# Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices
## 3 The Proposed Method
\begin{array}{rl}\phi_{1}=B\,e^{iW\tau_{0}}+\overline{B}e^{-iW\tau_{0}},\\ \\ v_{1}=\frac{\,\tilde{R}_{m}\omega_{1}\,l\,\tilde{\gamma}_{1}}{\,\tilde{R}_{m} \varphi_{1}\tilde{\varepsilon}_{p-1}}\,A\,e^{i\tau_{0}}+\frac{\,\tilde{R}_{m} \omega_{1}\,\tilde{\gamma}_{1}}{\,\tilde{R}_{m}\varphi_{1}\tilde{\varepsilon}_ {p+1}}\overline{A}\,e^{-i\tau_{0}},\\ \\ q_{1}=\frac{\,\omega_{1}\,l\,\tilde{\gamma}_{2}}{\,\omega_{1}\,l_{m}-i\,\tilde{ R}_{m}}\,A\,e^{i\tau_{0}}+\frac{\,\omega_{1}\,l\,\tilde{\gamma}_{2}}{\,\omega_{1} \,l_{m}+i\,\tilde{R}_{m}}\overline{A}\,e^{-i\tau_{0}},\end{array}  \tag{22}\]  
where \(A\) and \(B\) represent unknown complex functions of the slow time scales \(\tau_{1}\) and \(\tau_{2}\), while \(\overline{A}\) and \(\overline{B}\) denote their counterpoint complex conjugates. Substitution of the solutions (22)-(24) into the higher-order Equations (16)-(18) yields secular terms. To eliminate these terms, the following conditions are employed  
\[\frac{\partial A}{\partial\tau_{1}}=0,\,\frac{\partial B}{\partial\tau_{1}}=0. \tag{23}\]  
Therefore, the following second-order solutions are yielded  
\[x_{2}\,=\,W^{2}B\,\overline{B}+\frac{3\,W^{2}B^{2}e^{2 iW\tau_{0}}}{2(4\,W^{2} -1)}+\frac{\,\tilde{r}\,p^{2}e^{i\,p\tau_{0}}}{2(1-p^{2})}+CC, \tag{24}\]  
\[ \begin{array}{rl}\varphi_{2}=&-\frac{\,W(W+2)A\,B\,e^{(W+1)\tau_{0}}}{(2\,W+ 1)}+\frac{\,W(W-2)\,\overline{A}\,B\,e^{(W-1)\tau_{0}}}{(2\,W-1)}\\ &+\frac{\,\tilde{r}\,p^{2}e^{i\,p\tau_{0}}}{2(p^{2}-W^{2})}+CC,\\ \\ v_{2}=-\frac{\,p^{2}\tilde{r}\,\tilde{R}_{m}\omega_{1}\,l\,\tilde{\gamma}_{1} \,\tilde{c}^{\,\prime}p\tau_{0}}{2(p^{2}-1)(\tilde{R}_{p}\omega_{1}\,l\,\tilde{ \varepsilon}_{p}-i)}+\frac{\,3\,W^{2}B^{2}\tilde{R}_{m}\omega_{1}\,l\,\tilde{ \gamma}_{1}\,\tilde{\varepsilon}_{p}^{2 iW\tau_{0}}}{(4\,W^{2}-1)(2\tilde{R}_{ p}\omega_{1}\,\tilde{\varepsilon}_{p}-i)}+CC,\\ \\ q_{2}=-\frac{\,p^{2}\tilde{r}\,\omega_{1}\,l\,\tilde{\gamma}_{2}\,\tilde{c}^{\, \prime}p\tau_{0}}{2(p^{2}-1)(\tilde{R}_{m}\omega_{1}\,l_{m}-i\,\tilde{R}_{m})}+ \frac{\,3\,W^{2}B^{2}\omega_{1}\,l\,\tilde{\gamma}_{2}\,\tilde{c}^{\,\prime}p \tau_{0}}{(4\,W^{2}-1)(2\tilde{R}\omega_{1}\,l\,\tilde{\varepsilon}_{p}-i)}+ CC\,,\end{array}  \tag{25}\]  
where \(CC\) denotes the conjugates of the previous terms. As per the previous method, the elimination of the secular terms required that \(A\) and \(B\) depended just upon \(\tau_{2}\). The solutions for Equations (19) and (20) can be obtained by elimination of the secular terms, which yields the following equations  
\[ \begin{array}{rl}i\,\tilde{c}_{1}A+2i\frac{\partial A}{\partial\tau_{2}}+3 \tilde{\alpha}\tilde{\epsilon}_{5}^{2}A+\frac{\,\tilde{R}_{m}\omega_{1}\,l\, \tilde{\gamma}_{1}\,\tilde{\mu}_{1}A}{(k_{p}\omega_{1}\,\tilde{\varepsilon}_{p} -i)}+\frac{\,6\,W^{2}(W^{2}-1)A\,B\,\overline{B}}{(4\,W^{2}-1)}=0.\\ \\ i\,\tilde{c}_{1}A+2i\frac{\partial A}{\partial\tau_{2}}+3\tilde{\alpha}\tilde{ \epsilon}_{5}^{2}A+\frac{\,\omega_{1}\,l\,\tilde{\gamma}_{2}\,\tilde{\mu}_{2}A}{( \omega_{1}\,l_{m}-i\,\tilde{R}_{m})}+\frac{\,6\,W^{2}(W^{2}-1)A\,B\,\overline{B} }{(4\,W^{2}-1)}=0.\end{array}  \tag{26}\]  
\[iW\,\tilde{c}_{2}B+2 iW\frac{\partial B}{\partial\tau_{2}}+\frac{\,W^{2}(W^{2}-1)}{ 2(4\,W^{2}-1)}B[12\,A\overline{A}-(8W^{2}+1)B\overline{B}]=0. \tag{27}\]Finally, we obtain the third-order solutions as follows  
\[