Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam in the Supercritical Regime

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 24, No. 5 (214) (19 pages) c World Scientific Publishing Company DOI: /S X Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam in the Supercritical Regime Hu Ding and Qiao-Yun Yan Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 272, P. R. China dinghu3@shu.edu.cn yanqiaoyun@shu.edu.cn Jean W. Zu Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, M5S 3G8, Canada zu@mie.utoronto.ca Received July 17, 213; Revised December 7, 213 This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM. Keywords: Axially moving beam; supercritical; chaotic; Galerkin method; differential quadrature. 1. Introduction Some mechanical systems, such as robotic manipulators, magnetic tapes, textile fibers and conveyer belts are technological examples of axially moving continua. Depending on different conditions, these systems can be modeled as axially moving strings [Chen, 25; Mockensturm & Guo, 25], beams [Wickert, 1992; Guo & Wang, 21], webs [Marynowski, 24, 26], and plates [Liu et al., 212; Tang & Chen, 212; Hu & Zhang, 213]. One notable problem in these mechanical systems is the occurrence of large transverse vibrations due to moving speed variations termed as parametric vibrations [Chen & Yang, 25a, 25b; Bağdatli et al., 213; Wang, 212]. Understanding the parametric vibrations of these systems is important for the design. Chaos and bifurcation are important issues of the parametric vibrations in axially moving continua, and the bifurcation diagram provides a

2 H. Ding et al. summary of essential dynamics. Therefore, much research has been done to study the nonlinear dynamical behaviors of these systems. The nonlinear dynamics and bifurcations of an axially moving beam subjected to an axial transport of mass have been analyzed [Pellicano & Vestroni, 2]. The bifurcation and chaos in transverse motion of axially accelerating viscoelastic beams have been revealed [Yang & Chen, 25; Chen & Yang, 26]. The dynamic behaviors of the axially moving beam with three-parameter Zener element were discovered [Marynowski & Kapitaniak, 27]. The chaotic oscillations for a parametrically excited viscoelastic moving belt have been analyzed [Zhang & Song, 27], and the condition for Hopf bifurcation for axially moving strings was investigated [Wang et al., 28]. The nonlinear dynamical behaviors of an axially accelerating viscoelastic beam with the material derivative viscoelastic constitution relation were identified [Ding & Chen, 29]. The bifurcations of the periodic solutions of the axially moving beam with the three-to-one internal resonance were analyzed [Huang et al., 211]. The planar nonlinear dynamics of an axially accelerating beam was examined [Ghayesh et al., 212a]. The bifurcations and chaos of an axially moving plate under external and parametric excitations were studied [Liu et al., 212]. The multipulse chaotic dynamics in nonplanar motion of parametrically excited viscoelastic moving belt was investigated [Yao et al., 212]. The nonlinear coupled global dynamics of an axially moving viscoelastic beam was examined [Ghayesh et al., 213]. Based on all of the above-mentioned studies, chaos and bifurcations have been identified in the nonlinear vibrations of axially moving continua under various situations. On the other hand, the nonlinear vibration of the axially high-speed moving beams has been studied in [Pellicano & Vestroni, 22; Ding & Chen, 21] and exhibited more complicated nonlinear properties than the dynamic response at low speed [Wickert, 1992; Ding et al., 212]. So far, very limited attention has been paid to the nonlinear dynamic behaviors of the beams in the supercritical regime. There are only some papers that focus on this subject. Furthermore, it should be noted that the Galerkin method is used in all these papers. Based on 1-term Galerkin truncation, the chaotic response of accelerating continuum in the supercritical regime was studied [Ravindra & Zhu, 1998]. Instead of the parametric vibrations, the nonlinear forced dynamical behaviors were investigated for an axially high-speed moving viscoelastic beam based on Galerkin truncation [Ghayesh et al., 212b; Ghayesh & Amabili, 212, 213]. However, the governing equation was truncated and only the first-order mode was retained for the parametric vibrations of the supercritical moving beams. Besides, so far there have been no direct numerical approaches, such as the differential/ integral quadrature method, applied to the supercritical nonlinear dynamics of axially moving beams. All the above-mentioned activities on the axially accelerating continua assume the continua s axial tension to be uniform. Nonetheless, the equal axial tension assumption is not accurate when the systems axially move in acceleration. The dynamic model of axially accelerating systems was modified, and the effect of the axially varying tension due to the acceleration on the steady-state periodic response was examined [Chen & Tang, 211]. However, the effect of the axial tension variation on the bifurcations and chaos in high-speed axially moving continua has not been clear. Besides, the effect of the finite axial support rigidity on the bifurcations and chaos of supercritically moving systems has not been understood. For the nonlinear vibration of an axially moving beam, it is impossible to obtain the exact solution. The Galerkin truncation method is a powerful tool for analyzing the nonlinear dynamic behaviors of these systems. Based on 1-term [Ravindra & Zhu, 1998; Yao et al., 212], 2-term [Yang & Chen, 25; Zhang & Chen, 25; Marynowski, 26; Ha et al., 27; Wang et al., 28; Huang et al., 211; Liu et al., 212], 3-term [Chen et al., 21], 4-term [Marynowski, 24; Chen & Yang, 26; Marynowski & Kapitaniak, 27; Zhang & Song, 27], 6-term [Ghayesh et al., 212b; Ghayesh et al., 212c; Ghayesh et al., 213], 8-term [Pellicano & Vestroni, 2], 1-term [Ghayesh & Amabili, 212, 213], and 12-term [Pellicano & Vestroni, 22] Galerkin truncations, the nonlinear vibration has been studied for the axially moving continua. Although the Galerkin method has been widely used for identifying the bifurcations and chaos in axially moving systems, the difference between the various truncation terms has not been studied. Furthermore, to the best knowledge of the authors, the investigations have not been reported in the literature on the comparison of the nonlinear dynamic

3 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam behaviors of the axially moving systems from different methods. In view of the lack of investigation for the nonlinear dynamic behaviors of the axially moving systems, the present paper studies the bifurcations and chaos of the supercritical accelerating beam by using the high-order Galerkin truncation and the DQM & IQM. Moreover, the dynamic behaviors are compared for the 2-term, 4-term, and 6-term model truncations and the DQM & IQM. This study reveals that the nonlinear vibration of the axially accelerating viscoelastic beam in the supercritical regime displays period-doubling motion and chaotic motion characteristics. The nonlinear dynamic behaviors predicted by the various numerical approaches are certainly different. 2. Equations of Motion Consider an axially moving continuum with the Young s modulus E, the cross-sectional area A, the density ρ, the moment of inertia I, initial tension P, and the dynamic coefficient viscosity α represents the energy dissipative in the beam structure, traveling with the time-dependent axial transport speed γ(t) between two fixed boundaries separated by length L, wheret is the time coordinate. In the present study, γ(t) is assumed to be a small simple harmonic cosine variation about the constant mean speed. That is γ = γ + γ 1 cos(ωt), (1) where γ is the mean axial speed, γ 1 and ω are, respectively, the amplitude and frequency of the axial speed variation. The schematic of the axially moving continuum with the bending stiffness and the simply supported boundary conditions is shown in Fig. 1, where P is the axial tension, the fixed axial coordinate x measures the distance from the left boundary, the transverse displacement is denoted by u(x, t). The dimensionless nonlinear integro-partial-differential equation and the boundary conditions are obtained for governing the transverse vibration of an axially accelerating viscoelastic beam from mixed Eulerian Lagrangian description and Newton s second law as in [Chen & Tang, 211, 212]. Instead of assuming the dynamic viscosity coefficient is higher order of small quantity, two more nonlinear terms with the dynamic viscosity coefficient are not neglected [Chen et al., 212]. Therefore, in conjunction with Eq. (33) of [Chen & Tang, 211] with Eq. (11) of [Chen et al., 212], the governing equations are obtained. As the derivation of the equations of motion are almost the same as [Chen & Tang, 211], here the equations of motion are presented directly as follows: u, tt +2γu, xt +[(1 η)γ 2 (x 1) γ 1]u, xx + k 2 f u, xxxx + α(u, xxxxt + γu, xxxxx ) = 1 2 k2 1u, xx u, 2 xdx + αk2 1 k 2 f u, xx (u, x u, xt + γu, x u, xx )dx, (2) u(,t)=u(1,t)=, u, xx (,t)=u, xx (1,t)=, (3) where the comma preceding x or t denotes partial differentiation with respect to x or t, the dotdenotes differentiation with respect to time t, the axial support s rigidity is defined as η, varying between (infinite rigidity) and 1 (no rigidity) and the dimensionless variables are defined as follows u u L, x x L, t t P ρal 2 (4) and the dimensionless parameters are defined as follows E I ρa ρa k f = P L 2, γ γ, γ γ, P P Fig. 1. Schematic representation of an axially accelerating continuum

4 H. Ding et al. γ 1 γ 1 ρa P, α ω ω ρal 2 P, Iα E A L 3, k 1 =, ρap P (5) where the dimensionless parameter, k 1, is called the nonlinear coefficient, representing the effect of nonlinearity. k 2 f represents the bending stiffness of the axially moving continuum. α is the viscosity coefficient. Therefore, the translating continuum is called an axially moving beam. It should be noted that the dimensionless boundary conditions for the moving beam are the same as the dimensional conditions and can be described by Eq. (3). Omitting the effect of the dynamic viscosity, the equilibrium solutions of Eq. (2) satisfy [(1 η)γ 2 1]û + k 2 f û = 1 2 k2 1û (û ) 2 dx (6) where the prime indicates differentiation with respect to x and the superscript indicates the sense of the equilibrium transverse displacement. The simply supported boundary conditions at both ends for Eq. (6) are û() = û(1) =, û () = û (1) =. (7) The trivial configuration is always a solution for Eq. (6). In addition, the following first-order critical speed and the pair of nontrivial equilibrium solutions are obtained 1 γ ck = 1 η (k2 π 2 k 2 f +1). (8) û k (x) =± 2 kπk 2 1 [(1 η)γ 2 1] (kπk f ) 2 sin(kπx). (9) It should be noted that the numerical simulations in this study are calculated when the moving speed is faster than the first critical speed and does not reach the second critical one. Moreover, this is a commonly occurring situation in belt vibration of automobile engineering. In the present study, the numerical solutions are considered since it is impossible to exactly solve the nonlinear gyroscopic system (2). Using the DQM & IQM and the Galerkin method separately, system (2) is integrated to identify the routes to chaos in the axially moving system under consideration. 3. Differential and Integral Quadrature Method (DQM & IQM) The DQM & IQM are determined to be as the simple and efficient tool for numerical solving (integro-) partial differential equations with very high order of accuracy [Bert & Malik, 1996]. Therefore, the quadrature methods have attracted appreciable attention over the past two decades. Furthermore, the DQM has been shown as an unconditionally stable algorithm for the elastodynamic problem [Tanaka & Chen, 21]. In addition, as a computational tool, the DQM has been found to be more efficient than the Galerkin method [Hamed & Ghader, 29]. The DQM has also pointed out that its accuracy is trustworthy and efficient to consider the vibration of conical shells [Daneshjou et al., 213]. To investigate the chaos and bifurcations of the accelerating beam, the DQM & IQM are used for space discretization system (2). Furthermore, the fourth-order Runge Kutta algorithm is applied to solve the set of resulting nonlinear second-order differential equations. In the domain of the neutral axis coordinate x, N is introduced as an unequally spaced Chebyshev Gauss Lobatto sampling point as [ 1 cos x i = 1 2 (i 1)π N 1 ], i =1, 2,...,N. (1) This kind of space discrete sampling points have proven that they can accelerate the convergence rate of the DQM & IQM in the majority of cases [Bert & Malik, 1996]. As a consequence, function u(x i,t) is desired on the sampling point x i.thevalues of the function derivatives/integral at sampling points are expressed as the linear weighted sum of the function values u i = u(x i,t) by the quadrature rule. Thus, an nth-order derivative at x i is written as [Bert & Malik, 1996] n u(x, t) x n = x=xi A (n) ij u(x j,t)= A (n) ij u j, (11)

5 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam where n is the order of derivative. The integral terms in system (2) are written as [Shu et al., 1995] N 2 u, 2 x dx = A (1), u, x (u, xt + γu, xx )dx = k=1 I k I k k=1 kj u j A (1) kj u j A (1) kj u j + γ A (2) kj u j, (12) where the integral weighting coefficients I k (k =1, 2,...,N) for integrals are solved from 1 1 I I 2 x 1 x 2 x N 1 x N. = (13) I N 1 1 x N 2 1 x N 2 2 x N 2 I N N 1 N 1 x N 2 N x N 1 1 x N 1 2 x N 1 N 1 x N 1 N 1 N For the first-order derivative, the quadrature weighting coefficients are determined as [Shu et al., 1995] N (x i x k ) A (1) ij = (x i x j ) ;j i k=1,k i N k=1,k j (x j x k ) for j i; 1 x i x j for j = i; i, j =1, 2,...,N. (14) For the second-order derivative, the quadrature weighting coefficients are obtained through the following recurrence relationship A (2) ij = k=1 A (1) ik A(1) kj for i, j =1, 2,...,N. (15) Higher-order derivatives in the boundary conditions are incorporated exactly by modifying the weighting coefficient matrices [Malik & Bert, 1996]. To implement the second-order derivatives in boundary conditions (3), the weighting coefficient matrices are modified, yielding à (2) ij = A (2) 21 A (2) 22 A (2) 2,N 1 A (2) 2N A (2) N 1,1 A (2) N 1,2 A (2) N 1,N 1 A (2) N 1,N. (16)

6 H. Ding et al. The higher order derivatives are obtained through the following relationship à (r) ij = A (1) ik Ã(r 1) kj for r =3, 4,..., and i, j =1, 2,...,N. (17) k=1 Substitution of Eqs. (11) and (12) into system (2) yields a series of ordinary differential equations u 1 =, ü i + {[(1 η)γ 2 (x i 1)γ 1 ω sin(ωt) 1]Ã(2) = k2 1 2 à (2) ij u j γ = γ + γ 1 cos(ωt), I k k=1 A (1) kj u j 2 ij + 2α k 2 f + k 2 fã(4) ij + αγã(5) A (1) kj u j ij }u i + [2γA (1) ij A (1) kj u j + γ + αã(4) ij ] à (2) kj u j, u i for j =2, 3, 4,...,N 1. u N =. For a set of given dimensionless parameters in Eq. (5), u i are numerically solved from Eq. (18) using the fourth-order Runge Kutta method via discretizing the temporal variables. Besides, the first calculation s initial conditions are the same for all the plots as given below u(x i, ) = D sin(π, x i ), u, t (x i, ) = for i =1, 2,...,N. (19) For an odd N, the transverse displacement and velocity of the beam center are u(.5,t)=u(x (N+1)/2,t), u, t (.5,t)= u(x (N+1)/2,t). (2) It should be mentioned that the unequally spaced sampling points are set as N = 13 in the following numerical quadrature simulations. 4. Galerkin Method The Galerkin truncation method is used to discretize the spatial differential operator [Pellicano & Vestroni, 2], then leading to a high dimensional model. To investigate the nonlinear dynamic behaviors of the supercritical accelerating beam and compare with the results obtained by the DQM & IQM, system (2) is numerically solved by the Galerkin truncation in conjunction with the fourth-order (18) Runge Kutta method. The eigenfunctions of a stationary tensioned beam under the boundary conditions (3) are chosen as the basis functions. That is to say, the approximate solutions for the transverse vibration are assumed to take the following series expansion u(x, t) = q r (t)sin(rπx) ± 2 πk 2 [(1 η)γ 2 1] (πk f ) 2 sin(πx), 1 (21) where q r (t) (r =1, 2,...,M) are a set of generalized coordinates for the transverse displacements. If a new generalized coordinate q 1 (t) is defined as q 1 (t) q 1 (t) ± 2 πk 2 [(1 η)γ 2 1] (πk f ) 2, 1 (22) Eq. (21) yields u(x, t) = q r (t)sin(rπx). (23) Substitution of Eq. (23) into the derivatives of the transverse displacements function yields

7 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam u, x = q r (t)rπ cos(rπx), u, tt = q r (t)sin(rπx), u, xt = q r (t)rπ cos(rπx) u, xx = q r (t)( r 2 π 2 )sin(rπx), u, xxxx = q r (t)(r 4 π 4 )sin(rπx) (24) u, xxxxt = q r (t)(r 4 π 4 )sin(rπx), u, xxxxx = q r (t)(r 5 π 5 )cos(rπx). Substituting Eq. (24) into the integral terms in system (2) yields [ 1 M ] 2 u, 2 x dx = q r (t)rπ cos(rπx) dx = 1 r 2 π 2 qr 2 2 (t) (u, x u, xt )dx = (γu, x u, xx )dx = [ M ][ M ] q r (t)rπ cos(rπx) q r (t)rπ cos(rπx) dx = 1 r 2 π 2 q r (t) q r (t) 2 [ M ][ M ] γ q r (t)rπ cos(rπx) q r (t)( r 2 π 2 )sin(rπx) dx. (25) Substituting Eqs. (24) and (25) into system (2), one obtains the residual R M (x, t) = q r sin(rπx)+2γ q r rπ cos(rπx)+[(1 η)γ 2 (x 1) γ 1] q r ( r 2 π 2 )sin(rπx) + k 2 f k2 1 + αk2 1 k 2 f [ M ] q r (r 4 π 4 )sin(rπx)+α q r (r 4 π 4 )sin(rπx)+γ q r (r 5 π 5 )cos(rπx) q r (r 2 π 2 )sin(rπx) q r (r 2 π 2 )sin(rπx) [ M 2 q r rπ cos(rπx)] dx {[ M ][ M ] q r rπ cos(rπx) q r rπ cos(rπx) [ M ][ M ]} + γ q r rπ cos(rπx) q r (r 2 π 2 )sin(rπx) dx. (26) In the Galerkin scheme, the residual (26) should satisfy the following relationship if the weighting functions are also chosen as the eigenfunctions of a stationary tensioned beam under the boundary conditions (3) R M (x, t)sin(mπx)dx =, m =1, 2,...,M. (27) For given M, inserting Eq. (26) into Eq. (27) yields the following second-order ordinary differential equations for the M-term Galerkin approximation

8 H. Ding et al. q m (t)+ r+m is odd + k 2 f m4 π 4 q m (t)+α [ mr m 2 r 2 8γ q mr 3 ] r(t) (m r) 2 (m + r) 2 8 γq r(t) [(1 η)γ ] γ 1 m 2 π 2 q m (t) [ m 4 π 4 q m (t)+4γ + αk2 1 k 2 m 2 π 4 q m (t) 1 r 2 q r (t) q r (t) γ f 2 r+m is odd where the nonlinear terms are coupled. In the results presented in this study, the first two (M = 2), four (M =4),andsix(M = 6) beam modes are, respectively, used in the analysis. In other words, the nonlinear dynamics of the axially accelerating beam is numerically identified based on a 2-term, 4-term, and 6-term Galerkin truncation. The Runge Kutta method is the classical and convenient technique for numerically solving nonlinear ordinary differential equations as in system (28). For the present study, the fourth-order Runge Kutta method is adopted to discretize the generalized temporal variables. In addition, the initial conditions for the first calculation are the same for all the following plots as given below q 1 () = D, q 1 () =, q r () =, q r () = for r =2, 3,...,M, (29) where D indicates the amplitude of the initial vibration, and D =.1 is used in all numerical examples if there is no special declaration. Substituting Eq. (29) into Eq. (23), one can find that the initial conditions (23) for the Galerkin method are the same with the initial conditions (19) for the DQM & IQM. For M =2,M =4,andM = 6, the transverse displacements of the accelerating beam s center are as follows u M=2 (x, t) = u M=4 (x, t) = 2 q r (t)sin(rπx) = q 1 (t), 4 q r (t)sin(rπx) = q 1 (t) q 3 (t), s+p is odd ] mr 5 π 5 m 2 r 2 q r(t) k2 1m 2 π 4 q m (t) spq s (t)q p (t) =, u M=6 (x, t) = r 2 q 2 r(t) 6 q r (t)sin(rπx) = q 1 (t) q 3 (t)+q 5 (t). (28) (3) 5. Chaos and Bifurcations The bifurcation diagram is the modern technique for identifying the dynamic behaviors in the analysis of nonlinear systems. The dynamics may be viewed globally over a range of parameter values, thereby allowing simultaneous comparison of periodic and chaotic vibrations. In this section, the nonlinear dynamic behaviors of the axially accelerating viscoelastic beam in the supercritical speed range are compared by various terms of Galerkin truncation and the DQM & IQM. Two special cases are considered: (a) by considering the amplitude of the axial speed variation, the dynamic viscosity, and the axial support s rigidity parameter, as the bifurcation parameter, the bifurcations of the accelerating beam are compared based on the 2-term, 4-term, and 6-term Galerkin truncation as well as the DQM & IQM in Sec. 5.1; (b) based on the two numerical approaches, the nonlinear dynamic behaviors of the system with the same parameters are examined in Sec In this study, if no other values are assigned, an axially accelerating beam is considered with the dimensionless nonlinear coefficient k 1 = , the bending stiffness k f =.8, the dynamic viscosity α =.2, the axial support s rigidity η =.5. The first-order critical speed is determined as [Chen & Tang, 211] γ c1 = 1 1 η (k2 f π2 +1)= (31)

9 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam In the following numerical examples, the supercritical mean moving speed is set as γ =4.2, and the frequency of the axial speed variation is set as ω =3.5. The numerical nonlinear dynamic response is obtained in the time interval of [, 5 2π/ω] dimensionless time units. The transient response has been removed by excluding the first 96% of the time histories. The bifurcation diagram is set in a way that the last 2 steps are plotted in each diagram. Furthermore, the displacement and velocity at the maximum amplitude in the last 4% of the time histories at each step are adopted as the initial conditions of the next step Bifurcation comparison By varying the amplitude γ 1 of the axial speed fluctuation for all other fixed system parameters, Fig. 2 shows the comparison of the bifurcations of the nonlinear transverse vibration by the Galerkin truncation and the DQM & IQM. Figure 2 provides a summary of essential dynamics. The numerical results in Fig. 2 indicate that the amplitude of the speed fluctuation is an important parameter for influencing the nonlinear dynamic behaviors of the axially moving viscoelastic beam in the supercritical regime. In the details, the comparisons in Fig. 2 indicate the following information: (I) the displacement and velocity are increased with the increasing amplitude of the speed fluctuation; (II) Figs. 2(a) 2(d) all show that the periodic motion of the moving beam alternates with the complicated motion (the chaotic or quasiperiodic motion) of that beam; (III) with further increase of the amplitude of the axial speed variation, the complicated motion (a) (b) (c) Fig. 2. The comparisons of the bifurcation diagrams of the 2, 4, and 6 terms Galerkin truncation and the DQM & IQM: the amplitude of the speed fluctuation. (a) 2-term: displacement, (b) 4-term: displacement, (c) 6-term: displacement and (d) DQM & IQM: displacement (d)

10 H. Ding et al. suddenly disappears, and the periodic motion occurs again; (IV) however, with the changing amplitude of the speed variation, the complicated motion and the periodic motion appear alternatingly. On the other hand, the comparisons in Fig. 2 show that there are obviously differences among the bifurcation diagrams of the 2-term, 4-term, and 6-term Galerkin truncation and the DQM & IQM. In the details, with the same amplitude of the speed fluctuation, the periodic motion and the chaotic motion appear by predicting with various terms of truncation and the DQM & IQM. Furthermore, this kind of difference occurs in both small and large regimes of the amplitude of the speed variation. Usually, the more term truncation for the axially moving continua leads to more accurate results. But it is not easy to find the tendency in truncated systems based on the stationary beam eigenfunctions in Fig. 2. Moreover, it is hard to tell which term truncated solution is more close to the results of the DQM & IQM. Compared with nonlinear dynamics in the subcritical speed range [Ding & Chen, 29], Fig. 2 shows that the nonlinear vibration of the accelerating beam in the supercritical regime is more unstable than vibration in the subcritical speed range. To be specific, the complicated motion occurs with a larger amplitude of the speed variation for subcritical speed (Fig. 1 in [Ding & Chen, 29]), while quite small excitation leads to the complicated motion in the supercritical regime. The threshold value of the amplitude of fluctuation required to cause chaos decreases [Ravindra & Zhu, 1998]. To identify the nonlinear dynamic behaviors of the complicated motion in Fig. 2, Fig. 3 shows the (a) (b) (c) (d) Fig. 3. Chaotic motion from 6 terms Galerkin truncation. (a) The time history, (b) the power spectrum, (c) the initial sensitive: the initial phase and (d) the initial sensitive: chaotic

11 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam dynamic characteristics of the transverse vibration of the accelerating beam with γ 1 =.55 and α =.1 based on 6-term Galerkin truncation. In Fig. 3, three standard indicators are used to characterize the long-term nonlinear dynamics, including the time history, the amplitude spectrum, and the sensitivity to initial conditions. Based on the numerical simulations of Eq. (26) with M = 6, Fig. 3(a) presents the time history of the transverse vibration displacement of the beam center. To gain a better insight into the characteristics of the transverse nonlinear response, Fig. 3(b) presents the corresponding amplitude spectral density of the time history in Fig. 3(a) by fast Fourier transforms. As shown in Fig. 7(c), the Poincaré map also depicts the chaotic motion of the transverse vibration. Furthermore, the numerical results in Figs. 3(c) and 3(d) show that the nonlinear dynamic response is very sensitive to the initial conditions, where the lines stand for the results of D =.1, the dots stand for the results of D = From the observation in Fig. 3(a), it is found that the nonlinear vibration does not show periodic motion characteristics. Moreover, Fig. 3(b) shows that the nonlinear vibration of the moving beam includes infinite frequencies. In a word, the three standard indicators all demonstrate that the complicated motion of the accelerating beam in Fig. 2 shows chaotic motion characteristics. To study the effects of the axial mean speed, the axial support s rigidity, and the dynamic viscosity on the nonlinear dynamic behavior, Figs. 4 6 show the bifurcations of the transverse vibration of the accelerating beam based on the various term Galerkin truncation and the DQM & IQM. Figure 4 illustrates that the straight equilibrium (a) (b) (c) (d) Fig. 4. The comparisons of the bifurcation diagrams of the 2, 4, and 6 terms Galerkin truncation and the DQM & IQM: the mean speed. (a) 2-term: displacement, (b) 4-term: displacement, (c) 6-term: displacement and (d) DQM & IQM: displacement

12 H. Ding et al. configuration of the moving beam becomes unstable and is replaced by the nontrivial equilibriums in the high-speed region. As it is seen from Figs. 4 6, the complicated motion and the periodic motion exchange alternately, and the amplitude of the transverse vibration increases gradually with the increasing axial mean speed and the decreasing axial support s rigidity. Besides, Fig. 4 illustrates that the 2, 4, and 6-term Galerkin truncation and the DQM & IQM all predict that there is large area periodic motion around γ = Moreover, the common feature in all cases of Fig. 5 is that the equilibrium loses its stability after the axial support s rigidity is less a certain value. This possibly stemmed from the larger axial support s rigidity leading to larger critical speed. On the other hand, the numerical results in Figs. 4 6 also show that there are some differences among various truncation terms and numerical methods. Generally speaking, the 4-term and the 6-term truncations predict the results closer to that of the DQM & IQM than the results of the 2-term truncation in some areas in the bifurcation diagrams, such as the complicated motion area based on the 2-term truncations in Fig. 4 being obviously smaller than others. Similarly, the 2-term truncation predicts the complicated motion in the areas close to zero in Fig. 5 when other cases all predict periodic motion. Usually, the more even term truncation leads to more accurate results because the gyroscopic coupling is taken into consideration only in even term truncations. Thus the 4-term and 6-term Galerkin (a) (b) (c) Fig. 5. The comparisons of the bifurcation diagrams of the 2, 4, and 6 terms Galerkin truncation and the DQM & IQM: the axial support s rigidity. (a) 2-term: displacement, (b) 4-term: displacement, (c) 6-term: displacement and (d) DQM & IQM: displacement. (d)

13 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam (a) (b) (c) Fig. 6. The comparisons of the bifurcation diagrams of the 2, 4, and 6 terms Galerkin truncation and the DQM & IQM: the dynamic viscosity. (a) 2-term: displacement, (b) 4-term: displacement, (c) 6-term: displacement and (d) DQM & IQM: displacement. (d) truncations give better results than 2-term truncation. However, quantitative convergence was not clearly observed Nonlinear dynamic behaviors comparison Since Sec. 5.1 has showed that there is something in common as well as something different in the bifurcations based on various terms of Galerkin truncation and the DQM & IQM, this section compares the nonlinear dynamic characteristics with the same system parameters based on the two numerical approaches. The Poincaré map is a convenient tool to identify the dynamical behavior, especially chaos [Cai et al., 213]. Figures 7 and 8, respectively, show the comparisons of the periodic motion and the chaotic motion based on the 2-term, 4-term, and 6-term Galerkin truncation and the DQM & IQM. In Fig. 7, the amplitude of the speed fluctuation and the dynamic viscosity are set as γ 1 =.196 and α =.2. In Fig. 8, the parameters are set as γ 1 =.55 and α =.1. As shown in Figs. 7 and 8, the nonlinear dynamic characteristics with the same system parameters are all the same based on the different numerical ways. That is to say, various term truncations and the DQM & IQM all predict the periodic motion in Fig. 7 and the chaotic motion in Fig. 8. Nevertheless, the differences among the numerical results are also clearly shown in Figs. 7 and 8. In the details, the 4-term

14 H. Ding et al. (a) (b) (c) (d) Fig. 7. The comparisons of the 2, 4, and 6 terms Galerkin truncation and the DQM & IQM: the periodic motion. (a) 2-term: phase portrait, (b) 4-term: phase portrait, (c) 6-term: phase portrait and (d) DQM & IQM: phase portrait. (a) (b) Fig. 8. The comparisons of the 2, 4, and 6 terms Galerkin truncation and the DQM & IQM: the chaotic motion. (a) 2-term: Poincaré map, (b) 4-term: Poincaré map, (c) 6-term: Poincaré map and (d) DQM & IQM: Poincaré map

15 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam (c) (d) Fig. 8. and 6-term truncation and the DQM & IQM predict quite similar form for the periodic motion as well as the chaotic motion. Meanwhile, there are discernible differences among the motion forms based on the 2- term truncation and other numerical ways for both periodic and chaotic motion. Figure 9 also compares the nonlinear dynamic characteristic with the same parameters for the 2- term, 4-term, and 6-term Galerkin truncation and the DQM & IQM. In Fig. 9, the dimensionless parameters are set as γ 1 =.573 and α =.2. Figures 9(a), 9(b), 9(e) and 9(f), respectively, demonstrate the phase portraits for numerical solutions of the 2-term, 4-term, and 6-term Galerkin truncation and the DQM & IQM. The comparisons of Fig. 9 show that there is qualitative disagreement (Continued) among the different term truncated systems. In the details, the 2-term and 4-term truncation both predict a periodic motion while the 6-term truncation predicts a chaotic motion. Furthermore, the numerical results in Figs. 9(e) and 9(f) show that the 6-term Galerkin truncation and the DQM and IQM both predict similar chaotic motion form. In order to determine the effect of truncation on longterm dynamical behaviors, the Poincaré mapisconstructed for the 6-term Galerkin truncation and the DQM & IQM. Figures 9(c) and 9(d), respectively, illustrate the Poincaré maps corresponding to the phase portrait in Figs. 9(a) and 9(b). Figures 9(g) and 9(h), respectively, demonstrate the Poincaré maps corresponding to the phase portrait in Figs. 9(e) and 9(f). Based on these Poincaré (a) Fig. 9. The comparisons of the 2, 4, and 6 terms Galerkin truncation and the DQM & IQM: the periodic and chaotic motion. (a) 2-term: phase portrait, (b) 4-term: phase portrait, (c) 2-term: Poincaré map, (d) 4-term: Poincaré map, (e) 6-term: phase portrait, (f) DQM & IQM: phase portrait, (g) 6-term: Poincaré map and (h) DQM & IQM: Poincaré map (b)

16 H. Ding et al. (c) (d) (e) (f) (g) (h) Fig. 9. (Continued)

17 Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam maps, some general remarks are made. The nonlinear dynamics behaviors, which are obtained by the 2-term and 4-term truncations, both show periodic motion characteristics, and which are predicted by the 6-term truncation and the DQM & IQM, both show chaotic motion characteristics. Furthermore, the comparisons in Figs. 9(g) and 9(h) demonstrate that the chaotic motion forms are similar. 6. Conclusions The bifurcations and chaos of the transverse vibration of an axially accelerating viscoelastic beam are numerically studied in the supercritical speed range. The high-order Galerkin method and the DQM & IQM are both first applied to study the system under consideration, simplifying the nonlinear governing equation by defining a set of nonlinear ordinary differential equations with the coupled nonlinear terms. After the numerical solutions of those differential equations are calculated, the Poincaré maps are constructed to classify the motions. Furthermore, the bifurcation diagrams are obtained in the case that the various system parameters are considered as the bifurcation parameter. Based on the bifurcation diagrams, numerical examples are compared with the predictions of the 2-term, 4-term, and 6-term Galerkin truncation and the DQM & IQM. Moreover, the nonlinear dynamic characteristics obtained by these numerical approaches are compared with the same system parameters. From the numerical investigations, this study has drawn the following major conclusions: (1) The 2-term, 4-term, and 6-term Galerkin truncation and the DQM & IQM all predict that the chaotic motion and the periodic motion exchange alternately with varying amplitudes of the axial speed fluctuation, axial mean speed, axial support s rigidity, and dynamic viscosity. (2) The nonlinear vibration of the accelerating beam above the first critical speed is more unstable than the nonlinear vibration below the critical speed. (3) There are discernible differences between the motion form predictions from the 2-term Galerkin truncation and the other numerical approaches, which include the higher term truncations and the DQM & IQM. (4) There is qualitative disagreement among the different term truncated systems, and the 6-term Galerkin truncation, and the DQM & IQM predict similar motion forms. Acknowledgments The authors gratefully acknowledge the support of the State Key Program of National Natural Science Foundation of China (Nos and ), the National Natural Science Foundation of China (No ), and Innovation Program of Shanghai Municipal Education Commission (No. 12YZ28). References Bağdatli, S. M., Özkaya, E. & Öz, H. R. [213] Dynamics of axially accelerating beams with an intermediate support, ASME J. Vibr. Acoust. 133, Bert, C. W. & Malik, M. [1996] Differential quadrature method in computational mechanics: A review, Appl. Mech. Rev. 49, Cai, M., Liu, W. F. & Liu, J. K. [213] Bifurcation and chaos of airfoil with multiple strong nonlinearities, Appl. Math. Mech. Engl. Ed. 34, Chen, L. Q. [25] Analysis and control of transverse vibrations of axially moving strings, Appl. Mech. Rev. 58, Chen, L. Q. & Yang, X. D. [25a] Steady-state response of axially moving viscoelastic beams with pulsating speed: Comparison of two nonlinear models, Int. J. Solids Struct. 42, Chen, L. Q. & Yang, X. D. [25b] Stability in parametric resonance of an axially moving viscoelastic beam with time-dependent velocity, J. Sound Vibr. 284, Chen, L. Q. & Yang, X. D. [26] Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation, Chaos Solit. Fract. 27, Chen, L. H., Zhang, W. & Yang, F. H. [21] Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations, J. Sound Vibr. 329, Chen, L. Q. & Tang, Y. Q. [211] Combination and principal parametric resonances of axially accelerating viscoelastic beams: Recognition of longitudinally varying tensions, J. Sound Vibr. 33, Chen, L. Q., Ding, H. & Lim, C.W. [212] Principal parametric resonance of axially accelerating viscoelastic beams: Multiscale analysis and differential quadrature verification, Shock Vib. 19, Chen, L. Q. & Tang, Y. Q. [212] Parametric stability of axially accelerating viscoelastic beams with the

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