## 3 Modeling of the Beam
### Parameter Analysis of the PZT Cantilever Beam System  
Tables 2 and 3 show the parameters of the aluminum cantilever beam and details regarding the properties and locations of the PZT (Lead Zirconate Titanate) actuators and sensors within the beam system, respectively, as sourced from [13]. These parameters and properties are essential for understanding the dynamic behavior and actuation/sensing capabilities of the beam system.  
In this work, we investigated the dynamic characteristics of a flexible beam, aiming to understand its behavior under different conditions. The resulting model was constructed based on three modes of shape, which illustrate various ways the beam can bend or vibrate. Moreover, we integrated experimental data from [15] regarding the beam's natural vibration frequencies, which were determined to be 95.3618 Hz for mode 1, 597.5569 Hz for mode 2, and 1714 Hz for mode 3. Additionally, the damping coefficient was found to be 0.01. These pivotal experimental findings played a significant role in shaping our model development and subsequent analysis.  
This study aims to understand the vibrational behavior of a cantilever beam with three modes of shape and analyze the collective influence of these modes on the beam system. First, we will derive the transfer function for each of the first three modes of shape and subsequently combine them to represent the beam's overall dynamic behavior. We substitute the experimental values into Equation (4) to get the following:  
Transfer function of first mode shape:  
\[\frac{V_{s1}(s)}{V_{s1}(s)}=\frac{-2.944}{s^{2}+1.95s+9506} \tag{5}\]  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Properties** & **Units** & **Beam** \\ \hline E (Young Modulus) & lb/in\({}^{2}\) & \(1.09\times 10^{7}\) \\ \hline \(\rho\) (density) & lb/in\({}^{3}\) & 0.0975 \\ \hline w (width) & In & 0.6 \\ \hline t (thickness) & In & 0.05 \\ \hline
1 (Length) & In & 11.8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of Aluminum 6064 Beam.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Properties** & **Units** & **Sensor** & **Actuator** \\ \hline \(d_{31}\) & m/V & \(-1.90\times 10^{-10}\) & \(-1.90\times 10^{-10}\) \\ \hline \(g_{31}\) & Vm/N & \(-1.16\times 10^{-2}\) & \(-1.16\times 10^{-2}\) \\ \hline \(k_{31}\) & & 0.35 & 0.35 \\ \hline \(b_{a}\) (Width) & in & 0.4 & 0.4 \\ \hline t (Thickness) & in & 0.0105 & 0.0105 \\ \hline L (Length) & in & 0.5 & 1 \\ \hline \(x_{1}(\text{LocationonBeam})\) & in & 1.5 & 0 \\ \hline \(x_{2}(\text{LocationonBeam})\) & in & 2 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of PZT PSI-4A4E.  
Transfer function of second mode shape:  
\[\frac{V_{s2}(s)}{V_{a2}(s)}=\frac{-37.11}{s^{2}+11.79s+3.475e5} \tag{6}\]  
Transfer function of third mode shape:  
\[\frac{V_{s3}(s)}{V_{a3}(s)}=\frac{80.56}{s^{2}+32.56s+2.651e6} \tag{7}\]  
Now, we combine the individual transfer functions obtained by summing them, providing insights into the integrated dynamic behavior of the beam.  
\[\frac{V_{s}(s)}{V_{a}(s)}=\frac{40.51\ s^{4}-304.5\ s^{3}-7.88e7\ s^{2}-2.65e8\ s-3. 381e12}{s^{6}+46.3\ s^{5}+3.008e6\ s^{4}+4.884e7\ s^{3}+9.498e11\ s^{2}+2.201e12\ s+8.757e15} \tag{8}\]