# Response Statistics of a Shape Memory Alloy Oscillator with Random Excitation
## 2 Description of the SMA Oscillator Model  
In this work, the polynomial constitutive model to characterize the behavior of the oscillator with shape memory is introduced. Firstly, we provide the restoring force model of the SMA spring. On the basis of [PERSON]'s theory, [PERSON] [26] proposed a polynomial constitutive model to describe the behavior of the SMA oscillator with shape memory,which is also a common model to describe the pseudoelasticity and the shape memory effect [3]. There is a SDOF oscillator model that considers a polynomial free energy that depends on temperature and strain; the free energy is given as follows:  
\[E(\epsilon,T)=\frac{1}{2}a_{1}(T-T_{M})\epsilon^{2}-\frac{1}{4}a_{2}\epsilon^{4 }+\frac{1}{6}a_{3}\epsilon^{6}, \tag{1}\]  
where \(a_{1}\), \(a_{2}\), and \(a_{3}\) are positive material constants; \(\epsilon\) is the strain; \(T\) is the temperature; and \(T_{M}\) is the critical temperature for martensitic transformation. The minima and maxima points of the free energy represent the stability and instability of each phase of the SMAs, respectively. For the SMAs, three phases are considered in general, namely, austenite, single-variant martensite, and multiple-variant martensite. At high temperatures (\(T>T_{A},\ T_{A}\) is the critical temperature for austenite transformation), the free energy has only one minimum at vanishing strain, representing the equilibrium of the austenite phase. At low temperatures (\(T<T_{M}\)), it must have two minima at vanishing strain, corresponding to the equilibrium of the martensite phases. At intermediate temperatures (\(T_{M}\leq T\leq T_{A}\)), it has three equilibrium point corresponding to both of the austenite and martensite phases. For the SMA, \(T_{A}\) and \(T_{M}\) are two important temperature parameters, which correspond to the temperature where the austenite phase and the martensitic phase are stable. The relationship between \(T_{A}\), \(T_{M}\), \(a_{1}\), \(a_{2}\), and \(a_{3}\) is expressed as  
\[T_{A}=T_{M}+\frac{a_{2}^{2}}{4a_{1}a_{3}}\;. \tag{2}\]  
Thus, the stress-strain of the SMA is given by  
\[\delta=\frac{\partial E}{\partial\epsilon}=a_{1}(T-T_{M})\epsilon-a_{2} \epsilon^{3}+\frac{{a_{2}}^{2}}{4a_{1}(T_{A}-T_{M})}\epsilon^{5}\;. \tag{3}\]  
Subsequently, the restoring force of the SMA spring that depends on the displacement \(u\) can be established as  
\[K=K(u,T)=\overline{a_{1}}(T-T_{M})u-\overline{a_{2}}u^{3}+\overline{a_{3}}u^{ 5}\;, \tag{4}\]  
in which \(\overline{a_{1}}=\frac{a_{1}A}{L}\), \(\overline{a_{2}}=\frac{a_{2}A}{L^{3}}\), and \(\overline{a_{3}}=\frac{a_{3}A}{L^{5}}\). \(A\) is the area of the SMA spring wire, and \(L\) is the original length of the SMA spring. Now, we consider a SDOF SMA oscillator model as shown in Figure 1, where a mass \(m\) is connected to a rigid support through of a viscous damping with coefficient \(c\) and a shape memory element, and the mass block is subjected to an external force \(F(t)\). Figure 1: Stochastic dynamic model of shape memory alloys oscillator. By [PERSON]'s second law, the motion equation of vibration of the system can be expressed as  
\[m\ddot{u}+c\dot{u}+\overline{a_{1}}(T-T_{M})u-\overline{a}_{2}u^{3}+\overline{a_{ 3}}u^{5}=-m\overline{\overline{u_{b}}}\, \tag{5}\]  
where \(u\) represents the displacement of the mass \(m\), the dot denotes a derivative with respect to the time \(t\), and \(\overline{\overline{u_{b}}}\) denotes the base acceleration. In order to obtain a dimensionless equation of motion, we introduced the following nondimensional transformations  
\[ \begin{array}{l}x=\frac{u}{L};\ x_{b}=\frac{\overline{u_{b}}}{L}\,\quad\tau=\omega_{0}t\ ; \ \theta=\frac{T}{M_{M}};\ \alpha=\theta-1\ ;\ \beta=-\frac{\alpha_{2}A}{mL \omega_{0}^{2}};\\ \gamma=\frac{a_{1}A}{mL\omega_{0}^{2}}\ ;\ \zeta=\frac{c}{mL\omega_{0}};\ \omega_{0}^{2}= \frac{a_{1}AT_{M}}{mL}.\end{array}  \tag{6}\]  
where \(x\), \(\tau\), \(\zeta\), and \(\theta\) are the dimensionless displacement, time, damping coefficient, and temperature, respectively.