# Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices
## 2 Dynamical Modeling  
This section is devoted to introducing a complete description of the considered models. Therefore, let us consider the motion of 2 DOF dynamical models consisting of nonlinear spring pendulums with the same linear and nonlinear stiffness coefficients \(k_{1}\) and \(k_{2}\), respectively. It is considered that the point of suspensions \(O\) moves in a circular path of radius \(R\), with an angular velocity \(\Omega\), in which it is subjected to a harmonic torque \(M(t)=M_{0}\,\cos\Omega_{2}t\). The pendulum's other end \(Q\) is excited by an external harmonic force \(F(t)=F_{0}\,\cos\Omega_{1}t\) that drives the mass \(m\) along the pendulum length; see Figure 1. Here, \(F_{0},M_{0}\), and \(\Omega_{1},\Omega_{2}\) are the amplitudes and frequencies of \(F(t)\) and \(M(t)\). Let \(C_{1}\) and \(C_{2}\) represent the damping coefficients of the viscous force \(C_{1}\,\dot{x}\) and viscous moment \(C_{2}\,\dot{\theta}\), respectively; \(l_{0}\) is the spring's normal length and \(g\) is the gravitational acceleration. Now, we consider a connection of two devices (piezoelectric and electromagnetic) separately with the considered model, in which the resistive loads of the piezoelectric and electromagnetic circuits are \(R_{p}\) and \(R_{m}\), respectively. Let \(c_{p}\) denote the capacitance of the piezoelectric, \(l_{m}\) the inductance of the coil, and \(\gamma_{j}\,(j=1,2)\) the linear coupling coefficients for the piezoelectric circuit and electromagnetic one. Based on the portrayed Figure 1, we can write the cartesian coordinates of the point \(Q\) as follows  
\[ \begin{array}{l}X=R\,\cos\,\Omega t\,+(l_{0}+x)\,\cos\,\theta,\\ Y=R\,\sin\,\Omega t\,+(l_{0}+x)\sin\,\theta.\end{array}  \tag{1}\]  
Henceforth, the potential and kinetic energies of the dynamical system can be written as  
\[ \begin{array}{l}V=\frac{1}{2}k_{1}\,x^{2}+\frac{1}{4}k_{2}\,x^{4}-m\,g\,[R \,\cos\,\Omega t\,+(l_{0}+x)\,\cos\,\theta],\\ T=m\bigg{\{}\frac{1}{2}\,[R^{2}\,\Omega^{2}+\dot{x}^{2}+(l_{0}+x)^{2}\dot{ \theta}^{2}]-\,R\,\Omega\,\dot{x}\,\sin(\Omega t-\theta)\\ +R\,\Omega\,(l_{0}+x)\dot{\theta}\cos(\Omega t-\theta)\bigg{\}},\end{array}  \tag{2}\]  
where the dot represents the differentiation with respect to time \(t\). In order to obtain the governing equations of motion (EOM) for the mentioned systems, the following [PERSON]'s equations are used  
\[ \begin{array}{l}\frac{\dot{d}}{dt}(\frac{\partial L}{\partial x})-(\frac{ \partial L}{\partial x})=Q_{x},\\ \frac{\dot{d}}{dt}(\frac{\partial L}{\partial\theta})-(\frac{\partial L}{ \partial\theta})=Q_{\theta}.\end{array}  \tag{3}\]  
Here \(L=T-V\) is the Lagrangian function, and \(Q_{x}\,\) and \(Q_{\theta}\,\) are the generalized forces corresponding to the generalized standing coordinates \(x\) and \(\theta\), respectively. Moreover,  
Figure 1: Two dynamical models: **(a)** a damped spring with a piezoelectric device, **(b)** a damped spring with an electromagnetic device. the mechanism's equations for the piezoelectric circuit and electromagnetic one can be expressed as follows  
\[c_{p}\,\dot{v}+\frac{v}{R_{p}}-\gamma_{1}\,\dot{x}=0 \tag{4}\] \[l_{m}\,\dot{q}+R_{m}\,l_{m}\,q-\gamma_{2}\,\dot{x}=0\]  
where \(v\) and \(q\) are the voltage and the current of the load resistances \(R_{p}\) and \(R_{m}\), respectively. The generalized forces \(Q_{x}\) and \(Q_{\theta}\) have the forms  
\[Q_{x} =F_{0}\,\cos\Omega_{1}t-C_{1}\dot{x}-PM,\] \[Q_{\theta} =M_{0}\,\cos\Omega_{2}t-C_{2}\dot{\theta},\]  
where \(PM\) represents a coupling term of the two models equals \(\gamma_{1}v\) for the piezoelectric circuit (model \(\mathbf{a}\)) or \(\gamma_{2}q\) for the electromagnetic circuit (model \(\mathbf{b}\)). Let us consider the following dimensionless parameters  
\[