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When describing the vibration characteristics of the transverse parameters of an axially moving beam, the effects of nonlinear factors need to be introduced. Under normal circumstances, it is difficult to solve the nonlinear vibration equations of horizontal and vertical coupling with mixed partial derivative terms of time and space using approximate analytical methods. The longitudinal displacement in the vibration process is very small and much smaller than the lateral displacement, so the equation is simplified to a nonlinear vibration governing equation only about the lateral displacement. However, in these studies, no specific numerical basis for the reason for the simplification is given. Therefore, this paper considers the coupling model of the Euler beam’s lateral and longitudinal vibrations, chooses the Galerkin truncation method to decouple and numerically solve it, and obtains the numerical solution of the lateral and longitudinal vibration displacements. Based on the calculation results of the direct multi-scale method and the differential quadrature method of the simplified model, and the solution results of the eight-order Galerkin truncation method of the coupled model, the steady-state response amplitudes of the coupled model and the simplified model are compared to verify the accuracy of the simplified model Therefore, it provides a strong theoretical basis for the simplified model that ignores the longitudinal vibration.

Transverse and longitudinal coupled nonlinear vibration of an axially moving Euler beam under variable tension