In this paper, the nonlinear governing equations of motion for viscoelastic moving belt are established by using the generalized Hamilton’s principle for the first time. Two kinds of viscoelastic constitutive laws are adopted to describe the relation between the stress and strain for viscoelastic materials. Moreover, the correct forms of elastic strain energy, kinetic energy, and the virtual work performed by both external and viscous dissipative forces are given for the viscoelastic moving belt. Using the generalized Hamilton’s principle, the nonlinear governing equations of three-dimensional motion are established for the viscoelastic moving belt. Neglecting the axial deformation, the governing equations of in-plane motion and transverse nonlinear oscillations are also derived for the viscoelastic moving belt. Comparing the nonlinear governing equations of motion obtained here with those obtained by using the Newton’s second law, it is observed that the former completely agree with the latter.
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.
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