Combining Vessel Rotations |
 
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Combining Vessel Rotations |
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When vessel rotations from a number of sources are specified, an important consideration is how these rotations are combined, and how the displacements of attached nodes resulting from the combined rotations are calculated. This section describes the default Flexcom procedures for these operations, which are based on a large angle theory. The program alternative approach based on classical small angle theory is detailed in the next section. The significance of the various user inputs with respect to these two conventions is then explained in subsequent sections.
The large angle rotations formulation is based on the Flexcom 3D kinematics algorithm. In this, rotations are represented as vectors according to a specific definition. However, because finite (large) rotations do not behave as vector quantities (since they are not commutative), the program does not perform operations directly on these vectors. Instead the following procedure is adopted.
Any rotation “vector” uniquely defines a transformation matrix corresponding to a finite rotation as follows. The transformation matrix defines a rotation about an axis that is given by the direction of the vector, while the actual angle of rotation is given by the magnitude of the rotation vector. So the procedure to combine rotation vectors is that i) the transformation matrix corresponding to each individual rotation is calculated; ii) the transformation matrices are multiplied in a manner consistent with the order of application of the rotations; and finally iii) the combined rotation vector is calculated from the matrix product in a reverse procedure to that used to calculate transformation matrix from rotation vector. How this procedure is used in practice to combine vessel rotations from the various sources available in Flexcom is described in the next section.
At each solution time Flexcom calculates a total transformation matrix corresponding to combined vessel rotations according to the above procedure. This is then used to calculate corresponding motions of an attached node on a riser as follows. If at any time 
we denote as 
the vector from the vessel reference point (the point for which the rotations are defined) to the attached node, and if the total transformation matrix is 
, then 
is defined by:
(1)
where 
is the vector from initial vessel reference point position to the initial position of the attached node. So the displacement 
of an attached node from its initial position due to vessel rotations is given by:
(2)
where 
is the identity matrix.
When vessel rotations from a number of sources are to be combined using the optional small angle formulation, the procedure is more straightforward. The assumption of small angles means that rotation vectors can indeed be treated as vectors, and so rotations can be combined by the simple addition of rotation vectors. So for example in an analysis with a 6 DOF vessel static offset, drift rotations, and first order rotations calculated from RAOs, the total yaw is the sum of the yaw components from the three sources, and the total roll and pitch are likewise the sums of individual roll and pitch components.
The calculation of attached node displacements corresponding to these rotations is also more straightforward. If we denote as 
the total rotation vector at time 
whose components are the cumulative vessel yaw, roll and pitch motions, then the attached node displacement vector 
is given by:
(3)
where 
represents the vector cross product.
There is however one circumstance where rotations calculations involving transformation matrices are still used in the small angle case. If a riser attached node is restrained in DOFs 4-6, then the rotations due to vessel motions in any combination at any time are given by 
. However, in many cases, it is necessary to superpose or add these rotations to mean static riser (rather than vessel) rotations. There is no requirement in general that these riser rotations be small, and so simply adding rotation vectors in this case could lead to errors. So in this case the two rotations vectors are combined via transformation matrices as previously described.
•*VESSEL,INTEGRATED is used to specify all information pertaining to a vessel or vessels. Specifically, the ANGLES= input is used to specify the theory used to combine vessel rotations.