With the first kind of Lagrange’s equations, this paper presents the dynamical equations of multibody system with friction constraints. The generalized forces of friction forces are described in the form of matrix. Considering numerical method is widespread to analyze the characteristics of multibody system dynamics, this paper compares the two friction laws for solving the multibody system problem with dry friction constraints. Using Baumgarte’s and augmentation method, the differential-algebraic equations are given in the form of differential equations matrix to raise calculating efficiency. The friction force for Coulomb’s friction law and the continuous friction law is denoted, which converts subsection smooth systems to continuous smooth systems. An example is given to evaluate the validity of continuous law of friction. The numerical simulation shows that continuous law of friction is an effective method to process multibody system friction problem. The work in this paper also provides a new direction to research the non-smooth multibody system dynamics equation.