# Motion Response of the Submersible Underwater Towed System as Cable Length Changes
## 2 Computational Theory
### Governing Equation  
The total length of the \(S\) towed cable is discretized into \(N\) segments by the lumped mass method, and then the dynamic model is established. The process entails dividing the towed cable into \(N\) segments by discretizing it from end to end into \(N+1\) nodes. The end at \(s=0\) corresponds to the \(i=0\) node, while the end at \(s=S\) corresponds to the \(i=N\) node. The arc-length coordinates of these nodes satisfy the condition  
Figure 2: Diagram of submersible underwater towed system. \(0=s_{0}<s_{1}<\ldots<s_{i}<s_{i+1}\ldots<s_{N}=S\). The equations of motion for the nonlinearity at node \(i\) of the towed cable are obtained using [PERSON]'s second law. \[\mathbf{M}_{i}\mathbf{a}_{i}=\Delta\mathbf{T}_{i}+\mathbf{B}_{i}+\mathbf{G}_{i}+\mathbf{F}_{Di} \tag{2}\]  
where \(\mathbf{a}_{i}=\ddot{\mathbf{x}}_{i}\) represents acceleration; \(\Delta\mathbf{T}_{i},\mathbf{B}_{i},\mathbf{G}_{i},\mathbf{F}_{Di}\) represent the tension, buoyancy, gravity, and fluid resistance of the towed cable at node \(i\); \(\mathbf{M}_{i}\) represents the mass matrix; \(\mathbf{M}_{a,i}\) represents the inertial mass of the towed cable node itself and its added mass; \(\mathbf{M}_{a}\) represents the added mass matrix of a simple node. The relationship between \(\mathbf{M}_{i}\) and \(\mathbf{M}_{a,i}\) is as follows:  
\[\mathbf{M}_{i}=(\mu_{i-1/2}l_{i-1/2}+\mu_{i+1/2}l_{i+1/2})\mathbf{I}/2+\mathbf{M}_{a,i}\]  
\[\mathbf{M}_{a,i}=(\mathbf{M}_{a,i-1/2}+\mathbf{M}_{a,i+1/2})/2\]  
\[\mathbf{M}_{a}=\rho l\sigma \begin{pmatrix}1-\sin^{2}\theta\cos^{2}\theta&-\cos \theta\sin\theta\cos^{2}\phi&-\sin\theta\cos\phi\sin\phi\\ -\cos\theta\sin\theta\cos^{2}\phi&1-\cos^{2}\theta\cos^{2}\phi&-\cos\theta \cos\phi\sin\phi\\ -\sin\theta\cos\phi\sin\phi&-\cos\theta\cos\phi\sin\phi&\cos^{2}\phi\end{pmatrix} \]  
\(\mathbf{I}\) is a \(3\times 3\) unit matrix; \(\mu\), \(l\), and \(\rho\) represent the mass per unit length, the length between nodes, and the cross-sectional area of the towed cable, respectively; the subscript \(i+1/2\) represents the physical quantity located between node \(i\) and node \(i+1\). 1. Tension \(\mathbf{T}\) can be treated elastically in terms of the stress-strain relationship, and its constitutive relationship can be represented by [PERSON]'s law. \[\Delta\mathbf{T}_{i}=\mathbf{T}_{i+1/2}-\mathbf{T}_{i-1/2}\] \[\mathbf{T}_{i+1/2}=E\sigma\varepsilon_{i+1/2}\mathbf{\tau}_{i+1/2}\] \[\varepsilon_{i+1/2}=\sqrt{(x_{i+1}-x_{i})^{2}+(y_{i+1}-y_{i})^{2}+(z_{i+1}- z_{i})^{2}}/l_{i+1/2}-1\]  
where \(\mathbf{\tau}\) represents the unit tangent vector in the direction of cable length; \(E\) represents the elastic modulus; \(\varepsilon\) represents the stress. 2. Buoyancy \(\mathbf{B}\) and gravity \(\mathbf{G}\). The buoyancy and gravity at node \(i\) can be expressed as follows: \[\mathbf{B}_{i}+\mathbf{G}_{i}=-\rho g[(l_{i-1/2}\sigma_{i-1/2}+l_{i+1/2}\sigma_{i+1/2 })-(l_{i-1/2}\mu_{i-1/2}+l_{i+1/2}\mu_{i+1/2})]/2\] where \(\mathbf{g}\) represents the acceleration of gravity. 3. Fluid drag force \(\mathbf{F}_{Di}\). The drag force on the towed cable can be separated into oblique component drag and normal component drag, taking into account the strain impact on the cable. The expression is as follows: \[\mathbf{F}_{Di}=(\mathbf{F}_{Di+1/2}+\mathbf{F}_{Di-1/2})/2\] \[\mathbf{F}_{Di}\approx-\rho l\sqrt{1+\varepsilon}(C_{n}|v_{n,b}|v_{n,b}+\pi C _{t}|v_{t}|v_{t})\] where \(v=\dot{\mathbf{x}}\) represents the velocity vector of the node; \(d\) represents the diameter of the cable; and \(C_{n}\) and \(C_{t}\) represent the normal and tangential resistance coefficients, respectively. The motion of the towed body is described using a mass motion model, and its equations of motion control can be obtained by [PERSON]'s second law:  
\[\mathbf{M}_{c}\ddot{\mathbf{A}}_{s}=\mathbf{Q} \tag{3}\]  
The non-zero elements of the mass matrix are as follows:  
\[\mathbf{M}_{c}= \begin{pmatrix}m+m_{cx}&0&0\\ 0&m+m_{cy}&0\\ 0&0&m+m_{cz}\end{pmatrix}