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Synchronous Motion of Two Vertically Excited Planar Elastic Pendula Synchronous motion of two vertically excited planar elastic pendula M. Kapitaniaka,b∗, P. Perlikowskia, T. Kapitaniaka ∗corresponding author: [email protected] a Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland b Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, AB24 3UE, Aberdeen, Scotland, United Kingdom Abstract The dynamics of two planar elastic pendula mounted on the horizontally excited platform have been studied. We give evidence that the pendula can exhibit synchronous oscillatory and rotation motion and show that stable in-phase and anti-phase synchronous states always co-exist. The complete bifurcational scenario leading from synchronous to asynchronous motion is shown. We argue that our results are robust as they exist in the wide range of the system parameters. Keywords: coupled oscillators, elastic pendulum, synchronization 1 Introduction The elastic pendulum is a simple mechanical system which comprises heavy mass suspended from a fixed point by a light spring which can stretch but not bend when moving in the gravitational field. The state of the system is given by three (spherical elastic pendulum) or two (planar elastic pendulum) coordinates of the mass, i.e. the system has three (spherical case) or two (planar case) degrees of freedom. The equations of motion are easy to write but, in general, impossible to solve analytically, even in the Hamiltonian case. The elastic pendulum exhibits a wide and surprising range of highly complex dynamic phenomena [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. For small amplitudes perturbation techniques can be applied, the system is integrable and approximate analytical solutions can be found. The first known study of the elastic pendulum was made by Vitt and Gorelik [17]. They considered small oscillations of the planar pendulum and identified the linear normal modes of two distinct types, vertical or springing oscillations in which the elasticity is the restoring force and quasi-horizontal swinging oscillations arXiv:1207.0593v1 [nlin.CD] 3 Jul 2012 in which the system acts like a pendulum. When the frequency of the springing and swinging modes are in the ratio 2:1, an interesting non-linear phenomenon occurs, in which the energy is transferred periodically back and forth between the springing and swinging motions [1, 2, 3, 4, 5, 6]. The most detailed treatment of small amplitude oscillations of both plane and spherical elastic pendula is presented in the works of Lynch and his collaborators [7, 10, 11, 12]. For large finite amplitudes the system exhibits different dynamical bifurcations and can show chaotic behavior [8, 9, 13, 14, 15, 16]. The dynamics of elastic pendulum attached to linear forced oscillator has been studied by Sado [18]. She has shown a one parameter bifurcation diagrams showing different behaviour of the systems (periodic, quasiperiodic and chaotic). According to our knowledge this is the only study of considered systems, but one can find a lot of papers concerning dynamics of classical pendulum attached to linear or non-linear oscillator. Hatwal et al. [19, 20, 21] gives approximate solutions in the primary parametric instability zone, which allows calculation of the separate regions of periodic solutions. Further analysis enables us to understand the dynamics around primary and secondary resonances [22, 23, 24, 25, 26]. Then the analysis was extended to systems with non-linear base where non- linearity is usually introduced by changing the linear spring into nonlinear one [27, 28, 29, 30] or magnetorheological damper [31]. Recently the complete bifurcation diagram of oscillating and rotating solutions has been presented 1 k c1 F() t M y k k2 2 ϕ m φ m x2 x3 Figure 1: Model of the system [32]. Dynamics of two coupled single-well Duffing oscillators forced by the common signal has been investigated in our previous papers [33, 34]. We have shown the detailed analysis of synchronization phenomena and compare different methods of synchronization detection. In this paper we study the dynamics of two planar elastic pendula mounted on the horizontally excited platform. Our aim is to identify the possible synchronous states of two pendula. We give evidence that the pendula can synchronize both in the oscillatory and rotational motion moreover in-phase and anti-phase synchronizations co- exist. Our calculations have been performed using software Auto–07p [35] developed for numerical continuation of the periodic solutions and verified by the direct integration of the equations of motion. We argue that our results are robust as they exist in the wide range of the system parameters. The paper is organized as follows. Sec. 2 describes the considered model. We derive the equations of motion and identify the possible synchronization states. In Sec. 3 we study the stability of different types of synchronous motion. Finally Sec. 4 summarizes our results. 2 Model of the system The analyzed system is shown in Fig. 1. It consists of two identical elastic pendula of length l0, spring stiffness k2 and masses m, which are suspended on the oscillator. The oscillator consists of a bar, suspended on linear spring with stiffness k1 and linear viscous damper with damping coefficient c1. The system has five degrees of freedom. Mass M is constrained to move only in vertical direction and thus is described by the coordinate y. The motion of the first pendulum is described by angular displacement ϕ and its mass by coordinate x2, that represent the elongation of the elastic pendulum. Similarly the second pendulum is described by angular displacement φ and its mass by coordinate x3. Both pendula are damped by torques with identical damping coefficient c2, that depend on their angular velocities (not shown in Fig. 1). The small damping, with damping coefficient c3 is also taken for pendula masses. The system is forced parametrically by vertically applied force F (t)= F0 cos νt, acting on the bar of mass M, that connects the pendula. Force F0 denotes the amplitude of excitation and ν the excitation frequency. 2 The equations of motion can be derived using Lagrange equations of the second type. The kinetic energy T , potential energy V and Rayleigh dissipation D are given respectively by: 1 2 1 2 1 2 ˙2 − ˙ T = 2 (M +2m)y ˙ + 2 mx˙ 3 + 2 m(l0 + ywst2 + x3) φ + my˙x˙ 3 cos φ my˙φ(l0 + ywst2 + x3) sin φ+ (1) 1 1 + mx˙ 2 + m(l + y + x )2ϕ˙ 2 + my˙x˙ cos ϕ − my˙ϕ˙(l + y + x ) sin ϕ 2 2 2 0 wst2 2 2 0 wst2 2 V = −mg(l0 + ywst2 + x2)cos ϕ − mg(l0 + ywst2 + x3)cos φ + mg(l0 + ywst2)+ mg(l0 + ywst2)+ (2) 1 1 1 + k (y + y )2 + k (y + x )2 + k (y + x )2 − (M +2m)gy 2 1 wst1 2 2 wst2 2 2 2 wst2 3 1 1 1 1 D = C ϕ˙ 2 + C φ˙2 + C x˙ 2 + C x˙ 2 (3) 2 2 2 2 2 3 2 2 3 3 c y (M+2m)g y mg where 3 is the damping coefficient of the pendulum mass and wst1 = k1 , wst2 = k2 represent static deflation of mass M and pendulums’ mass m respectively. The system is described by five second order differential equations given in the following form: 2 m(l0 + ywst2 + x2) ϕ¨ +2m(l0 + ywst2 + x2)ϕ ˙x˙ 2 − my¨(l0 + ywst2 + x2) sin ϕ+ (4) +mg(l0 + ywst2 + x2) sin ϕ + C2ϕ˙ =0 2 m(l0 + ywst2 + x3) φ¨ +2m(l0 + ywst2 + x3)φ˙x˙ 3 − my¨(l0 + ywst2 + x3) sin φ+ (5) +mg(l0 + ywst2 + x3) sin φ + C2φ˙ =0 2 mx¨3 + my¨cos φ − mφ˙ (l0 + ywst2 + x3) − mg cos φ + k2(ywst2 + x3)+ C3x˙ 3 =0 (6) 2 mx¨2 + my¨cos ϕ − mϕ˙ (l0 + ywst2 + x2) − mg cos ϕ + k2(ywst2 + x2)+ C3x˙ 2 =0 (7) 2 (M +2m)¨y + mx¨3 cos φ − 2mx˙ 3φ˙ sin φ − m(l0 + ywst2 + x3)φ¨sin φ − m(l0 + ywst2 + x3)φ˙ cos φ+ (8) 2 +mx¨2 cos ϕ − 2mx˙ 2ϕ˙ sin ϕ − m(l0 + ywst2 + x2)¨ϕ sin ϕ − m(l0 + ywst2 + x2)ϕ ˙ cos ϕ+ −(M +2m)g + k1(y + ywst1)+ C1y˙ − F0 cos νt =0 In the numerical calculations we use the following values of parameters: M = 10 [kg], m =0.2 [kg], l0 =0.24849 [m], N N Ns Ns k1 = 1642.0 [ m ], k2 = 19.7 [ m ], c1 = 13.1 [ m ] , c2 = 0.00776 [Nms], c3 = 0.49 [ m ], ywst1 = 0.062 [m], ywst2 = 0.1 [m]. 2 k1 Introducing dimensionless time τ = ω1t, where ω1 = M+2m is the natural frequency of mass M with the attached pendula, we obtain dimensionless equations of motion written as: 2β β2 sin Ψ α ¨ 2 ˙ χ − 1 γ 2 ˙ Ψ+ Ψ ˙ 2 ¨ sin Ψ + + 2 Ψ=0 (9) (1 + y2st + χ2) (1 + y2st + χ2) (1 + y2st + χ2) (1 + y2st + χ2) 3 2β β2 sin Φ α ¨ 2 ˙ χ − 1 γ 2 ˙ Φ+ Φ ˙ 3 ¨ sin Φ + + 2 Φ=0 (10) (1 + y2st + χ3) (1 + y2st + χ3) (1 + y2st + χ3) (1 + y2st + χ3) β2 y χ 1 − 1+ 2st + 3 ˙ 2 − 1 χ¨3 + 2 γ¨ cosΦ 2 Φ 2 cosΦ+ yst2 + χ3 + α3χ˙ 3 =0 (11) β2 β2 β2 β2 y χ 1 − 1+ 2st + 2 ˙ 2 − 1 χ¨2 + 2 γ¨ cosΨ 2 Ψ 2 cosΨ + yst2 + χ2 + α3χ˙ 2 =0 (12) β2 β2 β2 2 2 β2 a 2β2a (1+y2st+χ3)a (1+y2st+χ3)a 2 β2 a γ¨ + 2 χ¨3 cosΦ − 2 χ˙ 3Φ˙ sin Φ − 2 Φ¨ sin Φ − 2 Φ˙ cosΦ + 2 χ¨2 cosΨ+ (13) β1 β1 β1 β1 β1 β a y χ a y χ a −2 2 ˙ − (1 + 2st + 2) ¨ − (1 + 2st + 2) ˙ 2 − 1 − 2 χ˙ 2Ψ sin Ψ 2 Ψ sin Ψ 2 Ψ cosΨ 2 + γ + y1st + α1γ˙ q cos µτ =0 β1 β1 β1 β1 2 k2 2 g ν ω1 ω2 m F0 C1 C2 where ω = , ω = , µ = , β1 = , β2 = , a = , q = 2 , α1 = , α2 = 2 , 2 m 4 l0 ω1 ω4 ω4 M+2m ω1 l0(M+2m) ω1(M+2m) mω4l0 C3 ywst1 ywst2 y y˙ y¨ x3 x˙ 3 x¨3 x2 α3 = 2 , y1st = , y2st = , γ = , γ˙ = , γ¨ = 2 , χ3 = , χ˙ 3 = , χ¨3 = 2 ,χ2 = , ml0ω2 l0 l0 l0 l0ω4 l0ω4 l0 l0ω2 l0ω2 l0 x˙ 2 x¨2 ϕ˙ ϕ¨ φ˙ φ¨ χ˙ 2 = , χ¨2 = 2 , Ψ= ϕ, Ψ=˙ , Ψ=¨ 2 , Φ= φ, Φ=˙ , Φ=¨ 2 l0ω2 l0ω2 ω4 ω4 ω4 ω4 The dimensionless parameters of the system have the following values: β1 =2, β2 =1.58, α1 =0.1, α2 =0.01, α3 =0.1, a =0.0192, y1st =0.25, y2st =0.4.
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