A new method for solving interaction with moving mass and structure with explicit precise integration method is proposed. When a vehicle runs on the track with the weld irregularity and the sleeper hanging, the wheel/rail force and the interaction force of the track parts are analyzed. The Duhamel integral of the inhomogeneous terms is calculated by the precise integration to avoid inversing state-space system matrix. The extrapolation formula of interaction load of the moving mass and the structure is obtained by the Lagrange interpolation polynomial. The dynamic response of the nonlinear interaction with moving mass and structure is solved without iteration. The repeated calculation of the exponential matrix is not required due to the time-independent system matrix generated. During integration step, the moving load is decomposed by the finite element shape function to represent the continuous movement of the load. The results indicate that the first peak of the wheel/rail force is induced by the weld irregularity. The sleeper hanging influences the second peak of the wheel/rail force. The hanging sleeper happens easily at rail weld. Once there is one unsupported sleeper, the adjacent sleeper may hang.