Abstract: The criterion of Lyapunov exponents is presented for dynamic analysis of non-smooth nonlinear dynamical systems by introducing n-dimensional non-smooth (rigid constraint and piecewise smooth) nonlinear dynamical systems, and by setting up some analytical relations between the spectrum of Lyapunov Exponents and Floquet characteristic multipliers of above systems by means of the Poincaré map method. Finally, the spectrum of Lyapunov exponents in a large range of parameters is calculated for a given nonlinear dynamical system with rigid constraints. The stability, bifurcations and chaos of this given system are analyzed by applying the criterion of Lyapunov exponents. The results are compared with the bifurcation diagram of this given system obtained by the Poincaré map method in the corresponding parameters range in order to validate correctness and validity of this criterion for dynamic analysis of non-smooth nonlinear dynamical systems.