A New Approximate Analytical Solution for Coupled Mathieu-Type Differential Equations in Bridge and Vibrating Vehicle Suspension Interactions

An energetic Lagrangian approach is applied to formulate coupled dynamic equations for unknown time-dependent displacements with respect to girder mid-span and vehicle suspension interaction, along with their rates and accelerations. The paper provides for a non-dimensional set of two linear non-homogeneous differential equations of a second order with variable coefficients, and a new analytical solution is proposed by implementing Lindstedt’s singular perturbation method. This allowed us to obtain algebraic equations for girder mid-span displacement dynamic factors and suspension transmissibility coefficients vs. dimensionless time, three ratios of the loading frequency with respect to the girder and vehicle, and combined interaction frequencies related to free vibration. It is demonstrated that the derived system of Mathieu-type differential equations represents the free vibration problem of girder–suspension interaction subjected to the motion of a weightless vehicle, while four non-trivial fundamental solutions to the excitation of free-vibrating girder–suspension and suspension–girder systems are proposed. Simplified approximate analytical solutions are also given for dynamic factors of girder mid-span displacement. The analytically predicted results of dynamic variables are also compared with those determined numerically.