Static calculation parameters
This editor contains all parameters available to configure the static calculation of a SIMO task.
1. Equilibrium calculation
The user can choose between two different algorithm in order to compute the equilibrium position of the system: transient method or NewtonRaphson method.
2. Transient method
This method is the most robust method, especially in case of systems with several bodies, and/or with strong nonlinearities (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:
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):

"Critical damping" force

Modified retardation function force, \(\mathrm {F_{ij}=4\Delta t\:h_{ij}(0)\dot {x}_j}\)

Hydrostatic stiffness force

Linear damping force

Quadratic damping force

Wind force (linear interpolation between directions)

Linear and quadratic current drag force (linear interpolation between directions)

Average wave drift force. The force will be calculated for the instantaneous heading the first and last time step. For other steps, corrections for instantaneous heading are done using first derivatives.

Small volume hydrodynamic force, wave particle velocities are set to zero.

Coupling element forces

Specified forces for
t = 0
. 
Positioning element forces

Thruster forces.
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

\(\mathrm {\delta x_i}\): 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:
\(\displaystyle \varepsilon _2=\mathrm {max}( \frac{\dot {x_i}}{\delta \dot {\alpha }},\frac{\ddot {x_i}}{\delta \ddot {\alpha }} ),\quad i=4,5,6\)
"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.
3. NewtonRaphson 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.
Multiple equilibrium calculations
This option is only available when using SIMO in SIMA.
When using the NewtonRaphson 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 Yaxes 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:

roll and pitch values

azimuth vector components: the azimuth vector X and Ycomponents are given in the bodyrelated coordinate system XR. The Zcomponent 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 Xaxis 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.
4. Eigenvalue calculation
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.
5. Eliminate degrees of freedom
5.1. Eliminate local degrees of freedom
Elimination of degrees of freedom is implemented by setting the
corresponding local acceleration components to zero during time domain
simulation (DYNMOD
) and equilibrium iteration (STAMOD
, when using
"Transient" algorithm).
5.2. Eliminate global degrees of freedom
This option only affect the static equilibrium calculation if the NewtonRaphson method is used. It does not affect the dynamic analysis.
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.
For userfriendlyness purpose, when restraining the rotational degrees of freedom, the following combinations are not unavailable:

restraining rotation about Xaxis only

restraining rotation about the Yaxis only
Those limitations are implemented to avoid confusing the user since for example, a rotation about Yaxis combined with a rotation about the Zaxis can actually trigger a change in roll angle (although the rotation about the Xaxis is locked). Restraining the rotation about the Zaxis only is made available since it is directly equal to the yaw angle of a body; the result is thus easier to understand for the user.
6. Results
The following forces are presented:

Average wave drift force calculated for the present heading, including wavecurrent interaction if specified

Wind force (linear interpolation between directions)

Current drag force based on linear and quadratic current coefficients (linear interpolation between directions)

Munk moment caused by current velocity and different magnitude of added mass specified in individual degree of freedom

Positioning element forces

Thruster forces

Coupling element forces

Forces from general line systems

Hydrostatic stiffness force

Gravity and buoyancy forces, also including small volume hydrodynamic force with wave particle velocities and accelerations set to zero. The forces contain contributions from current, gravity force and any static soil reaction forces.

Gravity forces due to time dependent added mass at time zero

Gravity and buoyancy force on slender elements and fixed body elements

Current drag forces acting on slender elements and fixed body elements

Specified forces at time equal to zero

External forces, i.e. forces from any special force subroutines linked into the program
If a dynamic positioning system is defined, thrust demand will be calculated according to the thrust demand equation.
7. Write file for visualization
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.
8. Symmetry codes
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.