# Response Statistics of a Shape Memory Alloy Oscillator with Random Excitation
## 3 Theoretical Analysis of the SMA Oscillator Model
 \tag{21}\]  
where \(D_{1}A\) stands for the derivative with the respect to the slow time scale \(T_{1}\), \(\overline{A}\) means the conjugate of \(A\), and  
\[hW(\tau)=hW(T_{0})=hW\left(\frac{T_{1}}{\varepsilon}\right)=\frac{h}{\sqrt{ \varepsilon}}W(T_{1}):=\overline{h}W(T_{1}). \tag{22}\]  
Substituting Equation (17) into Equation (21) and eliminating the secular terms, we yield  
\[-2i\Omega D_{1}A-i\zeta\Omega A+\sigma A-3\beta A^{2}\overline{A}-10\gamma A ^{3}\overline{A}^{2}+\frac{1}{2}fe^{iW(T_{1})}=0. \tag{23}\]  
The complex amplitude in the polar form is expressed as follows:  
\[A(T_{1})=\frac{1}{2}a(T_{1})e^{i\varphi(T_{1})},\ \overline{A}(T_{1})=\frac{1}{2}a (T_{1})e^{-i\varphi(T_{1})}.\]  
Then, we can obtain  
\[i\Omega D_{1}a-\Omega aD_{1}\varphi+\frac{1}{2}i\zeta\Omega a-\frac{1}{2} \sigma a+\frac{3}{8}\beta a^{3}+\frac{5}{16}\gamma a^{5}-\frac{1}{2}fe^{i[ \overline{h}W(T_{1})-\varphi]}=0. \tag{24}\]  
For convenience, let  
\[\phi=\overline{h}W(T_{1})-\varphi, \tag{25}\]  
\[i\Omega D_{1}a-\Omega aD_{1}\varphi+\frac{1}{2}i\zeta\Omega a-\frac{1}{2} \sigma a+\frac{3}{8}\beta a^{3}+\frac{5}{16}\gamma a^{5}-\frac{1}{2}fe^{i \phi}=0. \tag{26}\]Separating real and imaginary parts for the Equation (26), we deduce two so-called modulation equations to  
\[\left\{ \begin{array}{c}D_{1}a=\frac{1}{2\Omega}f\sin\phi-\frac{1}{2}\zeta a,\\ aD_{1}\varphi=-\frac{1}{2\Omega}a\varphi+\frac{3}{8\Omega}\beta a^{3}+\frac{5}{ 16\Omega}\gamma a^{5}-\frac{1}{2\Omega}f\cos\phi.\end{array} \right. \tag{27}\]  
Due to \(D_{1}\phi=\overline{h}D_{1}W(T_{1})-D_{1}\varphi\), that is, \(aD_{1}\phi=a\overline{h}D_{1}W(T_{1})-aD_{1}\varphi\), Equation (27) can be rewritten as  
\[\left\{ \begin{array}{c}D_{1}a=\frac{1}{2\Omega}f\sin\phi-\frac{1}{2}\zeta a, \\ aD_{1}\phi=a\overline{h}D_{1}W(T_{1})+\frac{1}{2\Omega}a\varphi-\frac{3}{8 \Omega}\beta a^{3}-\frac{5}{16\Omega}\gamma a^{5}+\frac{1}{2\Omega}f\cos\phi.\end{array} \right. \tag{28}\]  
Accordingly, the amplitude \(a\) and the phase \(\varphi\) can be calculated from Equation (28). Then the first-order approximate solution of the SMA oscillator model (13) is formulated as  
\[x=a\cos(\Omega\tau+\varphi)+O(\varepsilon), \tag{29}\]  
where \(O(\varepsilon)\) represents the high-order terms of small parameter \(\varepsilon\). Next, we concentrate on the steady-state solutions of Equation (13). **Case 1:** The deterministic case, i.e., the intensity of noise \(\overline{h}=0\). Equation (13) is reduced to a deterministic differential equation with harmonic excitation in this case. Considering the steady-state situation, i.e., \(D_{1}a=D_{1}\phi=0\), one obtains  
\[\left\{ \begin{array}{c}\frac{1}{2\Omega}f\sin\phi-\frac{1}{2}\zeta a=0,\\ \frac{1}{2\Omega}a\varphi-\frac{3}{8\Omega}\beta a^{3}-\frac{5}{16\Omega} \gamma a^{5}+\frac{1}{2\Omega}f\cos\phi=0.\end{array} \right. \tag{30}\]  
Then, according to \(\sin^{2}\phi+\cos^{2}\phi=1\), we obtain the following frequency-amplitude equation  
\[f^{2}=(\zeta\Omega a)^{2}+\bigg{(}-a\sigma+\frac{3}{4}\beta a^{3}+\frac{5}{8} \gamma a^{5}\bigg{)}^{2}. \tag{31}\]  
**Case 2:** The stochastic case, i.e., the intensity of noise \(\overline{h}\
eq 0\). When \(\overline{h}\
eq 0\), the effects of the noise have to be considered. The solution of Equation (28) cannot be given analytically, and therefore a perturbation technique is applied here. We suppose that  
\[a=a_{0}+a_{1},\ \phi=\phi_{0}+\phi_{1}, \tag{32}\]  
where \(a_{0}\) and \(\phi_{0}\) are the solutions of the deterministic case, and \(a_{1}\) and \(\phi_{1}\) are small disturbances under the deterministic situations mentioned above, which can be addressed as small parameters. Substituting Equations (30) and (32) into Equation (28) and eliminating the high-order small items, we obtain  
\[\left\{