# On Efficiency of Two-Degree-of-Freedom Galloping Energy Harvesters with Two Transducers
## 3 Efficiency of Two-Degree-of-Freedom System
 \tag{22b}\]  
where \(\delta_{Dij}=\kappa_{i}\frac{r^{2}\omega_{i}^{2}}{1+r^{2}\omega_{i}^{2}}\) and \(\varepsilon_{Dij}=\kappa_{i}\frac{r}{1+r^{2}\omega_{i}^{2}}\) represent the piezoelectric stiffness and electric damping of the \(i\)-th piezoelectric for vibration with the \(j\)-th frequency. Harmonic balancing of \(\text{cos}(\omega_{1}t)\) and \(\text{sin}(\omega_{1}t)\) of the algebraic equation obtained by substituting the solutions of Equation (21) into Equation (20c) and using the identities Equation (22) leads to the following system of equations:  
\[A_{2}=B_{1}p_{1}+A_{1}q_{1}, \tag{23a}\] \[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]  
where  
\[p_{1}=\frac{\left(k_{2}+\theta\delta_{D21}\right)\left(c_{2}+\theta^{2} \varepsilon_{D21}\right)\omega_{1}-\left(c_{2}+\theta\varepsilon_{D21}\right) \left(k_{2}+\theta^{2}\delta_{D21}-M\omega_{1}^{2}\right)\omega_{1}}{\left(c_{2 }+\theta^{2}\varepsilon_{D21}\right)^{2}\omega_{1}^{2}+\left(k_{2}+\theta^{2} \delta_{D21}-M\omega_{1}^{2}\right)^{2}}, \tag{23c}\]\[q_{1}=\frac{(c_{2}+\theta\varepsilon_{D21})\big{(}c_{2}+\theta^{2}\varepsilon_{D2 1}\big{)}\omega_{1}^{2}+(k_{2}+\theta\delta_{D21})\big{(}k_{2}+\theta^{2}\delta_ {T21}-M\omega_{1}^{2}\big{)}}{(c_{2}+\theta^{2}\varepsilon_{D22})^{2}\omega_{1} ^{2}+\big{(}k_{2}+\theta^{2}\delta_{D22}-M\omega_{1}^{2}\big{)}^{2}}. \tag{23d}\]  
By substituting the above identities into the harmonic balance equations for \(\cos(\omega_{1}t)\) and \(\sin(\omega_{1}t)\) of expression (20a), the following identities can be formulated:  
\[A_{1}\alpha_{1}+B_{1}\beta_{1}-A_{1}\frac{3\rho a_{3}\omega_{1}\big{(}\big{(} A_{2}^{2}+B_{2}^{2}\big{)}\omega_{1}^{2}+2\big{(}G_{2}^{2}+H_{2}^{2}\big{)} \omega_{2}^{2}\big{)}}{4u}=0, \tag{24a}\] \[A_{1}\beta_{1}-B_{1}\alpha_{1}+B_{1}\frac{3\rho a_{3}\omega_{1}\big{(}\big{(} A_{2}^{2}+B_{2}^{2}\big{)}\omega_{1}^{2}+2\big{(}G_{2}^{2}+H_{2}^{2}\big{)} \omega_{2}^{2}\big{)}}{4u}=0, \tag{24b}\]  
where  
\[\alpha_{1}=\Big{(}p_{1}\Big{(}k_{2}+\theta^{2}\delta_{D21}\Big{)}+(c_{1}+ \varepsilon_{T11}+(c_{2}+\varepsilon_{D21})(1-\theta q_{1}))\omega_{1}-u\rho \omega_{1}a_{1}\Big{)}, \tag{25a}\] \[\beta_{1}=1+\delta_{T11}+(\delta_{D21}\theta+k_{2})(1-\theta q_{1})-\omega_{1 }\Big{(}p_{1}\Big{(}c_{2}+\theta^{2}\varepsilon_{D21}\Big{)}+\omega_{1}\Big{)}. \tag{25b}\]  
Adapting the above procedure to balance the harmonics \(\cos(\omega_{2}t)\) and \(\sin(\omega_{2}t)\) of Equations (20c) and (20a) allows for the derivation of the following identities:  
\[G_{2}=H_{1}p_{2}+G_{1}q_{2}, \tag{26a}\] \[H_{2}=H_{1}q_{2}-G_{1}p_{2},\] (26b) \[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]0 (26c) \[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]1  
where  
\[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]2 \[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]3 (27b) \[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]4 (27c) \[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]5  
By adding Equation (24a) multiplied by \(B_{1}\) to Equation (24b) multiplied by \(A_{1}\), an expression was obtained that allows the frequency \(\omega_{1}\) to be explicitly formulated:  
\[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]6  
Similarly, by adding Equation (26c) multiplied by \(H_{1}\) to Equation (26d) multiplied by \(G_{1}\), an equation was obtained from which the frequency \(\omega_{2}\) can be derived:  
\[B_{2}=B_{1}q_{1}-A_{1}p_{1}, \tag{23b}\]7  
Note that in each pair of parameters (\(p_{1}\), \(p_{2}\)), (\(q_{1}\), \(q_{2}\)), (\(\varepsilon_{D21}\), \(\varepsilon_{D22}\)), (\(\delta_{D11}\), \(\delta_{D12}\)), (\(\delta_{D21}\), \(\delta_{D22}\)), the only difference is the frequency in their definition--\(\omega_{1}\) or \(\omega_{2}\). Considering the similarity between Equations (28) and (29), one can conclude that the frequencies \(\omega_{1}\) and \(\omega_{2}\) must be equal, thereby excluding the possibility of polymodal vibrations in the system. The expression describing the vibration frequency \(\omega_{2}=\omega_{1}\) can therefore be derived by solving only one of the above equations.