4.1 Restoring forces for underwater vehicles

According to the SNAME(1950) it is standard to express the submerged weight of the body and buoyance force as:

$$W=mg$$ $$B=\rho g\nabla$$

$\rho$ = water density $\nabla$ = volume of fluid displaced by the vehicle m = mass of the vessel including water in free flooding space g = acceleration of gravity

Gravitational force on vector form:

$$f_g^n=\left[\begin{array}{c}0\\ 0\\ W\end{array}\right]$$

Buoyancy force on vector form:

$$f_b^n=-\left[\begin{array}{c}0\\ 0\\ B\end{array}\right]$$

These forces can be transformed from NED to BODY by the following transformation:

$$f_g^b=R^T({\Theta }_{nb})f_g^n$$ $$f_b^b=R^T({\Theta }_{nb})f_b^n$$

Generalized restoring force in BODY with coordinate origin CO

$$g(\eta )=\left[\begin{array}{c}{f}_g^b+f_b^b\\ r_{bg}^b\times f_g^b+r_{bb}^b\times f_b^b\end{array}\right]$$

where

$$r_{bg}^b:={\left[\begin{array}{c}{x}_g,y_g,z_g\end{array}\right]}^T$$ $$r_{bb}^b:={\left[\begin{array}{c}{x}_b,y_b,z_b\end{array}\right]}^T$$

center of buoyance with respect to the CO center of gravity with respect to the CO

Multiplying out the generalized restoring force we get:

$$g(\eta )=\left[\begin{array}{c}(W-B)\sin \theta \\ -(W-B)\cos \theta \sin \varphi \\ -(W-B)\cos \theta \cos \varphi \\ -(y_{g}W-y_{b}B)\cos \theta \cos \varphi +(z_{g}W-z_{b}B)\cos \theta \sin \varphi \\ (z_{g}W-z_{b}B\sin \theta )+(x_{g}W-x_{b}B)\cos \theta \cos \varphi \\ -(x_{g}W-x_{b}B)\cos \theta \sin \varphi -(y_{g}W-y_{b}B)\sin \theta \end{array}\right]$$

It is possible to simplify this matrix by assuming the underwater vehicle is neutrally buoyant W=B:

$$g(\eta )=\left[\begin{array}{c}0\\ 0\\ 0\\ -B{G}_{y}W\cos \theta \cos \varphi +B{G}_{z}W\cos \theta \sin \varphi \\ B{G}_{z}W\sin \theta +B{G}_{x}W\cos \theta \cos \varphi \\ -B{G}_{x}W\cos \theta \sin \varphi -B{G}_{y}W\sin \theta \end{array}\right]$$

An even simpler representation is obtained for vehicles where CG and CB are located vertically on the z-axis, that is $x_b=x_g$ and $y_g=y_b$

$$g(\eta )={\left[\begin{array}{cccccc}0& 0& 0& B{G}_{z}W\cos \theta \sin \varphi & B{G}_{z}W\sin \theta & 0\end{array}\right]}^T$$