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Lower mating units model

Model and assumptions

The \(LMU\) model used is based in the Saipem UK Ltd. note Numerical Modelling of Installation Aids for Platform Installation by Peter S. K. Lay and Xavier Chevalley.

The model is based on the assumption that the relative orientations of the pin and bucket of the \(LMU\) are negligible and no moments are created at the contact location (i.e. only forces are transmitted between the pin and bucket of the \(LMU\)); the forces generated by the contact between pin and bucket are concentrated on the two \(LMU\) control nodes \(P\) and \(Q\) (see sketch below).

Moreover, damping and friction effects are not taken into account.

The model used is applicable to both the \(LMU\) configurations in the following sketch, for which it may be easily realized that the formulations to be adopted are the same.

The orientation of the \(LMU\) is space is determined by the orientation of the control node \(P\), the control node \(Q\) is assumed oriented as \(P\) (the relative orientation is supposed zero, as anticipated above).

In order to better understand the formulations used, a third point \(R\) can be introduced. \(P\) is part of the upper structure, \(Q\) is part of the bottom structure and \(R\) is linked to the upper structure through three springs which determine the \(LMU\) stiffness, as shown in the figure below. It should be therefore understood that the upper cone is not rigidly connected to the upper structure but linked to it through the \(LMU\) springs and can move within the upper structure.

From this sketch is can be easily seen that when the relative position of \(Q\) with respect to \(P\) is such that there is contact, point \(R\) moves. In particular, when there is also lateral contact, point \(R\) moves both laterally and vertically because of the cones geometry and so that the global force is perpendicular to the cones lateral surface; the extent of point \(R\) movement in the vertical and horizontal direction depends on the springs stiffness.

Formulations

Let the relative position of \(Q\) with respect to \(P\) be determined by \(dx\), \(dy\) and \(dz\) . The total horizontal distance between \(P\) and \(Q\) is:

\(dh=\sqrt{dx^2+dy^2}\)

The cone angle is given by:

\(\theta=arctan\lbrack\frac{h}{0.5(d_2-d_1)}\rbrack\) where \(h\) is the cone height and \(d_1\) and \(d_2\) the smaller and bigger cone diameters respectively.

In the case \(dz<-h\) the two cones are outside each other and no contact is possible.

If \(dz>0\) then \(P\) is lower than \(Q\) and there is vertical compression due to a vertical deflection \(z_{defl,z}\) and:

\(z_{defl,z}=dz\)

When the cones are inside each other but \(dz<0\), the allowable horizontal gap before contact may be determined by:

\(dh_{all}= \displaystyle \frac{- dz}{tan(\theta)}\)

Of course, if \(dz>0\) then \(dh_{all}=0\), in other words if there is vertical deflection the allowable gap before contact is null.

The effective horizontal deflection may therefore be calculated as:

\(dh_{tot}=dh-dh_{all}\)

If \(dh=0\), \(P\) and \(Q\) are perfectly aligned and there is no horizontal deflection, \(f_x\) and \(f_y\) are null and \(f_z\) is calculated by linear interpolation from the \(LMU\) force - deflection curve using the \(z_{defl,z}=dz\) deflection.

As stated before and shown in the second sketch, when there is horizontal deflection, because of the cone shape and because the global reaction force must be perpendicular to the cone lateral surface, both a lateral and a vertical force are produced. Depending on the springs stiffness the internal cone structure (point \(R\)) slides upwards as a consequence of a lateral deflection.

The problem is then to determine the effective lateral compression (\(h_{def}\)) and the vertical compression (\(z_{def,k}\)) due to the horizontal deflection.

Known \(h_{def}\), \(z_{def,k}\) can be calculated as:

\(z_{def,k}=(h_{tot}-h_{def})tan(\theta)\)

The calculation of \(h_{def}\) is done iteratively with the target condition:

\(arctan\left( \frac{f_v}{f_k} \right)\)

where \(f_v\) and \(f_k\) are the vertical and horizontal forces due to the \(z_{def,k}\) and \(h_{def}\) deflections. In particular:

\(f_v= \displaystyle \frac{z_{def,k}}{z_{def,z}+z_{def,k}}f_z\)

\(f_k=\sqrt{f_x^2+f_y^2}\)

\(f_z\), \(f_y\) and \(f_x\) are determined by linear interpolation from the \(LMU\) force-deflection curves using the followingdeflections:

\(z_{def}=z_{def,z}+z_{def,k}\) (total vertical compression due to both the vertical and horizontal deflections)

\(x_{def}=h_{def}\frac{dx}{dh}\)

\(y_{def}=h_{def}\frac{dy}{dh}\)