# Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices
## 5 Steady-State Solutions  
In this section, we focus our attention on investigating the solutions at the steady-state case of the considered dynamical models. This case corresponds to the zero values of  
Figure 5: The phase plane portraits of: (**I**) \(x(\tau)\) and (**II**) \(\theta(\tau)\).  
Figure 6: The comparison between the numerical and the approximate solutions for: (**I**) \(x(\tau)\) and (**II**) \(\theta(\tau)\).  
\(\frac{d\theta_{1}}{dT}\cdot\frac{d\theta_{2}}{dT}\), \(\frac{d\theta}{dT}\), and \(\frac{d\theta}{dT}\)[43]. Therefore, we can use Equation (39) to obtain the following form of algebraic equations  
\[\begin{array}{l}f_{1}\cos\theta_{1}-\frac{a}{2}\left\{\frac{3W^{2}(W^{2}-1) \,b^{2}}{(4W^{2}-1)}+2[3a\zeta_{s}^{2}+2(\,H_{1}-\sigma_{1})]\right\}=0,\\ f_{1}\sin\theta_{1}-a\left(c_{1}\,+\,2H_{2}\right)=0,\\ \frac{f_{2}}{W}\cos\theta_{2}+b\Big{\{}2\sigma_{2}+\frac{W(W^{2}-1)}{8(4W^{2} -1)}[(8W^{2}+1)b^{2}-12a^{2}]\,\Big{\}}=0,\\ \frac{f_{2}}{W}\sin\theta_{2}-c_{2}b=0.\end{array} \tag{39}\]  
Removing the modified phases \(\theta_{1}\) and \(\theta_{2}\) from Equation (40) produces the relationships between both amplitudes and the frequency through the introduced detuning parameters  
\[\begin{array}{l}f_{1}^{2}=a^{2}\Big{\{}\frac{3W^{2}(W^{2}-1)\,b^{2}}{2(4W^{ 2}-1)}+3a\zeta_{s}^{2}+2(H_{1}-\sigma_{1})\big{]}^{2}+[c_{1}+2\,H_{2}]^{2} \Big{\}},\\ f_{2}^{2}=W^{2}b^{2}\Big{\{}2\sigma_{2}+\frac{W(W^{2}-1)}{8(4W^{2}-1)}((8W^{2 }+1)b^{2}-12a^{2})\,\big{]}^{2}+c_{2}^{2}\Big{\}}.\end{array} \tag{40}\]  
Investigation of stability considers one of the fatal parts of the vibrations at a steady state. The behavior of the system was explored in the neighborhood region to the fixed points. To fulfil this target, let us assume  
\[\begin{array}{l}a=a_{10}+a_{11},\,b=b_{10}+b_{11},\\ \theta_{1}=\theta_{10}+\theta_{11},\,\theta_{2}=\theta_{20}+\theta_{21},\end{array} \tag{41}\]  
where \(a_{10},\theta_{10},b_{10},\) and \(\theta_{20}\) represent the steady-state solutions, while \(a_{11},\theta_{11},b_{11},\) and \(\theta_{21}\) denote the tiny perturbations comparing with \(a_{10},\theta_{10},b_{10},\) and \(\theta_{20}\).  
Substituting (42) into (39), we can obtain the following system of ordinary differential equations after linearization  
\[\begin{array}{l}a_{10}\frac{d\theta_{11}}{dT}=-\frac{f_{2}}{2W}\sin\theta_ {20}\theta_{21}+b_{11}[\sigma_{2}+\frac{3W(W^{2}-1)}{16(4W^{2}-1)}((8W^{2}+1 )b_{10}^{2}-4a_{10}^{2})\,]\\ \hskip 142.26378 pt-\frac{3W(W^{2}-1)\,a_{10}b_{10}\,b_{11}}{2(4W^{2}-1)},\\ \hskip 142.26378 pt\frac{db_{11}}{dT}=\frac{f_{2}}{2W}\cos\theta_{20}\theta_{2 1}-\frac{1}{2}c_{2}b_{11}.\end{array} \tag{42}\]  
Considering that \(a_{11},\theta_{11},\)\(b_{11},\) and \(\theta_{21}\) are the unknown perturbed functions of the linear system (39), we can express their solutions in the form \(k_{s}\,e^{\lambda\tau}\) in which \(k_{s}\,(s=1,2,3,4)\) are constants, and \(\lambda\) is the eigenvalue of the unknown perturbation. Consequently, the fixed points in (43) are asymptotically stable when the real parts of the roots of the following characteristic equation of (43),  
\[\lambda^{4}+\Gamma_{1}\lambda^{3}+\Gamma_{2}\lambda^{2}+\Gamma_{3}\lambda+ \Gamma_{4}=0, \tag{43}\]