Analytical solution for triply coupled torsional-flexural forced vibrations in asymmetric thin-walled beams under harmonic moving loads

Asymmetric components, such as asymmetric U-shaped beams and steel rails, are used in track engineering and bridge construction, often resulting from construction errors, mechanical wear, or specific engineering requirements. This asymmetry can increase instability of components under moving loads and further endanger structural safety. However, there are almost no effective analytical solutions for triply coupled torsional-flexural forced vibrations of asymmetric thin-walled beams (thin-walled beams with asymmetric cross sections) due to their complexity. To address this, the study established a comprehensive analytical method for obtaining solutions for triply coupled torsional-flexural dynamic responses (beam motion caused by dynamic loads on structures). Triply coupled torsional-flexural dynamic equations were developed, considering the effects of rotary inertia and warping. The equations were then transformed into linear equations through a combination of Fourier finite integral transformation and Laplace transformation, enabling decoupling and resolution of the equations. Analytical solutions for vertical, lateral, and torsional dynamic responses were derived from this approach. To validate these analytical solutions, a new numerical method was proposed, combining the Galerkin approach with the precise integration method. On this basis, the study further explored the triply coupled torsional-flexural resonance mechanism and the influence of the warping effect on dynamic response. The results suggest that high-frequency harmonic moving loads can cause significant resonance in all three directions, while medium- and low-frequency moving loads predominantly affect one or two directions. Moreover, the warping effect's influence on dynamic responses cannot be overlooked, particularly under high-frequency or low-speed harmonic moving loads. Additionally, the effects of load velocity and frequency of harmonic moving loads on the dynamic response of this beam are interconnected.