External and internal resonances of thin-walled curved beams under three-directional moving harmonic loads

Abstract

This paper deals with the internal and external resonances of simply supported curved beams subjected to three-directional harmonic moving loads. Firstly, based on the differential transformation methods and the Glerkin methods, the analytical solutions are newly proposed for vertical, torsional, radial and axial response of thin-walled curved beams under three-directional harmonic moving loads. Then, in the proposed analytical solutions, this study attains the conditions for the external resonance for in-plane and out-of-plane vibrations by letting the denominator of the proposed analytical solutions equal to zero. Besides, the semi-analytical solution for the internal resonance of curved beams is first proposed in this paper by the coincidence of both two mentioned-above resonant frequencies of beams. The internal resonance will occur when the beam is designed with the critical length for the given cross-section. The derived condition for the internal resonance can offer a basis for structural design to avoid the resonant appearance. All the analytical solutions are in good agreement with the results addressed in existing literature and numerical ones obtained from the finite element method (FEM). The results reveal that the corresponding motion of curved beams diverges, reaching a maximum value during the duration of the harmonic moving loads, when in the external resonance. Moreover, under the internal resonance condition, the curved beams are in repetitive transition between the out-of-plane mode and the in-plane mode without any external energy input. Additionally, in this scenario, the excitation frequency can coincide with the two resonance frequencies of beams resulting in simultaneous occurrences of out-of-plane and in-plane resonances on curved beams. The aim of this study highlights the potential resonant appearances on beams and provide reference in the design phase to avoid the internal resonance appearance and ensure the safety of structures.