Equilibrium calculation

When Calculate Equilibrium is on, SIMO will iterate to find a static equilibrium position for bodies. it is possible to choose between two different algorithms in order to compute the equilibrium position of the system: Transient method or Newton-Raphson method.

When calculating equilibrium position the option Require Successful Calculation is also available. When this option is checked the calculation will terminate with an error if the equilibrium calculation procedure is unable to find an equilibrium position. This means if any timedomain simulation is started it will terminate after the static calculation. If this option is unchecked, failure to find an equilibrium position will only be issued as a warning. In this case any consecutive timedomain simulation will be run with the last, unconverged, static position as initial position.

1. Transient method

This method is the most robust method, especially in case of systems with several bodies, and/or with strong non-linearities (like fenders, DP systems, …). It is recommended to use the "transient" method as the default method.

The following parameters are specified interactively:

  • \(\mathrm {T}\): Typical (maximum) natural period

  • \(\mathrm {\delta x}\): Position tolerance

  • \(\mathrm {\delta \alpha }\): Direction tolerance

  • \(\mathrm {\delta t}\): Time step

From position and direction tolerances, the following tolerances for velocities and acceleration are calculated:

  • \(\mathrm {\omega =2\pi /T}\)

  • \(\mathrm {\delta \dot {x}=\omega \delta x}\)

  • \(\mathrm {\delta \dot {\alpha }=\omega \delta \alpha }\)

  • \(\mathrm {\delta \ddot {x}=\omega ^2\delta x}\)

  • \(\mathrm {\delta \ddot {\alpha }=\omega ^2\delta \alpha }\)

By default, critical damping for each degree of freedom can be calculated:

\[c_i=2M_i\omega ,\quad i=1,6\]

where \(\mathrm {M_i}\) are the diagonal elements of the total mass matrix.

If a dynamic positioning system is defined, the thruster forces are calculated from the thruster allocation algorithm described in the Theory Manual. The thrust demand is

\[F_{d,i}=\delta x_iG_{il}F_{wi,i}F_{wa,i}F_{cu,i}F_{li,i}F_{co,i}\:,\quad i=1,2,6\]

where

  • \(\mathrm {\delta x_i}\): Positioning error in local coordinate system

  • \(\mathrm {G_{il}}\): Position and heading gains in linear controller matrix

  • \(\mathrm {F_{wi}}\): Static wind force

  • \(\mathrm {F_{wa}}\): Static wave drift force

  • \(\mathrm {F_{cu}}\): Current force

  • \(\mathrm {F_{li}}\): Positioning element force

  • \(\mathrm {F_{co}}\): Coupling force

The equations of motion are stepped the specified number of steps with the modified Euler algorithm. At the last step, relative position and direction "errors" are calculated as:

\[\displaystyle \varepsilon _1=\mathrm {max}( |\frac{\dot {x_i}}{\delta \dot {x}}|,|\frac{\ddot {x_i}}{\delta \ddot {x}}| ),\quad i=1,2,3 \\\\ \displaystyle \varepsilon _2=\mathrm {max}( |\frac{\dot {x_i}}{\delta \dot {\alpha }}|,|\frac{\ddot {x_i}}{\delta \ddot {\alpha }}| ),\quad i=4,5,6\]

"Equilibrium position" is found when both \(\mathrm {\varepsilon _1}\) and \(\mathrm {\varepsilon _2}\) are less than 1.

Note that static positions are updated when calculation of equilibrium is selected, even if equilibrium position is not found.

2. Newton-Raphson method

This method is generally faster when considering a system with only one body. The calculation is based on a linearized model of the system:

  • the force vector is evaluated at the position of the system

  • the stiffness matrix of the system is calculated using excursions values specified by the user

  • the position of the system is corrected according to the force vector and the stiffness matrix computed previously

  • the algorithm iterates until the values of the force vector are smaller than the force/moment tolerance values specified by the user. The corresponding position is then validated as the equilibrium position.

The system is solved using the coordinate systems XGB for each body, which allows the user to restrain "global" degrees of freedom (see Elimination of global degrees of freedom), which is often more convenient than restraining local degrees of freedom.

3. Multiple equilibrium calculations

When using the Newton-Raphson method, the user can use the option "Multiple equilibrium calculations". The user has to specify a body to analyse and a grid of roll and pitch values. For each point of the grid, SIMO will compute the equilibrium of the system, applying the roll and pitch values of the grid point to the desired body and locking its rotations about the X- and Y-axes of the XGB coordinate system. The results of each calculation (position and total force) can be found in the prs.lis file, but is not available directly in SIMA as results in the present version.

The grid of roll and pitch values can be given in terms of:

  • roll and pitch values

  • azimuth vector components: the azimuth vector X- and Y-components are given in the body-related coordinate system XR. The Z-component of the azimuth vector is always zero. The direction of the azimuth vector define the axis of rotation. The norm of the azimuth vector defines the magnitude of the rotation about the axis of rotation. The body is rotated about the rotation axis and the resulting roll and pitch values are used for the analysis at that grid point. The resulting yaw is ignored.

  • azimuth angle and rotation angle: the azimuth angle is the angle between the rotation axis and the X-axis of the XR coordinate system of the body. It defines the direction of the rotation axis. The body is rotated about the rotation axis and the resulting roll and pitch values are used for the analysis at that grid point. The resulting yaw is ignored.

In practice, this option can be used to quickly identify if an equilibrium position found by a static calculation is stable or unstable. For example, let’s assume that the user has run a static equilibrium calculation and found an equilibrium position with roll and pitch values of \(\mathrm {\phi }\) and \(\mathrm {\theta }\), respectively. The user can define a grid of 4 points around the equilibrium position by using the roll values \(\mathrm {{\phi -1deg,\phi +1deg}}\) and the pitch values \(\mathrm {{\theta -1deg,\theta +1deg}}\). Once the "Multiple equilibrium calculation" is run, the total force of each grid point can be found in the prs file. The restoring moments (in roll and pitch) address the stability in roll and pitch of the equilibrium position; if all the 4 points has restoring moments that tend to bring the vessel back to the equilibrium position, then the equilibrium is stable. It is unstable otherwise.

3.1. Eliminate global degrees of freedom

This option only affect the static equilibrium calculation if the Newton-Raphson method is used. It does not affect any consecutive dynamic simulation.

Elimination of global degrees of freedom is implemented by excluding the desired translations/rotations about the axes of the XGB coordinate system when computing the equilibrium position.

When restraining the rotational degrees of freedom, the following combinations are not available:

  • restraining rotation about X-axis only

  • restraining rotation about the Y-axis only

Those limitations are implemented to avoid confusion since for example, a rotation about Y-axis combined with a rotation about the Z-axis can actually trigger a change in roll angle (although the rotation about the X-axis is locked). Restraining the rotation about the Z-axis only is made available since it is directly equal to the yaw angle of a body.