|
Vessel types: Stiffness, added mass and damping |
|
|
Vessel types: Stiffness, added mass and damping |
The origin to which the stiffness, added mass and damping matrices all refer, with respect to the vessel axes.
The Z height of the reference origin above the mean water level and the heel and trim (relative to horizontal) when the vessel is in the datum position that was used to calculate its displaced volume, centre of buoyancy and hydrostatic stiffness matrix.
We use Z above mean water level, rather than Z relative to global axes, so that the vessel type data are independent of mean water level and of the choice of position of global origin.
The displacement method is appropriate for most vessels. This method uses Archimedes' principle, and requires that the wet hull surface is closed. For vessels such as catamarans, or platforms with distinct floating columns, each component of the hull must be closed.
The sectional method is a more general method which has no requirement for a closed hull. By using this method a single hull can be divided into multiple sectional vessels which, individually, may have open-ended wet hull surfaces. This method can also be used for vessels with a closed wet hull surface, since the two flavours of stiffness-load theory become equivalent in that case.
Displaced volume is the volume of water displaced by the vessel. The position of the centre of buoyancy is specified relative to vessel axes, when the vessel is in the datum position. These data are used together with the hydrostatic stiffness matrix to calculate the net weight and buoyancy load on the vessel. There are two distinct vessel configurations influencing these values, as follows.
If the datum position is a free-floating equilibrium position of the vessel, then the net weight and buoyancy load for the datum position will be zero since the two are equal and opposite. In this case, you may set the displaced volume and all three coordinates of the centre of buoyancy to '~', or you may give their explicit values.
Datum positions that are not free-floating equilibrium positions are defined by setting the displaced volume and/or the centre of buoyancy coordinates explicitly (i.e. not '~'). This is necessary if the vessel position used when calculating the hydrostatic stiffness data (i.e. in a diffraction analysis) is a disturbed position in which the weight and buoyancy forces do not balance. For example, the datum position for a TLP is likely to be the pulled-down position due to its tethers.
| Note: | The form of the buoyancy calculation in OrcaFlex depends upon whether the centre of buoyancy is given explicitly or as (~,~,~). The vessel type stiffness matrix linearises the effect of any shift in the moment arm between the centre of mass and the centre of buoyancy: this approximation can, however, be eliminated by OrcaFlex if the explicit position of the centre of buoyancy is known. |
The hydrostatic stiffness matrix describes the way in which the net weight + buoyancy load varies with changes in position from the datum position. Only the heave, roll and pitch rows of the matrix are specified; the rows for the surge, sway and yaw degrees of freedom are all zero. The units of the data and details of the theory are given in the theory section.
The hydrostatic stiffness matrix must be given with respect to axes through the reference origin in the conventions directions.
The hydrostatic stiffness only affects the vessel position in the static analysis if the vessel's include in static analysis option is set to 6 DOF. It only affects the vessel motion in a dynamic simulation if the vessel's primary motion is set to one of the calculated options.
The datum buoyancy load is the load (force and moment) due to water pressure evaluated at the reference origin, when the vessel is in the datum position.
The hydrostatic stiffness matrix is specified in the same way as for the displacement method, with one key difference: all six rows of the matrix are specified for a sectional body. The units of the data and details of the theory are given in the theory section.
The added mass and damping data represent the fluid loads on the vessel due to wave radiation effects. These loads will only influence the motion of the vessel if the primary motion is set to one of the calculated modes, and added mass and damping is an included effect.
| Note: | There are other sources of damping that can be important. See damping effects on vessel slow drift for further information. |
The added mass and damping matrices are given in all 6 degrees of freedom (or 6N degrees of freedom for a multibody group, where N is the number of vessels in the group). The matrices must be given with respect to axes through the reference origin in the conventions directions. The units of the added mass and damping matrices, and details of the theory, are given in the theory section.
OrcaFlex will only accept symmetric added mass and damping matrices. For vessels or multibody groups with zero forward speed, the matrices must be symmetric on theoretical grounds. In the case of significant non-zero forward speed, OrcaFlex cannot include the effect of any asymmetry in the added mass and damping due to that forward speed.
If manoeuvring load is included, then the constant added mass matrix or (if frequency-dependent data are in use) the longest-period added mass matrix, will be used in order to calculate the manoeuvring load.
If you choose constant for the added mass and damping method, then single-valued added mass and damping matrices are used.
If you choose frequency dependent, then you may define a number of added mass and damping matrices, each pair corresponding to a particular given frequency or period. Whether you specify period or frequency values is determined by the waves are referred to by vessel type convention.
For the constant (i.e. frequency independent) method, you should choose values for added mass and damping that are appropriate to the frequency of vessel motion you expect. To calculate slow drift motion of the vessel it is usual to enter low frequency values; otherwise, values corresponding to the dominant wave frequency would be more appropriate. Clearly, if the vessel experiences a wide range of frequencies, the frequency-dependent method is more suitable and would be expected to give better results.
The frequency dependent method requires you to give both the added mass and damping matrices, for a range of frequencies. The data should be consistent, in the sense that they obey the Kramers-Kronig relations – see consistent added mass and damping for details.
The calculation of frequency-dependent added mass and damping involves computing a convolution integral whose range of integration stretches all the way back to the start of the simulation. This would cause long simulations to run more and more slowly as the simulation progresses. OrcaFlex therefore provides two mechanisms to speed up the computation of this integral.
Firstly, the integration time step can be specified explicitly, so as to give control over how many integration steps are computed in the convolution integral. The default value for this time step is '~', which means to evaluate the convolution integral with an integration time step equal to the time step used in the simulation (the constant time step when using implicit time domain; the outer time step when using explicit time domain). In certain circumstances, when the impulse response function and the vessel motion vary slowly, it may be appropriate to set a larger integration time step, which would result in a faster calculation. Internally within OrcaFlex, the integration time-step specified will be rounded up to the next whole multiple of the simulation time step.
Secondly, the impulse response function in the integral decays to zero as the time lag approaches infinity, so there comes a point beyond which the rest of the integral can safely be neglected. OrcaFlex allows a cutoff time to be specified, which is the point at which it truncates the convolution integral.
Selecting a value for the cutoff time means balancing the approximation errors induced by small values against the longer calculation times resulting from larger values.
To make this easier, you may set the cutoff time to ~ and give a damping level approximation cutoff tolerance, as a percentage error relative to the largest damping level given. OrcaFlex will then automatically calculate the cutoff time that gives this level of approximation of the damping levels; the approximation error on the added mass will be similar.
You can judge the effect of the cutoff time and tolerance by adjusting them on the impulse response, added mass and damping graphs form. The damping graphs show two curves as functions of frequency: one shows the idealised (user-specified) damping, the other the damping realised by truncating the convolution integral at the given cutoff time or tolerance. When you are happy with the level of approximation, click the OK button to transfer the cutoff values to the original vessel type or multibody group and close the graph form; alternatively, click cancel to ignore any changes you have made on the graph form.
Of course there are a lot of such graphs, but it will normally be sufficient to look only at the diagonal components (surge-surge, sway-sway, etc.) for the vessel degrees of freedom that you expect to be excited. You can also look specifically at the frequencies of oscillation that are relevant to your simulations. For example, for slow drift simulations you can zoom in on the low frequency part of the damping curves, to see the effect of cutoff on the low frequency radiation damping level.
All of the above data can generally be obtained from the results of a diffraction program. OrcaFlex can import these data directly from the output files of some specific programs (AQWA and WAMIT) and from generic text files with OrcaFlex-specific markers added.
This form is opened by clicking on the show graphs button: for a single vessel, the button is on the vessel types form, stiffness, added mass, damping page (assuming you have selected frequency-dependent added mass and damping), and for a multibody group it is on the multibody groups form, added mass and damping page. The form shows graphs of:
Note that the added mass and damping data are matrix-valued quantities. These will be $6{\times}6$ matrices for a solitary vessel, the rows and columns corresponding to the surge, sway, heave, roll, pitch and yaw degrees of freedom (DOFs). An entry in a particular matrix represents the load generated in one degree of freedom, the row DOF, due to motion of the vessel in another degree of freedom, the column DOF. This means that there will be one set of graphs for each pair of row and column DOFs (so 36 for a solitary vessel). All combinations of row DOF and column DOF can be selected via the drop-down lists of the same name. In the case of vessels that belong to a multibody group, the added mass and damping matrices will be $6N{\times}6N$, where N is the number of interacting bodies in the group. This is because it is possible for added mass and damping loads on one vessel to be induced by the motion of another vessel. For an n-body multibody group there will therefore be 36N2 sets of graphs. To deal with this, two extra drop-down lists are visible: row vessel and column vessel. The graphs displayed will therefore correspond to the effect on the row vessel's row DOF due to motion in the column vessel's column DOF.
The scale of the time lag axis for the IRF is determined by the cutoff time. This graph may be useful in judging the effect of this cutoff time: a shorter cutoff time will allow for faster calculation, but too short may mean significant IRF values are discarded and accuracy lost as a result.
The IRF graph shows two curves, differing by the time step with which the integral used to calculate the IRF is discretised. The idealised curve uses a fine time step, while the model time step or specified time step curve indicates the actual IRF which will be realised by OrcaFlex, using the time step value described above. Note that this second curve will not be shown if implicit integration with variable time step has been selected, since OrcaFlex does not allow this combination. For particularly long time steps, you may find that the realised IRF curve departs significantly from the idealised one, in which case you should consider a shorter time step to improve resolution and avoid losing accuracy in the IRF calculation.
The added mass and damping graphs plot against frequency the values of the original data given for the vessel type or multibody group.
The added mass graph has superimposed on it a horizontal intercept representing the value of the infinite frequency added mass. If the infinite frequency added mass has been specified, then that is the value displayed. Otherwise, the calculated value is shown. In this case, so long as the data are given to sufficiently high frequency, this is an indicator of the consistency of the added mass and damping data.
The damping graph, similarly to the IRF graph, displays two curves, idealised and realised at cutoff time. As described above, these graphs will help you to decide on a suitable cutoff time or tolerance.
If you have selected the frequency domain solution method, the graphs available to you are more restricted. Impulse response is a time domain concept which has no meaning in the frequency domain, so the impulse response graph is not available. Cutoff time is meaningless likewise, so the cutoff time and cutoff tolerance controls are not shown, and the damping graph shows only a single curve representing the data. Frequency domain analysis is not supported for multibody groups, so the show graphs button is disabled on the multibody groups data form and none of the graphs are available.