# Dynamic Analysis of Underwater Torpedo during Straight-Line Navigation
## 2 Derivation of Munk Moment
\[\dot{\theta}=\omega_{y}\sin\phi+\omega_{z}\cos\phi \tag{10}\]\[\dot{\psi}=\omega_{y}\sec\theta\cos\phi-\omega_{z}\sec\theta\sin\phi \tag{11}\]  
\[\dot{\phi}=\omega_{x}-\omega_{y}\tan\theta\cos\phi+\omega_{z}\tan\theta\sin\phi \tag{12}\]  
\[\dot{x}_{0}=v_{x}\cos\theta\cos\psi+v_{y}(\sin\psi\sin\phi-\sin\theta\cos\psi \cos\phi)+v_{z}(\sin\psi\cos\phi-\sin\theta\cos\psi\sin\phi) \tag{13}\]  
\[\dot{y}_{0}=v_{x}\sin\theta+v_{y}\cos\theta\cos\phi-v_{z}\cos\theta\sin\phi \tag{14}\]  
\[\dot{z}_{0}=-v_{x}\cos\theta\sin\psi+v_{y}(\cos\phi\sin\phi+\sin\theta\sin\psi \cos\phi)+v_{z}(\cos\phi\cos\phi-\sin\theta\sin\psi\sin\phi) \tag{15}\]  
\[v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2} \tag{16}\]  
\[\alpha=-\arctan\bigl{(}v_{y}/v_{x}\bigr{)} \tag{17}\]  
\[\beta=\arctan\Bigl{(}v_{z}/\sqrt{v_{x}^{2}+v_{y}^{2}}\Bigr{)} \tag{18}\]  
\[\sin\Theta=\sin\theta\cos\alpha\cos\beta-\cos\theta\cos\phi\sin\alpha\cos\beta -\cos\theta\sin\phi\sin\beta \tag{19}\]  
\[\sin\Psi\cos\Theta= \begin{array}{l}\sin\psi\cos\theta\cos\alpha\cos\beta+ \cos\psi\sin\phi\sin\alpha\cos\beta+\\ \sin\psi\sin\theta\cos\phi\sin\alpha\cos\beta-\cos\phi\cos\phi\sin\beta+\sin \psi\sin\theta\sin\phi\sin\beta\end{array}  \tag{20}\]  
\[\sin\Phi_{c}\cos\Theta=\sin\theta\cos\alpha\sin\beta-\cos\theta\cos\phi\sin \alpha\sin\beta+\cos\theta\sin\phi\cos\beta \tag{21}\]  
The meaning of the above symbols is shown in Abbreviations. The full kinematic parameters for solving the spatial motion of the torpedo can be obtained from the above 21 equations. The Munk moment is added to the torpedo dynamics equations in the form of dimensionless coefficients. The force of the ideal fluid on the torpedo can be divided into three parts: the force of the fluid, the moment generated by the steady motion of the torpedo with constant rotation and steady motion, and the moment generated by the unsteady motion of the torpedo. The Munk moment is such a moment generated by the unstable motion of the torpedo. \[ \begin{cases}M_{\text{a}ix}=(\lambda_{22}-\lambda_{33})v_{y}v_{z}-\lambda_{3 2}v_{y}^{2}+\lambda_{23}v_{z}^{2}+\bigl{(}\lambda_{21}v_{z}-\lambda_{31}v_{y }\bigr{)}v_{x}\\ M_{\text{a}iy}=(\lambda_{33}-\lambda_{11})v_{z}v_{x}-\lambda_{13}v_{z}^{2}+ \lambda_{31}v_{x}^{2}+(\lambda_{32}v_{x}-\lambda_{12}v_{z})v_{y}\\ M_{\text{a}iz}=(\lambda_{11}-\lambda_{22})v_{x}v_{y}-\lambda_{21}v_{x}^{2}+ \lambda_{12}v_{y}^{2}+\bigl{(}\lambda_{13}v_{y}-\lambda_{23}v_{x}\bigr{)}v_{z }\end{cases}  \tag{22}\]  
\[ \begin{cases}M_{\text{a}ix}=(\lambda_{22}-\lambda_{33})v_{y}v_{z}\\ M_{\text{a}iy}=(\lambda_{33}-\lambda_{11})v_{z}v_{x}\\ M_{\text{a}iz}=(\lambda_{11}-\lambda_{22})v_{x}v_{y}\end{cases}  \tag{23}\]  
As the angle of attack \(\alpha=-\arctan\frac{v_{x}}{v_{y}}\), the sideslip angle \(\beta=-\arctan\frac{v_{z}}{\sqrt{v_{x}^{2}+v_{y}^{2}}}\), assuming that the angle of attack and the sideslip angle are both small, the Munk moment can be rewritten as the following:  
\[[M]=\frac{1}{2}v^{2} \begin{bmatrix}\sin 2\beta&0&0\\ 0&\sin 2\beta&0\\ 0&0&\sin 2\alpha\end{bmatrix} \cdot \begin{bmatrix}\lambda_{22}-\lambda_{33}&0&0\\ 0&\lambda_{33}-\lambda_{11}&0\\ 0&0&\lambda_{11}-\lambda_{22}\end{bmatrix}  \tag{24}\]