# Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices
## 6 Stability Analysis  
The investigated models were analyzed using the nonlinear stability approach [45]. Based on the above, the dynamical motion was considered under the influence of external harmonic force \(F(t)\) and moment \(M(t)\). The criteria of stability were implemented besides the simulations of the nonlinear system of Equation (39). It is observed that various parameters in the stability criterion, such as the frequency \(W\), coefficients of damping \(c_{j}\left(j=1,2\right)\), and the detuning parameters \(\sigma_{j}\), play a crucial destabilizing impact. Appropriate policies with varied system (39) parameters were considered to plot the corresponding diagrams of the system's stability. For distinct parametric regions, time histories of the amplitudes \(a\) and \(b\) are displayed, and their attributes are presented through the trajectories of the phase plane. The probably fixed points were drawn. Figures 8-23 explore the variation of these points with the detuning parameters \(\sigma_{j}\). In other words, Figure 8 is calculated when \(\sigma_{2}=0,W=0.448\), and \(c_{2}=0.011\) at various values of \(c_{1}\), in which the stable fixed points are represented by solid curves while the unstable ones are drawn by the dashed curves. The stability and instability areas lie in the ranges \(-0.05\leq\sigma_{1}<0.001\) and \(0.001\leq\sigma_{1}\leq 0.05\), respectively. Here, one critical fixed point is observed at the points \((0.0014,0.0157)\),\((0.0014,0.0236)\),\((0.0014,0.0414)\) and \((0.00149,0.0142)\),\((0.00149,0.0141)\), \((0.00149,0.013)\) for Figure 8,II, respectively. Moreover, Figures 9 and 10 illustrate the frequency response of both \(a(\sigma_{1})\) and \(b(\sigma_{1})\) at \(\sigma_{2}=0.001\) and \(\sigma_{2}=-0.001\), respectively. The comparison between Figures 8-10 shows the slight variation of the critical fixed points as seen in parts (I) of these figures, while the change of the locus of the corresponding fixed points becomes very clear as observed in parts (II) of the same figures. Figure 12: The frequency response at \(\sigma_{1}=-0.001\), \(W=0.448\), and \(c_{2}=0.011\) of: \(a(\sigma_{2})\) when (**I**) \(c_{1}=0.001\), (**II**) \(c_{1}=0.002\), (**III**) \(c_{1}=0.003\), and (**IV**) for \(b(\sigma_{2})\). Figure 14: The frequency response of: (**I**) \(a(\sigma_{2})\) and (**II**) \(b(\sigma_{2})\) when \(\sigma_{1}=0\), \(W=0.448\), and \(c_{1}=0.001\). Figure 16: The frequency response at \(\sigma_{2}=-0.001\), \(W=0.448\), and \(c_{1}=0.001\) for: \(a(\sigma_{1})\) as shown in (I), and \(b(\sigma_{1})\) as it is in (II–IV) when (\(c_{2}=0.011\),\(\sigma_{2}=0.022\),\(c_{2}=0.033\) ). Figure 17: The frequency response of: (**I**) \(a(\sigma_{2})\) and (**II**) \(b(\sigma_{2})\) when \(\sigma_{1}=-0.001\), \(W=0.448\) and \(c_{1}=0.001\). Figure 18: The frequency response at \(\sigma_{2}=0.001\), \(c_{1}=0.001\), and \(c_{2}=0.011\) of: \(a(\sigma_{1})\) as seen in (**I**), and \(b(\sigma_{1})\) as shown in (**II**) at \(W=0.1\), (**III**) at \(W=0.2\), and (**IV**) at \(W=0.3\). Figure 21: The frequency response at \(\sigma_{2}=-0.001\), \(c_{1}=0.001\), and \(c_{2}=0.011\) of: \(a(\sigma_{1})\) as seen in (**I**), and \(b(\sigma_{1})\) as shown in (**II**) at \(W=0.1\), (**III**) at \(W=0.2\), and (**IV**) at \(W=0.3\). Parts of Figure 11 illustrate the variation of \(c_{1}\) values on the behavior of the amplitudes \(a\) and \(b\) at \(\sigma_{1}=0,W=0.448\), and \(c_{2}=0.011\). It is shown that the stability and the instability areas are \(-0.002<\sigma_{2}\,\leq 0.05\) and \(-0.05\leq\sigma_{2}\,\leq-0.002\), respectively, with one critical stable fixed point. Figure 11 is replotted when \(\sigma_{1}=-0.001\) to reveal the variation of \(\sigma_{1}\) values on the frequency response of each \(a(\sigma_{2})\) and \(b(\sigma_{2})\) with the same other data as shown in Figure 12. On the other hand, according to the used data, Routh-Hurwitz conditions (45) are not satisfied when \(\sigma_{1}=0.001\).