Nonlinear dynamic characteristics of SMA gripper under bounded noise

. A kind of constitutive model of SMA is proposed in this paper, and the nonlinear dynamic response of a SMA gripper under bounded noise is studied. The harmonic driving signals and the random disturbance made up of bounded noise. The dynamic model of the system is established by Hamilton principle. The numerical and experimental results show that there is stochastic resonance in the system; the system’s vibration amplitude reaches the most when the outside excitation is moderate.


Introduction
Shape memory alloy (SMA) is a type of smart material, which has shape memory effect. SMA gripper is used in medical field widely [1]. To enhance the accuracy of SMA gripper, its dynamic characteristics should be studied. Many researchers have studied SMA gripper [1][2][3][4][5][6][7]. Kohl et al. studied a SMA gripper's dynamic response and control [2]. Just et al. applied position control to a SMA gripper and obtained high control accuracy [3]. To SMA materials, Graesser et al. proposed a three-dimensional SMA constitutive model [4]. Ivshin et al. developed a SMA thermo-mechanical model [5]. Although many achievements have been reported, most of them focused on the constitutive model, and the results of dynamic response of SMA gripper are absent.
SMA gripper used in medicine are controlled by harmonic currents to achieve the opening and closing action. However, SMA gripper are usually under stochastic excitation in the working process. Although the stochastic excitation is weak, it will affect the gripper's motion. The harmonic control force and the stochastic excitation generate a bounded noise, which cause the different dynamic characteristics from the harmonic system. The experimental results of SMA are shown in Figure 1. The length of Ti-Ni SMA film is 7mm, the width is 1.5mm, and the thickness is 0.1mm. The SMA's austenite finish temperature is 34℃. Thus, the hysteretic phenomenon is induced by the superelastic behavior of SMA. Zhu

SMA constitutive model
where  is the stress,  is the strain. To SMA shown in Figure 1     The results of prediction test to Eq. (1) are shown in Figure 4, and the mechanical model of a SMA gripper under bounded noise is shown in Figure 5, where is the bounded noise. The Hamilton's variational principle is: Thus, the dynamic model of a SMA gripper is: The equation of the system's response are:

Nonlinear dynamic characteristics of a SMA gripper
When the noise intensity 0   , the outside excitation becomes harmonic excitation, and the dynamic model can be shown as follows: ) sin( ) 2 ( 4 4 2 3 The solution of Eq. (7) is: The system's averaged equation is: Fig. 6. Stationary probability density of the system's response.
The numerical results of the system's response are presented in Figure 6, and the experimental results of SMA gripper under bounded noise are shown in Figures 7-9, where the frequency  =30Hz. Ti-Ni alloy is chosen as SMA film. The length of the micro gripper is 10cm, and its width is 1cm. The length of SMA thin film is 3cm, its width is 1cm, and its thickness is 0.4 mm. The stochastic resonance phenomenon occurs in the process.

Conclusion
A kind of constitutive model of SMA is proposed in this paper, and the nonlinear dynamic response of a SMA gripper under bounded noise is studied. The harmonic driving signals and the random disturbance made up of bounded noise. The dynamic model of the system is established by Hamilton principle. The numerical and experimental results show that there is stochastic resonance in the system; the system's vibration amplitude reaches the most when the outside excitation is moderate.