Parametric excitation of subharmonic oscillations

Subharmonic oscillations of order one-half for a single-degree-of-freedom system with quadratic, cubic, and quartic nonlinearities under parametric excitation are investigated. Two approximate methods (multiple scales and generalized synchronization) are used for comparison. The modulation equations (reduced equations) of the amplitudes and the phases are obtained. Steady-state solutions (periodic solutions) and their stability are determined. Numerical solutions are carried out, and graphical representations of the results are presented and discussed. The results obtained by the two methods are in excellent agreement.