## 3 Modeling of the Beam
### Dynamic Modeling of the Beam  
Figure 3 shows a piezoelectric cantilever beam. At one end, the beam is fixed, whereas at the other, it is free. As indicated in Figure 3, two piezoelectric actuator patches and one piezoelectric sensor are utilized. The beam's dynamic modeling is the transfer function relating the input voltage of the actuator to the voltage induced by the piezoelectric sensor. The beam's dynamic model is shown in the following Equations [15].  
A model of the structure is obtained through a modal analysis procedure. This approach finds a solution to the partial differential equation describing the dynamics of the flexible structure [17]:  
\[\rho_{b}A_{b}\frac{\partial^{2}w}{\partial^{2}t}+E_{b}I_{b}\frac{\partial^{4} w}{\partial^{4}x}=M_{a}\frac{\partial^{2}R(x)}{\partial x^{2}} \tag{1}\]  
where  
\(\rho_{b}\) is the mass density of the beam sensor material;  
\(A_{b}\) is the cross-sectional beam sensor area;  
\(E_{b}\) is the Young's Modulus of the beam sensor;  
\(I_{b}\) is the first moment of inertia, \(I_{b}=\frac{b_{a}I_{b}^{3}}{3}\);  
\(I_{b}\) is the thickness of the beam sensor;  
\(b_{a}\) is the width of the beam sensor;  
W is the beam transverse displacement;  
\(M_{a}\) is the bending moment acting on the beam;  
\(R(x)\) is \(H(x-x_{a1})-H(x-x_{a2})\);  
\(R\) is the generalized location function;  
\(H\) is the Heaviside function;  
Figure 2: Assumption that the Plane Sections Remain Plane.  
\(x_{a1}\) and \(x_{a2}\) are the locations of the actuators.  
The equation of the beam actuator dynamics system is:  
\[\ddot{\eta_{i}}(t)+2\xi_{i}\omega_{ni}\ddot{\eta_{i}}(t)+\omega_{ni}{}^{2}\eta_{i} (t)=\ k_{a}\left[\phi^{\prime}{}_{i}(x_{a2})-\phi^{\prime}{}_{i}(x_{a1})\right]V _{a}(t) \tag{2}\]  
where  
\(\eta_{i}(t)\) represents the temporal functions;  
\(\xi_{i}\) is damping;  
\(\omega_{ni}\) is the natural frequency.  
\[k_{a}=\frac{\frac{1}{2}E_{a}d_{31}b_{a}(h_{b}+h_{a})}{\rho_{a}A_{a}}\]  
where  
\(E_{a}\) is the Young's Modulus of the beam actuator;  
\(d_{31}\) is the electric charge constant;  
\(b_{a}\) is the width of the actuator;  
\(h_{a}\) is the thickness of the actuator;  
\(\rho_{a}\) is the mass density of the beam actuator material;  
\(A_{a}\) is the cross-sectional beam actuator area;  
\(\phi_{i}\) is the mode shape;  
\(V_{a}(t)\) is the actuator voltage;  
(.) denotes the time derivative and (\({}^{\prime}\)) denotes the space derivative.  
Taking the Laplace transform of the beam sensor dynamics system gives:  
\[V_{s}(s)=k_{s}{\sum_{i=1}^{N}}\frac{\left[\dot{\phi}_{i}(x_{s2})-\dot{\phi}_{i }(x_{s1})\right]}{C_{s}b_{s}(x_{s2}-x_{s1})} \tag{3}\]  
Figure 3: Cantilever Beam with Two PZT Actuators and Sensor.  
The expression for \(V_{s}(s)\) in terms of the input voltage \(V_{a}(s)\) gives:  
\[\frac{V_{\mathbf{s}}(\mathbf{s})}{V_{\mathbf{a}}(\mathbf{s})}=\sum_{i=1}^{ \mathbf{N}}\frac{k_{s}k_{a}\Big{[}\dot{\boldsymbol{\phi}}_{i}(\boldsymbol{x}_{ \mathbf{s}2})-\dot{\boldsymbol{\phi}}_{i}(\boldsymbol{x}_{\mathbf{s}1})\Big{]} \Big{[}\dot{\boldsymbol{\phi}}_{i}(\boldsymbol{x}_{\mathbf{a}2})-\dot{ \boldsymbol{\phi}}_{i}(\boldsymbol{x}_{\mathbf{a}1})\Big{]}}{\boldsymbol{C}_{ \mathbf{s}}b_{\mathbf{s}}(\boldsymbol{x}_{\mathbf{s}2}-\boldsymbol{x}_{ \mathbf{s}1})(\mathbf{s}^{2}+2\dot{\boldsymbol{\phi}}_{i}\boldsymbol{\omega}_{ i}\mathbf{s}+\boldsymbol{\omega}_{m}{}^{2})} \tag{4}\]