensie fore ( ). One ay note that in both ases, free body diara reain unhaned. Now if one uses Larane prinipe, The Kineti enery = T = + and (3) Potentia enery = U = + ( ) + 3 (4) So, the Laranian L = T U = + + ( ) + 3 The equation of otion for this free vibration ase an be found fro the Larane prinipe (5) d L L = 0, (6) dt q q and notin that the eneraized oordinate q = and q = whih yieds + ( + ) = 0 + ( + ) = 0 (8) 3 Sae as obtained before usin d Aebert prinipe. (7) Now writin the equation of otion in atri for = 0 +. (9) Here it ay be noted that for the present two dereeoffreedo syste, the syste is dynaiay unouped but statiay ouped.
4 Eape. Consider a athe ahine, whih an be odeed as a riid bar with its enter of ass not oinidin with its eoetri enter and supported by two sprins,,. ( ) C J e G ( + e ) ( + ) Fiure Fiure 3: Free body diara of the syste In this eape, it wi be shown, how the use of different oordinate systes ead to stati and or dynai ouped or unouped equations of otion. Ceary this is a twodereeof freedo syste and one ay epress the oordinate syste in any different ways. Fiure 3 shows the free body diara of the syste where point G is the enter of ass. Point C represents a point on the bar at whih we want to define the oordinates of this syste. This point is at a distane fro the eft end and between points C and G is e. Assuin fro riht end. Distane is the inear dispaeent of point C and the rotation about point C, the equation of otion of this syste an be obtained by usin d Aeber s prinipe. Now suation of a the fores, viz. the sprin fores and the inertia fores ust be equa to zero eads to the foowin equation. + e + ( ) + ( + ) = 0 (0) Aain tain oent of a the fores about point C J + ( + e ) e ( ) + ( + ) = 0 () G Notin J J e = G +, the above two equations in atri for an be written as e + 0 e J + = () + 0
5 Now dependin on the position of point C, few ases an are studied beow. Case : Considerin written as e = 0, i.e., point C and G oinides, the equation of otion an be ( ) ( + ) J + G = (3) + 0 So in this ase the syste is statiay ouped and if =, this oupin disappears, and we obtained unouped and vibrations. Case : If, =, the equation of otion beoes e e J + 0 =. (4) + 0 Hene in this ase the syste is dynaiay ouped but statiay unouped. Case 3: If we hoose otion wi beoe = 0, i.e. point C oinide with the eft end, the equation of e + 0. (5) e J + = 0 Here the syste is both statiay and dynaiay ouped. Nora Mode Vibration Aain onsiderin the probe of the sprinass syste in fiure with =, = 3, = = =, the equation of otion (9) an be written as + ( ) + = 0 ( ) + = 0 (6) 3
6 We define a nora ode osiation as one in whih eah ass underoes haroni otion of the sae frequeny, passin siutaneousy throuh the equiibriu position. For suh otion, we et iωt = Ae, = Ae (7) i ω t Hene, ( ω ) A A = 0 A + ( ω ) A = 0 or, in atri for ω A 0 = ω A 0 Hene for nonzero vaues of A and A (i.e., for nontrivia response) (8) (9) ω = 0. (0) ω Now substitutin 3 ω = λ, equation 6.. yieds λ (3 ) λ+ ( ) = 0 () Hene, λ = 3 ( 3) = and 3 λ = ( + 3) =.366 So, the natura frequenies of the syste are ω = λ = and ω =.366 Now fro equation ()., it ay be observed that for these frequenies, as both the equations are not independent, one an not et unique vaue of and A. So one shoud A find a noraized vaue. One ay noraize the response by findin the ratio of A to A. Fro the first equation (9) the noraized vaue an be iven by A = = A λ ω and fro the seond equation of (9), the noraized vaue an be iven by () A ω λ = = (3) A Now, substitutin ω = λ = in equation () and (3) yieds the sae vaues, as both these equations are ineary dependent. Here, 4
7 A A λ λ = = 0.73 (4) and siiary for ω = λ =.366 A = A λ λ =.73 (5) It ay be noted Equation (9) ives ony the ratio of the apitudes and not their absoute vaues, whih are arbitrary. If one of the apitudes is hosen to be or any nuber, we say that apitudes ratio is noraized to that nuber. The noraized apitude ratios are aed the nora odes and desinated by φ ( ). n Fro equation (4) and (5), the two nora odes of this probe are: φ( ) φ( ) = = In the st nora ode, the two asses ove in the sae diretion and are said to be in phase and in the nd ode the two asses ove in the opposite diretion and are said to be out of phase. Aso in the first ode when the seond ass oves unit distane, the first ass oves 0.73 units in the sae diretion and in the seond ode, when the seond ass oves unit distane; the first ass oves.73 units in opposite diretion. Free vibration usin nora odes When the syste is disturbed fro its initia position, the resutin freevibration of the syste wi be a obination of the different nora odes. The partiipation of different odes wi depend on the initia onditions of dispaeents and veoities. So for a syste the free vibration an be iven by = φ Asin( ωt+ ψ ) + φ Bsin( ω t+ ψ ) (7) 5
8 Here A and B are part of partiipation of first and seond odes respetivey in the resutin free vibration and ψ and ψ are the phase differene. They depend on the initia onditions. This is epained with the hep of the foowin eape. Eape: Let us onsider the sae sprinass probe (fiure 4) for whih the natura frequenies and nora odes are deterined. We have to deterine the resutin free vibration when the syste is iven an initia dispaeent (0) = 5, (0) = and initia veoity (0) = (0) = 0. Fiure 4 Soution: Any free vibration an be onsidered to be the superposition of its nora odes. For eah of these odes the tie soution an be epressed as: 0.73 = sin ωt.73 = sin ωt.00 The enera soution for the free vibration an then be written as: = A sin( ωt+ ψ) + B sin( ωt+ψ).00 where A and B aow different aounts of eah ode and ψ and ψ aows the two odes different phases or startin vaues. Substitutin: 6
9 (0) = = A sinψ+ B sinψ (0) (0) = = A os + B os (0) 0 ω ψ ω ψ osψ = osψ = 0 => ψ = ψ = 90 0 Substitutin in st set: = A + B 0.73A.73B= 5 A=.33 A+B = B=.33 Hene the resutin free vibration is =.33 osωt.33 osωt Nora odes fro eienvaues The equation of otion for a twodereeof freedo syste an be written in atri for as M + K = 0 (8) where M and K are the ass and stiffness atri respetivey; is the vetor of eneraized oordinates. Now preutipyin ay et I M K M in both side of equation 6.. one + = 0 (9) or, I + A= 0 (30) Here = A M K is nown as the dynai atri. Now to find the nora odes, iωt = Xe, = Xe, the above equation wi redue to i ω t [ A λi] X 0 = (3) 7
10 where { } T X = and λ= ω. Fro equation (3) it is apparent that the free vibration probe in this ase is redued to that of findin the eienvaues and eienvetors of the atri A. Eape: Deterine the nora odes of a doube penduu. Soution Kineti enery of the syste = T = + ( + + os( )) Potentia enery of the syste = ( os ) { ( os ) ( os )} {( ) ( os ) ( os )} U = + + = + + So Laranian of the syste = L= T U = + ( + + os( )) ( + )( os ) + ( os ) So usin Larane prinipe, and assuin sa ane of rotation, the equation of otion an be written in atri for as ( + ) ( + ) = 0 0 Fiure 5 { } Now onsiderin a speia ase when = = and = =, the above equation beoes = or, + = Now A= = 0 To find eienvaues of A, 8
11 λ A λi = 0 = 0 λ Or, 4 4 λ+ λ = 0 Or, λ 4 λ+ = 0 4 ± 4 8 Or, λ = = ( ± ) Hene natura frequenies are ω = , ω =.8478 The nora odes an be deterined fro the eienvaues. The orrespondin prinipa odes are obtained as = = ( ) = = ( ) λ= λ + λ= λ It ay be noted that whie in the first ode Both the penduu oves in the sae diretion, Fiure 6 In the seond ode they ove in opposite diretion One ay sove the sae probe by tain and Here is the horizonta distane oves by ass ass T as the eneraized oordinates. and. Fiure 7 show the free body diara of both the asses. T y y Fiure 7 T is the distane ove by 9
12 Fro the free body diara of ass, T os = Tsin = Aso fro the free body diara of ass, T os T os = T sin T sin + = 0 Assuin and to be sa, sin = tan = = / and sin = tan = = ( ) / Hene T =,and T = ( + ) ( + ) = 0 + = Hene in atri for 0 ( + ) = 0 Considerin the ase in whih = = and = =, the above equation beoes = 0 0
13 3 A = 3 λ and A λi = 0 = 0 Hene λ Or, λ 4 λ+ = 0 ( ) and λ ( ) or, λ = = + Sae as those obtained by tain and as the eneraized oordinates. Now X X λ= λ X X λ= λ = = = = λ = = = =.44 3 λ Fiure The different odes are as shown in the above fiure. Eape Deterine the equation of otion if the doube penduu is started with initia onditions (0) = (0) = 0.5, (0) = (0) = 0. Soution: The resutin free vibration an be onsidered to be the superposition of the nora odes. For eah of these odes, the tie soution an be written as
14 X X = sinω t = sinω t X X The enera soution for the free vibration an be written as = Asin( ωt+ ψ) + Bsin( ωt+ ψ) where Aand B are the aounts of first and seond ode s partiipation and ψ andψ are the startin vaues or phases of the two odes. Substitutin the initia onditions in the above equation = Asinψ+ Bsinψ 0.5 and = Aω osψ+ Bωosψ 0 0 For the seond set of equations to be satisfied, osψ = osψ = 0, so that ψ = ψ = 90. Hene A = and B = So the equation for free vibration an be iven by = osωt0.036osωt Dapedfree vibration of twodof systes Consider a two derees of freedo syste with dapin as shown in fiure 3 3 Fiure 9 Now the equation of otion of this syste an be iven by = + + (3) 0 3 3
15 As in the previous ase, here aso the soution of the above equations an be written as st = Ae and = Ae (33) st where A, A and s are onstant. Substitutin (33) in (3), one ay write s + ( + ) s+ + s A 0 = s s + ( + 3) s+ A Now for a nontrivia response i.e., for nonzero vaues of their oeffiient atri ust vanish. Hene A and A (34), the deterinant of s + ( + ) s+ + s s s + ( + 3) s+ + 3 = 0 or, ( s + ( + ) s+ + )( s + ( + ) s+ + ) + ( s+ ) = whih is a fourth order equation in s and is nown as the harateristi equation of the syste. This equation is to be soved to et four roots. The enera soution of the syste an be iven by st st st 3 s4t = Ae + Ae + Ae + Ae 3 4 = A e + A e + A e + A e st st st 3 s4t 3 4 Here A,,,3,4 i i = onditions and the oeffiients fro equation (34) as (35) (36) (37) are four arbitrary onstants to be deterined fro the initia A,,,3,4 i i = are reated to A i and an be deterined A s + = A s + + s + + i i i i ( ) i For a physia syste with dapin, the otion wi die out with tie. For a stabe syste, a the four roots ust be either rea neative nubers or ope nuber with neative rea parts. It ay be reaed that, if the roots ontain ope onjuate nubers, the otion wi be osiatory. (38) Eape: Find the response of the syste as shown in fiure 9 onsiderin = = 3 = and = 3 = 0and =. Soution. In this ase the harateristis equation beoes ( s s )( s s ) ( s ) = 0 = =, 3
16 ( s + s + ) ( s + ) = or, s + s + (4 + ) s + (4 ) s + 4 = 0 4 3, = 0 or s s s s, ( ) + ( ) = 0 or s s s s s,( + )( ) = 0 or s s s,( + )( ) = 0 or s s s Hene the roots are s, =± i and s3,4 = ± 3 So the syste has a pair of ope onjuate SEMIDEFINITE SYSTEMS The systes with have one of their natura frequenies equa to zero are nown as seidefinite or deenerate systes. One an show that the foowin two systes are deenerate systes. I I Fiure 0 Fiure Fro fiure 0 the equation of otion of the syste is 0 0 = (39) iωt iωt Assuin the soution = Ae and = Ae (40) ω A 0 = A ω 0 (4) So for nonzero vaues of A, A, 4
17 ω = 0 ω ( ω )( ω ) or, = 0 4 or, ( ) ω ω 0 (4) (43) + + = (44) or, ω ( ω ( + )) = 0 (45) ω = 0, and, ω = ( + ) Hene, the syste is a seidefinite or deenerate syste. Correspondin to the first ode frequeny, i.e., ω = 0, A = A. So the syste wi have a riidbody otion. For the seond ode frequeny A = = = = A ω ( + ) apitude ratio is inversey proportiona to the ass ratio the syste. Siiary one ay show for the tworotor syste, I = (48) I the ratio of ane of rotation inversey proportiona to the oent of inertia of the rotors. (46) (47) Fored haroni vibration, Vibration Absorber Consider a syste eited by a haroni fore F sinωtepressed by the atri equation F sinωt + = 0 (49) Sine the syste is undaped, the soution an be assued as X = sinωt X Substitutin equation (50) in equation (49), one obtains ω ω X F sinωt = sin t X ω ω ω 0 X ω ω F or, = X ω ω 0 (50) (5) 5
18 X ω F ω = X 0 ω ω Hene X ω + F ω 0 + ω ω = ω ω ( ω ), ω ω F = (53) Z( ω) [ Z ω ] where ( ) X = ( ω ) ω ω = ω ω F Z( ω) Eape Consider the syste shown in fiure where the ass fore Fsinω t. Find the response of the syste when = and = = 3. (54) (5) is subjeted to a F sinω t 3 Fiure Soution: The equation of otion of this syste an be written as 0 + F sinωt 0 + = Fsinωt 0 + = 0 So assuin the soution 6
19 X = sinωt and proeedin as epained before X [ Z( ω) ] ω = ω 4 4 Z( ω) = ( ω ) = ω 4ω + 3 = ( ω 4 ω + 3 ) or, Z( ω ) = ( ω )( 3 ) ( )( ) ω = ω ω ω ω where, ω = and ω = 3 are nora ode frequenies of this syste. Hene, X X = ( ω ) F ω ω ω ω = ( )( ) F ( )( ) ω ω ω ω So it ay be observed that the syste wi have aiu vibration when ω = ω or, ω = ω. Aso it ay be observed that X = 0, when ω = /. Tuned Vibration Absorber Consider a vibratin syste of ass, stiffness, subjeted to a fore Fsinω t. As studied in ase of fored vibration of sinederee of freedo syste, the syste wi have a steady state response iven by Fsin t = ω,where ωn / = ( ωn ω ) (55) whih wi be aiu when ω = ω n. Now to absorb this vibration, one ay add a seondary sprin and ass syste as shown in fiure 3. Fsinωt Fiure 3 7
20 The equation of otion for this syste an be iven by 0 + Fsinωt 0 + = 0 Coparin equation (49) and (56), (56) =, = 0, = 0, = = +, =, =, and =., Hene, Z( ω) = = ω ω ω + ω 4 + ω ω = ( λω )( λ ω ) (57) where λ and λ are the roots of the harateristi equation Z( ω ) = 0 of the freevibration of this syste., whih an be iven by λ, = ± Now fro equation (53) and (54) X X ( ω ) ( ω ), (58) F F = = (59) Z( ω) Z( ω) F Z( ω) = (60) Fro equation (59), it is ear that, X = 0, when ω =. Hene, by suitaby hoosin the stiffness and ass of the seondary sprin and ass syste, vibration an be opetey eiinated fro the priary syste. For Z( ω ) = + = + = F F and X = = ω =, (6) (6) 8
21 Centrifua Penduu Vibration Absorber The tuned vibration absorber is ony effetive when the frequeny of eterna eitation equas to the natura frequeny of the seondary sprin and ass syste. But in any ases, for eape in ase of an autoobie enine, the eitin torques are proportiona to the rotationa speed n whih ay vary over a wide rane. For the absorber to be effetive, its natura frequeny ust aso be proportiona to the speed. The harateristis of the entrifua penduu are ideay suited for this purpose. Pain the oordinates throuh point O, parae and nora to r, the ine r rotates with anuar veoity ( + φ ) ĵ î r R O O The aeeration of ass ˆ a os sin ( ) sin os ( ) ˆ = R φ + R φ r + φ i + R φ + R φ + r + φ j (63) Sine the oent about O is zero, MO = R sinφ + R os φ + r( + φ) r = 0 (64) Assuin φ to be sa, osφ =, sinφ = φ, so R R+ r r r (65) φ + φ = If we assue the otion of the whee to be a steady rotation osiation of frequeny ω, one ay write n pus a sa sinusoida nt sinωt 0 (66) nt osωt 0 (67) =ω 0 sinωt (68) Substitutin the above equations in equation (65) yieds, 9
22 R R+ r sin 0 t r r ω (69) φ + n φ = ω Hene the natura frequeny of the penduu is R ω n = n (70) r and its steadystate soution is ( R+ r)/ r = t (7) ω + ( Rn / r) φ ω 0 sin ω It ay be noted that the sae penduu in a ravity fied woud have a natura frequeny of r. So it ay be noted that for the entrifua penduu the ravity fied is repaed by the entrifua fied Rn. Torque eerted by the penduu on the whee With the ĵ oponent of a equa to zero, the penduu fore is a tension aon r, iven by ties the î oponent of a. ( osφ ˆ sinφ ˆ) osφ sin φ ( φ) T = R i + R j R R r iˆ + + =Rφ Rω 0 sinωt sinφrn rn r φ r φ Now assuin sa ane of rotation T = ( R+ r) n Rφ (73) Now substitutin the (73) in (7), R( R + r) n / r T = ( Rn / r) ω ω 0 sinωt R ( + r) = = J eff (74) rω / Rn Hene the effetive inertia an be written as J eff R ( + r) R ( + r) = = rω / Rn / ( ω ω ) n whih an be at its natura frequeny. This possesses soe diffiuties in the desin of the penduu. For eape to suppress a disturbin torque of frequeny equa to four ties the natura speed n, the penduu ust eet the requireent ω = (4 n) = n R/ r. Hene, as the enth of the penduu r = R/6 beoes very (7) (75) 30
23 sa it wi be diffiut to desin it. To avoid this one ay o for Chiton bifiar desin. Eerise probes. In a ertain refrieration pant, a setion of pipe arryin the refrierant vibrated vioenty at a opressor speed of 3 rp. To eiinate this diffiuty, it was proposed to ap a antiever sprin ass syste to the pipe to at as an absorber. For a tria test, for a 905. Absorber tuned to 3 p resuted in two natura frequenies of 98 and 7 p. If the absorber syste is to be desined so that the natura frequenies ie outside the reion 60 to 30 p, what ust be the weiht and sprin stiffness?. Derive the nora odes of vibration of a doube penduu with sae enth and ass of the penduu. 3. Deveop a atab ode for deterination of freevibration of a enera twoderee of freedo syste. 4. Derive the equation of otion for the doube penduu shown in fiure p in ters of and usin Larane prinipe. Deterine the natura frequenies and ode shapes of the systes. If the syste is started with the foowin initia onditions: (0) = (0) = X, v (0) =v (0)=0, (v and v are veoity) deterine the equation of otion. If the ower ass is iven an ipuse F 0 δ (t), deterine the response in ters of nora odes. L L Fiure P 5. A entrifua pup rotatin at 500 rp is driven by an eetri otor at 00 rp throuh a sine stae redution earin. The oents of inertia of the pup ipeer and the otor are 600. and 500. respetivey. The enths of the pup shaft and the otor shaft are 450 and 00, and their diaeters are 00 and 50 respetivey. Neetin the inertia of the ears, find the frequenies of torsiona osiations of the syste. Aso deterine the position of the nodes. 3
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