The effect of phase transformation on free and forced vibrations of SMA actuators: GDQ modeling of Timoshenko beam

This article aims at studying the free and forced vibrations of shape memory alloy beam ‎actuators. The constitutive equations are derived based on a three dimensional ‎phenomenological model which accounts for pseudoelastically and shape memory effect by ‎considering phase transformation strains in the formulation. In the present study, the ‎pseudoelastic deformations are studied. Assuming Timoshenko beam model for the structures ‎with moderate thickness, equations of motion are derived through Hamilton principle, and the ‎obtained partial differential equations are discretized by applying the Generalized Differential ‎Quadrature approach and then are solved incrementally using Newmark integration method. ‎The return mapping algorithm, considering Newton-Raphson iteration procedure, is employed ‎to update phase transformation strains and any related parameters during solution in each time ‎steps. Results show that energy dissipation occurs due to hysteric behavior of SMA beams ‎during vibrations for both free and forced cases. In most cases, the reduction in amplitudes ‎continues till the material behaves like the elastic solids within austenite phase where no ‎dissipation is appeared. Also, it can be observed from the results that the amplitude of ‎damped vibrations is constant for SMA structure. Therefore, it exhibits more stable behavior ‎in comparison with elastic beam with similar harmonic actuations. Consequently, it is ‎concluded that the SMA materials can be applicable to adjust response frequency in addition ‎to their various applications such as in dampers.