# Response Statistics of a Shape Memory Alloy Oscillator with Random Excitation
## 2 Description of the SMA Oscillator Model
As the SMAs exhibit different properties depending on the temperature \(\theta\), we only considered the temperature where the phase is stable in the alloy, that is, \(\theta>1\) in the present study. Then, the dimensionless equation of nonlinear random vibration of the SMA oscillator systems can be rewritten as  
\[\ddot{x}+\zeta\dot{x}+\alpha x+\beta x^{3}+\gamma x^{5}=-\ddot{x_{b}}. \tag{7}\]  
The dimensionless potential function \(U(x)\) of the SMA oscillator can be expressed as the following general form:  
\[U(x)=\frac{1}{2}\alpha x^{2}+\frac{1}{4}\beta x^{4}+\frac{1}{6}\gamma x^{6}. \tag{8}\]  
Depending on parameters of the nonlinear terms \(\alpha\), \(\beta\), and \(\gamma\), the potential function \(U(x)\) has three kinds of different potentials, as shown in Figure 2. According to the activation temperature \(\theta_{A}=1.9\), the SMA oscillator can be divided into the following three categories [27]:  
1. \(\theta\leq 1\) (low temperature). The potential function has two minima and one maximum, that is, three fixed points. 2. \(1<\theta<\theta_{A}\) (medium temperature). The potential function has three stable equilibria and two unstable saddle points. Switching among the three wells or restricted in one of three potential wells depending on the total energy and the initial condition. 3. \(\theta=\theta_{A}\). The potential has three minima. Figure 2: Potential function of the SMA oscillator system (7) for various relative temperatures \(\theta\) with the system parameters: \(\alpha=\theta-1\), \(\beta=-1.3\times 10^{3}\), \(\gamma=4.7\times 10^{5}\). \(\theta>\theta_{A}\) (high temperature). The potential function has only one minimum (potential well). In the present study, we considered \(-\ddot{x_{b}}=\eta(\tau)\), where \(\eta(\tau)\) is a random process used to characterize random disturbances of the external environment, which is expressed as follows:  
\[\eta(\tau)=f\cos[\Omega\tau+hW(\tau)], \tag{9}\]  
where \(f\) and \(\Omega\) denote the amplitude and the central frequency of the random excitation, respectively; \(W(\tau)\) is the standard Wiener process; and \(h\geq 0\) is the noise intensity. Obviously, the random excitation \(\eta(\tau)\) will degenerate into a deterministic periodic force if \(h=0\). The corresponding power spectrum density of the random process \(\eta(\tau)\) is given as follows [19; 21]:  
\[S_{\eta}(\omega)=\frac{1}{2}\frac{f^{2}h^{2}\Big{(}\Omega^{2}+\omega^{2}+h^{4 }/4\Big{)}}{\Big{(}\Omega^{2}-\omega^{2}+h^{4}/4\Big{)}^{2}+\omega^{2}h^{4}}, \tag{10}\]  
We uncovered that, when the noise intensity \(h\) is small, the spectrum \(S_{\eta}(\omega)\) concentrates on a narrow-band region, which can be regarded as a narrow-band process. In the present paper, only a narrow-band noise case is considered, i.e., the noise intensity \(h\) is assumed to be a small parameter. In this investigation, to realize the theoretical analysis, we assumed that the SMA oscillator has a weak nonlinearity. Hence, a small parameter \(0<\varepsilon\ll 1\) was introduced to rescale the following parameters as  
\[\zeta\rightarrow\varepsilon\zeta,\ \ \beta\rightarrow\varepsilon\beta\,\ \gamma \rightarrow\varepsilon\gamma\,\ f\rightarrow\varepsilon f.\]  
Thus, the governed equation of the established SMA oscillator model can be rewritten as  
\[\ddot{x}+\varepsilon\zeta\dot{x}+\alpha x+\varepsilon\beta x^{3}+\varepsilon \gamma x^{5}=\varepsilon f\cos[\Omega\tau+hW(\tau)]. \tag{11}\]