Improved algorithms applying the numerical Laplace method for response analyses of Timoshenko beam subjected to typical external loads

摘要

This study develops Su and Ma's theory regarding the stationary load case by considering the structural damping (which can be arbitrarily proportional or non-proportional) and various boundary constraints. Original derivations are presented based on Timoshenko's beam model and the Kelvin Voigt damping model. Numerical Laplace inversion, i.e., the Durbin method, is applied to obtain the time-domain solutions. Further, an extension to the general moving load case is performed by means of the finite element method, and an algorithm by virtue of mathematical fast transformation is exploited to improve the computational efficiency. The normal mode superposition method is introduced to validate the numerical solutions. In case studies, the accuracy of the Laplace method is first verified through contrastive studies. Based on that analysis, numerical experiments regarding the moving load case are conducted while considering the implications of moving velocities and non-proportional damping. The results demonstrate that the numerical method is effective and efficient for typical boundary conditions, although this approach suffers from certain algorithm-based instabilities. The load velocity and the structural damping are fundamental influential factors on the dynamical performance, in which even divergence appears in a damped system, and regular piecewise-linear rules are concluded in non-proportional damping variations. Since damping is a fundamental parameter for structural dynamics and the stationary and moving loads consist of the basic excitations, the Laplace method has a strong application prospect.