A non-linear formulation is presented for computing the lowfrequency motions of semisubmersibles, Expressions for the potential forces are derived using second-order perturbation theory employing an application of Green's second identity to enable their evaluation without the need to determine the second-order potential explicitly, First-order quantities are evaluated using the 3-D souce distribution technique, The quadratic viscous forces are expressed in terms of the empirical drag coefficient and the tri-chromatic relative velocities. The second-order equa' tions of motion are solved in the frequency domain to give the Quadratic Transfer Functions. 1.
Semi submersibles are designed such that their natural freiquencies are well below the frequencies which have significant 'wave energy. However, model tests (Lundgren and Berg, 1982), and full scale measurements (MacLeod, 1986) have conclusively shown that a significant component of their motions occur at these low frequencies. Such motions can only be the result of a non-linear mechanism. In present day engineering practice only the linear responses are taken into design consideration - the often large motions associated with the natural frequencies are not accounted for. It is the aim of the present study to provide a means of evaluating the second-order low-frequency motions of semi submersibles in terms of the Quadratic Transfer Function, from which spectral responses may then be calculated. This paper is particularly concerned with the vertical motions of roll and pitch, The horizontal motions associated with the natural frequencies of the mooring system have recently received much interest. These studies have generally relied on approximations for the second-order low-frequency drift forces (Newman, 1974), (Bowers, 1976), (Pinkster, 1979). However, such approximations tend to break down for higher frequencies and/or for the vertical motions. Matsui (1989) has computed the exact second-order forces and his approach is adopted here, as described in section 2.