# Dynamic Analysis of Underwater Torpedo during Straight-Line Navigation
## 2 Derivation of Munk Moment  
A torpedo sailing in an unsteady current will undergo an unstable moment called the Munk moment. In order to calculate the Munk moment, it is necessary to make some assumptions. The torpedo is regarded as a 6 DOF body, and the mass and mass distribution remain unchanged during navigation. The shape of the torpedo is a revolving body, symmetrical in the middle-longitudinal section and middle-transverse section. The product of inertia is not taken into account, which means \(J_{xy}=J_{yx}=J_{zx}=J_{zx}=J_{yz}=J_{zy}=0\). During torpedo navigation, the density and pressure of the fluid remain unchanged. The position coordinates and linear velocity are defined as follows:  
\[r=[x_{0},y_{0},z_{0}]^{T},\;v=\left[v_{x},v_{y},v_{z}\right]^{T},\;v_{0}=\left[ v_{x0},v_{y0},v_{z0}\right]^{T} \tag{1}\]  
The attitude angle and angular velocity are defined as follows:  
\[\Omega=\left[\theta,\psi,\phi\right]^{T},\;\omega=\left[\omega_{x},\omega_{y},\omega_{z}\right]^{T} \tag{2}\]  
The rudder angle and trajectory angle are defined as follows:  
\[\delta=\left[\delta_{e},\delta_{r},\delta_{d}\right]^{T},\;\angle=\left[ \Theta,\Psi,\Phi_{c}\right]^{T} \tag{3}\]  
The motion equations of the torpedo include kinetic equations and kinematic equations. According to the theorem of momentum, the kinetic equation of the torpedo can be obtained, as depicted in Equations (4)-(9) (the torpedo conducts small maneuvering motion, neglects the second-order terms of the torpedo's motion parameters, and regards the position of the center of mass as a small first-order quantity). \[(m+\lambda_{11})\dot{v}_{x}=T-C_{x5}\frac{1}{2}\rho v^{2}S-\Delta G\sin\theta \tag{4}\]  
\[(m+\lambda_{22})\dot{v}_{y}+(mx_{c}+\lambda_{26})\dot{\omega}_{z}+mv_{x} \omega_{z}=\frac{1}{2}\rho v^{2}S\left(C_{y}^{\delta}\alpha+C_{y}^{\delta_{z} }\delta_{e}+C_{y}^{\overline{\omega}_{z}}\overline{\omega}_{z}\right)-\Delta G \cos\theta\cos\phi \tag{5}\]  
\[(m+\lambda_{33})\dot{v}_{z}-(mx_{c}+\lambda_{35})\dot{\omega}_{y}-mv_{x} \omega_{y}=\frac{1}{2}\rho v^{2}S\left(C_{z}^{\delta}\alpha+C_{z}^{\delta_{z} }\delta_{r}+C_{z}^{\overline{\omega}_{y}}\overline{\omega}_{y}\right)+\Delta G \cos\theta\sin\phi \tag{6}\]  
\[ \begin{array}{c}(J_{xx}+\lambda_{44})\dot{\omega}_{x}-mv_{x}\big{(}y_{c} \omega_{y}+z_{c}\omega_{z}\big{)}=\frac{1}{2}\rho v^{2}SL\left(m_{x}^{\beta} \beta+m_{x}^{\delta_{x}}\delta_{r}+m_{x}^{\delta_{x}}\delta_{d}+m_{x}^{ \overline{\omega}_{x}}\overline{\omega}_{x}+\right.\\ \left.\begin{array}{c}m_{x}^{\overline{\omega}_{y}}\overline{\omega}_{y} \end{array}\right)+G\cos\theta(y_{c}\sin\phi+z_{c}\cos\phi)+\Delta M_{xp} \end{array}  \tag{7}\]  
\[\begin{array}{c}\left(J_{yy}+\lambda_{55}\right)\dot{\omega}_{y}-(mx_{c}- \lambda_{35})\dot{v}_{z}+mx_{c}v_{x}\omega_{y}=\frac{1}{2}\rho v^{2}SL\left(m_ {y}^{\beta}\beta+m_{y}^{\delta_{y}}\delta_{r}+m_{y}^{\overline{\omega}_{x}} \overline{\omega}_{x}+\right.\\ \left. \begin{array}{c}m_{y}^{\overline{\omega}_{y}}\overline{\omega}_{y} \end{array} \right)-G(x_{c}\cos\theta\sin\phi+z_{c}\sin\theta)\\ \left. \begin{array}{c}(J_{zz}+\lambda_{66})\dot{\omega}_{z}+(mx_{c}+\lambda_{ 26})\dot{v}_{y}+mx_{c}v_{x}\omega_{z}=\frac{1}{2}\rho v^{2}SL\left(m_{x}^{ \alpha}\alpha+m_{z}^{\delta_{x}}\delta_{e}+m_{z}^{\overline{\omega}_{z}} \overline{\omega}_{z}\right)+\\ G(y_{c}\sin\theta-x_{c}\cos\theta\cos\phi)\end{array}  \tag{9}\]  
The kinematic equations can be obtained based on the transition matrix, as depicted in Equations (10)-(21).