Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds

• Published in: Journal of sound and vibration 2012-03, Vol.331 (7), p.1612-1623
• Main Authors: Zhang, Guo-Ce, Ding, Hu, Chen, Li-Qun, Yang, Shao-Pu
• Format: Article
• Language: English
• Subjects: Beams (radiation); Exact sciences and technology; Excitation; Forced vibration; Fundamental areas of phenomenology (including applications); Galerkin methods; Mathematical analysis; Mathematical models; Nonlinearity; Physics; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Vibration; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2011.12.004

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated. ► Nonlinear forced vibration of supercritically moving viscoelastic beams is studied. ► Galerkin method is used to analyze the governing equation. ► Frequency–response curves are obtained via the multiple-scale method. ► Approximate results are numerically verified via the finite different schemes.