ﺛﺮ اﻧﺘﻘﺎل ﻓﺎز ﺑﺮ ارﺗﻌﺎﺷﺎت آزاد و اﺟﺒﺎري ﻣﺤﺮﮐﻬﺎي ﺣﺎﻓﻆ ﺷﮑﻞ ﺑﺎ ﻣﺪﻟﺴﺎزي ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮ

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

در اﯾﻦ ﻣﻘﺎﻟﻪ، ارﺗﻌﺎﺷﺎت آزاد و اﺟﺒﺎري ﻣﺤﺮك ﻫﺎي از ﺟﻨﺲ آﻟﯿﺎژﻫﺎي ﺣﺎﻓﻆ ﺷﮑﻞ ﺑﺎ در ﻧﻈﺮ ﮔﺮﻓﺘﻦ اﺛﺮات اﻧﺘﻘﺎل ﻓﺎز و اﺷﺒﺎع ﺗﻮﺳﻂ ﻣﺪل ﺗﯿﺮ ﺗﯿﻤﻮﺷﻨﮑﻮ ﺑﺎ ﺷﺮاﯾﻂ ﻣﺮزي ﯾﮑﺴﺮ ﮔﯿﺮدار ﺑﺮرﺳﯽ ﺷﺪه اﺳﺖ. ﭘﺲ از اﺳﺘﺨﺮاج ﻣﻌﺎدﻻت ﺣﺮﮐﺖ از اﺻﻞ ﻫﻤﯿﻠﺘﻮن، روش ﻣﺮﺑﻌﺎت دﯾﻔﺮاﻧﺴﯿﻠﯽ ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ ﺑﺮاي ﮔﺴﺴﺘﻪ ﺳﺎزي ﻣﻌﺎدﻻت ﺑﮑﺎر رﻓﺘﻪ و ﺑﻪ دﻟﯿﻞ ﻏﯿﺮ ﺧﻄﯽ ﺑﻮدن از اﻟﮕﻮرﯾﺘﻢ ﻧﮕﺎﺷﺖ ﺑﺎزﮔﺸﺘﯽ، روش ﻧﯿﻮﺗﻦ راﻓﺴﻮن و روش ﻧﯿﻮﻣﺎرك ﺣﻞ زﻣﺎﻧﯽ آﻧﻬﺎ ﺑﺪﺳﺖ آﻣﺪه اﺳﺖ. ﻧﺘﺎﯾﺞ ﻧﺸﺎن ﻣﯽ دﻫﻨﺪ ﺑﻪ دﻟﯿﻞ ﺗﺸﮑﯿﻞ ﺣﻠﻘﻪ ﻫﺎي ﻫﯿﺴﺘﺮزﯾﺲ ارﺗﻌﺎﺷﺎت ﻣﯿﺮا ﺷﺪه و در ﺑﺴﯿﺎري از ﺣﺎﻻت ﺑﻪ ﻧﺎﺣﯿﻪ اﻻﺳﺘﯿﮏ در ﻓﺎز آﺳﺘﻨﯿﺖ ﻣﺤﺪود ﻣﯽ ﺷﻮﻧﺪ.

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.