## 2 Theoretical Background
### Vibrational Mechanics  
Vortex induced vibration of a plate is a form of response from a forced harmonics excitation. In order to simplify the calculation, the plate is assumed to vibrate linearly in single degree of freedom. Considering the structure is an energy harvesting piezoelectric structure, a piezoelectric component should be considered [49]. Therefore, the general forced vibration response for a linear equation can be expressed as followed:  
\[[M]\{\ddot{y}(t)\}+[C]\{\dot{y}(t)\}+[K]\{y(t)\}+[\theta]\{V(t)\}=\{F(t)\} \tag{7}\]  
where \(y\) is the forced response, \([M]\) is the mass matrix, \([C]=\alpha[M]+\beta[K]\) is the proportional damping matrix, \([K]\) is the stiffness matrix, \(\theta\) is the coupling coefficient and \(V(t)\) is electrode voltage and \(\{F(t)\}\) is the force excitation vector. According to [PERSON] et al., the term \([\theta]\{V(t)\}\) is related to electrical damping and stiffness when the electrodes are connected to a resistive load, \(R\). Equation (8) shows the electromechanical output when a resistive load, \(R\) is connected where \(I\) is current [50].  
\[-V(t)=\ IR=\ R\theta\dot{y}(t)+RC_{p}\dot{V}(t) \tag{8}\]  
Equation (7) is re-written as:  
\[[M]\{\ddot{y}(t)\}+[C+C_{e}]\{\dot{y}(t)\}+[K+K_{e}]\{y(t)\}=\{F(t)\} \tag{9}\]  
Noted that the coupling coefficient, \(\theta\) and the electrical capacitance, \(C_{p}\) for a given dimension of piezoelectric plate, (i.e., \(L\) is length, \(W\) is width, and \(T\) is thickness) are defined as:  
\[\theta=\frac{3d_{31}LW}{4\epsilon_{33}^{T}T} \tag{10}\]  
\[C_{p}=\frac{\epsilon_{33}^{S}LW}{T} \tag{11}\]The electrical damping, \(C_{e}\) at frequency, \(\omega\) could be expressed as:  
\[C_{e}=\frac{RC_{p}\left(\frac{\theta^{2}}{C_{p}}\right)}{1+\left(RC_{p}\omega \right)^{2}} \tag{12}\]  
and electrical stiffness, \(K_{e}\) at frequency, \(\omega\) could be expressed as:  
\[K_{e}=\frac{\left(RC_{p}\omega\right)^{2}\left(\frac{\theta^{2}}{C_{p}}\right) }{1+\left(RC_{p}\omega\right)^{2}} \tag{13}\]  
The flutter and lift force by vortex is undergoing harmonic excitation the force excitation matrix, \(\{F(t)\}\) can be expressed in terms of excitation frequency, \(\omega\) and time, \(t\) shown in Equation (14):  
\[\{F\}=\{P\}\cos(\omega t) \tag{14}\]  
A solution in terms of location amplitude vector, \(y(t)\) hence can be solved from Equation (9) as:  
\[\{y(t)\}=\ \frac{\{P\}\cos(\omega t)}{\left(K+K_{e}-\omega^{2}M\right)+j\omega(C+ C_{e})} \tag{15}\]  
The average power produced from a PVEH plate can be expressed as:  
\[P_{avg}=\frac{R\omega^{2}\theta^{2}{y_{t}}^{2}}{2\left(1+\left(RC\omega\right) ^{2}\right)} \tag{16}\]  
When considering \(y_{0}\) as a function of load, \(R\) based on Equation (15) above and the average power equation, as shown in Equation (16), the optimum power with respect to \(R\) is obtained as:  
\[R_{opt}=\ 2\zeta/C_{p}\omega\sqrt{4\zeta^{2}+\theta^{4}} \tag{17}\]  
Such a system is too complicated to be useful for an analytical design. In addition, the coupling effect of \(\theta\) is assumed to be relatively small and weak and Equation (17) can be simplified as:  
\[R_{opt}=\ 1/C_{p}\omega. \tag{18}\]