# On Efficiency of Two-Degree-of-Freedom Galloping Energy Harvesters with Two Transducers
## 3 Efficiency of Two-Degree-of-Freedom System  
According to the information provided in Section 1, the presence of additional mass in systems with two degrees of freedom implies the necessity to consider the placement of the electromechanical transducer. It may be positioned between the main mass and the stationary base (Figure 3a), between the masses (Figure 3b), or in both locations (Figure 3c). To evaluate the influence of piezoelectric placement on efficiency, the efficiency characteristics of the variant with both transducers will be derived. Subsequently, the consequence of the absence of one of them will be examined. Figure 3: Subvariants of two-degree-of-freedom system: (**a**) transducer between base and lower mass, (**b**) transducer between masses, (**c**) two transducer. The dynamics of devices within the discussed category can be described by a general dimensionless mathematical model of the form Equation (20). In addition to the identities in Equation (3), the following also holds:  
\[ \begin{array}{ll}y_{i}=\frac{\ddot{z}_{i}}{h},\ v_{i}=\xi\frac{\dot{\theta}_{i}} {\dot{m}_{i}\ \dot{\overline{\dot{\alpha}}}_{\dot{\overline{\dot{\alpha}}}_{\dot{ \overline{\dot{\alpha}}}}}^{2}},\ c_{i}=\frac{\dot{c}_{i}}{\dot{m}_{1}\dot{ \alpha}_{\overline{\dot{\alpha}}}},\ k_{2}=\frac{\dot{k}_{2}}{\dot{k}_{1}},\ \kappa=\frac{\dot{\theta}_{i}^{2}}{C_{p\ i\ m_{1}} \ \dot{\omega}_{\overline{\dot{\alpha}}}^{2}},\\ \theta=\frac{\dot{\theta}_{1}}{\dot{\theta}_{2}},\ M=\frac{\dot{m}_{2}}{\dot{ m}_{1}},\ i=1,2\end{array}  \tag{19}\]  
\[ \begin{array}{ll}\ddot{y}_{1}+y-k_{2}(y_{2}-y_{1})+c_{1}\dot{y}_{1}-c_{2} \big{(}\dot{y}_{2}-\dot{y}_{1}\big{)}-v_{1}+v_{2}=\rho\Bigg{(}a_{1}u\dot{y}_{ 1}+a_{3}\frac{\dot{y}_{1}^{3}}{u}\Bigg{)},\end{array}  \tag{20a}\] \[\kappa_{1}\dot{v}_{1}+\frac{v_{1}}{r_{1}}+\dot{y}_{1},\] (20b) \[M\ddot{y}_{2}+k_{2}(y_{2}-y_{1})+c_{2}\big{(}\dot{y}_{2}-\dot{y}_{1}\big{)}-v_{ 2}=0,\] (20c) \[\kappa_{2}\dot{v}_{2}+\frac{v_{2}}{r_{2}}+\dot{y}_{2}-\dot{y}_{1}=0. \tag{20d}\]  
To derive the solutions of the GEH2D mathematical model, the procedure outlined in [28] was adapted. It was assumed that the approximate solution of the model Equation (20) will have the following form:  
\[y_{1}=A_{1}\text{cos}(\omega_{1}t)+B_{1}\text{sin}(\omega_{1}t)+G_{1}\text{ cos}(\omega_{2}t)+H_{1}\text{sin}(\omega_{2}t), \tag{21a}\] \[v_{1}=n_{1}\text{cos}(\omega_{1}t)+n_{2}\text{sin}(\omega_{1}t)+n_{3}\text{ cos}(\omega_{2}t)+n_{4}\text{sin}(\omega_{2}t),\] (21b) \[y_{2}=A_{2}\text{cos}(\omega_{1}t)+B_{2}\text{sin}(\omega_{1}t)+G_{2}\text{ cos}(\omega_{2}t)+H_{2}\text{sin}(\omega_{2}t),\] (21c) \[v_{2}=\dot{\theta}_{1}\text{cos}(\omega_{1}t)+\dot{\theta}_{2}\text{sin}( \omega_{1}t)+\dot{\theta}_{3}\text{cos}(\omega_{2}t)+\dot{\theta}_{4}\text{ sin}(\omega_{2}t), \tag{21d}\]  
Substituting the above identities into Equations (20b) and (20d) and then balancing the harmonics \(\text{cos}(\omega_{1}t)\), \(\text{sin}(\omega_{1}t)\), \(\text{cos}(\omega_{2}t)\), \(\text{sin}(\omega_{2}t)\) allows for the derivation of the relationships between the amplitudes of voltage and the amplitudes of vibration. \[ \begin{array}{ll}n_{1}=B_{1}\omega_{1}\varepsilon_{D11}-A_{1}\delta_{D11}, &n_{2}=-A_{1}\omega_{1}\varepsilon_{D11}-B_{1}\delta_{D11},\\ n_{3}=H_{1}\omega_{2}\varepsilon_{D12}-G_{1}\delta_{D12},&n_{4}=-G_{1} \omega_{2}\varepsilon_{D12}-H_{1}\delta_{D12},\end{array}  \tag{22a}\] \[ \begin{array}{ll}\dot{\theta}_{1}=\theta(\varepsilon_{D21}(\theta B_{2}-B_{1 })\omega_{1}+\delta_{D21}(A_{1}-\theta A_{2})),\\ \dot{\theta}_{2}=\theta(\varepsilon_{D21}(A_{1}-\theta A_{2})\omega_{1}+ \delta_{D21}(B_{1}-\theta B_{2})),\\ \dot{\theta}_{3}=\theta(\varepsilon_{D22}(\theta H_{2}-H_{1})\omega_{2}+ \delta_{D22}(G_{1}-\theta G_{2})),\\ \dot{\theta}_{4}=\theta(\varepsilon_{D22}(G_{1}-\theta G_{2})\omega_{2}+ \delta_{D22}(H_{1}-\theta H_{2})).\end{array}