A kinetic model of the piecewise-linear nonlinear suspension system that consists of a dominant spring and an assistant spring is established. Bifurcation of the resonance solution to a suspension system with two degrees of freedom is investigated with the singularity theory. Transition sets of the system and 40 groups of bifurcation diagrams are obtained. The local bifurcation is found, and shows the overall characteristics of bifurcation. Based on the relationship between parameters and the topological bifurcation solutions, motion characteristics with different parameters are obtained. The results provides a theoretical basis for the optimal control of vehicle suspension system parameters.