# Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices
## 3 The Proposed Method  
In the present section, the AMS is applied to achieve the asymptotic solutions of the above Equations (7)-(9). Therefore, we examine the dynamics of considered systems in a small neighborhood of their static equilibrium position [41]. Consequently, we express the amplitudes of all oscillations in terms of a small parameter \(0<\epsilon<<1\) as follows  
\[ \begin{array}{ll}z(\tau)=\epsilon\,\widetilde{x}(\tau;\epsilon),\,\theta( \tau)=\epsilon\,\widetilde{\varphi}(\tau;\epsilon),\\ v(\tau)=\epsilon\,\widetilde{v}(\tau;\epsilon)\,,\,q(\tau)=\epsilon\, \widetilde{q}(\tau;\epsilon).\end{array}  \tag{7}\]  
Consider employment of the following variables and parameters  
\[ \begin{array}{ll}r=\epsilon^{2}\,\widetilde{r},\,\mu_{1}=\epsilon^{2}\, \widetilde{l}_{1},\,\mu_{2}=\epsilon^{2}\,\widetilde{l}_{2},\,l_{m}=\epsilon^{ 2}\widetilde{l}_{m}\\ \gamma_{1}=\epsilon^{2}\,\widetilde{\gamma}_{1},\,\gamma_{2}=\epsilon^{2}\, \widetilde{\gamma}_{2},\,\alpha=\epsilon^{2}\,\widetilde{\alpha},\\ c_{1}=\epsilon^{2}\,\widetilde{c}_{1},\,f_{1}=\epsilon^{3}\,\widetilde{l}_{1},\,f_{2}=\epsilon^{3}\,\widetilde{l}_{2},\\ R_{p}=\widetilde{R}_{p}/\epsilon^{2}\,,\,c_{p}=\epsilon^{2}\widetilde{c}_{p}, \,R_{m}=\epsilon^{2}\widetilde{R}_{m}.\end{array} \]  
We looked for the desired solutions \(\widetilde{x},\,\widetilde{\varphi},\widetilde{v}\), and \(\widetilde{q}\) as power series of \(\epsilon\) according to \[ \begin{array}{l}\widetilde{x}=\sum_{k=1}^{3}\varepsilon^{k}x_{k}(\tau_{0},\,\tau_ {1},\,\tau_{2})+O(\varepsilon^{4}),\\ \widetilde{\varphi}=\sum_{k=1}^{3}\varepsilon^{k}\varphi_{k}(\tau_{0},\,\tau_ {1},\,\tau_{2})+O(\varepsilon^{4}),\\ \widetilde{v}=\sum_{k=1}^{3}\varepsilon^{k}v_{k}(\tau_{0},\,\tau_{1},\,\tau_ {2})+O(\varepsilon^{4}),\\ \widetilde{q}=\sum_{k=1}^{3}\varepsilon^{k}q_{k}(\tau_{0},\,\tau_{1},\,\tau_ {2})+O(\varepsilon^{4}),\end{array}  \tag{9}\]  
where \(\tau_{n}=\varepsilon^{n}\tau\) (\(n=0,\,1,\,2\)) are different time scales. The system of Equations (7)-(9) was transformed to a set of partial differential equations (PDE), i.e., we used the following operators of time derivatives  
\[ \begin{array}{l}\frac{d}{d\tau}=\frac{\partial}{\partial\tau_{0}}+ \varepsilon\frac{\partial}{\partial\tau_{1}}+\varepsilon^{2}\frac{\partial}{ \partial\tau_{2}},\\ \frac{d^{2}}{d\tau^{2}}=\frac{\partial^{2}}{\partial\tau_{0}^{2}}+2\varepsilon \frac{\partial^{2}}{\partial\tau_{0}\partial\tau_{1}}+\varepsilon^{2}(\frac{ \partial^{2}}{\partial\tau_{1}^{2}}+2\frac{\partial^{2}}{\partial\tau_{0} \partial\tau_{2}})+O(\varepsilon^{3}).\end{array}  \tag{10}\]  
Terms of \(O(\varepsilon^{3})\) and higher orders in the operators (12) are neglected. Employing a procedure of splitting with respect to perturbation parameter \(\varepsilon\), the following nine linear PDE are obtained  
(_i_) Order of (\(\varepsilon\))  
\[\frac{\partial^{2}x_{1}}{\partial\tau_{0}^{2}}+x_{1}=0, \tag{11}\]  
\[ \begin{array}{l}\frac{\partial^{2}\phi_{1}}{\partial\tau_{0}^{2}}+W^{2} \phi_{1}=0,\\ \frac{\partial\tau_{1}}{\partial\tau_{0}}+\frac{v_{1}}{\omega_{1}\tilde{v}_{ \widetilde{v}}R_{p}}=\frac{l\widetilde{\tau}_{1}}{\tilde{v}_{p}}\frac{ \partial x_{1}}{\partial\tau_{0}},\\ \frac{\partial\eta_{1}}{\partial\tau_{0}}+\frac{q_{1}\widetilde{R}_{m}}{ \omega_{1}\tilde{l}_{m}}=\frac{l\widetilde{\tau}_{1}}{l_{m}}\frac{\partial x _{1}}{\partial\tau_{0}}.\end{array}  \tag{13}\]  
(_ii_) Order of (\(\varepsilon^{2}\))  
\[ \begin{array}{l}\frac{\partial^{2}x_{2}}{\partial\tau_{0}^{2}}+x_{2}=\frac {1}{2}p^{2}\,\widetilde{r}e^{i\,p\,\tau_{0}}-\frac{1}{2}W^{2}\varphi_{1}^{2}+ \big{(}\frac{\partial\varphi_{1}}{\partial\tau_{0}}\big{)}^{2}-2\frac{ \partial^{2}x_{1}}{\partial\tau_{0}\partial\tau_{1}},\end{array}  \tag{14}\]  
\[