# On Efficiency of Two-Degree-of-Freedom Galloping Energy Harvesters with Two Transducers
## 3 Efficiency of Two-Degree-of-Freedom System
The relationship between the vibration amplitude and the system frequency remains unknown. The first of the equations necessary to determine this relationship was obtained by subtracting Equation (24b) multiplied by \(B_{1}\) from Equation (24a) multiplied by \(A_{1}\). The second one is the difference of Equation (26c) multiplied by \(G_{1}\) and Equation (26d) multiplied by \(H_{1}\):  
\[ \begin{split}& p_{1}\big{(}k_{2}+\theta^{2}\delta_{D21}\big{)}+(c_{1}+ c_{2}(1-q_{1})+\varepsilon_{D11}+\varepsilon_{D21}\theta(1-\theta q_{1})) \omega_{1}\\ &-u\rho\omega_{1}a_{1}-\frac{3\omega_{1}\big{(}A_{y}^{2}\omega_{1 }^{2}+2C_{y}^{2}\omega_{2}^{2}\big{)}^{2}a_{3}}{4u}=0,\\ & p_{2}\big{(}k_{2}+\theta^{2}\delta_{D22}\big{)}+(c_{1}+c_{2}(1- q_{1})+\varepsilon_{D12}+\varepsilon_{D22}\theta(1-\theta q_{2}))\omega_{2}\\ &-u\rho\omega_{2}a_{1}-\frac{3\omega_{2}\big{(}2A_{y}^{2}\omega_ {1}^{2}+C_{y}^{2}\omega_{2}^{2}\big{)}^{2}a_{3}}{4u}=0,\end{split}  \tag{30a}\] \[ \begin{split}& p_{2}\big{(}k_{2}+\theta^{2}\delta_{D22}\big{)}+(c_{1}+ c_{2}(1-q_{1})+\varepsilon_{D12}+\varepsilon_{D22}\theta(1-\theta q_{2})) \omega_{2}\\ &-u\rho\omega_{2}a_{1}-\frac{3\omega_{2}\big{(}2A_{y}^{2}\omega_ {1}^{2}+C_{y}^{2}\omega_{2}^{2}\big{)}^{2}a_{3}}{4u}=0,\end{split}  \tag{30b}\]  
where \(A_{y}^{2}=A_{1}^{2}+B_{1}^{2}\) and \(G_{y}^{2}=G_{1}^{2}+H_{1}^{2}\). The set of Equation (30) has three non-trivial solutions in terms of \(A_{y}^{2}\) and \(G_{y}^{2}\), which, after recalling the \(\omega_{2}=\omega_{1}\) identity, take the following form:  
\[A_{y}^{2}=\frac{4u\big{(}p_{1}\big{(}k_{2}+\theta^{2}\delta_{D21}\big{)}+(c_{ 1}+c_{2}(1-q_{1})+\varepsilon_{D11}+\varepsilon_{D21}\theta(1-\theta q_{1}))- u\rho a_{1}\big{)}}{G_{y}^{2}=0}, \tag{31a}\] \[A_{y}^{2}=0,\] (31b) \[A_{y}^{2}=\frac{4u\big{(}3p_{1}\big{(}k_{2}+\theta^{2}\delta_{D21}\big{)} \omega_{1}+c_{1}+c_{2}(1-q_{1})+\varepsilon_{D11}+\varepsilon_{D21}\theta(1- \theta q_{1})\big{)}-u\rho a_{1}}{9\rho\omega_{1}^{2}a_{3}},\] (31c) \[G_{y}^{2}=\frac{4u\big{(}-3p_{1}\big{(}k_{2}+\theta^{2}\delta_{D21}\big{)} \omega_{1}+c_{1}+c_{2}(1-q_{1})+\varepsilon_{T11}+\varepsilon_{D21}\theta(1- \theta q_{1})\big{)}-u\rho a_{1}}{9\rho\omega_{1}^{2}a_{3}}.\]  
These expressions, along with the previously derived identities, enable the explicit formulation of solutions for the system Equation (20) in the form Equation (21). It should be noted that the identities given by Equations (31a) and (31b) correspond to the same solution in the form of Equation (21); thus, only one of them, namely Equation (31a), will be further analyzed. Moreover, no set of system parameters and initial conditions has been found that would lead to the excitation of vibrations with amplitude Equation (31c). Therefore, this solution was considered unstable, and the efficiency has been derived based only on Equation (31a):  
\[\eta_{D}=\frac{\frac{1}{T_{c}}\int_{0}^{T_{c}}\Big{(}\frac{\eta_{1}^{2}(t)}{ \tau_{1}\kappa_{1}}+\frac{\upsilon_{2}^{2}(t)}{\tau_{2}\kappa_{2}}\Big{)}dt}{ \rho u^{3}}=\frac{2\lambda\big{(}\mu-u\rho a_{1}\big{)}}{3u^{2}\rho^{2}a_{3}}, \tag{32}\]  
where  
\[\lambda=\Big{(}\varepsilon_{D11}+\varepsilon_{D12}\theta^{2}\Big{(}1-2\theta q _{1}+\theta^{2}\Big{(}{p_{1}}^{2}+{q_{1}}^{2}\Big{)}\Big{)}\Big{)}, \tag{33a}\] \[\mu=(c_{1}+(c_{2}+\theta\varepsilon_{D11})(1-\theta q_{1}))+\frac{p_{1}\big{(}k_ {2}+\theta^{2}\delta_{TD21}\big{)}}{\omega_{1}}. \tag{33b}\]  
The critical speed \(u_{cr}^{D}\) and nominal speed \(u_{p}^{D}\) of the GEH2D are given by the following expressions:  
\[\eta_{D}=\gamma_{1}\frac{2\lambda(\mu-u\rho a_{1})}{3u^{2}\rho^{2}a_{3}}=0 \Rightarrow u=u_{cr}^{D}=\frac{\mu}{\rho a_{1}}, \tag{34}\]  
\[\frac{\delta\eta_{D}}{\delta u}=0\Rightarrow u=u_{p}^{D}=2\frac{\mu}{\rho a_{ 1}}=2u_{cr}^{D}. \tag{35}\]Now it is possible to derive the peak efficiency of the system:  
\[\eta_{p}^{D}=\eta^{D}\Big{(}u_{p}^{D}\Big{)}=\frac{-\lambda\ a_{1}^{2}}{6\gamma_{2 }a_{3}}. \tag{36}\]  
To fully determine the features of the analyzed variant, it is necessary to define its high-efficiency bandwidth \(B^{D}\).