Analytical and Numerical Validation of a Moving Modes Method for Traveling Interaction on Long Structures

This work is devoted to the validation of a computational dynamics approach previously developed by the authors for the simulation of moving loads interacting with flexible bodies through arbitrary contact modeling. The method has been applied to the modeling and simulation of the coupled dynamics of railroad vehicles moving on deformable tracks with arbitrary undeformed geometry. The procedure presented makes use of a fully arbitrary Lagrangian–Eulerian (ALE) description of the long flexible solid (track) whose mechanical properties may be captured using a dynamics-preserving selection of modes, e.g., via a Padé approximation of a transfer function. The modes accompany the contact interaction rather than being referred to a fixed frame, as it occurs in the finite-element floating frame of reference formulation. In the method discussed in this paper, the mesh, which moves through the long flexible solid, is defined in the trajectory coordinate system (TCS) used to describe the dynamics of the set of bodies (vehicle) that interact with the long flexible structure. For this reason, the selection of modes can be focused on the preservation of the dynamics of the structure instead of having to ensure the structure's static displacement convergence due to the motion of the load. In this paper, the validation of the so-called trajectory coordinate system/moving modes (TCS/MM) method is performed in four different aspects: (a) the analytical mechanics approach is used to obtain the equations of motion in a nonmaterial volume, (b) the resulting equations of motion are compared to the classical discretization procedures of partial differential equations (PDE), (c) the suitability of the moving modes (MM) to describe deformation due to variable-velocity moving loads, and (d) the capability of the finite nonmaterial volume to describe the dynamics of an infinitely long flexible body. Validation (a) is completely general. However, the particular example of a moving load applied to a straight beam resting on a Winkler foundation, with known semi-analytical solution, is used to perform validations (b), (c), and (d).