Abstract: In this study, a powerful analytical method, called He’s Parameter-Expanding Method (PEM) is used to obtain the exact solutions of nonlinear free vibrations of a mass grounded by linear and nonlinear springs. Based on a single equation of motion in terms of relative displacement variable, a qualitative analysis is completed and some interesting dynamic behaviors are discovered. The ranges of oscillations are determined and expressions of exact periods for symmetric and asymmetric oscillations are established. It is shown that one term in series expansions is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. Moreover, the numerical solution based on shooting method and fourth order Runge Kutta method have been developed. Comparison of the obtained solution with those obtained using numerical method shows that this method is effective and convenient for solving this problem. This method introduces a capable tool for solving this kind of nonlinear problems.