Nonlinear dynamics of high-dimensional models of in-plane and out-of-plane vibration in an axially moving viscoelastic beam

Abstract

Nonlinear dynamics of high-dimensional models of an axially moving viscoelastic beam with in-plane and out-of-plane vibration with combined parametric and forcing excitations are investigated by the incremental harmonic balance (IHB) method in this paper. Governing equations of transverse in-plane and out-of-plane and longitudinal vibration are obtained basing on the Hamilton's principle. The Galerkin method is used to separate time variable and spatial variable to obtain a set of multi-order differential equations. The IHB method with the fast Fourier transform (FFT) is used to solve periodic response of high-dimensional models of the beam for which convergent mode is reached. Stability of the steady-state periodic solutions is analyzed using the multivariable Floquet theory. Particular attention is paid to in-plane and out-of-plane vibration on convergent mode of the beam with combined parametric and forcing excitations. Multiple solutions are observed, and jump phenomena between in-plane and out-of-plane vibration with different transverse cross sections are discovered.