Natural frequencies of axially moving beams with coupled longitudinal–transversal vibrations: Comparison of three methods

Abstract This paper analyzes the coupled vibration problem of viscoelastic beams in axial motion using the Euler-Bernoulli beam theory. A physical model is established that includes parameters such as density, cross-sectional area, elastic modulus, viscoelastic coefficient, support length, and initial axial tension. Based on the generalized Hamiltonian principle, a set of motion equations in coupled transverse and longitudinal forms is derived. The equations are simplified using dimensionless variables and applying simply supported boundary conditions. Numerical simulations are conducted using the Differential Quadrature (DQ) method, the Modified Weight Coefficient method, and the Galerkin truncation method. The first two frequencies are compared for the three methods at transverse and longitudinal truncation orders M=N=8. It is found that the first frequency matches well, while the second frequency shows deviations in the DQ method. Adjusting the order improves the results.