# On Efficiency of Two-Degree-of-Freedom Galloping Energy Harvesters with Two Transducers
## 2 Efficiency of Reference Variant
Substituting solutions of the form Equation (6) into the model Equation (1) leads to the transformation of the system of differential equations into a system of algebraic equations:  
\[A_{y}(1-\omega^{2})\text{cos}(\omega t)+A_{y}^{3}\frac{\rho\omega^{3}a_{3}}{u} \text{sin}^{3}(\omega t)+A_{y}\omega(u\rho a_{1}-c)\text{sin}(\omega t)-A_{v} \text{cos}(\omega t+\varphi)=0, \tag{7a}\] \[A_{v}\left(\frac{\text{cos}(\omega t+\varphi)}{r}-\omega\text{sin}(\omega t +\varphi)\right)-A_{y}\kappa\text{ }\omega\text{sin}(\omega t)=0. \tag{7b}\]  
This condition will be satisfied for every moment of time \(t\) if and only if the sum of the coefficients with corresponding time functions equals zero. From Equation (7b), the following was deduced:  
\[\frac{A_{v}(\text{cos}(\varphi)-r\omega\text{sin}(\varphi))}{r}=0, \tag{8a}\] Figure 1:0  
Equation (8a) shows that  
Figure 1:1  
Based on Equation (8b) and Equation (9), the expression describing the relationship between the vibration amplitude and the voltage amplitude can be derived:  
Figure 1:2  
By balancing the harmonics of Equation (7a), the remaining two algebraic equations necessary to determine an approximate solution of the device model are obtained. Afterthe previously derived relations Equations (9) and (10) are taken into account, they take the form  
Figure 1:3 Figure 1:4  
where \(k_{e}^{L}=\kappa\frac{r^{2}\omega^{2}}{1+r^{2}\omega^{2}}\) and \(e^{L}=\kappa\frac{r}{1+r^{2}\omega^{2}}\) are the piezoelectric stiffness and electrical damping of the linear system. Based on the above system of equations, it can be shown that  
Figure 1:5 Figure 1:6  
Equation (12a) shows that \(\omega_{2}^{2}<0\), regardless of the system parameters. Therefore, in the following part of the work, the notation \(\omega_{1}=\omega\) has been adopted. Returning now to the general definition of efficiency Equation (5) and substituting the expressions Equations (6b), (10), and (12) into it, we obtain  
Figure 1:7  
Figure 2 illustrates the efficiency characteristics of the system. The efficiency is depicted as the ratio \(\eta^{L}/\eta_{p}^{L}\), and this representation is maintained throughout the study. The figure also includes the analogous characteristic obtained numerically for initial conditions \(y(0)=0.1\), \(\dot{y}(0)=0\), \(n(0)=0\), utilizing the fourth-order Runge-Kutta method. In this figure, it can be observed that the system undergoes Hopf bifurcation at a certain speed, from now on referred to as the critical flow speed \(u_{\sigma}\). Its value results directly from Equation (13):  
Figure 1:8  
Another noteworthy observation from the same set of graphs is that irrespective of the parameter set, there exists a specific nominal flow speed \(u=u_{p}^{L}\) for which the efficiency \(\eta^{L}\)\(\eta_{p}^{L}\) attains a maximum value, denoted as the peak efficiency \(\eta_{p}^{L}\). These quantities are given by the expressions  
Figure 1:9  
\[\ddot{y}+c\dot{y}+y-v=\rho\left(a_{1}u\dot{y}+a_{3}\frac{\dot{y}^{3}}{u}\right), \tag{4a}\]0  
The aforementioned identities were previously derived in [3, 4]. However, the remainder of the article discusses entirely original content. An unexplored property of GEHs, which can be inferred from Figure 2 or deduced from the identity \(u_{p}^{L}=2u_{cr}\), is noteworthy. A system with a low critical speed \(u_{cr}\) will experience a more pronounced decline in efficiency due to the deviation of the flow speed \(u\) at which it operates from the nominal speed \(u_{p}^{L}\). Let the measure of this phenomenon be the flow speed bandwidth in which the system efficiency \(\eta^{L}\) does not fall below \(90\%\) of the maximum efficiency \(\eta_{p}^{L}\), hereafter referred to as the high-efficiency band \(B^{L}\).