Convergence of iterative vehicle-bridge interaction schemes

The modeling of Vehicle-Bridge Interaction (VBI) requires the coupling of two dynamic subsystems, which is often achieved through iterative numerical schemes. This contribution investigates the convergence characteristics of such iterative methods by analyzing a simplified system consisting of a single-degree-of-freedom oscillator crossing a bridge modeled via a single modal coordinate. The governing differential equations are formulated and discretized to obtain the iteration matrix. The convergence of the scheme is evaluated through the spectral radius, which can be analytically determined as a function of the dimensionless mass ratio, stiffness ratio and time step ratio. A parametric study shows that the spectral radius depends strongly on the governing dimensionless parameters and may be either below or above unity. Both unconditionally and conditionally convergent regions are identified, separated by a distinct convergence boundary. For certain combinations of the problem parameters, the iterative scheme does not converge. These results provide a theoretical foundation questioning the robustness of iterative solvers for VBI.