# Frequency Sensitivity and Mode Characteristics of Body Structure for a Piezoelectric-Based UUV
## 3 Dynamic Model and Equations  
Figure 3 shows the dynamic model of the UUV body structure, where, \(u\), \(v\), and \(w\) represent the axial, tangential, and normal displacements of the middle surface, \(\theta\) is the tangential angle, and \(l\) is the length of the body structure, while \(\delta\) and \(R_{m}\) are the wall thickness and the middle surface radius, respectively. It is assumed that the dynamic displacements are \(\Delta u\), \(\Delta v\), and \(\Delta w\). Hence, from [PERSON]'s theory [17; 18; 19], the dynamic equations of the UUV's body can be expressed by:  
\[\left\{ \begin{array}{l}\frac{\partial^{2}\Delta u}{\partial x^{2}}+\frac{1- \mu}{2}\frac{\partial^{2}\Delta u}{R_{m}^{2}\partial\theta^{2}}+\frac{1+\mu}{ 2}\frac{\partial^{2}\Delta v}{R_{m}\partial\alpha\partial\theta}+v\frac{ \partial\Delta v}{R_{m}\partial x}+\frac{\partial^{\delta}}{K}\omega^{2}\Delta u =0\\ \frac{1+\mu}{2}\frac{\partial^{2}\Delta u}{R_{m}\alpha\partial\theta}+\frac{1 -\mu}{2}\frac{\partial^{2}\Delta v}{\partial x^{2}}+\frac{\partial^{2}\Delta v }{R_{m}^{2}\partial\theta^{2}}+\frac{\partial\Delta v}{R_{m}^{2}\partial \theta}+\\ k\left[\frac{1-\mu}{2}\frac{\partial^{2}\Delta v}{\partial x^{2}}+\frac{ \partial^{2}\Delta v}{R_{m}^{2}\partial\theta^{2}}-\frac{\partial^{2}\Delta v }{\partial x^{2}\partial\theta}-\frac{\partial^{2}\Delta v}{R_{m}^{2}\partial \theta}\right]+\frac{\partial^{\delta}}{K}\omega^{2}\Delta v=0\\ \frac{12}{\delta^{2}}\left[\mu\frac{\partial\Delta u}{R_{m}\alpha\partial x}+ \frac{\partial\Delta u}{R_{m}^{2}\partial x}+\frac{\Delta v}{R_{m}^{2}}-k \left(\frac{\partial^{3}\Delta v}{\partial x^{2}\partial\theta}+\frac{ \partial^{3}\Delta v}{R_{m}^{2}\partial\theta}\right)\right]+\
abla^{2} \
abla^{2}\Delta w-\frac{\partial^{\delta}}{D}\omega^{2}\Delta w=0\end{array}  \right., \tag{1}\]  
where, \(K\) and \(D\) are the shell stiffness and bending stiffness, respectively, \(K=\frac{E\delta}{1-\mu^{2}}\), \(D=\frac{E\delta^{2}}{12(1-\mu^{2})}\); \(\mu\) is the Poisson ratio; \(\rho\) is the material density of the body structure; \(\omega\) is the natural frequency of the system; \(\
abla\) is the Laplace operator, \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{R_{m}^{2} \partial\theta^{2}}\); \(k\) is a constant coefficient, \(k=\frac{\delta^{2}}{12R_{m}^{2}}\). Assuming the axial displacement and rotation angle at \(x=0\) and \(l\) are both zero, the boundary conditions are:  
\[\left\{ \begin{array}{l}\left.\begin{array}{l}\left.\Delta u(x,\theta) \right|_{x=0}=\left.\Delta u(x,\theta)\right|_{x=l}=0\\ \left.\frac{\partial\Delta v(x,\theta)}{\partial x}\right|_{x=0}=\left.\frac{ \partial\Delta v(x,\theta)}{\partial x}\right|_{x=l}=0\end{array}\right.,\end{array} \right. \tag{2}\]  
Figure 3: Dynamic model of the UUV body structure. Figure 2: Piezoelectric ceramics and shunt circuit. (**a**) Annular piezoelectric ceramics; (**b**) shunt circuit. Thus, the dynamic displacement of the system can be written as:  
\[\left\{ \begin{array}{l}\Delta u(x,\theta)=A\sin(\lambda x)\sin(n\theta)\\ \Delta v(x,\theta)=B\cos(\lambda x)\cos(n\theta)\\ \Delta w(x,\theta)=C\cos(\lambda x)\sin(n\theta)\\ \end{array} \right., \tag{3}\]  
where, \(\lambda=m\pi/l\), \(m\) and \(n\) are the axial and circumferential wave number, respectively. Substituting Equation (3) into Equation (1), the following can be obtained:  
\[\left\{ \begin{array}{l}\left(\Omega^{2}-G_{11}\right)A+G_{12}B+G_{13}C=0\\ G_{21}A+\left(\Omega^{2}-G_{22}\right)B+G_{23}C=0\\ G_{31}A+G_{32}B+\left(\Omega^{2}-G_{33}\right)C=0\\ \end{array} \right., \tag{4}\]  
where, \(G_{11}=\left(\lambda R_{m}\right)^{2}+\frac{1-\mu}{2}n^{2}\), \(G_{12}=G_{21}=\frac{1+\mu}{2}(\lambda R_{m})n\), \(G_{13}=G_{31}=-\mu\lambda R_{m}\), \(G_{22}=(1+k)\left[\frac{1-\mu}{2}(\lambda R_{m})^{2}+n^{2}\right]\), \(G_{23}=n+kn\left[\left(\lambda R_{m}\right)^{2}+n^{2}\right]\), \(G_{33}=1+k\left[\left(\lambda R_{m}\right)^{2}+n^{2}\right]^{2}\), \(\Omega^{2}=\frac{\theta^{6}}{K}R_{m}^{2}\omega^{2}\).