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Results: Hydrostatics |
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Results: Hydrostatics |
These results are given for each body on the hydrostatics sheet of the results tables. The same results are also given in the mesh details spreadsheet, which is available before running a calculation.
The data specified for each body determines the available results. In particular, many quantities are not available for sectional bodies because they are permitted to be open-ended. E.g. there is no meaningful definition of displaced volume or water plane area for sectional bodies.
Volumetric quantities are available for displacement bodies but not for sectional bodies.
The volume of water displaced by the body, inferred from the mesh \begin{equation} \begin{aligned} |\VInt| & = -\int_{\SB}n_i x_i \ud S & \textrm{for }i & =1,2 \textrm{ or }3 \end{aligned} \end{equation}
The centre of buoyancy (in body coordinates $\Bxyz$), inferred from the mesh \begin{equation} \begin{aligned} \left(\vec{x}_b\right)_i & = -\frac{1}{2|\VInt|}\int_{\SB}n_i x_i^2 \ud S & \textrm{for }i & =1,2,3 \end{aligned} \end{equation}
Water plane quantities are available for displacement bodies if they pierce the sea surface and have a waterline. They are not available for sectional bodies.
The total area, $|\SI|$, enclosed by all the body's waterlines, as detected in the mesh.
The centre of floatation (in body coordinates $\Bxyz$), inferred from the body waterlines detected in the mesh \begin{equation} \begin{aligned} \left(\vec{x}_f\right)_i & = \frac{1}{|\SI|}\int_{\SI}x_i \ud S & \textrm{for }i & =1,2 \end{aligned} \end{equation}
Water plane moments (in body coordinates $\Bxyz$), inferred from the body waterlines detected in the mesh \begin{equation} \begin{aligned} L_{xx} & = \int_{\SI}x^2 \ud S, & L_{xy} & = \int_{\SI}xy \ud S, & L_{yy} & = \int_{\SI}y^2 \ud S \end{aligned} \end{equation}
The mass, $M$, of the body, specified or implied by the inertia data.
The centre of mass, $\vec{x}_m$, of the body (in body coordinates $\Bxyz$), specified or implied by the inertia data.
The full inertia matrix $M_{ij}$ (in body coordinates $\Bxyz$), implied by the inertia data. The units of the $3{\times}3$ blocks of the inertia matrix are \begin{equation} \nonumber \left[ \begin{matrix} M & ML \\ ML & ML^2 \end{matrix} \right] \end{equation} where $M$ and $L$ denote the units of mass and length, respectively.
The mean hydrostatic load (in body coordinates $\Bxyz$). This is the sum of the weight and the load due to hydrostatic water pressure in the absence of waves.
| Tip: | For a body in free-floating equilibrium, these results should be zero. For a body that is not in simple hydrostatic (free-floating) equilibrium, these results will be non-zero and the additional effects that contribute to the body's position must be captured in the constraints data. |
The hydrostatic stiffness matrix $K_{ij}$ (in body coordinates $\Bxyz$). Each value $K_{ij}$ gives the constant of proportionality between the $i$ component of the load on the body and the $j$ component of its motion. Hence the units of the $3{\times}3$ blocks of the stiffness matrix are \begin{equation} \nonumber \left[ \begin{matrix} \dfrac{F}{L} & \dfrac{F}{\rad} \\ \dfrac{FL}{L} & \dfrac{FL}{\rad} \end{matrix} \right] \end{equation} where $F$ and $L$ denote the force and length units respectively.
For displacement bodies the hydrostatic stiffness matrix is given by \begin{equation} \rho g \left[ \begin{matrix} % row 1 0 & 0 & 0 & % cols 1-3 0 & 0 & 0 \\ % cols 4-6 % row 2 0 & 0 & 0 & % cols 1-3 0 & 0 & 0 \\ % cols 4-6 % row 3 0 & 0 & \int_{\SB} n_3 \ud S & % cols 1-3 \int_{\SB} n_3 y \ud S & -\int_{\SB} n_3 x \ud S & 0 \\ % cols 4-6 % row 4 0 & 0 & \int_{\SB} n_3 y \ud S & % cols 1-3 \int_{\SB} n_3 y^2 \ud S + q_3 & -\int_{\SB} n_3 xy \ud S & -q_1 \\ % cols 4-6 % row 5 0 & 0 & -\int_{\SB} n_3 x \ud S & % cols 1-3 -\int_{\SB} n_3 xy \ud S & \int_{\SB} n_3 x^2 \ud S + q_3 & -q_2 \\ % cols 4-6 % row 6 0 & 0 & 0 & % cols 1-3 0 & 0 & 0 % cols 4-6 \end{matrix} \right] \end{equation} where we define the following quantity to condense the notation \begin{equation} \vec{q} = |\VInt| \vec{x}_b - M \rho^{-1} \vec{x}_m \end{equation}
For sectional bodies the hydrostatic stiffness matrix is given by \begin{equation} \left[ \begin{matrix} % row 1 0 & 0 & q^a_1 & % cols 1-3 q^b_{12} & -q^b_{11} + q^b_{33} & -q^b_{23} \\ % cols 4-6 % row 2 0 & 0 & q^a_2 & % cols 1-3 q^b_{22} - q^b_{33} & -q^b_{21} & q^b_{13} \\ % cols 4-6 % row 3 0 & 0 & q^a_3 & % cols 1-3 q^b_{32} + q^b_{23} & -q^b_{31} - q^b_{13} & 0 \\ % cols 4-6 % row 4 0 & 0 & q^b_{32} - q^b_{23} & % cols 1-3 q^c_{12} - M g z_m & -q^c_{11} + q^c_{33} & -q^c_{23} + M g x_m \\ % cols 4-6 % row 5 0 & 0 & q^b_{13} - q^b_{31} & % cols 1-3 q^c_{22} - q^c_{33} & -q^c_{21} - M g z_m & q^c_{13} + M g y_m \\ % cols 4-6 % row 6 0 & 0 & q^b_{21} - q^b_{12} & % cols 1-3 q^c_{32} + q^c_{23} & -q^c_{31} - q^c_{13} & 0 % cols 4-6 \end{matrix} \right] \end{equation} where we define the following quantities, for $1 \le i, j \le 3$, to condense the notation \begin{equation} \begin{aligned} q^a_i & = \rho g \int_{\SB} n_i \ud S \\ q^b_{ij} & = \rho g \int_{\SB} n_i x_j \ud S \\ q^c_{ij} & = \rho g \int_{\SB} (\vec{x} \times \vec{n})_i x_j \ud S \end{aligned} \end{equation}