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

• Published in: Applied Mathematical Modelling 2020-03, Vol.79, p.161-179
• Main Authors: Tang, J.L., Liu, J.K., Huang, J.L.
• Format: Article
• Language: English
• Subjects: Axially moving viscoelastic beam; Convergence; Differential equations; Dimensional stability; Excitation; Fast Fourier transformations; Fourier transforms; Galerkin method; Hamilton's principle; High-dimensional models; Incremental harmonic balance method with the FFT; Jump phenomenon; Mathematical models; Nonlinear dynamics; Nonplanar vibration; Stability analysis; Vibration; Viscoelasticity
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2019.10.028

•Nonlinear dynamics of high-dimensional nonlinear models of an axially moving beam are analyzed.•The IHB method with the FFT is used to analyze dynamics response of the beam.•Stability and bifurcation for steady-state periodic solution are analyzed by Floquet theory.•Jump phenomena between in-plane and out-of-plane vibration of the beam are investigated in detail. 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.