Studies on Bifurcation and Chaos of a String-Beam Coupled System with Two Degrees-of-Freedom

Abstract

In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system. Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions. It is found from numerical simulations that there are obvious jumping phenomena in the resonant response-frequency curves. It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions.

Nonlinear study of the dynamic behavior of a string-beam coupled system under combined excitations

“…[ [27][28][29] regarding the behavior of vibrations and occurrence of jump phenomena and multi-valued amplitudes. (7) From the force response curve we concluded that, the region of multi-valued amplitude is increased for increasing values of ω 1 , ω 2 , α 14 , α 21 , σ and disappear for increasing μ 1 , μ 2 .…”

“…[27,28], we can obtain the following nonlinear ordinary differential governing equations (ODE) of motion for the string-beam coupled system under external, parametric and tuned excitation forces with two degrees of freedom…”

“…(6) and (7) and more details can be found in Refs. [27,28]. The damping coefficients, nonlinear parameters and the forcing amplitudes are assumed to be 11 , εα 13 , εα 14 , εα 21 , εα 22 , εα 23 , εα 24 , εβ 21 , εβ 22 , εβ 23 , εf 2 , εf 11 , εf 12 , εf 3 }, (8) where ε is a small perturbation parameter and 0 < ε ≤ 1.…”

“…Zhang et al [26] investigated the global bifurcations and chaotic dynamics in a rotoractive magnetic bearings system with time-varying stiffness using a global perturbation method. Zhang and Cao [27,28] investigated the global bifurcations and chaotic dynamics of a string-beam coupled system subjected to parametric and external excitations. Phase portrait, waveform, and Poincare map were applied to study the periodic and chaotic motions.…”

“…In comparison with the previous works of a stringbeam [27,28], the global bifurcations and chaotic dynamics of a string-beam coupled system subjected to parametric and external excitations are investigated with 1 : 2 internal resonances. The bifurcation analysis indicates that the string-beam coupled system can undergo pitchfork bifurcation, homoclinic bifurcations and the Silnikov type singlepulse homoclinic orbit.…”