## Vessel theory: Stiffness, added mass and damping |

The stiffness load, and constant frequency added mass and damping loads, are calculated using the formulae given below.

All these loads are applied at the stiffness, added mass and damping reference origin unless otherwise stated.

The hydrostatic stiffness matrix specifies how the net weight and buoyancy load (force and moment) on the vessel varies with heave, roll and pitch relative to the vessel in its datum configuration. Only the heave, roll and pitch components of the matrix are specified; the components for the surge, sway and yaw directions are all zero.

The hydrostatic stiffness matrix, $\mat{K}$, encodes how the weight and buoyancy loads on the vessel change due to small changes in the vessel's position and orientation. Two effects contribute to $\mat{K}$: the change in load due to the change in the submerged volume of the vessel as it heaves, rolls and pitches (often referred to as waterplane area effects), and the change in moment caused by movement of the vessel's centre of mass and centre of buoyancy as it rolls and pitches (moment arm effects). The hydrostatic stiffness matrices output by most diffraction packages (e.g. WAMIT, AQWA) account for both of these contributions. As stated, the hydrostatic stiffness matrix is only applicable for small changes in the vessel's position and orientation; it is a linearisation of what is, in general, a nonlinear problem. Whilst OrcaFlex cannot account for the waterplane area effects in a nonlinear fashion (since it does not have detailed information about the shape of the vessel hull), it *is* able to capture the moment arm effects during a simulation by simply applying the weight and buoyancy forces at the instantaneous positions of the centres of mass and buoyancy respectively (under the assumption that the volume of water displaced by the vessel remains constant). To do this, we must subtract the linearised approximation to the moment arm effects from the user's stiffness matrix. This is only possible if vessel's centre of buoyancy has been specified explicitly: if it has instead been input as (~,~,~), meaning that it is at an unspecified position on the same vertical line as the centre of mass, then we do not know the length of the moment arm needed to adjust K. OrcaFlex therefore has two modes of calculation:

The net weight and buoyancy load is calculated using the matrix equation \begin{equation} \vec{l} = \vec{l}_0 - \mat{K}\vec{p} \end{equation} where

$\vec{l} = [\fz,\mx,\my]^T$ is a column 3-vector containing the heave, roll and pitch components of the net weight + buoyancy load that acts on the vessel at the reference origin. The moments $M_x$ and $M_y$ are about the vessel diffraction frame x- and y-directions, while the force $F_z$ acts along the diffraction frame z-axis. The superscript ${}^T$ denotes transpose, simply to turn the row-vector written into a column vector.

$\vec{l}_0$ is the value of $\vec{l}$ when the vessel is in its datum position. This will be zero if the datum position is a free-floating equilibrium position of the vessel when no other objects are connected. Otherwise, when the mass of water displaced by the vessel in its datum position is not equal to structural mass of the vessel, $\vec{l}_0$ will be a pure force acting vertically.

$\mat{K}$ is the user-specified hydrostatic stiffness matrix.

$\vec{p}$ is a column 3-vector containing the heave position and roll and pitch angles (in radians) at the reference origin, relative to the user-specified datum Z, heel and trim of that reference origin.

So $p_1 = \text{Z coordinate of reference origin} - (\text{mean surface Z} + \text{datum Z relative to surface})$. And $p_2$ and $p_3$ are the roll and pitch (in radians) needed to rotate the vessel from its datum heel and trim orientation to its current orientation.

The angular offsets in $\vec{p}$ are in radians, so the units of the stiffness matrix are \begin{equation} \nonumber \left[ \begin{matrix} \dfrac{F}{L} & \dfrac{F}{\rad} & \dfrac{F}{\rad} \\ \dfrac{FL}{L} & \dfrac{FL}{\rad} & \dfrac{FL}{\rad} \\ \dfrac{FL}{L} & \dfrac{FL}{\rad} & \dfrac{FL}{\rad} \end{matrix} \right] \end{equation} where $F$ and $L$ denote the force and length units respectively.

Alternatively, if the centre of buoyancy is given, then the buoyancy load arising from waterplane area effects is calculated from \begin{equation} \vec{l} = -\mat{K'}\vec{p} \end{equation} where $\mat{K'}$ is the user's stiffness matrix, $\mat{K}$, after the removal of the appropriate moment arm contributions. The moment arm contributions are removed by subtracting a term of the form $(m_wz_b - mz_g)g$ from the roll-roll and pitch-pitch components of $\mat{K}$, where $m$ is the structural mass of the vessel, $m_w$ the mass of displaced water and $z_b$ and $z_g$ are the vertical distances of the centre of buoyancy and centre of mass above the reference origin when the vessel is in its datum position.

In addition, OrcaFlex applies a weight force vertically at the vessel's centre of mass \begin{equation} f_g = -mg \end{equation} and a buoyancy force vertically at the vessel's centre of buoyancy \begin{equation} f_b = m_wg \end{equation} These two forces are applied by OrcaFlex at every time-step of the simulation, which means that the nonlinear effects of the shift in the moment arm between the centres of mass and buoyancy will be captured precisely. Conversely the change in the buoyancy load associated with the waterplane area effect can only be captured linearly because OrcaFlex has no knowledge of the vessel's geometry above and below the waterline.

Allowing the centre of buoyancy to be specified in this way enables OrcaFlex to model systems where the waterplane area effect is relatively small, e.g. spars. In such a system, allowing for the nonlinear shift in the centre of buoyancy moment arm will result in an accurate buoyancy load even at large angles. However, for some systems where the waterplane area effect is significant, the inconsistency between the linear treatment of the waterplane area effect and the nonlinear treatment of the moment arm may become undesirable at large angles.

For a vessel using filtering calculation mode, a choice is offered for the calculation of the angles provided to hydrostatic stiffness in $\vec{p}$. The option named **orientation** is decomposition of an orientation matrix. The orientation matrix that describes the rotational offset between the vessel diffraction frame and the vessel primary frame is converted into Euler angles, and those roll and pitch angles are used in $\vec{p}$.

The option named **subtraction** is to subtract the vessel type datum heel and trim angles from the instantaneous roll and pitch angles of the primary frame. The rotational offset between the vessel primary frame and the global axes is converted into Euler angles, and then the datum angles are subtracted from these.

These two options will give the same results for a vessel type with zero datum trim angle. For cases with significant non-zero datum trim, we recommend the orientation calculation method, because we believe it to be more rigorous.

The theory here gives details of the added mass and damping loads when using the **constant** added mass and damping method. The calculation for frequency-dependent added mass and damping loads is more complex, and is described in impulse response and convolution, but the units given below apply in both cases.

When using the **constant** added mass and damping method, the damping load is equal to $-\mat{D}\vec{v}$, where $\mat{D}$ is the specified constant damping matrix and $\vec{v}$ is the (6 degree of freedom) vector of vessel velocity and angular velocity relative to the earth at the specified reference origin.

The damping load is calculated from the matrix equation \begin{equation} \vec{l} = -\mat{D}\vec{v} \end{equation} where

$\vec{l}=[\fx,\fy,\fz,\mx,\my,\mz]^T$ is the column 6-vector containing the components, in the vessel diffraction frame directions, of the damping force $(\fx,\fy,\fz)$ and moment $(\mx,\my,\mz)$ that act on the vessel at the reference origin.

$\mat{D}$ is the user-specified damping matrix.

$\vec{v} = [v_\mathrm{x},v_\mathrm{y},v_\mathrm{z},\omega_\mathrm{x},\omega_\mathrm{y},\omega_\mathrm{z}]^T$ is the column 6-vector containing the components, in the vessel diffraction frame directions, of the velocity $(v_\mathrm{x},v_\mathrm{y},v_\mathrm{z})$ at the specified reference origin, and the angular velocity $(\omega_\mathrm{x},\omega_\mathrm{y},\omega_\mathrm{z})$ of the vessel, relative to earth.

Note: | All components referred to here are in the directions specified by the conventions data. |

The angular velocities here are in $\text{radians}/T$, so the units of the $3{\times}3$ blocks of the damping matrix are \begin{equation} \nonumber \left[ \begin{matrix} \dfrac{F}{L/T} & \dfrac{F}{\rad/T} \\ \dfrac{FL}{L/T} & \dfrac{FL}{\rad/T} \end{matrix} \right] \end{equation} where $F$, $L$ and $T$ denote the units of force, length and time, respectively.

When using the **constant** added mass and damping method, the added mass load calculation is analogous to the damping load described above.

The added mass and damping load (force and moment) at the reference origin is therefore equal to $-\mat{M_\mathrm{a}}\vec{a}$, where $\mat{M_\mathrm{a}}$ is the given added mass matrix and $\vec{a}$ is the column 6-vector containing the components, in the vessel diffraction frame directions, of the vessel translational and angular acceleration relative to earth at the reference origin.

The angular accelerations are expressed in $\text{radians}/T^2$, so the units of the $3{\times}3$ blocks of the added mass matrix are \begin{equation} \nonumber \left[ \begin{matrix} M & ML \\ ML & ML^2 \end{matrix} \right] \end{equation} where $M$, $L$ and $T$ denote the units of mass, length and time, respectively.

Note: | If the vessel's primary motion is set to calculated (3 DOF) then reduced added mass and damping matrices are used, having only surge, sway and yaw components, and the damping and added mass loads are not calculated in the heave, roll and pitch directions. The stiffness matrix, having only heave, roll and pitch components, is not used at all in this case. |