Abstract: In nonsmooth multibody system dynamics, conventional modeling and numerical approaches for linear complementarity problems are difficult to accurately capture the true physical properties. In this study, the Newton restitution coefficient and the Moreau-Jean midpoint algorithm are combined. By retaining the nonlinear terms of the kinetic energy, potential energy, and contact constraints in the Lagrange’s equations, an LCP algorithm based on Lagrange’s equations is developed. Meanwhile, the piecewise friction coefficient model is introduced to more realistically reflect the nonlinear nature of friction. The results show that the LCP algorithm based on the Lagrange’s equations has superior numerical accuracy. The piecewise friction coefficient model enables this system to reach a stable state more quickly, in accordance with the energy dissipation law. This study is crucial for comprehending and predicting collision dynamics under complex contact conditions.