Load on the Risers

. (A.32)

Hence, for α1 = 0, λi is proportional to ωi and for α2 = 0, λi is inverted pro-portional to ω_i. If we know the overall damping level for two frequencies we can determine α₁ and α₂ by

α₁ = 2ω₁ω₂ ω₂²− ω1²

(λ₁ω₂− λ2ω₁) , (A.33a) α₂ = 2 (λ₂ω₂− λ1ω₁)

ω₂²− ω1²

. (A.33b)

The inertia term leads to damping from rigid body motions in the same way as these motions give inertia forces. This damping contribution is not wanted since structure damping is mainly caused by strains in the material of the structure.

Hence, the inertia term in Rayleigh damping should not be included. The stiffness term α₂K is directly related to strains in the structure and can model interior structural damping adequately. The damping ratio will, however, increase with increasing frequency. The damping level, λ_i, must therefore be correct at the actual response frequency. The influence from damping at higher frequencies is without significance for the solution, and may in fact contribute to a more smooth time history since false oscillations caused by the numerical method might be suppressed.

A.3 Load on the Risers

The risers will experience load in the global horizontal direction due to currents.

The current loads will induce forces on the riser, which we will find by using Morison’s equation. The algorithm for loads, forces and displacements for the static initialization procedure could be summarized as follows:

1. Decompose the current from the global frame to the local frame.

2. Calculate the load at each node for each element locally.

3. Calculate the forces at each node for each element locally.

4. Transform the forces back to the global system.

5. Add the forces for each node to find the total force in the global frame.

6. Introduce load terms from for the specified motions from the TLP.

7. Find the displacement vector by r = K⁻¹R.

Below each point is explained in detail.

Decompose the current from the global frame to the local frame. First the current, given in the global frame, has to be decomposed into the local frame of the node at which the current attacks. It is assumed that the global current velocity vector in each node i, v_cur,i^f , only has a horizontal component, such that the v_z^f-component is zero. The flow is linear, i.e. there are no vortices in the flow.

The decomposed current in i-frame is then

v^f_cur,i =

Calculate the load at each node for each element locally. The loads in each element’s two nodes are found by Morison’s equation for the axial and transverse direction

where f_ext,xⁱ is the external force in x_i-direction for element i. When a force, stiffness matrix etc. is described in the i-frame, it is implicitly for element i only.

ρ_w is the density of the displaced fluid, D_i the diameter of element i, and C_Dn and CDt the drag force coefficients in normal and tangential direction. v_xⁱ and v_zⁱ are the current velocities in the local i-frame. The load in each direction is given by

This gives the load in each of the two nodes of an element to be q_xi,1 = 1

Calculate the forces at each node for each element locally. Assuming linearly varying transverse and axial load, the load on a bar is found in Fylling et al. (2005) and is given as

where f_nⁱ is the normal and f_tⁱ the tangential forces. The sum of forces acting on each local element in its frame is then given by

fⁱ = f_tⁱ+ f_nⁱ. (A.48)

This could be simplified to include forces in the x_i-direction only since these are far larger than the forces in the z_i-direction. The forces in z_i-directions will not have major impact on the dynamics, hence they could be neglected. The resulting

external force in each element is then given in the same direction as the load as

Transform the forces back to the global system. Before adding the forces acting at each node, they have to be represented in the same frame. The forces are transformed from its local frame to the global frame the same way as the displacements

f_i^f=T^f_ifⁱ, (A.50)

where f_i^f is the external force in the two nodes in element i given in the global f -frame.

Add the forces for each node to find the total force in the global frame.

After the forces at all nodes are transformed back to the global coordinate system, the forces and moments are summarized in each node

f_x,i^f = f_x^f_1,i+ f_x^f_2,i−1, (A.51) f_z,i^f = f_z1,i^f + f_z2,i−1^f . (A.52) After summarizing the forces at each node, the total force vector in f -frame due to the current forces is called f_drag.

Introduce load terms for the specified motions from the TLP. The specified motions for the quasi-static case are accounted for by correction terms in the load vector. These are found from

f_corr= f_drag− kspex_{T LP}, (A.53) where f_corr is the corrected force vector, k_spe is the specified stiffness vector and x_{T LP} is the scalar TLP motion in surge. f_corr is a vector due to the two-dimensional model, and k_spe is a vector due to the specified motion at the top node in surge only. All vectors are described in the global f -frame.

Find the displacement vector by r = K⁻¹R. The displacements of the riser for the quasi-static case is then found to be

r = K⁻¹R, (A.54)

where R is the force vector. Upper case is used here since this is most com-mon within the field of structural mechanics for load and force vectors. See the following iteration algorithm for equilibrium between internal and external forces.