Modeling and Dynamical Behavior of Rotating Composite Shafts with SMA Wires

A dynamical model is developed for the rotating composite shaft with shape-memory alloy (SMA) wires embedded in. The rotating shaft is represented as a thin-walled composite of circular cross-section with SMA wires embedded parallel to shaft’s longitudinal axis. A thermomechanical constitutive equation of SMA proposed by Brinson is employed and the recovery stress of the constrained SMA wires is derived. The equations of motion are derived based on the variational-asymptotical method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin method. The model incorporates the transverse shear, rotary inertia, and anisotropy of composite material. Numerical results of natural frequencies and critical speeds are obtained. It is shown that the natural frequencies of the nonrotating shaft and the critical rotating speed increase as SMA wire fraction and initial strain increase and the increase in natural frequencies becomes more significant as SMA wire fraction increases. The initial strain of SMA wires appears to have marginal effect on dynamical behaviors of the shaft. The actuation performance of SMA wires is found to be closely related to the ply-angle. 1. Introduction Composite materials have found the increased applications for replacement of the conventional metallic materials in the rotating flexible shaft employed for drive shafts of helicopters, steam, and gas turbines. This is likely attributed to high stiffness and strength/weight ratios of composite shaft compared with its metallic counterparts. The development trend in design of light-weight composite shafts is towards higher operating speeds, which gives rise to the problems of high vibration amplitude and stability. Seeking the solution of these problems has caused great research effort [1] in the dynamic of composite rotor. A review on the literature in this area has shown that composite shafts have high whirling resistance capability and are less susceptible to dynamic instability associated with metallic shafts [2]. Several attempts to develop mathematical models of spinning composite shafts are reported in the literature. These models include the shaft models based on shell theories [3], or beam theories combined with the strain—displacement relations of the shell theories [4], or a thin-walled beam theory [5]. Song et al. [5] developed the composite thin-walled shaft model based on a thin-walled beam theory of Rehfield [6]. This model was used to investigate the natural frequencies and