A careful analysis of the motion of an offshore structure such as a tension 1eg platform (TLP) reveals that it is governed by Mathieu-Hill differential equations assuming small displacements. Only a simplified analysis is presented which reduces the motion to one degree of freedom. This paper discusses the hydrodynamic effects which produce the time-dependency of the coefficients in the equation.
As a result of these, response components at other than the wave period result. Attention is focused on the steady state solution due to rectification and the implication of this that the effective stiffness of the system is dependent upon the wave period and height. Surge force-displacement curves are produced which include the pendulum stiffening and it is shown that the static offset angle of a TLP can be reduced by as much as 50% due to the wave effect.
An additional steady drift force due to rectification is identified.
The linear dynamic analysis of floating vessels in waves is based on the claim that displacements in the degrees of freedom are sufficiently small that induced forces which involve products of these displacements or their derivatives may be neglected. In addition, the wave theory is developed on the assumption of infinitesimally small wave height and the essentially nonlinear viscous drag forces, if included, are linearised in accordance with some theory invoking "equiva1ence". These approximations have the benefit of permitting a frequency domain solution which is inexpensive, rapid and provides a great deal of insight to the behaviour of the system.
In the main, satisfactory correlation can apparently be obtained between calculated and measured responses and mooring forces for small scale models of, for example, semi-submersibles and tension leg platforms tested in wavetanks. However, with theories which depend so strongly on the empirical nature of Morison's wave force formula it is not sufficient just to get good agreement - it must be known to be for the correct reasons. The use of hydrodynamic mass and drag coefficients determined from similar experiments should not surprisingly provide good results but in doing so could be making important effects arising for totally different reasons. This problem is partly circumvented by the use of diffraction wave force models but then discrepancies are then blamed on the absence of viscous drag in the mathematical model.
There is a growing awareness of the complexity of the dynamics of TLP's due largely to the observed phenomena of subharmonic responses and dynamic instabilities. These cannot be explained by such simplified theories. Extended linear theories are discussed in References 1, 2, and 3 and are pursued herein. However, the natural modification to the response calculations for design purposes has been to include all the known nonlinear effects in a time-domain solution. This has the disadvantage of being cumbersome and expensive, but more importantly it results in the user losing the understanding of the mechanisms at play. This is sufficient justification for the investigation and development of the extended linear theories.