Subharmonic resonance and transition to chaos of nonlinear oscillators with a combined softening and hardening nonlinearities

The concern of this work is the 1/3 sub-harmonic resonance response and transition to chaos in the response of harmonically driven single degree of freedom nonlinear oscillators with a combined static and inertia nonlinearities. Approximate analytical solutions to the 1/3 sub-harmonic resonance curves are obtained using the harmonic balance (HB) method and the multiple scales (MMS) perturbation method. Stability analyses of the obtained approximate solutions were used to determine zones of chaotic behavior in the primary frequency response curve. The obtained analytical results were verified for selected values of system parameters using computer simulations and with the aid of time histories, phase planes, Poincare map, FFT and Lyapunov exponents.