Chaos and subharmonic bifurcation of a composite laminated buckled beam with a lumped mass

With analytical and numerical methods, dynamic response of a composite laminated buckled beam with a lumped mass and an axial excitation is investigated in this paper. The dynamic equation of the first mode for this model, which is a perturbation of a Hamilton system with both quadratic and cubic nonlinearities as well as external and parametric excitations, is obtained with the Galerkin method. The distribution of the equilibriums, homoclinic orbits and periodic orbits for the unperturbed system under different system parameters are obtained analytically. Chaotic vibrations arising from homoclinic intersections are studied with Melnikov method. Chaotic feature on the location of the lumped mass is investigated in detail. It is presented that there exist controllable frequencies decreasing monotonously with the location of the lumped mass for this system. Subharmonic bifurcation is also studied with the subharmonic Melnikov method. It is numerically proved that the system may undergo chaotic motions through finite subharmonic bifurcations. Numerical simulations including the phase portraits, time histories, Poincar e ´ sections, Lyapunov spectrums and bifurcation diagram are given, which confirm the analytical results.