# Vibration Energy Harvesting from Plates by Means of Piezoelectric Dynamic Vibration Absorbers
## 2 Mathematical Model
### Plate Model  
The equation of the forced vibrations of a plate is:  
\[\rho\ddot{w}(x,y,t)+C_{plate}\dot{w}(x,y,t)+D\bigg{(}\frac{\partial^{4}w(x,y,t) }{\partial x^{4}}+2\frac{\partial^{4}w(x,y,t)}{\partial x^{2}\partial y^{2}}+ \frac{\partial^{4}w(x,y,t)}{\partial y^{4}}\bigg{)}=F(x,y,t) \tag{1}\]  
where \(w\) is the transversal displacement of any point of the plate, which is a function of time and space, \(F(x,y,t)\) is a force distribution, \(\rho\) is the surface density of the plate, \(D\) is the flexural rigidity of the plate, and \(C_{plate}\) is a damping coefficient; proportional damping is assumed. The plate has homogeneous thickness \(h\), and the flexural rigidity can be calculated from the modulus of elasticity \(E\) and from Poisson's ratio \(\
u\) as \(D=Eh^{3}/(12(1-\
u^{2}))\). Vibration modes of the plate can be found through the associated homogeneous Equation (1). According to the Rayleigh method, when the \(i\)-th mode of the plate is considered, which has \(m\) nodal lines in the \(x\) direction and \(n\) nodal lines in the \(y\) direction, the transversal displacement of each point of the rectangular plate can be expressed by the function:  
\[\Phi_{i}(x,y)\equiv\Phi_{mn}(x,y)=X_{m}(x)Y_{n}(y) \tag{2}\]  
Figure 1: Energy harvesting system (**top**). Bottom and section views of the plate equipped with the CPDVA (**bottom**). where \(X_{m}(x)\) and \(Y_{n}(y)\) are the fundamental mode shapes along the \(x\) and \(y\) directions, respectively, and depend on the boundary conditions of the plate. For a rectangular plate clamped at \(x=0\) and \(x=a\) and free at \(y=0\) and \(y=b\), the \(X_{m}(x)\) functions are:  
\[X_{m}(x)= \begin{cases}\cos\Bigl{(}\gamma_{1}\cdot\Bigl{(}\frac{x}{a}-\frac{1}{ 2}\Bigr{)}\Bigr{)}+\frac{\sin(\gamma_{1}/2)}{\sinh(\gamma_{1}/2)}\cosh\Bigl{(} \gamma_{1}\cdot\Bigl{(}\frac{x}{a}-\frac{1}{2}\Bigr{)}\Bigr{)},&m=2,4,6,\dots \\ \sin\Bigl{(}\gamma_{2}\cdot\Bigl{(}\frac{x}{a}-\frac{1}{2}\Bigr{)}\Bigr{)}- \frac{\sin(\gamma_{2}/2)}{\sinh(\gamma_{2}/2)}\sinh\Bigl{(}\gamma_{2}\cdot \Bigl{(}\frac{x}{a}-\frac{1}{2}\Bigr{)}\Bigr{)},&m=3,5,7,\dots\end{cases}  \tag{3}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) are, respectively, the roots of:  
\[\tan\Bigl{(}\frac{\gamma_{1}}{2}\Bigr{)}+\tanh\Bigl{(}\frac{\gamma_{1}}{2} \Bigr{)}=0 \tag{4}\]  
\[\tan\Bigl{(}\frac{\gamma_{2}}{2}\Bigr{)}-\tanh\Bigl{(}\frac{\gamma_{2}}{2} \Bigr{)}=0 \tag{5}\]  
The functions \(Y_{n}(y)\) are defined as:  
\[Y_{n}(y)= \begin{cases}1,&n=0\\ 1-\frac{2y}{b},&n=1\\ \cos\Bigl{(}\gamma_{1}\cdot\Bigl{(}\frac{y}{b}-\frac{1}{2}\Bigr{)}\Bigr{)}- \frac{\sin(\gamma_{1}/2)}{\sinh(\gamma_{1}/2)}\cosh\Bigl{(}\gamma_{1}\cdot \Bigl{(}\frac{y}{b}-\frac{1}{2}\Bigr{)}\Bigr{)},&n=2,4,6,\dots\\ \sin\Bigl{(}\gamma_{2}\cdot\Bigl{(}\frac{y}{b}-\frac{1}{2}\Bigr{)}\Bigr{)}+ \frac{\sin(\gamma_{2}/2)}{\sinh(\gamma_{2}/2)}\sinh\Bigl{(}\gamma_{2}\cdot \Bigl{(}\frac{y}{b}-\frac{1}{2}\Bigr{)}\Bigr{)},&n=3,5,7,\dots\end{cases}  \tag{6}\]  
The following orthogonality conditions hold for plate modes:  
\[\int_{0}^{a}\int_{0}^{b}\rho\Phi_{j}(x,y)\Phi_{i}(x,y)dxdy=m_{i}\delta_{ij} \tag{7}\]  
\[\int_{0}^{a}\int_{0}^{b}\Phi_{j}(x,y)\
abla^{4}\Phi_{i}(x,y)dxdy=\omega_{i}^{2 }\delta_{ij} \tag{8}\]  
where \(\delta_{ij}\) is equal to 1 if \(i=j\) and 0 otherwise; \(m_{i}\) and \(\omega_{i}\) are the modal mass and the natural angular frequency of the mode \(i\), respectively; and the operator \(\
abla^{4}\) is defined as \(\
abla^{4}=\frac{\partial^{4}}{\partial x^{4}}+2\frac{\partial^{4}}{\partial x ^{2}\partial y^{2}}+\frac{\partial^{4}}{\partial y^{4}}\). The parameters of the plate considered in this study are summarized in Table 1. It is worth noting that the properties of the material influence the natural frequencies of the plate. However, the mathematical model is still valid with different materials of the plate. From the plate parameters, the natural frequency and shape of each mode can be determined (for the calculation of natural frequencies, refer to [25]).