Appendix F: Mass and inertia summary of RIFLEX elements

1. Overview

A summary of the mass, centre of mass, and inertia tensor, can be determined for a user specified number of lines, relative to a user specified reference frame. The reference frame can be specified in terms of its origin’s coordinates in the global frame and a z-rotation, or as the reference frame of a named SIMO body. The mass, centre of mass and inertia tensor can be calculated for elements in their initial (stress-free, before static analysis) or final (deformed, after static analysis) configurations.

2. Restrictions

The following restrictions currently apply:

  • Only beam and bar elements are allowed

  • An element is ignored if it has a nodal component attached to it

  • Properties of SIMO bodies attached to nodes are ignored

  • Only structural mass (including marine growth, excluding contents) is taken into account

  • If the output reference frame is specified as a named SIMO body, its orientation in the final static configuration is always used.

3. Procedure

An overview of the procedure for calculating mass and inertia properties is described for a single element, using the notation given in the figure. Colours are used to clearly indicate the frame in which a vector quantity is expressed.

m cog inertia

  1. Calculate the inertia tensor in a frame located in an element’s centre of mass, aligned with the local frame. \(\overline{\overline{I}}_e^c=\begin{bmatrix} I_{xx} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\), where \(I_{xx} = m r_{gyr.}^2\)

  2. Translate the inertia tensor \(\overline{\overline{I}}_e^c\) from the mass centre to \(\overline{\overline{I}}_e^e\) in the element’s local frame, where

    \(\overline{\overline{I}}_e^e = \overline{\overline{I}}_e^c + m([-\overline{c}_e^e,-\overline{c}_e^e]) - 2m([-\overline{c}_e^e,-\overline{0}_e^e])\)

  3. Rotate \(\overline{\overline{I}}_e^e\) to \(\overline{\overline{I}}_e^r\), i.e. a frame parallel to the output reference frame, with its origin in the local element frame

    \(\overline{\overline{I}}_e^r = \overline{\overline{A}}_e^r \overline{\overline{I}}_e^e (\overline{\overline{A}}_e^r)^T\)

    Also create offset vectors from the local element origin to the output reference frame origin and local centre of mass, both expressed in the output reference frame

    \(\overline{r}_e^r = (\overline{\overline{A}}_r^g)^T(\overline{p}_r^g - \overline{p}_e^g)\)

    \(\overline{c}_e^r = \overline{\overline{A}}_e^r \overline{c}_e^e = (\overline{\overline{A}}_r^g)^T \overline{\overline{A}}_e^g \overline{c}_e^e\)

  4. Translate the inertia tensor \(\overline{\overline{I}}_e^r\) from the element origin to \(\overline{\overline{I}}_r^r\) in the output reference frame origin

    \(\overline{\overline{I}}_r^r = \overline{\overline{I}}_e^r + m([\overline{r}_e^r,\overline{r}_e^r]) - 2m([\overline{r}_e^r, \overline{c}_e^r])\)

Once these steps have been completed for all relevant elements, contributions are combined, and a summary is written

Note: In the steps outlined above, it was assumed that given vectors \(\overline{a}\) and \(\overline{b}\), the notation \([\overline{a}]\) indicates forming of a skew symmetric matrix, i.e. \( [\overline{a}] = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix}\)

and that \([(\overline{a}, \overline{b})] = -\frac{1}{2}\left([\overline{a}][\overline{b}] + [\overline{b}][\overline{a}]\right)\)