# Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices
## 3 The Proposed Method
\begin{array}{l}\frac{\partial^{2}\varphi_{2}}{\partial\tau_{0}^{2}}+W^{2} \varphi_{2}=\frac{1}{2}i\,p^{2}\,\widetilde{r}e^{i\,p\,\tau_{0}}-W^{2}x_{1} \varphi_{1}-2\frac{\partial x_{1}}{\partial\tau_{0}}\frac{\partial\varphi_{1 }}{\partial\tau_{0}}\\ -2\frac{\partial^{2}\varphi_{1}}{\partial\tau_{0}\tau_{1}}-2\,x_{1}\,\frac{ \partial^{2}\varphi_{1}}{\partial\tau_{0}^{2}},\end{array}  \tag{15}\]  
\[ \begin{array}{l}\frac{\partial\tau_{0}}{\partial\tau_{0}}+\frac{v_{2}}{ \omega_{1}\tilde{v}_{\widetilde{v}}R_{p}}=\frac{l\widetilde{\tau}_{1}}{ \tilde{v}_{p}}\big{(}\frac{\partial x_{1}}{\partial\tau_{1}}+\frac{\partial x _{2}}{\partial\tau_{0}}\big{)}-\frac{\partial v_{1}}{\partial\tau_{1}},\\ \frac{\partial q_{2}}{\partial\tau_{0}}+\frac{q_{2}\widetilde{R}_{m}}{\omega_{1 }l_{m}}=\frac{l\widetilde{\tau}_{2}}{l_{m}}\big{(}\frac{\partial x_{1}}{ \partial\tau_{1}}+\frac{\partial x_{2}}{\partial\tau_{0}}\big{)}-\frac{ \partial q_{1}}{\partial\tau_{1}}.\end{array}  \tag{16}\]  
(_iii_) Order of (\(\varepsilon^{3}\))  
\[ \begin{array}{l}\frac{\partial^{2}x_{3}}{\partial\tau_{0}^{2}}+x_{3}&=\frac {1}{2}\widetilde{f}_{1}e^{i\,p_{1}\,\tau_{0}}-\frac{1}{2}i\,p^{2}\,\widetilde{r} \varphi_{1}e^{i\,p\,\tau_{0}}-W^{2}\varphi_{1}\varphi_{2}-\frac{\partial^{2}x_ {1}}{\partial\tau_{1}^{2}}-\widetilde{c}_{1}\frac{\partial x_{1}}{\partial \tau_{0}}\\ &-2\big{(}\frac{\partial^{2}x_{1}}{\partial\tau_{0}\partial\tau_{2}}+\frac{ \partial^{2}x_{2}}{\partial\tau_{0}\partial\tau_{1}}\big{)}+2\big{(}\frac{ \partial\varphi_{1}}{\partial\tau_{0}}\frac{\partial\varphi_{1}}{\partial\tau_{1}} +\frac{\partial\varphi_{1}}{\partial\tau_{0}}\frac{\partial\varphi_{1}}{ \partial\tau_{0}}\big{)}\\ &+x_{1}\big{(}\frac{\partial\varphi_{1}}{\partial\tau_{0}}\big{)}^{2}- \widetilde{\mu}_{1}\,v_{1}(\sigma\,\widetilde{r}_{2}\,q_{1}\,)-3\,\widetilde{ \alpha}\,\zeta_{s}^{2}x_{1},\end{array}  \tag{17}\]\[ \begin{array}{rl}\frac{\partial^{2}\varphi_{3}}{\partial\tau_{0}^{2}}+W^{2} \varphi_{3}&=\frac{1}{2}(\tilde{f}_{2}e^{i\,p_{2}\,\tau_{0}}-i\,p^{2}\,\tilde{r} _{1}e^{i\,p\,\tau_{0}}-p^{2}\,\tilde{r}\varphi_{1}e^{i\,p\,\tau_{0}})\\ &-W^{2}(\varphi_{2}x_{1}+\varphi_{1}x_{2})\,+\frac{1}{6}W^{2}\varphi_{1}^{3}- \frac{\partial^{2}\varphi_{1}}{\partial\tau_{1}^{2}}-\tilde{c}_{2}\frac{ \partial\varphi_{1}}{\partial\tau_{0}}\\ &-2\frac{\partial x_{1}}{\partial\tau_{0}}(\frac{\partial\varphi_{1}}{ \partial\tau_{0}}+\frac{\partial\varphi_{2}}{\partial\tau_{0}})-2\frac{ \partial\varphi_{1}}{\partial\tau_{0}}(\frac{\partial x_{1}}{\partial\tau_{ 1}}+\frac{\partial x_{2}}{\partial\tau_{0}})\\ &-2x_{1}(\frac{3x_{1}}{\partial\tau_{0}}\frac{\partial\varphi_{1}}{ \partial\tau_{0}}+2\frac{\partial x_{1}}{\partial\tau_{0}\varphi_{1}}+\frac{ \partial^{2}\varphi_{2}}{\partial\tau_{0}})-\frac{\partial\varphi_{1}}{ \partial\tau_{0}^{2}}\\ &\times(x_{1}^{2}+2x_{2})-2(\frac{\partial^{2}\varphi_{1}}{\partial\tau_{0}}+ \frac{\partial^{2}\varphi_{2}}{\partial\tau_{0}\varphi_{1}}),\\ &\frac{\partial\varphi_{3}}{\partial\tau_{0}}+\frac{v_{3}}{ \omega_{1}\tilde{c}^{2}R_{\varphi}}=\frac{l\,\tilde{\gamma}_{1}}{\,c_{p}}( \frac{\partial x_{1}}{\partial\tau_{2}}+\frac{\partial x_{2}}{\partial\tau_{ 1}}+\frac{\partial x_{3}}{\partial\tau_{0}})-\frac{\partial\varphi_{1}}{ \partial\tau_{2}}-\frac{\partial\varphi_{1}}{\partial\tau_{1}},\\ &\frac{\partial\alpha_{3}}{\partial\tau_{0}}+\frac{q_{3}\,\tilde{R}_{m}}{ \omega_{1}\tilde{l}_{m}}=\frac{l\,\tilde{\gamma}_{2}}{\,l_{m}}(\frac{\partial x _{1}}{\partial\tau_{2}}+\frac{\partial x_{2}}{\partial\tau_{1}}+\frac{ \partial x_{3}}{\partial\tau_{0}})-\frac{\partial\varphi_{1}}{\partial\tau_{ 2}}-\frac{\partial\varphi_{1}}{\partial\tau_{1}}.\end{array}  \tag{19}\]  
These equations can be solved sequentially. Therefore, the solutions of Equations (13)-(15) are as follows  
\[x_{1}\,=A\,e^{i\tau_{0}}+\overline{A}\,e^{-i\tau_{0}}, \tag{20}\]  
\[