Effects of Non-homogeneous Boundaries on Nonlinear Vibrations of an Axially Moving Belt with the Timoshenko Theory

Purpose: An axially moving belt wrapped around two fixed wheels is described by the Timoshenko beam model. This study explores the necessity of the Timoshenko beam model, as well as the impact of non-homogeneous boundaries on the nonlinear forced vibration of axially moving belts. Methods: The governing equation with geometric nonlinearity is formulated based on the generalized Hamilton’s principle. Equilibrium deformations and natural frequencies of the axially moving Timoshenko beam model are determined by using the differential and integral quadrature methods (DIQMs). The Galerkin truncation method (GTM) and the harmonic balance method (HBM) are applied to analyze the amplitude-frequency responses of the Euler–Bernoulli beam model and Timoshenko beam model with non-homogeneous boundaries. Conclusion: According to these analysis results, the amplitude-frequency responses at all positions on the Timoshenko beam are significantly affected by wheels. In terms of amplitude-frequency response, the Timoshenko beam model shows a significantly stronger nonlinear behavior compared to the Euler–Bernoulli beam model. In general, the impact of shear force and the rotary inertia on the amplitude-frequency response characteristics of beam models can be amplified. Therefore, the Timoshenko beam is more effective in representing complex nonlinear behaviors of the forced vibration. The introduction of non-homogeneous boundaries intensifies the sensitivity of models to parameters. Moreover, the distinction between the Euler–Bernoulli beam and the Timoshenko beam becomes notable when influenced by non-homogeneous boundaries.