7. A Cross-Iteration Algorithm for Vehicle–Track Nonlinear Coupling Vibration Analysis

The dynamic finite element equation of the vehicle-track nonlinear coupling system is a large, high-order and nonlinear coupled second-order differential equation set, the efficiency of the conventional algorithm for solving the large and complicated differential equation set is not satisfied. In this chapter, the cross-iterative algorithm for solving the dynamic finite element equation of nonlinear vehicle-track coupling system is described, and the algorithm verification and convergence analysis are carried out by given examples. In simulation analysis, the whole system is divided into two subsystems, i.e., the vehicle subsystem considered as a vehicle element with a primary and secondary suspension system, and the track or the track-bridge subsystem regarded as a three elastic beam model. Coupling of the two systems is achieved by equilibrium conditions for wheel-rail nonlinear contact forces and geometrical compatibility conditions. In order to accelerate the iterative convergence, a relaxation technique is introduced to modify the wheel-rail contact forces. The cross-iteration algorithm has the shining advantages of higher efficiency, better precision, and simple programming. Based on the proposed model and the algorithm, the source code, “Train-track-continuous bridge coupling system dynamics calculation program VTBDYN_1.0,” developed using MATLAB mathematical tools, is provided in Appendix C.