Research paperControllable parametric excitation effect on linear and nonlinear vibrational resonances in the dynamics of a buckled beam

•High values of the parametric force amplitude reduce the number of resonances.•After a threshold value of parametric force amplitude, no resonance appears.•Maximum amplitude remains constant before the critical parametric amplitude.•The threshold value corresponds to the second parametric resonance value.•No resonance occurs for low values of the parametric force frequency.

In this study, the effect of a controllable parametric excitation on both linear and nonlinear vibrational resonances on the dynamic of a buckled beam excited by a combination of uncontrollable low- and high-frequency periodic forces are investigated. First of all, the beam dynamic is assumed to be constrained by two periodic and independent ambient solicitations, such as wind and earthquake. An axial load of the beam represented by a periodic and parametric excitation is used to control the vibrational resonance phenomenon, induced by the presence of the two external excitations. Approximate analytical expressions for the linear response and the high-frequency force amplitude at which linear vibrational resonance occurs are obtained. An analytical expression of the amplitude of the nonlinear response at the superharmonic equal to the double of the low-frequency, is obtained. For all these expressions, we show the effect of the parametric excitation. We compare all the obtained results with the ones of the case where, the parametric force is absent. It is shown that, the presence of the parametric excitation permit the suppression of both linear and nonlinear vibrational resonances. Moreover, the vibration amplitudes of the buckled beam are significantly reduced, around certain threshold values for the amplitude and the frequency of the parametric excitation.