Nonlinear vibration of a post-buckled beam subjected to external and parametric excitations

An analytical investigation of the nonlinear vibration of a post-buckled beam subjected to harmonic excitations is presented. The system can be modelled as a nonlinear oscillator with parametric and external excitations having frequencies Omega(1) and Omega(2). The method of multiple scales is used to determine to second order the amplitude and phase-modulation equations. Attention is focused on subharmonic resonances (Omega(1) congruent to 2 omega(0), Omega(2) congruent to 2 omega(0); Omega(1) congruent to 3 omega(0), Omega(2) congruent to 3 omega(0); Omega(1) congruent to 4 omega(0), Omega(2) congruent to 4 omega(0); Omega(1) congruent to 5 omega(0) and Omega(1) congruent to 6 omega(0)) where omega(0) is the natural frequency. Steady-state amplitude for each case is plotted as a function of a detuning parameter showing the influences of the several parameters. Stability is performed on figures by solid and broken lines. There exist multivalued solutions which increase or decrease by the variation of some parameters. The response amplitude is not affected by increasing and decreasing the parameters F-1 and alpha(3) for the cases of subharmonics of order one-half and one-third. The solution loses stability on increasing the parameters alpha(4), alpha(5) and F-1 in the case of subharmonic resonance of order one-fifth and also for decreasing the parameters mu and alpha(4) in the case of subharmonic resonance of order one-sixth.