# Frequency Sensitivity and Mode Characteristics of Body Structure for a Piezoelectric-Based UUV
## 3 Dynamic Model and Equations
The coefficients of Equation (4) have a nonzero solution, so the frequency equations can be written as:  
\[\left| \begin{array}{ccc}\Omega^{2}-G_{11}&G_{12}&G_{13}\\ G_{21}&\Omega^{2}-G_{22}&G_{23}\\ G_{31}&G_{32}&\Omega^{2}-G_{33}\\ \end{array} \right|=0 \tag{5}\]  
Hence, the natural frequency of the system can be expressed by:  
\[\omega_{imm}=\frac{1}{3R_{m}}\sqrt{\frac{-6E\sqrt{P_{1}^{2}-3P_{2}}\cos\left[ \alpha+\frac{2\pi}{3}(i-1)\right]-3 EP_{1}}{\rho(1-\mu^{2})}}(i=1,2,3;\ m,n=1,2,3,\cdots)} \tag{6}\]  
where, \(P_{1}=-\left(G_{11}+G_{22}+G_{33}\right)\), \(P_{2}=G_{11}G_{22}+G_{22}G_{33}+G_{33}G_{11}-\left(G_{12}^{2}+G_{23}^{2}+G_{3 1}^{2}\right)\), \(\alpha=\frac{1}{3}\cos^{-1}\left[\frac{2P_{13}^{2}-9P_{1}P_{2}+27P_{3}}{2\left( P_{1}^{2}-3P_{2}\right)^{3/2}}\right]\), \(P_{3}=G_{11}G_{23}^{2}+G_{22}G_{31}^{2}+G_{33}G_{12}^{2}+2G_{12}G_{23}G_{31}- G_{11}G_{22}G_{33}\). For an arbitrary parameter \(\kappa\), the sensitivity of the natural frequencies can be written as:  
\[S_{imm}=\frac{\partial\omega_{imm}}{\partial\kappa} \tag{7}\]  
The frequency sensitivity with the middle surface radius \(R_{m}\) can be expressed by:  
\[ \begin{array}{l}S_{R_{m}-imm}=-\frac{1}{3R_{m}^{2}}\left\{\frac{-6E\sqrt{P_ {1}^{2}-3P_{2}}\cos\left[\alpha+\frac{2\pi}{3}(i-1)\right]-3 EP_{1}}{\rho(1- \mu^{2})}\right\}^{\frac{1}{4}}-\frac{1}{6R_{m}}\left\{\frac{-6E\sqrt{P_{1}^{2 }-3P_{2}}\cos\left[\alpha+\frac{2\pi}{3}(i-1)\right]-3 EP_{1}}{\rho(1-\mu^{2}) }\right\}^{-\frac{1}{2}}\\ \times\frac{1}{\rho(1-\mu^{2})}\left\{-3E\left(P_{1}^{2}-3P_{2}\right)^{-\frac{ 1}{2}}\left(2P_{1}\frac{\partial P_{1}}{\partial\lambda}-3\frac{\partial P_{2 }}{\partial\lambda}\right)\cos\left[\alpha+\frac{2\pi}{3}(i-1)\right]-6E\left( P_{1}^{2}-3P_{2}\right)^{\frac{1}{2}}\sin\left[\alpha+\frac{2\pi}{3}(i-1) \right]\frac{\partial\lambda}{\partial R_{m}}\right\}\end{array}  \tag{8}\]  
Meanwhile, the frequency sensitivities with wall thickness \(\delta\), body length \(l\), and material density \(\rho\) are given as follows:  
\[ \begin{array}{l}S_{\delta-imm}=\frac{1}{6R_{m}\sqrt{\rho(1-\mu^{2})}}\left\{ -6E\sqrt{P_{1}^{2}-3P_{2}}\cos\left[\alpha+\frac{2\pi}{3}(i-1)\right]-3 EP_{1} \right\}^{-\frac{1}{2}}\left\{-6E\left(P_{1}^{2}-3P_{2}\right)^{\frac{1}{2}} \sin\left[\alpha+\frac{2\pi}{3}(i-1)\right]\frac{\partial\lambda}{\partial \lambda}\right.\\ \left.-3E\left(P_{1}^{2}-3P_{2}\right)^{-\frac{1}{2}}\left(2P_{1}\frac{\partial P _{1}}{\partial\lambda}-3\frac{\partial P_{2}}{\partial\lambda}\right)\cos \left[\alpha+\frac{2\pi}{3}(i-1)\right]\right\}\\ S_{l-imm}=\frac{1}{6R_{m}\sqrt{\rho(1-\mu^{2})}}\left\{-6E\sqrt{P_{1}^{2}-3P_{2}} \cos\left[\alpha+\frac{2\pi}{3}(i-1)\right]-3 EP_{1}\right\}^{-\frac{1}{2}} \left\{-6E\left(P_{1}^{2}-3P_{2}\right)^{\frac{1}{2}}\sin\left[\alpha+\frac{2 \pi}{3}(i-1)\right]\frac{\partial\lambda}{\partial\lambda}\right.\\ \left.-3E\left(P_{1}^{2}-3P_{2}\right)^{-\frac{1}{2}}\left(2P_{1}\frac{\partial P _{1}}{\partial\lambda}-3\frac{\partial P_{2}}{\partial\lambda}\right)\cos\left[ \alpha+\frac{2\pi}{3}(i-1)\right]\right\}\\ \end{array}  \tag{10}\]\[S_{p-imn}= \frac{1}{6R_{n}\sqrt{\rho(1^{-p})^{2}}}\Big{\{}-6E\sqrt{p_{1}^{2}-3P_{ 2}}\cos\left[\alpha+\frac{2\pi}{3}(i-1)\right]-3E{P_{1}}\Big{\}}^{-\frac{1}{2}} \Big{\{}-6E\left(P_{1}^{2}-3P_{2}\right)^{\frac{1}{3}}\sin\left[\alpha+\frac{2 \pi}{3}(i-1)\right]\frac{\alpha_{0}}{M} \tag{11}\] \[-3E\left(P_{1}^{2}-3P_{2}\right)^{-\frac{1}{2}}\Big{(}2P_{1}\frac {3P_{1}}{M}-3\frac{3P_{2}}{M}\Big{)}\cos\left[\alpha+\frac{2\pi}{3}(i-1)\right] +\frac{4E\pi}{P^{2}}\sqrt{P_{1}^{2}-3P_{2}}\cos\left[\alpha+\frac{2\pi}{3}(i-1 )\right]+\frac{3E{P_{1}}}{P^{2}}\Big{\}}\]  
For a \(\omega_{imn}\), letting coefficient \(C=1\), the following can be obtained:  
\[\left\{