Response of Articulated Towers to Waves and Current
Authors: Kirk C.L. , Jain R.K. 
Published: December 2007

Abstract
The dynamic response of articulated towers to noncollinear Airy waves and steady current has been investigated, where the wave and current forces have been computed by a modified form of Morison's equation. The two equations of motion obtained by Lagrange's method describe the response in terms of meridional and circumferential angles. These equations are highly nonlinear and are solved numerically by the block integration method for various wave parameters. The predicted response is a complex whirling motion of the tower around a vertical axis.

Introduction
There has been an increasing use of mobile offshore systems in the North Sea for storing and loading oil into attendant tankers, particularly for fields that have a limited production capability or are too remote from refining or terminal facilities to warrant laying a pipeline. Mobile loading platforms are also used as an interim measure platforms are also used as an interim measure during pipeline laying for large fields and later as a backup system in case of pipeline failure.A typical type of mobile loading and storage system is the articulated buoyant loading tower, which may have either a single universal joint at the sea bed or a second joint nearer the surface. The tower is designed for a maximum tilt angle of 20 deg. under extreme tanker mooring conditions caused by wind, waves, and current. In assessing the performance characteristics and the strength of an performance characteristics and the strength of an articulated tower under severe environmental conditions, both with and without a tanker, it is essential to determine its dynamic response by theoretical methods, by model testing in a wind/ current/wave tank, and by measuring response of real structures on site.This paper analyzes the motion of a single articulated tower without a tanker under the combined action of forces resulting from current and a train of regular linear waves. The problem is of interest to both the operators and the designers of loading towers because it is important to estimate motion in moderate seas for the case of mooring an approaching tanker, as well as the extreme deflections that would occur under the 100-year design wave.The dynamic response of the tower is obtained by formulating the equations of motion by Lagrange's method. The wave forces are determined using a modified form of Morison's equation that accounts for the relative motion of the water particles with respect to the structure. The equations of motion are highly nonlinear and analytical solutions are not possible; thus, a numerical solution has been selected in which the block integration method has been used. The cases considered areorthogonal waves and current,collinear waves and current having the same or opposite directions of propagation, andthe directions of propagation inclined propagation, and (3) the directions of propagation inclined at an angle of 45 deg.

DESCRIPTION OF PROBLEM
In the schematic of the typical articulated tower shown in Fig. 1, the orthogonal fixed-axis reference system is chosen so that the X and Z axes are taken in a horizontal plane parallel to the sea bed. During motion of the tower the OZ'axis is perpendicular to the Y axis in the plane containing the perpendicular to the Y axis in the plane containing the OY axis and the tower axis OC, while OX'is normal to the YOZ'plane. The instantaneous position of the tower is completely determined by the coordinates psi, and theta, where psi is the angle between the planes psi, and theta, where psi is the angle between the planes YOC and YOZ, and theta is the meridional angle made by OC and OY in the instantaneous position of the plane YOZ'. plane YOZ'. The tower is subjected to the action of linear waves propagating in the direction of the X axis and a steady current of velocity v that may vary with depth. The direction of the current flow is at an angle a to the X axis. In the absence of waves, the tower will be in a static equilibrium position specified by coordinates (theta, pi/2-a). Under the combined action of waves and current the structure will oscillate around the OY axis. The purpose of this paper is to formulate and solve the equations of motion of the tower subjected to a variety of wave lengths, wave heights, and a current of constant velocity but with variable direction.
SPEJ
P. 283