Periodic and Chaotic Responses of an Axially Accelerating Viscoelastic Beam Under Two-Frequency Excitations

Abstract

This study focuses on the steady-state periodic response and the chaotic behavior in the transverse motion of an axially moving viscoelastic tensioned beam with two-frequency excitations. The two-frequency excitations come from the external harmonic excitation and the parametric excitation from harmonic fluctuations of the moving speed. A dynamic model is established to include the finite axial support rigidity, the material derivative in the viscoelastic constitution relation, and the longitudinally varying tension due to the axial acceleration. The derived nonlinear integro-partial-differential equation of motion possesses space-dependent coefficients. Applying the differential quadrature method (DQM) and the integral quadrature method (IQM) to the equation of the transverse motion, a set of nonlinear ordinary differential equations is obtained. Based on the Runge-Kutta time discretization, the time history of the axially moving beam is numerically solved for the case of the primary resonance, the super-harmonic resonance, and the principal parametric resonance. For the first time, the nonlinear dynamics is studied under various relations between the forcing frequency and the parametric frequency, such as equal, multiple, and incommensurable relationships. The stable periodic response and its sensitivity to initial conditions are determined using the bidirectional frequency sweep. Furthermore, chaotic motions are identified using different methods including the Poincaré map, the maximum Lyapunov exponent, the fast Fourier transforms, and the initial value sensitivity. Numerical simulations reveal the characteristics of the periodic, quasiperiodic, and chaotic motion of a nonlinear axially moving beam under two-frequency excitations.

Primary resonance of traveling viscoelastic beam under internal resonance

“…The transverse vibration of the traveling beam with the external harmonic excitation is governed Fig. 1 Traveling beam with simply supported ends by [23] ρA(u, tt +γu,…”

Periodic responses of a pulley−belt system with one-way clutch under inertia excitation

“…[26,27] have discovered rich nonlinear phenomena, as well as Awrejcewicz et al have shown that continuous mechanical systems may possess various bifurcations, different types of chaos, quasi-periodic orbits [47,48], the nonlinear dynamics of the pulleyÀbelt system with one-way clutch under two-excitation should be an interesting further research direction.…”

Kinematic Aspects in Modeling Large-Amplitude Vibration of Axially Moving Beams

“…By the extension of Hamilton's principle, Behdinan et al [1997] obtained the governing equations for flexible sliding beams with a constant velocity deployed or retrieved through a rigid channel based on the assumptions of Euler-Bernoulli beam theory. Many researchers applied directly extended Hamilton's principle to study the nonlinear vibration of axially moving beams and plates with simple geometric conditions such as Ghayesh and Farokhi [2015], Tang et al [2008], Seddighi and Eipakchi [2016], Ding and Zu [2013], Ali and Elham [2017]. Based on the extended Hamilton principle, Zhang et al obtained many remarkable results for the nonliner vibration of the axially moving beam or plate [Cao and Zhang, 2006, Chen et al, 2007, Chen et al, 2010, Yao et al, 2012, Yang et al, 2012, Yang and Zhang, 2014 But, these works did not point out clearly what description has been used.…”