Structural Mass

1. Centre of Gravity

The location of the centre of gravity is specified using three coordinates:

  • X: X-coordinate of centre of gravity, \(\mathrm {[m]}\)

  • Y: Y-coordinate of centre of gravity, \(\mathrm {[m]}\)

  • Z: Z-coordinate of centre of gravity, \(\mathrm {[m]}\)

The coordinates and all other body data should be given in the body-fixed coordinate system.

2. Mass coefficients

The body mass, moment of inertia and negative products of inertia:

  • Mass: Mass, \(\mathrm {[kg]}\)

  • Ixx: Moment of inertia about X-axis \(\mathrm {[kg\cdot m^2]}\), defined as

    \(\mathrm {\iiint_{m}(y^{2}+z^{2})dm}\)

  • Iyx: Negative product of inertia in Y-X \(\mathrm {[kg\cdot m^2]}\), defined as

    \(\mathrm {-\iiint_{m}yxdm}\)

  • Iyy: Moment of inertia about Y-axis \(\mathrm {[kg\cdot m^2]}\), defined as

    \(\mathrm {\iiint_{m}(x^{2}+z^{2})dm}\)

  • Izx: Negative product of inertia in Z-X \(\mathrm {[kg\cdot m^2]}\), defined as

    \(\mathrm {-\iiint_{m}zxdm}\)

  • Izy: Negative product of inertia in Z-Y \(\mathrm {[kg\cdot m^2]}\), defined as

    \(\mathrm {-\iiint_{m}zydm}\)

  • Izz: Moment of inertia about Z-axis \(\mathrm {[kg\cdot m^2]}\), defined as

    \(\mathrm {\iiint_{m}(x^{2}+y^{2})dm}\)

Note that the moments of inertia and negative products of inertia are to be specified in the local body-fixed coordinate system and about the origin of the body-fixed system. This means that \(\mathrm {(x,y,z)}\) in the volume integrals above should be understood as the position of the infinitesimal mass particle in the body-fixed coordinate system.

In case that the moments of inertia and negative products of inertia are known at centre of gravity they transform to the body-fixed origin by the following expressions:

  • \(\mathrm {IXX}=I^{cog}_{xx}+\mathrm {M}\cdot (\mathrm {YCOG^2+ZCOG^2})\)

  • \(\mathrm {IYY}=I^{cog}_{yy}+\mathrm {M}\cdot (\mathrm {XCOG^2+ZCOG^2})\)

  • \(\mathrm {IZZ}=I^{cog}_{zz}+\mathrm {M}\cdot (\mathrm {XCOG^2+YCOG^2})\)

  • \(\mathrm {IYX}=I^{cog}_{yx}-\mathrm {M}\cdot \mathrm {XCOG\cdot YCOG}\)

  • \(\mathrm {IZX}=I^{cog}_{zx}-\mathrm {M}\cdot \mathrm {XCOG\cdot ZCOG}\)

  • \(\mathrm {IZY}=I^{cog}_{zy}-\mathrm {M}\cdot \mathrm {YCOG\cdot ZCOG}\)

The applied symmetrical structural mass matrix can be written as:

\[\begin{bmatrix}\mathrm {M}&0&0&0&\mathrm {M\cdot ZCOG}&\mathrm {-M\cdot YCOG}\\0&\mathrm {M}&0&\mathrm {-M\cdot ZCOG}&0&\mathrm {M\cdot XCOG}\\0&0&\mathrm {M}&\mathrm {M\cdot YCOG}&\mathrm {-M\cdot XCOG}&0\\0&\mathrm {-M\cdot ZCOG}&\mathrm {M\cdot YCOG}&\mathrm {IXX}&\mathrm {IYX}&\mathrm {IZX}\\\mathrm {M\cdot ZCOG}&0&\mathrm {-M\cdot XCOG}&\mathrm {IYX}&\mathrm {IYY}&\mathrm {IZY}\\\mathrm {-M\cdot YCOG}&\mathrm {M\cdot XCOG}&0&\mathrm {IZX}&\mathrm {IZY}&\mathrm {IZZ}\\\end{bmatrix}\]

A mass matrix has to be positive definite to ensure that kinetic energy is always positive. SIMO will give an error if the mass matrix is not positive definite besed on the following requirements:

  • \(\mathrm {M}\geq 0\)

  • \(\mathrm {I_{xx}^{cog}\geq 0}\)

  • \(\mathrm {I_{yy}^{cog}\geq 0}\)

  • \(\mathrm {I_{zz}^{cog}\geq 0}\)

  • \(\mathrm {I_{xx}^{(cog)}I_{yy}^{(cog)}-(I_{yx}^{(cog)})^{2}\geq 0}\)

  • \(\mathrm {I_{xx}^{(cog)}I_{yy}^{(cog)}I_{zz}^{(cog)}-I_{xx}^{(cog)}(I_{zy}^{(cog)})^{2}-I_{yy}^{(cog)}(I_{zx}^{(cog)})^{2}-I_{zz}^{(cog)}(I_{yx}^{(cog)})^{2}+2I_{yx}^{(cog)}I_{zx}^{(cog)}I_{zy}^{(cog)}\geq 0}\)

Additional mass may be given by specifying

  • Time dependent mass

  • Distributed element force

2.1. Gravity force

For floating vessels the gravity and buoyancy forces are normally modelled by use of the linear hydrostatic stiffness matrix, where the reference point for stiffness is taken as the position where the buoyancy and weight are in equilibrium.

Optionally, the gravity force due to the structural mass and acting at the centre of gravity may be computed. When gravity force is applied the model must include an explicit buoyancy force to balance the gravity force, and the restoring matrix must be modified so that the effect of gravity is not included.

When using Nonlinear Hydrostatics, it is often relevant to include the gravity force, since the hydrostatic force model will only return the buoyancy force.

Activating gravity forces will not affect gravity forces due to the mass term of distributed element forces or time dependent mass.

2.2. Gravity force included

To activate the gravity force computation, check the Gravity force included checkbox in the Body editor.