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

Abstract

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 freeoscillation 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, Poincaré map, and Lyapunov exponents.

Manipulation of wave motion in smart nonlinear phononic crystals made of shape memory alloys

“…Shape memory alloys (SMAs) are a group of smart materials with the ability to recover their original shape upon heating/cooling cycles. Due to their fascinating stress-and temperature-dependent characteristics, SMAs have received increasing attention over the past years [44][45][46]. Along with the experimental investigations, some mathematical formulations were also developed to predict the behavior of SMAs in various working phases [47].…”

Force transmissibility characteristics of a pseudoelastic oscillator

“…Machado et al (2009) estimated the Lyapunov exponents of a force excited SMA oscillator operating in isothermal and non-isothermal environments. Piccirillo et al (2010) developed a semi-analytical technique for inspecting subharmonic resonance, superharmonic resonance and stability in SMA oscillator. Numerical simulations were carried out to verify the approximate results obtained.…”

Bogdanov–Takens Bifurcation in a Shape Memory Alloy Oscillator with Delayed Feedback

“…System (28) as a universal unfolding [30,31] with codimension-2 has been well studied. We can acquire the complete bifurcation diagram and topological classification of the trajectory of system (28), and the bifurcation diagram of system (28) on the perturbation parameter v 1 and v 2 planes is shown in Figure 1 [17,21]. Furthermore, we give a concise form to list the conclusion as follows:…”