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

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Journal of Vibration and Shock ›› 2024, Vol. 43 ›› Issue (21) : 237-244.

TANG Youqi1, ZHOU Xingyu1, CHEN Ling2, TAN Xia1, MAO Yongheng1

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Abstract

A transverse and longitudinal coupling mathematical model for an axially accelerating viscoelastic beam is constructed, considering the effects of the longitudinally varying tension, the external viscous damping coefficient, and the support stiffness coefficient. The steady-state response of the beam under the axial tension variation is numerically analyzed. The Galerkin method is used to transform the continuous model into a series of ordinary differential equations which can be truncated indefinitely in theory. The errors in the relevant literature are corrected. The running time of different truncation orders and their effects on the final result are compared. The numerical solution of the steady-state response is obtained based on the 8th-order Galerkin truncation method and the 4th-order Runge-Kutta method. Then, the vibration amplitude results of the coupled model and the simplified model are compared and analyzed under different numerical methods and the approximate analytic methods in the literature. Through the time history, the phase portrait, and the frequency spectrum analyses, the nonlinear vibration characteristics of the beam under subharmonic parameter resonance are revealed. From the spectrum analysis of longitudinal vibration and transverse vibration, it can be detected that the system has 3:1 internal resonance. The results show that it is feasible to ignore the higher order term of longitudinal displacement in order to simplify the model under certain conditions.

Key words

axially moving beam / Galerkin truncation / internal resonance / steady state response

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TANG Youqi1, ZHOU Xingyu1, CHEN Ling2, TAN Xia1, MAO Yongheng1. Nonlinear transverse-longitudinal coupled vibration of axially moving beam using Galerkin method and truncation technique[J]. Journal of Vibration and Shock, 2024, 43(21): 237-244

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