# On Efficiency of Two-Degree-of-Freedom Galloping Energy Harvesters with Two Transducers
## 4 Comparison and Discussion  
As suggested by Equations (34), (35), and (38), critical speeds, nominal speed, and high-efficiency bandwidths are interrelated in the same manner in both analyzed GEH \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(M\) & 0.5 \\ \(k_{2}\) & 0.3 \\ \(c_{1}\) & 0.1 \\ \(c_{2}\) & 0.25 \\ \(k_{1}\) & 0.9 \\ \(k_{2}\) & 1.8 \\ \(r_{1}\) & 3 \\ \(r_{2}\) & 2 \\ \(\theta\) & 1.3 \\ \(a_{1}\) & 2.3 \\ \(a_{3}\) & \(-18\) \\ \(\rho\) & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: GEH2D system parameter values. Figure 4: Efficiency characteristic of GEH2D for parameter values presented in Table 2.  
variants. Consequently, comparing the efficiency of these systems can only be achieved by comparing their peak efficiencies. Since the efficiency of both devices depends on their nominal speeds, it is necessary to compare systems with the same nominal speeds. The electrical damping of the linear system \(e^{L}\) will therefore be selected in such a way that this condition is satisfied. In the following derivation, it is assumed that the structural damping of the linear system c is equal to the structural damping \(c_{1}\) of the damper connecting the lower mass of the GEH2D with the base, i.e., \(c=c_{1}\). The shape of the flowing body, represented by the coefficients \(a_{1}\) and \(a_{3}\), and the density of the fluid remain the same for all variants. \[u_{p}^{I}=u_{p}^{L}, \tag{39a}\] \[2\frac{\mu+\gamma_{2}}{\rho a_{1}}=2\frac{e^{L}+c_{1}}{\rho a_{1}},\] (39b) \[e^{L}=\gamma_{2}+\mu-c_{1}. \tag{39c}\]  
According to Equations (16) and (39c), the efficiency of a system with one degree of freedom with a nominal speed equal to the nominal speed of the GEH2D is equal to  
\[\eta^{L}\Big{(}u_{p}^{D}\Big{)}=-\frac{(\mu-c_{1})a_{1}^{2}}{6\mu a_{3}}. \tag{40}\]  
The ratio of peak efficiencies of compared systems with the same nominal speed is therefore given by  
\[\frac{\eta^{L}\Big{(}u_{p}^{D}\Big{)}}{\eta^{D}\Big{(}u_{p}^{D}\Big{)}}=\frac {s_{1}+s_{2}}{s_{1}+s_{3}}, \tag{41}\]  
where  
\[s_{1}=\varepsilon_{T11}\bigg{(}\Big{(}k_{2}+\theta^{2}\delta_{D21}\Big{)} \Big{(}k_{2}+\theta^{2}\delta_{T21}-2M\omega_{1}{}^{2}\Big{)}+\Big{(}c_{2}+ \theta^{2}\varepsilon_{D21}\Big{)}^{2}\omega_{1}{}^{2}+M^{2}\omega_{1}{}^{4} \bigg{)}, \tag{42a}\] \[s_{2}=M\omega_{1}{}^{2}\Big{(}(k_{2}\varepsilon_{D21}-c_{2}\delta_{D21})(\theta -1)\theta+M\omega_{1}{}^{2}(c_{2}+\theta\varepsilon_{D21})\Big{)},\] (42b) Figure 4:0  
Clearly, according to Equation (41), the discussed system with two degrees of freedom will have greater efficiency than the reference system if and only if \(s_{3}>s_{2}\). For variant with only one lower transducer (Figure 3a), where \(\varepsilon_{D21}=\delta_{D21}=0\), this inequality takes the form  
Figure 4:1  
which implies that such a device cannot be more efficient than the reference system, regardless of its parameters. This conclusion contradicts the results presented in [25]. The reason for this discrepancy lies in the fact that in the cited work, the operating conditions of the compared systems were not standardized--the devices had different critical speeds and, consequently, different nominal speeds. Despite the unquestionable value of this article, the conclusion stated therein can be subject to questioning. In the case of the variant with only the upper transducer (Figure 3b), where \(\varepsilon_{T11}=\delta_{T11}=0\), the condition for efficiency improvement takes the following form:  
Figure 4:2  
which indicates that this variant does not offer efficiency enhancement. Similarly, an identical and impossible-to-satisfy requirement is associated with the special case of the third variant, where there are two identical transducers, meaning \(\varepsilon_{D11}=\varepsilon_{T21}\), \(\delta_{D11}=\delta_{D21}\)  
administration, F.S. ; funding acquisition, R.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the Polish Ministry of Higher Education, grant number 0612/SBAD/3628.