# Vibration Energy Harvesting from Plates by Means of Piezoelectric Dynamic Vibration Absorbers
## 2 Mathematical Model
### Coupled Equations  
The aim of this section is to calculate the frequency response functions (FRFs) between applied loads, generated voltage and plate displacement, since they make possible the calculation of the response of the system in the presence of harmonic, periodic, transient and random loads. To generalize the analysis, the following loads acting on the plate are considered: a uniform distributed load \(q_{z}(t)\), a concentrated force \(F_{c}(t)\) applied at the point of coordinates \((x_{c},y_{c})\) and the forcing term due to the CPDVA \(F_{a}(t)\). The uniform distributed load can represent both the action of a fluid in contact with the plate and an inertial load resulting from the acceleration of the two clamped sides of the plate \((\tilde{z}_{f}(t))\) due to environmental vibrations. In this case:  
\[q_{z}(t)=-\rho\tilde{z}_{f}(t) \tag{18}\]  
The concentrated force is a reference load condition that simulates the effect of impulsive tests carried out by means of hammers for modal testing. The force due to the CPDVA is the sum of two terms. The first term is the inertial force related to the oscillation of the harvester base, which has mass \(M_{a}\) and is rigidly fixed to the plate. The second term is related to the oscillation of the cantilever with a tip mass. The second term can be obtained by isolating the damping and the stiffness terms in Equation (9). Therefore, the following expression is found:  
\[F_{a}(t)=-M\bigg{(}\ddot{u}_{r}(t)+\bigg{(}\alpha+\frac{M_{a}}{M}\bigg{)} \ddot{w}(x_{a},y_{a},t)\bigg{)} \tag{19}\]  
In order to obtain the equations of motion of the system composed of the plate and the CPDVA, the distributed and the concentrated forces are inserted in (1), and the resulting equation is combined with (9):  
\[ \begin{cases}\rho\dot{w}(x,y,t)+C_{plate}\dot{w}(x,y,t)+D\
abla^{4}w(x,y,t)=q_{ z}(t)+F_{c}(t)\delta(x-x_{c})\delta(y-y_{c})+F_{a}(t)\delta(x-x_{a})\delta(y-y_{a})\\ M\ddot{u}_{r}(t)+C\dot{u}_{r}(t)+Ku_{r}(t)=-\alpha M\ddot{w}(x_{a},y_{a},t)\\ \end{cases}  \tag{20}\]  
where \(\delta\) is the Dirac delta function. The transversal displacement of the plate is expressed as a linear combination of the modes of the plate. In particular, the modes of vibration of the clamped-clamped-free-free plate described in Section 2.1 are adopted:  
\[w(x,y,t)=\sum\limits_{i=1}^{\infty}\Phi_{i}(x,y)\eta_{i}(t) \tag{21}\]  
where \(\eta_{i}(t)\) is the modal coordinate of mode \(i\). In practical simulations, the summation in (21) is truncated by taking into account only the first \(N\) modes of the plate in the frequency band of interest. Substituting (19) and (21) in (20) gives:  
\[ \begin{cases}\rho\sum\limits_{i=1}^{N}\[PERSON]_{i}(x,y)\ddot{\eta}_{i}(t)+C_{plate }\sum\limits_{i=1}^{N}\[PERSON]_{i}(x,y)\dot{\eta}_{i}(t)+D\
abla^{4}\[PERSON]_{i}(x,y) \eta_{i}(t)=q_{z}(t)+F_{c}(t)\delta(x-x_{c})\delta(y-y_{c})-\\ M\bigg{(}\ddot{u}_{r}(t)+\Big{(}\alpha+\frac{M_{a}}{M}\Big{)}\sum\limits_{i=1 }^{N}\Phi_{i}(x_{a},y_{a})\dot{\eta}_{i}(t)\bigg{)}\delta(x-x_{a})\delta(y-y_{ a})\\ M\ddot{u}_{r}(t)+C\dot{u}_{r}(t)+Ku_{r}(t)=-\alpha M\sum\limits_{i=1}^{N}\Phi_{i}(x_{a},y_{a}) \ddot{\eta}_{i}(t)\\ \end{cases}  \tag{22}\]  
Then, the orthogonality conditions (7) and (8) are exploited by multiplying (22) for the generic mode \(\Phi_{i}(x,y)\) and integrating on the whole plate. In this way, the following system of \(N+1\) differential equations in modal coordinates is obtained:\[ \begin{cases}m_{i}\dot{\eta}_{i}(t)+2m_{i}\zeta_{i}\omega_{i}\dot{\eta}_{i}(t)+m_{ i}\omega_{i}^{2}\eta_{i}(t)=\int\limits_{0}^{a}\int\limits_{0}^{b}\Phi_{i}(x,y)q_{z}(t) dxdy+\Phi_{i}(x_{c},y_{c})F_{c}(t)-\\ M\Phi_{i}(x_{a},y_{a})\bigg{(}\ddot{u}_{r}(t)+\left(\alpha+\frac{M_{a}}{M} \right)\sum\limits_{i=1}^{N}\Phi_{i}(x_{a},y_{a})\ddot{\eta}_{i}(t)\bigg{)}\\ M\ddot{u}_{r}(t)+Cu_{r}(t)+Ku_{r}(t)=-\alpha M\sum\limits_{i=1}^{N}\Phi_{i}(x_ {a},y_{a})\ddot{\eta}_{i}(t)\\ \end{cases}  \tag{23}\]  
where the first equation holds for \(i=1,2,\ldots,N\).