Static calculation parameters

The user can choose between two different algorithm in order to compute the equilibrium position of the system: transient method or Newton-Raphson 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:

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

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

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

The equilibrium condition is found by stepping the equations of motion. The following force models will be included (if they are defined):

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

where

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:

\(\displaystyle \varepsilon _2=\mathrm {max}( |\frac{\dot {x_i}}{\delta \dot {\alpha }}|,|\frac{\ddot {x_i}}{\delta \ddot {\alpha }}| ),\quad i=4,5,6\)

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

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

It is advised to check the equilibrium position by calculating static forces and printing new initial positions.

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 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.

Multiple equilibrium calculations

This option is only available when using SIMO in SIMA.

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 in the present version. This will be updated in a future version of SIMO/SIMA.

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

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.

Natural periods and eigen vectors are calculated for selected modes. The calculation is based on a linearized model of the system. Mass and stiffness matrices are calculated, and the system solved by a standard Jacobian solver. The effect of damping is not included in the analysis.

The mass matrix is built from the virtual mass matrix, including both structural and hydrodynamic mass. The frequency dependency of the latter is not accounted for in the analysis as asymptotic values are used. For bodies of type 1 (see the chapter Body) this means added mass at infinite frequency, while the added mass at zero frequency is used for bodies of type 2.

The secant stiffness for the user selected modes is calculated at the initial position. The user may choose to specify the secant excursion used in the calculation for each mode. The stiffness matrix is made symmetric.

For each natural period, the eigen vectors expresses the relative contribution from each mode. Translations are given in the length unit \(\mathrm{[L]}\) and rotations in degrees \(\mathrm{[deg]}\).

In the interpretation of the results, it should be kept in mind that damping is not included in the analysis. Modes that are heavily damped may thereby not be recognized in time domain simulation results.

Observe also that environmental forces are not included. Thus, the yaw motion of a turret moored or a single point moored vessel can not be checked by an eigenvalue analysis.

The following forces are presented:

If a dynamic positioning system is defined, thrust demand will be calculated according to the thrust demand equation.

A file for 3D visualisation can be written. This means that the system can be visualized before or after equilibrium condition calculation has been made.This can be advantageous as a preliminary control of the system geometry.

Wind force coefficients, current force coefficients, wave drift force coefficients and transfer functions are given as functions of directions relative to the body.

In the calculations, extrapolation on directions are not allowed. In order to avoid problems with program termination due to attempted extrapolation, it is advised that both 0 degrees and 360 degrees should be represented in the range of directions on the input file.

If symmetry codes are used, this will automatically be taken care of. For example, if symmetry code is 2 and the directions 20 degrees and 70 degrees are specified, they will be mirrored to -20, 20, 70, 110, 160, 200, 250, 290, 340 and 380 degrees.

NB: It is strongly advised that values are specified for the symmetry lines. For the example above, 0 deg and 90 deg should also be included as input.