# Vibration Energy Harvesting from Plates by Means of Piezoelectric Dynamic Vibration Absorbers
## 2 Mathematical Model
### Coupled Equations
For the computation of the frequency response functions (FRFs), the following harmonic solutions are assumed:  
\[\eta_{i}(t) =\eta_{i,0}\cdot e^{i\omega t} \tag{24}\] \[u_{r}(t) =u_{r,0}\cdot e^{i\omega t}\] (25) \[q_{z}(t) =q_{z,0}\cdot e^{i\omega t}\] (26) \[F_{c}(t) =F_{c,0}\cdot e^{i\omega t} \tag{27}\]  
where \(i\) is the imaginary unit. Introducing (24)-(27) in (23), this system of linear equations holds:  
\[ \begin{cases}-\omega^{2}m_{i}\eta_{i,0}+2i\ \omega m_{i}\zeta_{i}\omega_{i} \eta_{i,0}+m_{i}\omega_{i}^{2}\eta_{i,0}(t)=\int\limits_{0}^{a}\int\limits_{0 }^{b}\Phi_{i}(x,y)q_{z,0}\ dxdy+\Phi_{i}(x_{c},y_{c})F_{c,0}+\\ \omega^{2}M\Phi_{i}(x_{a},y_{a})\bigg{(}u_{r,0}+\left(\alpha+\frac{M_{a}}{M} \right)\sum\limits_{i=1}^{N}\Phi_{i}(x_{a},y_{a})\eta_{i,0}\bigg{)}\\ \\ -\omega^{2}Mu_{r,0}+i\ \omega Cu_{r,0}+Ku_{r,0}=\omega^{2}\alpha M\sum\limits_{i=1}^{N} \Phi_{i}(x_{a},y_{a})\eta_{i,0}\end{cases}  \tag{28}\]  
where the first equation holds for \(i=1,2,\ldots,N\). System (28) is rearranged by letting the unknown \(\eta_{1,0}\), \(\eta_{2,0}\),..., \(\eta_{N,0}\) and \(u_{r,0}\) be on the left side and moving the assigned forcing terms to the right side. Moreover, for brevity, the following symbols are adopted:  
\[ \begin{split}\Phi_{i}^{a}&=\[PERSON]_{i}(x_{a},y_{a})\\ \[PERSON]_{i}^{c}&=\[PERSON]_{i}(x_{c},y_{c})\\ A_{i}&=-m_{i}\omega^{2}+2 im_{i}\zeta_{i}\omega_{i} \omega+m_{i}\omega_{i}^{2}\\ & B_{i}=M\omega^{2}\[PERSON]_{i}^{a}\Big{(}\alpha+\frac{M_{a}}{M} \Big{)}\\ & P_{i}=\int\limits_{0}^{a}\int\limits_{0}^{b}\[PERSON]_{i}(x,y)dxdy\\ D&=-M\omega^{2}+iC\omega+K\end{split}  \tag{29}\]  
Therefore:  
\[ \begin{cases}\left(A_{i}-\[PERSON]_{i}^{a}B_{i}\right)\eta_{i,0}-\[PERSON]_{i}^{a}\sum \limits_{j=1}^{n}B_{j}\eta_{j,0}(1-\delta_{ij})-B_{i}\frac{M}{\alpha M+M_{a}}u_ {r,0}=P_{i}q_{z,0}+\[PERSON]_{i}^{c}F_{c,0}\\ -\alpha M\omega^{2}\sum\limits_{i=1}^{n}\[PERSON]_{i}^{a}\eta_{i,0}+Du_{r,0}=0\\ \end{cases}  \tag{30}\]  
where the first equation holds for \(i=1,2,\ldots,N\). Considering the uniform distributed load amplitude \(q_{z,0}\) or the concentrated force amplitude \(F_{c,0}\) as inputs, Equation (30) can be solved for a range of forcing frequencies \(\omega\) and amplitudes \(\eta_{1,0}\), \(\eta_{2,0}\),..., \(\eta_{N,0}\), and \(u_{r,0}\) can be obtained at each frequency. The amplitude of the plate displacement \(w_{0}\) can be calculated from the modal coordinates by means of:  
\[w_{0}(x,y,\omega)=\sum_{i=1}^{N}\Phi_{i}(x,y)\eta_{i,0}(\omega) \tag{31}\]  
Therefore, FRFs between the plate displacement (and acceleration) at the generic point \(x_{p},y_{p}\) and the loads can be calculated using the following definitions:  
\[FRF_{w_{p},q_{z}}(\omega)=\frac{w_{0}(x_{p},y_{p},\omega)}{q_{z,0}}\qquad\qquad FR _{\varpi_{p},q_{z}}(\omega)=-\omega^{2}\frac{w_{0}(x_{p},y_{p},\omega)}{q_{z,0}} \tag{32}\]  
\[FRF_{w_{p},F_{c}}(\omega)=\frac{w_{0}(x_{p},y_{p},\omega)}{F_{c,0}}\qquad\qquad FR _{\varpi_{p},F_{c}}(\omega)=-\omega^{2}\frac{w_{0}(x_{p},y_{p},\omega)}{F_{c,0}} \tag{33}\]  
where (32) are used for the uniform distributed load, and (33) are used for the concentrated load. The amplitude of the open-circuit (OC) voltage can be retrieved by exploiting Equation (17); the following expression holds:  
\[v_{\infty,0}=\frac{\theta}{C_{p}}u_{r,0} \tag{34}\]  
Finally, the FRFs between the OC voltage and applied loads are:  
\[FRF_{v,q_{z}}(\omega)=\frac{v_{\infty,0}(\omega)}{q_{z,0}} \tag{35}\]  
\[FRF_{v,F_{c}}(\omega)=\frac{v_{\infty,0}(\omega)}{F_{c,0}} \tag{36}\]  
for cases with distributed and concentrated loads, respectively. The mathematical model was implemented in Matlab, and in particular, the modes of vibration were integrated using the symbolic toolbox.