A method is developed to predict the critical harmonic excitation of systems undergoing nonlinear oscillations. The method is based on the total energy approach which limits the system responses within a region bounded by a critical total energy in the phase space. Three one-degree-of-freedom nonlinear systems are investigated. Their governing ordinary differential equations are associated with a quadratic nonlinearity and/or a cubic nonlinearity. The study also is extended to a two-degree-of-freedom nonlinear system. The harmonic balance method is the analytical technique used in solving the nonlinear ordinary differential equations. In comparison with the approximate analytical solutions, numerical approaches are implemented. / Master of Science