The simple wire coupling is modelled as a linear spring according to :

where :

The effective axial stiffness is given by :

where :

Knowing the position of each line end, the elongation and thereby the tension may be determined.

Material damping is included as:

The possibility of coupling several wire segments in a common branch point is incuded. In this way wire systems including slings may be modelled.

All wire segments will have one end fastened in a body and the other in the common branch point. Axial stiffness properties of each segment, \(\mathrm {\boldsymbol{k_{i}}}\) , are found in the same way as for the single wire coupling. To determine the location of the branch point, an iteration procedure must be used.

The model is identical to that described as station-keeping force. As a coupling force, any force-elongation relationship may be specified. The curves for increasing and decreasing elongation may be different in order to model hysteresis effects. Damping may be specified as a force proportional to any exponent of the relative velocity of the end points. This model is also used for passive motion compensators.

The lift line model gives a fast and quite accurate model for prediction of the total static and dynamic line end forces and load offset. This coupling type requires that the line is mainly vertical. The net weight of the body at the lower end of the line will be applied as a force on the line. It is therefore not possible to have two coupling lines down to the same body. The simplified lift line is modelled by the nominal Young’s modulus (\(\mathrm {E}\) ), diameter, transverse and longitudinal drag coefficient and the mass per meter length. The model gives zero compressive forces. The basic model is a straight line. A modification to the straight-line assumption is made to take into account the transfer of horizontal motion and forces from the crane to the lifted body due to WF motion, LF motion and transient motions (moving vessel).

Current forces and the damping forces of the line acting on the body and the load are calculated along a straight line. The line weight and buoyancy and the dynamic line mass forces are also calculated along this line.

The load offset is calculated using the cross-flow principle for the drag force components, \(\mathrm {q_{lw,d,i}}\) normal to the line:

and similarly in y- and z-direction. The density of water is denoted \(\mathrm {\rho }\), \(\mathrm {C_d}\) is the drag coefficient and \(\mathrm {\boldsymbol{v_r}=[\nu_{rx},\nu_{ry}\nu_{rz}]^T}\) is the relative velocity between the line and the water particles. The cross-flow principle is based on the normal flow velocity. The velocity is decomposed into a normal velocity and a tangential velocity along the lift line, hence vertical forces will be present for horizontal velocities on non-vertical lift lines even for relative small deflection angles from the vertical axis. Initially the lift line stretching is taken into account by a linear force elongation relationship assuming uniform strain along the lift line:

where \(\mathrm {L}\) is the total line length, and \(\mathrm {\Delta L}\) is the elongation of the line.

A straight-line model overestimates the transfer of the horizontal motion from the surface vessel to the load. A modification to the straight-line assumption is made in order to take into account the transfer of the horizontal motion and forces from the crane to the lifted load due to the WF motion. A vibrating string analogy is used in order to estimate the length of the lift line that is influenced by the crane vessel WF motion, and to express that this length increases with increasing tension. By using the vibrating string analogy, the wavelength \(\mathrm {\lambda_1}\) of the first order motion mode is given by:

where \(q_{lw,\mathrm {top}}\) is the top tension, and \(\mathrm {m'}\) , is the mass per unit length of the lift line. The first order mode period \(\mathrm {T_1}\) is substituted by the zero-crossing period of the wave spectrum \(\mathrm {T_z}\) . Part of the line or the whole submerged line length is exposed to motions according to the vibrating string analogy. \(\mathrm {\lambda_1/4}\) is used as the basis for the calculation of the forces on the part of the submerged line length that is influenced by the WF crane vessel motion. The vessel-induced WF velocity of the line from the vessel decreases linearly down along the lift line. Hence, the drag force due to this motion is quadratically decreased due to the quadratic nature of the drag forces. These velocities and the other velocity components acting on the line are superposed. Alternatively, due to the vibrating string analogy it can be argued that the motions should have a periodical shape (a quarter of a cosine wavelength), but the linearly decreasing velocity is assumed here. The assumed shape of the vessel induced WF motion neglects any other velocity components acting on the line, a highly questionable but useful method.

This simplified model has been extended to include the weight and buoyancy of the line and also calculate the dynamic top and bottom tension due to line mass forces. Lower end of the line is considered as a free end. The top tension \(q_{lw,w,\mathrm {top}}\) due to the weight and buoyancy for a line fully submerged in water and with uniform weight distribution along the line, as shown in Line elongation and tension due to weight and buoyancy for a fully submerged line with uniform weight distribution., is:

where \(\mathrm {g}\) is the acceleration of gravity and \(\mathrm {w_w'}\) is the unit weight in water. The line elongation \(\Delta L_w\) due to the weight and buoyancy is:

In Line elongation and tension due to weight and buoyancy for a partly submerged line with uniform weight distribution., the line is partly submerged, and the top tension and the line elongation due to the weight and buoyancy is then:

where \(\mathrm {w_a'}\) is the unit weight in air of the line and \(\mathrm {L_w}\) is the submerged line length.

Considering linearly distributed acceleration along the line (local z-axis from the line end), as shown in Figure 1, the tension due to the accelerations of the line from \(\mathrm {\ddot z_C}\) (crane acceleration) and \(\mathrm {\ddot z_L}\) (load acceleration) is:

The elongation \(\mathrm {\Delta L_a}\) due to the accelerations of the line is then:

The rachet coupling can take tension or compressive force. The force will start to act after a specified time. The force consists of 3 components: a constant force, a stiffness force, and a damping force. The constant force component (which can be set to zero) acts from \(\mathrm {t=0}\) . For a ratchet with compressive force, the damping will only act when the distance is decreasing and for a "tension ratchet" the damping will only act when the distance is increasing.

An example with a tension ratchet is shown in Figure 1. From \(\mathrm {t=0}\) to point A, the force in the ratchet will be equal to the specified constant force. The distance in point A, will be used as a reference distance for the stiffness force until point C is reached. The distances in point C and D are new distance references. In the green part of the curve, the force is equal to the constant force component.

It is possible to specify an event of ratchet failure. It will then fail if a specified force is exceeded. When the element fails, the tension is set to zero.

If zero is specified both for the stiffness and the (constant) minimum force, the ratchet element can be used to model hydraulic/pneumatic cylinders with direction-dependent damping.

An example of a ratchet coupling is shown in Figure 2.

Body A has a forced sinusoidal motion with amplitude \(1\mathrm {m}\) and period \(10\mathrm {s}\) .

Body B can slide without friction along the x-axis. Body B has a mass of \(1000\,\mathrm {tons}\) and a linear damping of \(10000\,\mathrm {kNs/m}\) .

The resulting motion and force is shown in Example of the motions and ratchet force in the simulation example.

Stiff couplings such as articulated joints and hinges may couple a body to the ground. Such elements may be modelled by extremely high stiffness.