1. Geometry & Rigid‑Body Inertia
Symbols:
Symbol |
Meaning |
Units (example) |
|---|
\(L\) |
Length |
\(m\) |
\(B\) |
Beam / width |
\(m\) |
\(T\) |
Height / depth |
\(m\) |
\(\rho\) |
Fluid density |
\(kg\,m^3\) (1000 for water) |
\(m\) |
Mass |
\(kg\) |
\(I_x,I_y,I_z\) |
Principal inertias about body axes |
\(kg \, m^2\) |
Mass
For a full solid of uniform density:
\[
m = \rho L B T.
\]
Moments of Inertia (through centroid)
For a rectangular prism (axes aligned with edges):
\[\begin{split}
I_x = \tfrac{1}{12} m (B^2 + T^2),\\
I_y = \tfrac{1}{12} m (L^2 + T^2),\\
I_z = \tfrac{1}{12} m (L^2 + B^2).\
\end{split}\]
Rigid‑body mass-inertia matrix (no products of inertia, CG at origin):
\[
\mathbf{M}-{\mathrm{RB}} = \mathrm{diag}(m,m,m, I_x, I_y, I_z).
\]
2. Added Mass & Damping
Added Mass
Diagonal approximation:
\[
\mathbf{M}_{\mathrm{A}} = \mathrm{diag}(X_{\dot u}, Y_{\dot v}, Z_{\dot w}, K_{\dot p}, M_{\dot q}, N_{\dot r}).
\]
Often parameterised by fractions of the corresponding rigid‑body terms (e.g. \(X_{\dot u} = \alpha_u m\), \(K_{\dot p}=\alpha_p I_x\), etc.).
Effective mass:
\[
\mathbf{M}_{\mathrm{eff}} = \mathbf{M}_{\mathrm{RB}} + \mathbf{M}_{\mathrm{A}}.
\]
Linear Damping
Coefficients collected in a diagonal matrix (coefficients only):
\[
\mathbf{D} = \mathrm{diag}(d_u, d_v, d_w, d_p, d_q, d_r).
\]
Applied force:
\[
\mathbf{D}\nu = [d_u u,\ d_v v,\ d_w w,\ d_p p,\ d_q q,\ d_r r]^\top
\]
(sign convention embedded in \(d_i\)).
3. Kinematics
State vectors:
\[
\eta = [x, y, z, \phi, \theta, \psi]^\top, \qquad
\nu = [u, v, w, p, q, r]^\top.
\]
Kinematic relation:
\[
\dot{\eta} = \mathbf{J}(\eta)\nu.
\]
Block structure:
\[\begin{split}
\mathbf{J}(\eta)=
\begin{bmatrix}
R_{\mathrm{lin}}(\phi,\theta,\psi) & \mathbf{0} \\
\mathbf{0} & T_{\mathrm{ang}}(\phi,\theta)
\end{bmatrix}
\end{split}\]
with standard Z-Y-X rotation composition \(R_{\mathrm{lin}} = R_z(\psi)R_y(\theta)R_x(\phi)\) and Euler‑rate mapping \(T_{\mathrm{ang}}\).
4. Dynamics
Full 6‑DOF equation (body frame):
\[
(\mathbf{M}_{\mathrm{RB}} + \mathbf{M}_{\mathrm{A}})\dot{\nu} + (\mathbf{C}_{\mathrm{RB}}(\nu) + \mathbf{C}_{\mathrm{A}}(\nu))\nu + \mathbf{D}\nu
= \tau + \tau_{\mathrm{ext}} + \mathbf{g}_{\mathrm{restoring}}(\eta).
\]
Coriolis / Centripetal Terms
Let \(\mathbf{v} = [u,v,w]^\top\), \(\omega = [p,q,r]^\top\), and \(S(\cdot)\) the skew operator.
Rigid‑body:
\[\begin{split}
\mathbf{C}_{\mathrm{RB}}(\nu) = \begin{bmatrix} \mathbf{0} & -m S(\omega) \\ -m S(\mathbf{v}) & - S(\mathbf{I}\omega) \end{bmatrix},\quad
\mathbf{I}\omega = [I_x p, I_y q, I_z r].
\end{split}\]
Added mass (partition \(\mathbf{M}_{\mathrm{A}}\) into linear / rotational blocks):
\[\begin{split}
\mathbf{C}_{\mathrm{A}}(\nu) = \begin{bmatrix} \mathbf{0} & - S(\mathbf{M}_{\mathrm{A,lin}}\mathbf{v}) \\ - S(\mathbf{M}_{\mathrm{A,lin}}\mathbf{v}) & - S(\mathbf{M}_{\mathrm{A,rot}}\omega) \end{bmatrix}.
\end{split}\]
Damping Force Vector
\[
\mathbf{D}\nu = [d_u u,\ d_v v,\ d_w w,\ d_p p,\ d_q q,\ d_r r]^\top.
\]
Hydrostatics (Heave)
Small displacement approximation:
\( F_z = - \rho g L B\, z. \)
Restoring (Roll / Pitch)
Small‑angle moments:
\[\begin{split}
\mathbf{g}_{\mathrm{restoring}}(\eta) = \begin{bmatrix}
0 \\ 0 \\ 0 \\ - m g\, \mathrm{GM}_{\phi} \, \phi \\ - m g \, \mathrm{GM}_{\theta} \, \theta \\ 0
\end{bmatrix}.
\end{split}\]
5. Time Integration
System:
\[
\dot{\eta} = \mathbf{J}(\eta)\nu,
\]
\[
\mathbf{M}_{\mathrm{eff}} \dot{\nu} &+ (\mathbf{C}_{\mathrm{RB}} + \mathbf{C}_{\mathrm{A}})\nu + \mathbf{D}\nu = \tau + \tau_{\mathrm{ext}} + \mathbf{g}_{\mathrm{restoring}}(\eta).
\]
Implemented with classical 4th‑order Runge-Kutta (RK4) over step \(\Delta t\):
Evaluate derivatives at current state.
Two midpoint evaluations (using half steps).
Final endpoint evaluation.
Weighted combination to advance \((\eta,\nu)\).