Parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity

In this paper the parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity are analyzed. The two-dimensional Lindstedt–Poincare perturbation technique applied for finding an approximate solution of linear parametrically excited systems is extended for analyzing a strong nonlinear oscillator. Based on the solution of a nonlinear differential equation with constant coefficients, an approximative solution is introduced. The transition curves and transient surfaces along which periodic solutions exist are obtained. Their strong dependence on the initial conditions is evident. To prove the analytical solution, the numerical experiment is done. For certain values initial conditions and parameter values, the time history diagrams for the oscillator are plotted.