Analytical periodic motions and bifurcations in a nonlinear rotor system

Transcription

1 Int. J. Dynam. Contol 4 2: DOI.7/s nalytical peiodic motions and bifucations in a nonlinea oto system Jianzhe Huang lbet C. J. Luo Received: 5 Octobe 3 / Revised: 26 Decembe 3 / ccepted: 26 Decembe 3 / Published online: 9 Febuay 4 Spinge-Velag Belin Heidelbeg 4 bstact In this pape, analytical solutions fo peiod-m motions in a nonlinea oto system ae discussed. This oto system with two-degees of feedom is one of the simplest oto dynamical systems, and peiodic excitations ae fom the oto eccenticity. The analytical expessions of peiodic solutions ae developed, and the coesponding stability and bifucation analyses of peiod-m motions ae caied out. nalytical bifucation tees of peiod- motions to chaos ae pesented. The Hopf bifucations of peiodic motions cause not only the bifucation tee but quasi-peiodic motions. Displacement obits of peiodic motions in nonlinea oto systems ae used to illustate motion complexity, and hamonic amplitude spectums give hamonic effects on peiodic motions of the nonlinea oto. Coexisting peiodic motions exist in the nonlinea oto. Keywods Peiod-m motions Nonlinea oto systems nalytical bifucation tees nalytical dynamics Stability and bifucation Hamonic balance method Intoduction The oto dynamics is about the vibation of otating shaft with disks, and the shaft is suppoted by beaings with seals. In industial application, flexible otos ae extensively used. J. Huang Depatment of Mechanical Engineeing, Washington Univesity in St. Louis, Saint Louis, MO 633, US. C. J. Luo B Depatment of Mechanical and Industial Engineeing, Southen Illinois Univesity Edwadsville, Edwadsville, IL , US aluo@siue.edu The high pefomance machines always opeate ove the fist citical speed. In 883, Gustav Delaval fo the fist time manufactued a gas tubine which can opeate ove the fist citical otation speed. In 99, Jeffcott fist developed equations of motion fo linea oto dynamics. Fo such a linea oto system, it can be easily analyzed. Howeve, the esults may not be adequate fo flexible otos with high opeation speed. Thus one consideed the beaing cleaance, squeezing film dampes, seals and fluid dynamics effects in flexible oto systems. In 974, Begg 2 investigated stability of a fiction-induced oto whil motions. In 982, Childs 3 applied a petubation method to investigate subhamonic esponses of a oto with a small nonlineaity. In 987, Choi and Noah 4 use the hamonic balance method M and fast Fouie tansfomation FFT to study subhamonic and suppehamonic esponses in a oto with a beaing cleaance. In 988, Ehich 5 numeically investigated highe ode suhhamonic esponses in such a oto system unde a high opeation speed. In addition, Day 6 used the multiplescale method to show the apeiodic motion. In 99, Kim and Noah 7 used the hamonic balance method to discuss the bifucation of peiodic motions in a modified Jeffcott oto with beaing cleaings. In 995, Choi and Noah 8 still used the hamonic balance method to investigate mode-locking motion and chaos in such a Jeffcott oto. The quasi-peiodic motions and stability fo such a modified Jeffcott oto was also pesented though the hamonic balance method in Kim and Noah 9. In 998, Chu and Zhang used the hamonic balance method to detemine peiodic motions and numeically show the bifucation scenaios. In fact, the modified Jeffcott oto is discontinuous. Thus, the hamonic balance method may not be an adequate method fo peiodic motions in such a modified oto with discontinuity, which can be only as a ough pediction. In, Jiang and Ulbich investigated stability of sliding whil in a nonlinea Jeffcott oto.

2 426 J. Huang,. C. J. Luo In 2, Luo 2 developed a genealized hamonic balance method to get the appoximate analytical solutions of peiodic motions and chaos in nonlinea dynamical systems. This method used the finite Fouie seies to expess peiodic motions, and the coefficients ae time-vaying. With the pinciple of vitual wok, a dynamical system of coefficients is obtained fom which the steady-state solution is achieved, and the coesponding stability and bifucation ae completed. Luo and Huang 4 used the genealized hamonic balance method to obtain appoximate analytical solutions of peiodic motions in the Duffing oscillato, and the analytical bifucation tees of peiodic motions to chaos ae obtained. Luo and Huang 5 employed the genealized hamonic balance method with finite tems to obtain the analytical solution of peiod- motion of the Duffing oscillato with a twin-well potential. Luo and Huang 6 used a genealized hamonic balance method to find analytical solutions of peiod-m motions in such a Duffing oscillato. The analytical bifucation tees of peiodic motions in the Duffing oscillato to chaos ae obtained also see, Luo and Huang 7 9. Such analytical bifucation tees show the connection fom peiodic solution to chaos analytically. To bette undestand nonlinea behavios in nonlinea dynamical systems, the analytical solutions fo the bifucation tees fom peiod- motion to chaos in a peiodically foced oscillato with quadatic nonlineaity wee pesented in Luo and Yu 22, and peiod-m motions in the peiodically foced, van de Pol equation was pesented in Luo and Laken 23. s afoementioned, the peiodic solutions fo one-degee of feedom nonlinea systems wee investigated. Heein, the new method will be used to investigate nonlinea dynamics of oto dynamical systems. In this pape, analytical solutions fo peiod-m motions in a nonlinea oto system will be developed, and the stability and bifucation analyses of peiod-m motions will be caied out. nalytical bifucation tees of peiod- motions to chaos will be pesented. Illustations of peiodic and quasi-peiodic motions will be completed to show nonlinea behavios in the nonlinea oto system. 2 nalytical solutions Conside a nonlinea oto system ẍ + δẋ + + γαx + βxx 2 + y 2 =e 2 cos t, ÿ + δẏ + αy + βyx 2 + y 2 = e 2 sin t whee δ is the linea damping coefficient. α and β ae linea and cubic sping coefficients, espectively. e and ae the eccentic distance and otational speed of the oto, espectively. γ is an asymmetic coefficient. The standad fom of Eq. is ẍ + f x, y, ẋ, ẏ, t =, ÿ + f 2 x, y, ẋ, ẏ, t = 2 whee f x, y, ẋ, ẏ, t = δẋ + + γαx + βxx 2 + y 2 e 2 cos t, f 2 x, y, ẋ, ẏ, t = δẏ + αy + βyx 2 + y 2 e 2 sin t. 3 In Luo 3, conside the analytical solution of peiod-m motion as N x t = a m t + b k/m t cos k m t k= + c k/m t sin k m t, N y t = a m t + b 2k/m t cos k m t k= + c 2k/m t sin k m t. Then the fist and second ode deivatives of x t and y t ae N ẋ t =ȧ m + ḃ k/m + k m c k/m cos k t m k= + ċ k/m k m b k/m sin k t m, N ẏ t =ȧ m + ḃ 2k/m + k m c 2k/m cos k t m k= + ċ 2k/m k m b 2k/m sin k t m ; N ẍ t =ä m + b k/m + 2 k m ċk/m k m 2 b k/m k= cos k m t+ c k/m 2 k m ḃk/m k m 2 c k/m sin k m t, N ÿ t =ä m + b 2k/m + 2 k m ċ2k/m k= k m 2 b 2k/m cos k m t 4 5

3 nalytical peiodic motions and bifucations in a nonlinea oto system 427 Define + c 2k/m 2 k m ḃ2k/m k 2 c 2k/m m k sin m t. 6 a m T, am, a m = b m = T b /m,...,b N/m ; b 2/m,...,b 2N/m = b m, b m 2 T, c m = T c /m,...,c N/m ; c 2/m,...,c 2N/m = c m ; c m T 2. Substitution of Eqs. 4 6 into Eq. 3 and aveaging all temsofcosk t/m and sink t/m tem gives a m ä m + Fm, b m, c m, ȧ m, ḃ m, ċ m = b k/m + 2 k k 2 m ċk/m b k/m m + F c k/m a m, b m, c m, ȧ m, ḃ m, ċ m =, c k/m 2 k k 2 m ḃk/m c k/m m + F s k/m ä m + Fm a m a m, b m, c m, ȧ m, ḃ m, ċ m = ;, b m, c m, ȧ m, ḃ m, ċ m =, b 2k/m + 2 k k 2 m ċ2k/m b 2k/m m + F c 2k/m a m, b m, c m, ȧ m, ḃ m, ċ m =, c 2k/m 2 k k 2 m ḃ2k/m c 2k/m m + F s 2k/m whee F m = mt F c k a m, b m, c m, ȧ m, ḃ m, ċ m =. a m, b m, c m, ȧ m, ḃ m, ċ m mt F x m, y m, t dt = δȧ m + + γ αa m + β f m, a m, b m, c m, ȧ m, ḃ m, ċ m = 2 mt mt k F x m, y m, t cos m t dt = δ ḃ k/m + k m c k/m + β f c k/m e 2 δk m, + + γ αb k/m F s k a m, b m, c m, ȧ m, ḃ m, ċ m = 2 mt = δ mt k F x m, y m, t sin m t dt ċ k/m k m b k/m + + γ αc k/m + β f s k/m. F m a m, b m, c m, ȧ m, ḃ m, ċ m F c 2k F s 2k = mt mt F 2 x m, y m, tdt = δȧ m + αa m + β f m, a m, b m, c m, ȧ m, ḃ m, ċ m = 2 mt mt k F 2 x m, y m, t cos m t dt ḃ 2k/m + k m c 2k/m = δ a m, b m, c m, ȧ m, ḃ m, ċ m mt + αb 2k/m + β f c 2k/m, k F 2 x m, y m, t sin m t dt = 2 mt = δ ċ 2k/m k m b 2k/m + αc 2k/m + β f s 2k/m e 2 δ m k The functions fo the fist oscillato ae f m = + f c k/m = 3 + f s k/m = 3 + a m 3 m + a q= l= j= i= a m a m 2 f m i, j, l, q 2bk/m + 2a m am b 2k/m + q= l= j= i= a m 9 f c k/m i, j, l, q 2ck/m + 2a m am c 2k/m + q= l= j= i= f s k/m i, j, l, q a m a m 2bk/m 2ck/m 2

4 428 J. Huang,. C. J. Luo with f m f m 3am i, j, l, = 2N b i/mb j/m δi j 3am i, j, l, 2 = 2N c i/mc j/m δi j f m i, j, l, 3 = 4 b i/mb j/m b l/m δ i j l + δ i j+l + δ i+ j l f m i, j, l, 4 = 3 4 b i/mc j/m c l/m f m f m f m f m δ i+ j l + δ i j+l δ i j l am i, j, l, 5 = 2N b 2i/mb 2 j/m δi j am i, j, l, 6 = 2N c 2i/mc 2 j/m δi j am i, j, l, 7 = N b i/mb 2 j/m δi j am i, j, l, 8 = N c i/mc 2 j/m δi j f m i, j, l, 9 = 4 b i/mb 2 j/m b 2l/m δ i j l + δ i j+l + δ i+ j l f m i, j, l, = 4 b i/mc 2 j/m c 2l/m δ i+ j l + δ i j+l δ i j l f m i, j, l, = 2 c i/mc 2 j/m b 2l/m δ i j l + δ i j+l δ i+ j l 3 f c k/m f c k/m f c am i, j, l, 7 = N b i/mb 2 j/m δ i k j + δk i+ j am i, j, l, 8 = N c i/mc 2 j/m δ i k j δk i+ j k/m i, j, l, 9 = 4 b i/mb 2 j/m b 2l/m δ i k j l + δk i j+l + δ i+ k j l + δk i+ j+l f c k/m i, j, l, = 4 b i/mc 2 j/m c 2l/m δ k i j+l f c + δ k i+ j l δk i j l δk i+ j+l k/m i, j, l, = 2 b 2i/mc 2 j/m c l/m δ i k j+l + δk i+ j l δ i k j l δk i+ j+l and f s k/m 3am i, j, l, = N b i/mc j/m δi+ k j sgni jδk i j f s k/m i, j, l, 2 = 4 c i/mc j/m c l/m sgni j +lδ k i j+l + sgni + j lδ k i+ j l sgni j lδ k i j l δk i+ j+l f s k/m i, j, l, 3 = 3 4 b i/mb j/m c l/m sgni j +lδ k i j+l + δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l and f c k/m f c k/m 3am i, j, l, = 2N b i/mb j/m δ i k j + δk i+ j 3am i, j, l, 2 = 2N c i/mc j/m δ i k j δk i+ j f c k/m i, j, l, 3 = 4 b i/mb j/m b l/m δ i k j l + δk i j+l + δ i+ k j l + δk i+ j+l f c k/m i, j, l, 4 = 3 4 b i/mc j/m c l/m δ i k j+l + δk i+ j l δ i k j l δk i+ j+l f c k/m f c k/m am i, j, l, 5 = 2N b 2i/mb 2 j/m δ i k j + δk i+ j am i, j, l, 6 = 2N c 2i/mc 2 j/m δ i k j δk i+ j 4 f s k/m f s k/m f s k/m am i, j, l, 4 = am i, j, l, 5 = am i, j, l, 6 = N b i/mc 2 j/m δi+ k j sgni jδk i j N c i/mb 2 j/m 5 δi+ k j + sgni jδk i j N b 2i/mc 2 j/m δi+ k j sgni jδk i j f s k/m i, j, l, 7 = 4 c i/mc 2 j/m c 2l/m sgni j +lδ k i j+l + sgni + j lδ k i+ j l sgni j lδ k i j l δk i+ j+l f s k/m i, j, l, 8 = 2 b i/mb 2 j/m c 2l/m sgni j +lδ k i j+l

5 nalytical peiodic motions and bifucations in a nonlinea oto system δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l f s k/m i, j, l, 9 = 4 b 2i/mb 2 j/m c l/m sgni j +lδ k i j+l + δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l The functions fo the second oscillato ae f m = f c 2k/m = 3 a m 3 + a m + + f s 2k/m = 3 whee f m f m + q= l= j= i= a m 2a m 2b2k/m +2a m q= l= j= i= a m 9 2c2k/m +2a m q= l= j= i= f m i, j, l, q am b k/m + a m 2b2k/m f c 2k/m i, j, l, q 6 am c k/m + f s 2k/m i, j, l, q 3am i, j, l, = 2N b 2i/mb 2 j/m δi j 3am i, j, l, 2 = 2N c 2i/mc 2 j/m δi j f m i, j, l, 3 = 4 b 2i/mb 2 j/m b 2l/m a m δ i j l + δ i j+l + δ i+ j l f m i, j, l, 4 = 3 4 b 2i/mc 2 j/m c 2l/m f m f m f m f m δ i+ j l + δ i j+l δ i j l am i, j, l, 5 = 2N b 2i/mb 2 j/m δi j 2c2k/m am i, j, l, 6 = 2N c i/mc j/m δi j 7 am i, j, l, 7 = N b i/mb 2 j/m δi j am i, j, l, 8 = N c i/mc 2 j/m δi j f m i, j, l, 9 = 4 b i/mb j/m b 2l/m δ i j l + δ i j+l + δ i+ j l f m i, j, l, = 4 b 2i/mc j/m c l/m δ i+ j l + δ i j+l δ i j l f m i, j, l, = 2 c 2i/mc j/m b l/m and f c 2k/m f c 2k/m δ i j l + δ i j+l δ i+ j l 3am i, j, l, = 2N b 2i/mb 2 j/m δ i k j + δk i+ j 3am i, j, l, 2 = 2N c 2i/mc 2 j/m δ i k j δk i+ j f c 2k/m i, j, l, 3 = 4 b 2i/mb 2 j/m b 2l/m δ k i j l + δ k i j+l + δk i+ j l + δk i+ j+l f c 2k/m i, j, l, 4 = 3 4 b 2i/mc 2 j/m c 2l/m δ k i j+l f c 2k/m f c 2k/m f c 2k/m f c 2k/m f c + δ k i+ j l δk i j l δk i+ j+l am i, j, l, 5 = 2N b i/mb j/m δ i k j + δk i+ j am i, j, l, 6 = 2N c i/mc j/m δ i k j δk i+ j am i, j, l, 7 = N b i/mb 2 j/m δ i k j + δk i+ j am i, j, l, 8 = N c i/mc 2 j/m δ i k j δk i+ j 8 2k/m i, j, l, 9 = 4 b i/mb j/m b 2l/m δ i k j l + δk i j+l + δ i+ k j l + δk i+ j+l f c 2k/m i, j, l, = 4 b 2i/mc j/m c l/m δ i k j+l + δk i+ j l δ i k j l δk i+ j+l f c 2k/m i, j, l, = 2 b i/mc j/m c 2l/m δ i k j+l + δk i+ j l δ i k j l δk i+ j+l and f s 2k/m 3am i, j, l, = N b 2i/mc 2 j/m δ k i+ j sgni jδk i j f s 2k/m i, j, l, 2 = 4 c 2i/mc 2 j/m c 2l/m sgni j +lδ k i j+l + sgni + j lδ k i+ j l sgni j lδ k i j l δk i+ j+l f s 2k/m i, j, l, 3 = 3 4 b 2i/mb 2 j/m c 2l/m sgni j+lδ k i j+l

6 43 J. Huang,. C. J. Luo f s 2k/m f s 2k/m f s 2k/m am i, j, l, 4 = am i, j, l, 5 = am i, j, l, 6 = +δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l N b i/mc 2 j/m δi+ k j sgni jδk i j N c i/mb 2 j/m δi+ k j + sgni jδk i j N b i/mc j/m δi+ k j sgni jδk i j 9 f s 2k/m i, j, l, 7 = 4 c i/mc j/m c 2l/m sgni j +lδ k i j+l +sgni + j lδ k i+ j l sgni j lδ k i j l δk i+ j+l f s 2k/m i, j, l, 8 = 2 b i/mb 2 j/m c l/m sgni j +lδ k i j+l +δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l f s 2k/m i, j, l, 9 = 4 b i/mb j/m c 2l/m sgni j +lδ k i j+l Define z m = a m, b m, c m T + δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l = a m, b /m,...,b N/m, c /m,...,c N/m ; z m a m, b 2/m,...,b 2N/m, c 2/m,...,c 2N/m T z m, z m 2,...,z m = ż m = = ȧ m, ḃ m, ċ m T 2N+ ; zm 2N+2, zm 2N+3,...,zm ȧ m, ḃ /m,...,ḃ N/m, ċ /m,...,ċ N/m ; T 4N+2 ȧ m T, ḃ 2/m,...,ḃ 2N/m, ċ 2/m,...,ċ 2N/m ż m, ż m 2,...,ż m 2N+ ;żm 2N+2, żm 2N+3,...,żm 4N+2 Equations 8 and 9 ae ewitten as ż m = z m and ż m = g m z m, z m T 2 whee g m z m, z m = whee F m zm, z m F c /m zm, z m 2k m ċ m + k 2 m 2 b m F s /m zm, z m + 2k m ḃ m + k 2 m 2 c m F m zm, z m F c 2/m zm, z m 2k m ċ m 2 + k 2 m 2 b m 2 F s 2/m zm, z m + 2k m ḃ m 2 + k 2 m 2 c m 2 22 k = diag, 2,...,N, k 2 = diag, 2 2,...,N 2, T, F c /m = F c /m, Fc 2/m N/m,...,Fs T, F s /m = F s /m, Fs 2/m N/m,...,Fs 23 F c 2/m = F c 2/m, Fc 22/m,...,Fc 2N/m T, F s 2/m = F s 2/m, Fs 22/m,...,Fs 2N/m fo N =, 2,...,. Setting y m z m, z m Thus, Eq. 2 becomes T and f m = z m, g m T, 24 ẏ m = f m y m. 25 The steady-state solutions fo peiodic motion can be obtained by setting ẏ m =, i.e., F m z m, z m = F c /m z m, z m 2b m + k 2 = m F s /m z m, z m 2c m + k 2 = m 26 F m z m, z m = F c 2/m z m, z m 2b m + k 2 2 = m F s 2/m z m, z m 2c m + k 2 2 = m The 4N + 2 nonlinea equations in Eq. 26 ae solved by the Newton Raphson method. In Luo 3, the lineaized equation at y m = z m, T is ẏ m = Df m y m y m 27 whee Df m y m = f m y m / y m y m 28

7 nalytical peiodic motions and bifucations in a nonlinea oto system 43 The coesponding eigenvalues ae detemined by Df m y m λi 42N+ 22N+ =. 29 whee Dfy m 22N+ 22N+ I = 22N+ 22N+ and G 22N+ 2N+ H 22N+ 22N+ G = gm z m = G, G c, G s, G, G 2c, G 2s T 3 3 G i = G i, G i,...,g i 4N+, T, G ic = G ic, G ic 2,...,G ic N 32 G is = G s, G is 2,...,G is T N fo i =, 2; and N =, 2,... with G ic k = G ic k, Gic k,...,gic k4n+, 33 G is k = G is k, Gis k,...,gis k4n+ fo k =, 2,...N. The coesponding components ae G G c = = +γ αδ +βg, k 2δ m k δ k m δ k+n αδk + βgc, G s k 2δ = m k+n + δ k m δ k G G 2c = αδ k+n + βgs, + γ + γ = αδ2n+ βg, k 2δ m k+2n+ δ k m δ k+3n+ αδk+2n+ βg2c, G 2s k 2δ = m k+3n+ + δ k m δ k+2n+ whee g αδ k+3n+ βg2s fo =,, 2,...,4N + = 3a m 2 δ + am 2 δ + 2am am δ 2N+ + g i, j, l, q q= l= j= i= with g i, j, l, = 3 2N b i/mb j/m δ + 2am b i/mδ j δ i j g i, j, l, 2 = 3 2N c i/mc j/m δ + 2am c i/mδ j+n δ i j g i, j, l, 3 = 3 4 b i/mb j/m δ l δ i j l + δ i j+l + δ i+ j l g i, j, l, 4 = 3 4 c j/mc l/m δi + 2b i/m c j/m δl+n δ i+ j l + δ i j+l δ i j l g i, j, l, 5 = 2N b 2i/mb 2 j/m δ + 2am b 2i/mδ j+2n+ δ i j g i, j, l, 6 = 2N c 2i/mc 2 j/m δ + 2am c 2i/mδ j+3n+ δ i j g i, j, l, 7 = N b i/mb 2 j/m δ2n+ + am b 2 j/mδi + a m b i/mδ j+2n+ δ i j 36 g i, j, l, 8 = N c i/mc 2 j/m δ2n+ + am c 2 j/mδi+n + a m c i/mδ j+3n+ δ i j g i, j, l, 9 = 4 b 2 j/mb 2l/m δi + 2b i/m b 2 j/m δl+2n+ δ i j l + δ i j+l + δ i+ j l g i, j, l, = 4 c 2 j/mc 2l/m δi + 2b i/m c 2 j/m δl+3n+ δ i+ j l + δ i j+l δ i j l g i, j, l, = 2 c 2 j/mb 2l/m δi+n + c i/mb 2l/m δ j+3n+ and g c + c i/m c 2 j/m δl+2n+ δi j l + δ i j+l δ i+ j l = 3a m 2 δ k + 6am b k/mδ + 2am b 2k/mδ + 2a m b 2k/mδ 2N+ + 2am am δ k+2n+ + 2a m b k/mδ2n+ + am 2 δk + g c i, j, l, q 37 q= l= j= i= with g c i, j, l, = 3 2N b i/mb j/m δ + 2am b i/mδ j δ i k j + δk i+ j g c i, j, l, 2 = 3 2N c i/mc j/m δ +2am c i/mδ j+n δ k i j δk i+ j g c i, j, l, 3 = 3 4 b i/mb j/m δl + δ k i+ j l + δk i+ j+l δ i k j l + δk i j+l g c i, j, l, 4 = 3 c j/m c l/m δi + 2b i/m c j/m δ l+n 4 δ i k j+l + δk i+ j l δk i j l δk i+ j+l

8 432 J. Huang,. C. J. Luo g c i, j, l, 5 = 2N b 2i/mb 2 j/m δ + 2am b 2i/mδ j+2n+ δ i k j + δk i+ j g c i, j, l, 6 = 2N c 2i/mc 2 j/m δ + am c 2i/mδ j+3n+ g c i, j, l, 7 = N δ i k j δk i+ j b i/m b 2 j/m δ2n+ + am + a m b i/mδ j+2n+ δk i j + δk i+ j g c i, j, l, 8 = c i/m c 2 j/m δ2n+ N + am b 2 j/mδi 38 c 2 j/mδ i+n + a m c i/mδ j+3n+ δk i j δk i+ j g c i, j, l, 9 = b2 j/m b 2l/m δi + 2b i/m b 2 j/m δ 4 l+2n+ δ i k j l + δk i j+l + δk i+ j l + δk i+ j+l g c i, j, l, = c2 j/m c 2l/m δi + 2b i/m c 2 j/m δ 4 l+3n+ δ i k j+l + δk i+ j l δk i j l δk i+ j+l g c i, j, l, = c 2 j/m c l/m δi+2n+ 2 + b 2i/mc l/m δ j+3n+ and g s with + b 2i/m c 2 j/m δl+n δ i k j+l + δk i+ j l δ i k j l δk i+ j+l = 3a m 2 δk+n + 6am c k/mδ + 2am c 2k/mδ + 2a m c 2k/mδ2N+ + 2am am δ k+3n+ + 2a m c k/mδ2n+ + am 2 δk+n + 9 q= l= j= i= g s i, j, l, q 39 g s i, j, l, = 3 N am c j/mδi + a m b i/mδ j+n + b i/m c j/m δ δi+ k j sgni jδk i j g s i, j, l, 2 = 3 4 c i/mc j/m δl+n + sgni + j lδ k i+ j l sgni j + lδ k i j+l sgni j lδ k i j l δk i+ j+l g s i, j, l, 3 = 3 4 b i/mb j/m δ l+n + 2b i/mc l/m δ j sgni j + lδ k i j+l + δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l g s i, j, l, 4 = N am c 2 j/mδi + a m b i/mδ j+3n+ + b i/m c 2 j/m δ 2N+ δ k i+ j sgni jδk i j g s i, j, l, 5 = N am b 2 j/mδi+n + am c i/mδ j+2n+ + c i/m b 2 j/m δ 2N+ δ k i+ j + sgni jδk i j g s i, j, l, 6 = N am c 2 j/mδi+2n+ + am b 2i/mδ j+3n+ + b 2i/m c yj/m δ δ i+ k j sgni jδk i j i, j, l, 7 = c2 j/m c 2l/m δi+n 4 + 2c i/mc 2 j/m δl+3n+ sgni j + lδ i k j+l + sgni + j lδk i+ j l sgni j lδ i k j l δk i+ j+l g s g s i, j, l, 8 = 2 b 2 j/mc 2l/m δi + b i/m c 2l/m δ j+2n+ g s + b i/m b 2 j/m δl+3n+ sgni j +lδ i k j+l +δk i+ j+l sgni + j lδ i+ k j l sgni j lδk i j l i, j, l, 9 = 4 2b 2 j/mc l/m δi+2n+ + b 2i/mb 2 j/m δl+n sgni j +lδ i k j+l +δk i+ j+l sgni + j lδk i+ j l sgni j lδ i k j l 4 and g with = 3a m 2 δ2n+ + am 2 δ2n+ + 2am am δ + g i, j, l, q 4 q= l= j= i= g i, j, l, = 3 2N b 2i/mb 2 j/m δ2n+ + 2am b 2i/mδ j+2n+ δ i j, g i, j, l, 2 = 3 2N c 2i/mc 2 j/m δ2n+ + 2am c 2i/mδ j+3n+ δ i j, g i, j, l, 3 = 3 4 b 2i/mb 2 j/m δl+2n+ δ i j l + δ i j+l + δ i+ j l, g i, j, l, 4 = 3 4 c 2 j/mc 2l/m δ i+2n+ + 2b 2i/mc 2 j/m δ l+3n+ δ i+ j l + δ i j+l δ i j l, g i, j, l, 5 = 2N b i/mb j/m δ2n+ +2am b i/mδ j δ i j, g i, j, l, 6 = 2N c i/mc j/m δ2n+ +2am c i/mδ j+n δ i j, g i, j, l, 7 = N b i/mb 2 j/m δ +am b i/mδ j+2n+ + a m b yj/mδ i δ i j, 42 g i, j, l, 8 = N c i/mc 2 j/m δ + am c i/mδ j+3n+ + a m c 2 j/mδ i+n δ i j,

9 nalytical peiodic motions and bifucations in a nonlinea oto system 433 g i, j, l, 9 = 4 2b j/mb 2l/m δi + b i/m b j/m δl+2n+ δi j l + δ i j+l + δ i+ j l, g i, j, l, = 4 c j/mc l/m δi+2n+ + 2b 2i/mc j/m δl+n δi+ j l + δ i j+l δ i j l, g i, j, l, = 2 c j/mb l/m δi+3n+ + c 2i/mb l/m δ j+n and g 2c + c 2i/m c j/m δ l δ i j l + δ i j+l δ i+ j l. = 3a m 2 δ k+2n+ + 6am b 2k/mδ 2N+ + 2am b k/mδ + 2a m b k/mδ 2N+ + 2am am δ k + 2am b 2k/mδ + a m 2 δk+2n+ + q= l= j= i= with g 2c i, j, l, = 3 b2i/m b 2 j/m δ2n+ 2N g 2c i, j, l, q + 2a m b 2i/mδ j+2n+ δk i j + δk i+ j g 2c i, j, l, 2 = 3 2N c 2i/mc 2 j/m δ 2N a m c 2i/mδ j+3n+ δk i j δk i+ j g 2c i, j, l, 3 = 3 4 b 2i/mb 2 j/m δl+2n+ δ i k j l + δk i j+l + δ i+ k j l + δk i+ j+l g 2c i, j, l, 4 = 3 c2 j/m c 2l/m δi+2n+ 4 +2b 2i/mc 2 j/m δl+3n+ δ i k j+l + δk i+ j l δk i j l δk i+ j+l g 2c i, j, l, 5 = bi/m b j/m δ2n+ 2N + 2a m b i/mδ j δk i j + δk i+ j g 2c i, j, l, 6 = ci/m c j/m δ2n+ 2N + 2a m c i/mδ j+n δk i j δk i+ j g 2c i, j, l, 7 = N b i/mb 2 j/m δ 44 + a m b 2 j/mδi + a m b i/mδ j+2n+ δ i k j + δk i+ j g 2c i, j, l, 8 = N c i/mc 2 j/m δ + am c 2 j/mδi+n + a m c i/mδ j+3n+ δ k i j δk i+ j g 2c i, j, l, 9 = 4 2b j/mb 2l/m δ i + b i/m b j/m δ l+2n+ δ i k j l + δk i j+l + δk i+ j l + δk i+ j+l g 2c i, j, l, = c j/m c l/m δi+2n b 2i/mc j/m δl+n δ i k j+l + δk i+ j l δk i j l δk i+ j+l g 2c i, j, l, = 2 c j/mc 2l/m δi and g 2s with + b i/m c 2l/m δ j+n + b i/mc j/m δl+3n+ δ i k j+l + δk i+ j l δk i j l δk i+ j+l = 3a m 2 δk+3n+ + 6am c yk/mδ2n+ + 2am c k/mδ + 2a m c k/mδ2n+ + 2am am δ k+n + 2am c 2k/mδ + a m 2 δ k+3n+ + 9 q= l= j= i= g 2s i, j, l, q 45 g 2s i, j, l, = 3 N am c 2 j/mδi+2n+ +am b 2i/mδ j+3n+ + b 2i/m c 2 j/m δ2n+ δi+ k j sgni jδk i j g 2s i, j, l, 2 = 3 4 c 2i/mc 2 j/m δl+3n+ + sgni+ j lδ k i+ j l sgni j lδ k i j l δk i+ j+l g 2s i, j, l, 3 = 3 4 b 2i/mb 2 j/m δl+3n+ sgni j+lδ k i j+l +2b 2i/m c 2l/m δ j+2n+ sgni j + lδ i k j+l +δk i+ j+l sgni + j lδ i k + j l sgni j lδ i k j l g 2s i, j, l, 4 = N am c 2 j/mδi + a m b i/mδ j+3n+ + b i/mc 2 j/m δ δi+ k j sgni jδk i j g 2s i, j, l, 5 = N am b 2 j/mδi+n + am c i/mδ j+2n+ + c i/m b 2 j/m δ δi+ k j + sgni jδk i j g 2s i, j, l, 6 = N am c j/mδi + a m b i/mδ j+n 46 + b i/m c j/m δ2n+ δi+ k j sgni jδk i j

10 434 J. Huang,. C. J. Luo g 2s i, j, l, 7 = 4 2c j/mc 2l/m δi+n +c i/m c j/m δl+3n+ sgni j + lδ i k j+l + sgni + j lδ k i+ j l sgni j lδ k i j l δk i+ j+l g 2s i, j, l, 8 = 2 b 2 j/mc l/m δi + b i/m c l/m δ j+2n+ g 2s + b i/m b 2 j/m δl+n sgni j + lδ i k j+l + δk i+ j+l sgni + j lδ i+ k j l sgni j lδ i k j l i, j, l, 9 = 4 2b j/mc 2l/m δi + b i/m b j/m δl+3n+ sgni j + lδ i k j+l and H = gm z m + δi+ k j+l sgni + j lδk i+ j l sgni j lδ i k j l = H, H c, H s, H, H 2c, H 2s T 47 whee H i = H i, H i,...,h i 4N+, T, H ic = H ic, H ic 2,...,H ic N 48 H is = H is, H is 2,...,H is T N fo i =, 2 and N =, 2,..., with H ic k = H ic k, H ic ic k,...,h k4n+, 49 H is k = H is k, H is is k,...,h k4n+ fo k =, 2,...N. The coesponding components ae H = δδ, H c = 2 k m δ k+n δδ k, H s = 2 k m δ k δδ k+n, 5 H = δδ 2N+, H 2c = 2 k m δ k+3n+ δδ k+2n+, H 2s = 2 k m δ k+2n+ δδ k+3n+ fo =,,...,4N +. Fom Luo 3, the eigenvalues of Df m y m ae classified as n, n 2, n 3 n 4, n 5, n 6 5 The coesponding bounday between the stable and unstable solutions is given by the saddle-node bifucation and Hopf bifucation. 3 Fequency-amplitude chaacteistics The cuves of hamonic amplitude vaying with excitation fequency ae illustated. The coesponding solution in Eq. 4 can be e-witten as x t = a m + N k= y t = a m + N k= k/m cos k m t ϕ k/m 2k/m cos k m t ϕ 2k/m,, 52 whee the hamonic amplitude and phase ae defined by ik/m bik/m 2 + c2 ik/m,ϕ ik/m = actan c ik/m 53 b ik/m The system paametes ae δ =5, α=.68, β=, γ =., e =.5 54 The aconyms U and ae used to epesent the saddle-unstable node and saddle-node bifucations, espectively. The aconyms U and ae used to epesent the unstable Hopf bifucation subcitical and stable Hopf bifucation supecitical, espectively. Solid and dashed cuves epesent stable and unstable peiod-m motions, espectively. 3. Peiod- motions Fom the above paametes, the fequency-amplitude cuves of peiod- motions in x-diection and y-diection of the oto ae pesented in Figs. and 2 that ae based on 3 hamonic tems. In Fig., the peiod- motion of the nonlinea oto in the x-diection is pesented. In Fig. i, the constant a vesus otation speed is pesented. Fo the symmetic peiod- motion, a = is obseved. Fo the asymmetic peiod- motion, the otation speed lies in the appoximate ange of.883, 2.7. Fom the symmetic to asymmetic

11 nalytical peiodic motions and bifucations in a nonlinea oto system 435 Constant Tem, a.8 6e e U 45 U U S i Hamonic mplitude,..8 S U.6 U U ii Hamonic mplitude, U.2. U U S Hamonic mplitude, U U U U S iii iv Hamonic mplitude, U 4 2 U S Hamonic mplitude, U U S U v vi Hamonic mplitude, 2 3.3e-3 2.2e-3.e-3 e-4 e-5 e U U U Hamonic mplitude, 3 e-2 e-3 e-4 e-5 U e-6 U U S e vii viii Fig. Peiod- motion of the x-diection in the nonlinea oto: fequency amplitude cuves of hamonic tems based on 3 hamonic tems 3: i a, ii viii k k =, 2,...5, 2, 3, δ = 2,α =.68,β =,γ =., e =.5

12 436 J. Huang,. C. J. Luo Constant Tem, a U U e-3 i U S.9 2. U Hamonic mplitude, U U ii S Hamonic mplitude, U.2 U S iii Hamonic mplitude, S U iv Hamonic mplitude, 24 Hamonic mplitude, U 24 2 U S v 4.e-4 e-5 U U 3.e-4 e-6 e-7 U 2.e e-4 U S vii Hamonic mplitude, 25 Hamonic mplitude, U U e-3 e-4 e-5 e-6 e-7 vi U viii U U S S Fig. 2 Peiod- motion of the y-diection in the nonlinea oto: fequency amplitude cuves of hamonic tems based on 3 hamonic tems 3: i a, ii viii 2k k =, 2,...5, 2, 3, δ = 2,α =.68,β =,γ =., e =.5

13 nalytical peiodic motions and bifucations in a nonlinea oto system 437 peiod- motion, the two saddle-node bifucations occu at.592, 2.7. The asymmetic peiod- motion possesses the unstable Hopf bifucation U, Hopf bifucation, and saddle-node bifucation, espectively. The unstable Hopf bifucations U of the asymmetic peiod- motion ae located at.687,.94865, 2.. The stable Hopf bifucations of the asymmetic peiod- motion occu at.895,.28,.58722,.79. The othe saddle-node bifucations of the asymmetic peiod- motion ae at.883,.74,.83. The ange of the constant is a <.8. The stable asymmetic peiod- motion has 6 segments and the unstable asymmetic peiod- motion possesses 5 segments. One of six segments is vey shot with.883,.895 and a 4 3, which is zoomed. Only positive constant of a > is pesented fom which the cente of the peiod- motion is on the positive x-axis. The constant of a < fo asymmetic peiod- motion will not be pesented. Such a constant has the same magnitude and the cente of the peiod- motion is on the negativex-axis. In Fig. ii, the hamonic amplitude vaying with otation speed is pesented. In addition to the stable and unstable Hopf bifucations and saddle-node bifucations fo the asymmetic peiod- motion, the Hopf bifucation and saddle-node bifucations fo symmetic peiod- motion can be detemined. The saddle-node bifucations fo symmetic peiod- motion ae at.426,.472. The Hopf bifucations of the symmetic peiod- motion occu at.572,.6, Fom such Hopf bifucation, the quasi-peiodic motions ae obseved. Based on the taditional analysis, one may obtain the appoximate fequency-amplitude cuves fo symmetic motion. The hamonic amplitudes fo the symmetic and asymmetic peiod- motions ae pesented with <.. In Fig. iii, the hamonic amplitude 2 vesus otation speed ae pesented. Only asymmetic peiod- motion exists as fo constant a with 2 <.4. Fo symmetic peiod- motion, 2 = is obseved. In Fig. iv, the hamonic amplitude 3 vaying with otation fequency is pesented. Fo asymmetic motion, its local view is zoomed fo the stability detail. The hamonic amplitude 3 < 2. possesses lagest values compaed to the pimay hamonic amplitude <.. The hamonic amplitude 4 vesus otation speed is pesented in Fig. v. The quantity level of 4 dops to fom In Fig. vi, the hamonic amplitude 5 vesus otation speed is pesented, and the quantity level of 5 with 4 is quite close to 5. To avoid abundant illustations, the hamonic amplitudes 2 and 3 ae pesented in Fig. vii, viii, espectively. The quantity levels of 2 and 3 ae close to 3. s usual, the moe hamonic tems should be included. The local view of asymmetic peiod- motion is zoomed in Fig. vii, and the logaithm scale is used to pesent the hamonic amplitude of 3 in Fig. viii. Fo this nonlinea oto system, the peiod- motion of the nonlinea oto in the y-diection is pesented in Fig. 2. In Fig. 2i, the constant tem a vesus otation speed is pesented. Only the asymmetic peiod- motion exists as in the x-diection, but the values of a lie in the ange of a.24,. Two local aeas ae zoomed to view the details. The positive a can be obtained with mio symmety as fo negative a. The hamonic amplitude 2 vaying with otation speed is pesented in Fig. 2ii, and the pimay hamonic amplitude in the y-diection is 2,.5 diffeent fom in the x-diection. The symmetic and asymmetic peiod- motions ae pesented. In Fig. 2iii,the hamonic amplitude 22 vesus otation speed is depicted only fo the asymmetic peiod-2 motion with 22 because the symmetic peiod- motion has 22 =. The hamonic amplitude 23 vesus otation speed is pesented in Fig. 2iv. The quantity level of 23 is about 23.3 much less than 3 2. The hamonic amplitudes 24 and 25 vaying with otation speed ae shown in Fig. 2v, vi, espectively. The quantity levels of the two hamonic amplitudes ae 24 5 and To avoid abundant illustations, the hamonic amplitudes 22 and 23 ae pesented in Fig. 2vii, viii, espectively. Thei quantity levels ae and Bifucation tees Fom the Hopf bifucations of symmetic peiod- motion, the quasi-peiodic motion o othe peiodic motions may exist. Howeve, fo asymmetic peiod- motion, its Hopf bifucation may cause peiod-2 motions. Fo the stable Hopf bifucation, the stable peiod-2 motion will be obtained. Fo the unstable Hopf bifucation, the unstable peiod-2 motion will be achieved. The analytical solutions of peiod-2 motions ae based on the 26 hamonic tems 26 in the Fouie seies solution. In Fig. 3, the fist banch of bifucation tee of the peiod- motion to peiod-2 motion of the nonlinea oto in the x- diection is pesented. The constant tem a m m =, 2 vesus otation speed is pesented in Fig. 3i. Two local aeas ae zoomed to show the bifucation chaacteistics. The saddle-node bifucations of peiod-2 motions occu at.548, 2.22 and the Hopf bifucation of peiod-2 motion occus at.587. The unstable Hopf bifucation of unstable asymmetic peiod- motion occus at The unstable peiod-2 motions appea fom the unstable Hopf bifucations. The positive constant a m lies in the ange of a m,.8. The bifucation elation of peiod- to peiod-2 motion is clealy illustated. Fo peiod-2 motion, hamonic amplitude /2 is pesented in Fig. 3ii. The appeaances of peiod-2 motions take place at

14 438 J. Huang,. C. J. Luo Constant Tem, a m Hamonic mplitude, Hamonic mplitude, U U U U U U i U U U iii.6.58 U U U v U Hamonic mplitude, /2 Hamonic mplitude, 3/2 Hamonic mplitude, U.588 U.5 U.7.9 U 2. ii U U U 2. iv.2.4 U U U 64 7 U vi Hamonic mplitude, 2.8e-4.2e-4 6.e-5 U.6e-6 U vii U Hamonic mplitude, 3 e-4 e-5 e-6 e-7 U U U U Fig. 3 Bifucation tee of peiod- motion to chaos of the x-diection in the nonlinea oto: fequency amplitude cuves of hamonic tems based on 26 hamonic tems: i a m, ii viii k/mk =, 2,...,4, 6, 24, 26, m = 2, δ = 2,α =.68,β =,γ =., e =.5 viii

15 nalytical peiodic motions and bifucations in a nonlinea oto system 439 the Hopf bifucation of peiod- motions, which is also the saddle-node bifucation of peiod-2 motion fo appeaance at The unstable Hopf bifucations U of peiod- motion give the unstable saddle-node bifucations U of peiod-2 motions fo the onset of unstable peiod- 2 motion. The unstable saddle-node bifucations U of peiod-2 motions ae at.5874,.94865, 2.. The quantity level of /2 4 2 is obseved and the peiod-2 motion fo this banch is in the ange of.5, 2.. The hamonic amplitude vesus otation speed is pesented in Fig. 3iii fo the bifucation tee of peiod- motion to peiod-2 motion via the Hopf bifucations. The quantity level of fo peiod-2 motion is in the ange of.7 fo.5, 2.. Compaed to /2, the hamonic amplitude 3/2 vesus otation speed is pesented in Fig. 3iv with 3/2 6 2 in.5, 2.. Compaed to the peiod- motion, the hamonic amplitudes 2 and 3 vesus otation speed ae illustated in Fig. 3v and vi, espectively. Thei quantity levels ae 2.4 and 3.2. In Fig. 3v, no symmetic peiod- motion exists, and both asymmetic peiod- motion and peiod-2 motion exist to show the bifucation tee much clealy. Such an illustation is like constant tem in Fig. 3i. Fo hamonic amplitude 3, symmetic and asymmetic peiod- motion and peiod-2 motion coexist. So the illustations will be vey cowded. So only the aea elated to peiod-2 motion is pesented in Fig. 3vi. To educe abundant illustations, the hamonic amplitudes 2 and 3 vesus otation speed ae illustated in Fig. 3vii, viii, espectively. The quantity levels fo the two hamonic amplitudes fo peiod-2 motions ae and 3 4. s in Fig. 3, the fist banch of bifucation tee of the peiod- motion to peiod-2 motion of the nonlinea oto in the y-diection is pesented in Fig. 4. InFig. 4i, the constant tem a m m =, 2 vesus otation speed is pesented. local aea is zoomed to show the bifucation chaacteistics. The bifucation points ae the same as discussed in Fig. 3. The negative constant a m lies in the ange of a m.24,. The bifucation elation of peiod- to peiod-2 motion is clealy illustated. Fo peiod-2 motion, the hamonic amplitude 2/2 in the y-diection is pesented in Fig. 4ii, which is diffeent fom the hamonic amplitude in the x-diection. The locations of bifucations ae thesameasinfig.3. The quantity level of 2/2 5 2 is obseved in the ange of.5, 2.. The hamonic amplitude 2 vaying with otation speed is pesented in Fig. 4iii fo the bifucation tee of peiod- motion to peiod-2 motion. The quantity level of 2 fo peiod- 2 motion is obseved fo.5, 2.. The hamonic amplitude 23/2 vesus otation speed is pesented in Fig. 4iv with 23/ The hamonic amplitudes 22 and 23 vesus otation speed ae illustated in Fig. 4v, vi, espectively. The quantity levels of 22 and 23.2 ae obseved. To educe abundant illustations, the hamonic amplitudes 2 and 3 vesus otation speed ae illustated in Fig. 4vii, viii, espectively. The quantity levels fo the two hamonic amplitudes fo peiod-2 motions ae and In Fig. 5, the second banch of bifucation tee of the peiod- motion to peiod-2 motion of the nonlinea oto in the x-diection is pesented in.3,.8, which is much simple than the fist banch of bifucation tee in.5, 2.. Only the asymmetic peiod- motion elative to peiod-2 motion is pesented heein. The constant tem a m m =, 2 vesus otation speed is pesented in Fig. 5i. The Hopf bifucation of peiod- motion occus at.7872 and the Hopf bifucation of peiod-2 motion occus at The unstable Hopf bifucation of unstable asymmetic peiod- motion occus at.689. The unstable peiod-2 motions appeas fom the unstable Hopf bifucation of peiod- motion. The positive constant a m lies in the ange of a m, 5. The bifucation elation of peiod- to peiod-2 motion is clealy illustated. Fo peiod-2 motion, hamonic amplitude /2 is pesented in Fig. 5ii. The onset of peiod-2 motions takes place at the Hopf bifucation of peiod- motions, which is the saddle-node bifucation of peiod-2 motion fo appeaance at The unstable Hopf bifucations of peiod- motion give the unstable saddle-node bifucation of peiod-2 motions fo the onset of peiod-2 motion. The unstable saddle-node bifucations U of peiod-2 motions ae at.689. The quantity level of /2 9 2 is obseved in.4,.75. The hamonic amplitude vesus otation speed is pesented in Fig. 5iii fo the bifucation tee of peiod- motion to peiod-2 motion via the stable and unstable Hopf bifucations. The quantity level of fo the peiod-2 motion is in the ange of.5 fo.4,.75. The hamonic amplitude 3/2 vesus otation speed is pesented in Fig. 5iv with 3/2 in.4,.75. The hamonic amplitudes 2 and 3 vesus otation speed ae illustated in Fig. 5v and vi, espectively. Thei quantity levels ae 2.3 and 3.6. To educe the abundant illustations, the hamonic amplitudes 2 and 3 vesus otation speed ae illustated in Fig. 5vii and viii, espectively. The quantity levels fo the two hamonic amplitudes fo peiod-2 motions ae and The second banch of bifucation tee of the peiod- motion to peiod-2 motion of the nonlinea oto in the y- diection is pesented in Fig. 6. InFig. 6i, the constant tem m =, 2 vesus otation speed is pesented. The pos- a m

16 44 J. Huang,. C. J. Luo Constant Tem, a m U U -2 U i U Hamonic mplitude, 2/ U U.5 U.7.9 U 2. ii Hamonic mplitude, 22 Hamonic mplitude, 2 Hamonic mplitude, U U U U iii U.654 v U U U e-4.2e-4 U 5.e U 8.e e-5 U U U vii U U Hamonic mplitude, 23/2 Hamonic mplitude, 23 Hamonic mplitude, U.588 U.5 U.7.9 U 2. iv U U e-4 e-5 e-6 e-7 U U vi U U viii U U U Fig. 4 Bifucation tee of peiod- motion to chaos of the y-diection in the nonlinea oto: fequency amplitude cuves of hamonic tems based on 26 hamonic tems: i a m, ii viii 2k/m k =, 2,...,4, 6, 24, 26, m = 2, δ = 2,α =.68,β =,γ =., e =.5

17 nalytical peiodic motions and bifucations in a nonlinea oto system Constant Tem, a m 5 U Hamonic mplitude, /2 6 3 Hamonic mplitude, i iii U Hamonic mplitude, 3/2 U ii U iv.6 Hamonic mplitude, 2.2. U Hamonic mplitude, U v vi Hamonic mplitude, 2 3.e-4 2.e-4.e-4 U vii Hamonic mplitude, 3 e-4 e-5 e-6 e-7 U viii Fig. 5 Bifucation tee of peiod- motion to chaos of the x-diection in the nonlinea oto: fequency amplitude cuves of hamonic tems based on 26 hamonic tems: i a m, ii viii k/m k =, 2,...,4, 6, 24, 26, m = 2,δ = 2,α =.68,β =,γ =., e =.5

18 442 J. Huang,. C. J. Luo Constant Tem, a m U i Hamonic mplitude, 2/ U ii.8 U.2 Hamonic mplitude, Hamonic mplitude, 23/ iii U iv 7 Hamonic mplitude, U Hamonic mplitude, U v vi.2e-4.e-4 Hamonic mplitude, 22 8.e-5 4.e vii U Hamonic mplitude, 23.e-5.e-6.e-7 U viii Fig. 6 Bifucation tee of peiod- motion to chaos of the x-diection in the nonlinea oto: fequency amplitude cuves of hamonic tems based on 26 hamonic tems: i a m, ii viii 2k/m k =, 2,...,4, 6, 24, 26, m = 2, δ = 2,α =.68,β =,γ =., e =.5

19 nalytical peiodic motions and bifucations in a nonlinea oto system Hamonic mplitude, /5 P i Hamonic mplitude, 3/5.2 P ii.8.2 Hamonic mplitude, P-5 Hamonic mplitude, P iii iv 9.e-4 Hamonic mplitude, P-5 Hamonic mplitude, 6.e-4 3.e-4 P v Rotational Fequency, Ω vi Fig. 7 Independent peiod-5 motion in the x-diection of the nonlinea oto: fequency amplitude cuves of hamonic tems based on 55 hamonic tems 55: i vi k/m k =, 3, 5,, 5, 55, m = 2, δ = 2 α =.68,β =,γ =., e =.5 itive constant a m lies in the ange of am.2,.. The bifucation elation of peiod- to peiod-2 motion is clealy illustated. Fo peiod-2 motion, the hamonic amplitude 2/2 in the y-diection is pesented in Fig. 6ii, The quantity level of 2/2 7 2 is obseved in.4,.75. The hamonic amplitude 2 vaying with otation speed is pesented in Fig. 6iii fo the bifucation tee of peiod- motion to peiod-2 motion. The quantity level of 2.8 fo peiod-2 motion is obseved fo.5, 2.. The hamonic amplitude 23/2 vesus otation speed is pesented in Fig. 6iv with 23/2.2. The hamonic amplitudes 22 and 23 vaying with otation speed ae pesented in Fig. 6v and vi, espectively. The quantity levels of 22 and 23 7 ae obseved. To educe abundant illustations, the hamonic amplitudes 2 and 3 vesus otation speed ae illustated in Fig. 6vii and viii, espectively. The quantity levels fo the two hamonic ampli-

20 444 J. Huang,. C. J. Luo.33.5 Hamonic mplitude, 2/5.22. P-5 Hamonic mplitude, 23/5. 5 P i ii.4 8 Hamonic mplitude, 2..8 P-5 Hamonic mplitude, P iii Rotational Fequency, Ω iv 2.4e-4 Hamonic mplitude, P-5 Hamonic mplitude, 2.6e-4 8.e-5 P Rotational Fequency, Ω v vi Fig. 8 Independent peiod-5 motion in the y-diection of the nonlinea oto: fequency amplitude cuves of hamonic tems based on 55 hamonic tems 55: i vi k/m k =, 3, 5,, 5, 55, m = 2, δ = 2 α =.68,β =,γ =., e =.5 tudes fo peiod-2 motions ae and Independent peiod-5 motion In the pevious section, the peiod- motion to peiod-2 motions is pesented via bifucations. Heein, the analytical solution of an independent peiod-5 motion is based on 55 hamonic tems, and the fequency-amplitude chaacteistics ae pesented in Figs. 7 and 8. This independent peiod-5 motion is symmetic with saddle-node bifucations only. In Fig. 7, the fequency-amplitude chaacteistics of peiod-5 motion fo the x-diection of the nonlinea oto ae pesented. Since the independent peiod-5 motion is symmetic, a 5 = and k/m = k = 2l, l =, 2,... and m = 5 is obtained. Thus, only k/m k = 2l +, l =, 2,...and m = 5 ae pesented. In Fig. 7i, the hamonic amplitude /5 vesus otation speed is pesented. The two saddle-node bifucations occu at = 2.485,

21 nalytical peiodic motions and bifucations in a nonlinea oto system Velocity, dx/dt T Time, t i.6 T Time, t ii Velocity, dy/dt T Time, t iii 3. T Time, t iv 2.2 Velocity, dx/dt.5 Velocity, dy/dt v Hamonic mplitude, k vi e e vii Fig. 9 Peiod- motion of a nonlinea oto = 2., 3: i x-displacement, ii x-velocity; iii y-displacement, iv y- velocity; v x-tajectoy and vi y-tajectoy; vii displacement obit, viii x-hamonic amplitude, and ix y-hamonic amplitude. a viii Initial condition x, ẋ = ,2.3936E-3 and y, ẏ = E-3,.359.δ = 2,α =.68,β =,γ =., e =.5 5

22 446 J. Huang,. C. J. Luo Hamonic mplitude, 2k e Fig. 9 continued ix The stable and unstable peiod-5 motions fom a closed loop in the fequency-amplitude cuve. The quantity levels of stable and unstable peiod-5 motions ae /5 8 and /5.24, espectively. The hamonic amplitude 3/5 vaying with otation speed is pesented in Fig. 7ii. The quantity levels of /5 and 3/5 ae quite simila. The pimay hamonic amplitude vesus otation speed is aanged in Fig. 7iii. The quantity level of is vey lage with. To educe abundant illustations, the hamonic amplitudes 3 and 5 ae pesented in Fig. 7iv, v, espectively. The coesponding quantity levels of the hamonic amplitudes ae 3.2 and 5 4, espectively. The th hamonic amplitude vaying with otation speed is pesented in Fig. 7vi. The quantity levels of hamonic amplitudes fo the stable and unstable peiod-5 motions ae 4 and 3, espectively. In Fig. 8, the fequency-amplitude chaacteistic of peiod- 5 motion fo the y-diection of the nonlinea oto is also pesented. In Fig. 8i, the hamonic amplitude 2/5 vesus otation speed is pesented. The quantity levels of stable and unstable peiod-5 motions ae 2/5.2 and 2/5.3, espectively. The hamonic amplitude 23/5 vaying with otation speed is pesented in Fig. 8ii. The quantity level of 23/5 educes to 23/5.5. The pimay hamonic amplitude 2 vesus otation speed is aanged in Fig. 8iii. The quantity level of 2 is vey lage with 2.5. To educe abundant illustations, the hamonic amplitudes 23 and 25 ae pesented in Fig. 8iv, v, espectively. The coesponding quantity levels of the hamonic amplitudes ae 23 8 and 25 2, espectively. The th hamonic amplitude vaying with otation speed is pesented in Fig. 8vi. The quantity levels of hamonic amplitudes fo the stable and unstable peiod-5 motions ae and , espectively. 4 Numeical illustations To illustate peiod-m motions in the nonlinea oto system, numeical and analytical solutions will be pesented. The initial conditions fo numeical simulations ae computed fom appoximate analytical solutions of peiodic solutions. In all plots, cicula symbols gives appoximate solutions, and solid cuves give numeical simulation esults. The aconym with a lage cicula symbol epesents initial condition fo all plots. The numeical solutions of peiodic motions ae geneated via the midpoint discete scheme. In Fig. 9, a peiod- motion based on 3 hamonic tems 3 is pesented fo = 2. with othe paametes in Eq. 54. The displacement and velocity esponses in the x-diection of the nonlinea oto ae pesented in Fig. 9i, ii, espectively. One peiod T fo the peiod- motion esponse is labeled in the two plots. Similaly, the displacement and velocity esponses in the y-diection of the nonlinea oto ae also pesented in Fig. 9iii, iv, espectively. The analytical and numeical solutions match vey well. The two tajectoies fo x and y-diections ae pesented fo ove 4 peiods in Fig. 9v, vi, espectively. the two tajectoies ae diffeent because of the inteaction. The initial conditions ae maked by lage cicula symbols and labeled by. In engineeing, one is inteested in displacement obits in oto dynamics. The displacement obit of oto in x and y-diections is pesented in Fig. 9vii. Fo bette undestanding of hamonic contibutions, the hamonic amplitude spectums of oto in x and y-diections ae pesented in Fig. 9viii, ix. The hamonic amplitude spectums ae computed fom analytical solutions. The main hamonic amplitudes of oto in the x-diection ae a 26626,.6859, , 3.556, and The othe hamonic amplitudes in the x-diection ae 5 5 3, k 3 k = 6, 7, 8 5 5, 9 4, 3 5, k 6 k =, 2, 3. Howeve, the main hamonic amplitudes of oto in the y-diection ae a , , , , and The othe hamonic amplitudes in the y-diection ae , k 4 k = 6, 8, , , 2 5, 2 3 6, , and In Fig., a peiod-2 motion based on 26 hamonic tems 26 ae pesented fo = 2. with othe paametes in Eq. 54. The time-histoies of displacement and velocity in the x-diection of the nonlinea oto ae pesented in Fig. i, ii, espectively. Compaed to the coexisting

23 nalytical peiodic motions and bifucations in a nonlinea oto system Velocity, dx/dt T Time, t i.6 2T Time, t ii Velocity, dy/dt T Time, t iii 3. 2T Time, t iv 2.2 Velocity, dx/dt.5 Velocity, dy/dt v vi / vii Hamonic mplitude, k/ e a /2 5/ /2 7/2 9/2 5 / viii Fig. Stable peiod-2 motion of a nonlinea oto = 2., 26: i x-displacement, ii x-velocity; iii y-displacement, v y-velocity; v x-tajectoy and vi y-tajectoy; vii displacement obit, viii x-hamonic amplitude, and ix y-hamonic amplitude. Initial conditions x, ẋ =.48228,59237 and y, ẏ = 8,.856. δ = 2, α =.68, β =, γ =., e =.5

24 448 J. Huang,. C. J. Luo Hamonic mplitude, 2k/ /2 6e /2 23/2 25/ /2 29/2 2/ Fig. continued 27 ix peiod- motion, the displacement and velocity of peiod- 2 motion cannot keep the peiod- motion pattens. Two peiods 2T fo the peiod-2 motion is labeled in the two plots. The time-histoies of displacement and velocity in the y-diection of the nonlinea oto ae also pesented in Fig. iii, iv, espectively. The two tajectoies fo x and y-diections ae pesented fo ove 4 peiods in Fig. v, vi, espectively. Compaed to peiod- motion, the peiod-doubling esponses ae clealy obseved. The initial conditions ae maked by lage cicula symbols and also labeled by. The displacement obit of oto in the x and y-diections is pesented in Fig. vii. To show hamonic contibutions on the peiod-2 motion, the hamonic amplitude spectums of the stable peiod-2 motion of the nonlinea oto in the x and y-diections ae pesented in Fig. viii, ix. The hamonic amplitude spectums of the stable peiod-2 motion ae given by analytical solutions. The main hamonic amplitudes in the x-diection ae a ,.68973, 2 347, , and Howeve, / , 3/ , 5/2 676, 7/ The othe hamonic amplitudes in the x-diection ae 9/2 2 4, 5 4 3, /2.2 4, 6 2 4, 3/2.5 3, 7 3, k/2 6, 4 k = 6, 7, Howeve, the main hamonic amplitudes in the y-diection ae a 2.3 3, , 22.26, , and Howeve, 2/2 66, 23/ , 25/2 2854, 27/ The othe hamonic amplitudes in the y-diection ae 29/ , , 2/ , , 23/ , , 2k/2 7, 4 k = 6, 7, Fo otation speed =.63, a stable peiod- motion, a stable peiod-2 motion and an unstable peiod-2 motion coexist, as shown in Fig.. The input data fo numeical simulation is listed in Table. To educe illustations, the time-histoies of displacements and velocities in the x and y-diections will not be pesented heein. The x and y- tajectoies, displacement obit and, x and y-hamonic spectums fo peiod- motion ae pesented in Fig. i v fo =.63, espectively. The analytical solutions ae based on 3 hamonic tems 3. Two tajectoies of the x and y- diections of the nonlinea oto possess a simple peiod- cycle. The analytical and numeical esults match vey well. The main hamonic amplitudes of the stable peiod- motion in the x-diection ae a ,.4684, , , 4 568, and The othe hamonic amplitudes of the stable peiod- motion in the x-diection ae k 6, 3 k = 6, 7,...,3. Howeve, the main hamonic amplitudes of the stable peiod- motioninthey-diection ae a 2.76, , , , , and The othe hamonic amplitudes in the y-diection ae 2k 6, 3 k = 6, 7,...,3. The x and y- tajectoies, displacement obit and, the x and y-hamonic spectums fo stable peiod-2 motion ae pesented in Fig. vi x fo =.63, espectively. The analytical solutions based on 26 hamonic tems 26. The two tajectoies fo peiod-2 motion do not simply epeat the tajectoies of the peiod- motion. The displacement obit becomes vey complicated, as shown in Fig. viii. The analytical and numeical esults ae in a good ageement. The x and y- hamonic spectums of the stable peiod-2 motion ae pesented in Fig. ix, x. The main hamonic amplitudes of the stable peiod-2 motioninthex-diection ae a ,.423, , 3 449, ,and Howeve, /2 3556, 3/ , 5/2.85, 7/ , and 9/ Compaed to the peiod-2 motion pesented befoe, the effects of hamonic amplitudes 2l/2 l =, 2,...on the peiod-2 motion becomes lage. The othe hamonic amplitudes in the x-diection ae k/2 6, 3 k = 6, 7, Howeve, the main hamonic amplitudes of the stable peiod- 2motioninthey-diection ae a 2.735, , , , , 25 84, and Howeve, 2/2 73, 23/2 3477, 25/2 7263, 27/ , 29/2 4. 3, and 2/ The othe