Nonlinear transverse-longitudinal coupled vibration of axially moving beam using Galerkin method and truncation technique

Here, considering longitudinal variation of tension, external viscous damping coefficient and support stiffness, a transverse-longitudinal coupled mathematical model was constructed to describe a viscoelastic beam with axial variable speed. Numerical analysis was performed to study steady-state response of the beam under axial variable tension conditions. Galerkin method was used to convert the continuous model into a series of ordinary differential equation systems which were theoretically infinite but could be truncated. Their accurate general expressions were given, and wrong terms in relevant literature were corrected. Furthermore, effects of different truncation orders on the final results were compared, and variations of computation time with truncation orders varying were analyzed. Based on 8th order Galerkin expansions truncated and 4th order Runge-Kutta numerical integration method, the steady-state response of the beam system was numerically solved. Then, vibration amplitude results of the coupled model and the simplified model were analyzed contrastively under the approximate analytical method and different numerical methods in literature. Nonlinear vibration characteristics of the moving beam under subharmonic parametric resonance were revealed in 3 aspects of time history diagram, phase diagram and frequency spectral analysis. The presence of 3 : 1 internal resonance in the beam system could be detected from frequency spectral analyses of both longitudinal and transverse vibrations. The study showed that under specific conditions, it is feasible to ignore high-order terms of longitudinal displacement for model simplification. © 2024 Chinese Vibration Engineering Society. All rights reserved.