Analytical study of the nonlinear behavior of a shape memory oscillator: Part II-resonance secondary

In Part II of this work, we investigated the dynamics of the shape memory oscillator, in the particular case of the secondary resonances. We used the equation of motion developed in Part I (Piccirillo et al., Nonlinear Dyn., 2009). The method of multiple scales is used to obtain an approximate solution to the governing equations of motion. To examine subharmonic and superharmonic resonances, we need to order the excitation so that it appears at same time as the free-oscillation part of the solution. Firstly, the analysis is made for the superharmonic resonance where we find the frequency-response curves and these curves show the influences of the damping, nonlinearity, and amplitude of the excitation. Results showed that it occurs in the jump phenomena, bifurcation saddle-node, and motions periodic the period-2. In the subharmonic resonance, we note that it does not occur in the jump phenomena, but on the other hand, we found the regions where the nontrivial solutions of the subharmonic resonance exit. The frequency-response curves show the behavior of the oscillator for the variation of the control parameters. Numerical simulations are performed and the simulation results are visualized by means of the phase portrait, Poincare map, and Lyapunov exponents.