A weakly nonlinear system under simultaneous sinusoidal external and parametric excitations is investigated. Quadratic and cubic nonlinearities are present in the governing equations. A general perturbation analysis, the Method of Multiple Scales (MMS), is performed for numerous resonance frequencies. Emphasis is initially placed on the response of the system under parametric excitation alone. The nonresonant external and parametric excitations are then considered. Finally, responses involving both parametric and external excitations are considered. The excitation frequencies are assumed to be from the same source. .

When the frequency of the_parametric and external excitations are different (λ≠Ω), many of the different resonances investigated have solvability conditions similar to those found in two preliminary works performed by Mook, Plaut and HaQuang. When the frequencies are nearly equal, numerous steady-state response curves are shown. Unlike the linear analysis, the frequency-response curves show many multi-valued responses. In some instances, as many as five amplitudes exist for a given frequency. Three are stable and two are unstable. In addition, multi-modal responses were found to exist under a single-mode excitation. This result is unique since no internal resonance was considered. For certain values of the coefficient of the nonlinear restoring forces, stable bimodal steady states were observed.

In order to verify some of the theoretical results obtained by MMS, a sixth-order Runge Kutta procedure was performed on the original governing equation. The numerically integrated results and the approximate solution of MMS show excellent agreement when the parameter ε is sufficiently small. However, when ε is sufficiently large, the MMS approximate solution breaks down. Interesting phenomena, such as periodic doubling and chaos, are observed.