# Frequency Sensitivity and Mode Characteristics of Body Structure for a Piezoelectric-Based UUV
## 4 Results and Discussion  
### Frequency Sensitivity Analysis  
Table 1 shows the system parameters of the proposed UUV structure. Substituting the parameters into Equation (6), the natural frequencies of the UUV body structure can be determined (see Table 2). Similarly, substituting the parameters into Equations (8)-(11), the frequency sensitivity with parameter changes can be obtained, which are presented in Figure 4. Since the laws for a mode (m, n) with the three frequencies (\(\omega_{1 mm}\), \(\omega_{2 nm}\) and \(\omega_{3 nm}\)) are similar, only modes B(2, j) with the lowest frequency are selected to investigate the frequency sensitivity characteristic of the proposed UUV body. From Table 2 and Figure 4, it can be observed that: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Values** & **Items** & **Values** \\ \hline wall thickness \(\delta\) (mm) & 8 & elasticity modulus \(E\) (GPa) & 210 \\ middle surface radius \(R_{m}\) (mm) & 146 & Poisson’s ratio \(\mu\) & 0.3 \\ length of body structure \(l\) (mm) & 1100 & density of the material \(\rho\) (Kg/m\({}^{3}\)) & 7850 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. (1) For a certain group of (\(m\), \(n\)), there are three frequencies that exist in the system. When \(m\) and \(n\) are not all zero values, the three frequencies are different from each other. Since \(m\) and \(n\) increase, the three frequencies also increase. When the axial wave number \(m\) has a constant value, the first frequency \(\omega_{1}\) decreases initially then increases as the circumferential wave number \(n\) increases. However, the second and the third frequencies (\(\omega_{2}\) and \(\omega_{3}\)) increase with \(n\) for all values. In addition, when the circumferential wave number \(n\) has a constant value, the frequencies increase with the increase of the axial wave number \(m\). But the increment of the first frequency \(\omega_{1}\) is larger than that of the other conditions when \(n\) is quite small. (2) The changing rate of the lower order frequencies is larger when the wave number changes. Hence, the lower order frequencies should be carefully examined and the lower order resonance should be avoided during the UUV operation. (3) The middle surface radius \(R_{m}\) has a great influence on the frequency sensitivity. As \(R_{m}\) increases, the sensitivity of B(2,1) decreases, whereas the sensitivities of B(2,2), B(2,3) and B(2,4) increase. In addition, the changing rate of B(2,4) is largest, and has a maximum value of 52.9%. In addition, the circumferential wave number \(n\) influences the frequency sensitivity as well and the bigger the \(n\), the larger the sensitivity. (4) The change of frequency sensitivity with the parameters wall thickness \(\delta\) and material density \(\rho\) are quite small. As \(\delta\) and \(\rho\) increase, the sensitivity increases slightly. (5) As the body length \(l\) increases, the frequency sensitivity also increases. With the increase of the circumferential wave number \(n\), the changing rate of the frequency sensitivity is reduced, which is different from the sensitivity of \(R_{m}\). Meanwhile, the effect of \(l\) on the frequency sensitivity is larger than that of \(\delta\) and \(\rho\). In summary, the natural frequencies are sensitive to the change of the middle surface radius \(R_{m}\) and body length \(l\), and these parameters should be carefully studied during the optimization of the structural dimensions of the UUV.