# On Efficiency of Two-Degree-of-Freedom Galloping Energy Harvesters with Two Transducers
## 2 Efficiency of Reference Variant  
While the primary focus of this study lies in analyzing the efficiency of systems with two degrees of freedom, understanding the characteristics of this class of devices can be facilitated by comparing them with a system possessing well-established properties--a linear device with one degree of freedom, hereafter referred to as the reference device (Figure 1). Therefore, it is justified to conduct an analysis of the efficiency of this system, particularly as we aim to highlight a certain intriguing feature of this variant, which has not been previously addressed in the literature. According to articles [3,4], the mathematical model of a GEH takes the form shown in Equation (1) with parameters detailed in Table 1.  
\[\hat{m}\ddot{z}+\hat{c}\,\dot{z}+\hat{k}z-\theta\xi=\hat{F}_{L}(\hat{u})=-\frac{ 1}{2}\hat{\rho}\hat{u}^{2}\hat{h}\left(a_{1}\frac{\dot{z}}{\hat{u}}+a_{3}\left( \frac{\dot{z}}{\hat{u}}\right)^{3}\right), \tag{1a}\] \[\hat{C}_{p}\dot{\xi}+\frac{\dot{\xi}}{\hat{R}}+\hat{\theta}\dot{x}=0. \tag{1b}\]  
The efficiency \(\eta\) of such a system is defined as the ratio of the average power generated by the device \(\hat{P}_{g}\) over a time period equal to the period \(\hat{T}\) of the voltage function \(\xi(\tau)\) to the surface flow power density \(\tilde{P}_{f}\):  
\[\eta^{L}=\frac{\tilde{P}_{g}}{\tilde{P}_{f}}=\frac{\frac{1}{\tau}\hat{J}_{0} ^{\hat{T}}\frac{\hat{c}^{2}(\tau)}{\hat{R}}d\tau}{\frac{1}{2}\hat{\rho}\hat{h }\hat{n}^{3}}, \tag{2}\]  
By introducing the following dimensionless quantities:  
\[ \begin{array}{l}y=\frac{\dot{z}}{\hat{h}},\ v=\xi\frac{\dot{\theta}}{\hat{h} \,\dot{\omega}_{n}^{2}},\ \dot{\omega}_{n}=\sqrt{\frac{\dot{k}}{\hat{m}}},\ c=\frac{\dot{c}}{\hat{m}\dot{ \omega}_{n}},\ u=\frac{\dot{u}}{\hat{h}\dot{\omega}_{n}},\ \rho=\dot{\rho}\frac{\dot{k}^{2}}{2\hat{m}},\ r=\hat{C}_{p}\hat{R}\dot{ \omega}_{n},\\ \kappa=\frac{\dot{\theta}^{2}}{\hat{c}_{p}\ \hat{n}\dot{\omega}_{n}^{2}},\ t=\tau\dot{ \omega}_{n},\end{array}  \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Unit** \\ \hline \(\hat{n}\) & Mass & kg \\ \(\hat{k}\) & Stiffness coefficient & N/m \\ \(\hat{c}\) & Damping coefficient & kg/s \\ \(\hat{u}\) & Flow speed & m/s \\ \(\hat{\rho}\) & Planar fluid density & kg/m\({}^{2}\) \\ \(\hat{h}\) & Characteristic dimension & m \\ \(\hat{\theta}\) & Piezoelectric coefficient & N/V \\ \(\hat{C}_{p}\) & Equivalent capacity & F \\ \(\hat{R}\) & Circuit electrical resistance & \(\Omega\) \\ \(\tau\) & Time & s \\ \(z=z(\tau)\) & Vibration vs. time function & m \\ \(\xi=\xi(\tau)\) & Voltage vs. time function & V \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of GEH parameters. Figure 1: Rodel of reference variant. the device dimensionless mathematical model can be written as  
\[\ddot{y}+c\dot{y}+y-v=\rho\left(a_{1}u\dot{y}+a_{3}\frac{\dot{y}^{3}}{u}\right), \tag{4a}\] \[\dot{v}+\frac{v}{r}+\kappa\dot{y}=0. \tag{4b}\]  
Assuming that \(T_{c}\) is the period of the function \(v(t)\), efficiency in terms of dimensionless parameters can be expressed as  
\[\eta^{L}=\frac{\frac{1}{T_{c}}\int_{0}^{T_{c}}\frac{v^{2}(t)}{\kappa\frac{t}{r }}dt}{\rho t^{3}}. \tag{5}\]  
To characterize the efficiency \(\eta^{L}\) of the system, it is essential to derive the voltage function \(v(t)\) generated by it, which can be accomplished using the harmonic balance method. Consequently, it was assumed that the limit cycle of the system would be described by a set of solutions in the following form:  
\[y=A_{y}\text{cos}(\omega t), \tag{6a}\] \[v=A_{y}\text{cos}(\omega t+\varphi), \tag{6b}\]  
where the four unknown quantities \(A_{y}\), \(A_{v}\), \(\omega=\frac{\dot{\omega}}{\dot{\omega}_{y}}\), and \(\varphi\) represent the dimensionless vibration amplitude, dimensionless voltage amplitude, dimensionless vibration frequency, and the phase shift between the oscillator vibrations and voltage oscillations, respectively. The parameter \(\hat{\omega}\) denotes the unknown dimensional frequency of the system's vibrations.