The single degree of freedom system used for the studies is a simplified structure behaving as a floating disk supported by a spring. The multidegree-of-freedom-system is a model of a riser (or tether) fixed at the sea bottom and free at the top. Consistent mass, stiffness and forces are used for the multidegree-of-freedom model. The wave loads given by Morison’s equation consist of two parts: the inertia term and the drag term. The drag force is a nonlinear, and thus, the equation of motion is also nonlinear. The multidegree-of-freedom system was analyzed computing the mass matrix, the stiffness matrix and the forces in both the undeformed and the deformed position of the structure. The former represents a linear structure subjected to nonlinear forces with the nonlinearities due to wave action only. The latter represents a structure with geometric nonlinearity subjected to wave and buoyancy forces and a pretension force at the top.
The Hybrid-frequency-time-domain procedure is applied linearizing first the equation of motion by passing the nonlinear terms to the right-hand side of the equation and considering them as part of the forces. One has then to assume values for the unknown displacements, velocities or accelerations included in the forces, solve the linear equation and iterate with the new results. The efficiency of the Hybrid-frequency-time-domain procedure is thus related to the number of iterations required to obtain accurate solutions. Ways to improve the convergence rate of the Hybrid-frequency-time-domain solutions are investigated in this study.
Since the drag force given by Morison’s equation involves the relative velocity, the behavior of the structure is that of a damped system. The linearized equation of motion is however that of an undamped system subjected to the forces. It is thus more realistic to add a fictitious damping into the linearized equation of motion and the same term to the right-hand side of the equation. This fictitious damping results in smaller unbalanced forces and a faster iterative process.
To calculate long duration responses, the time-history is generally divided into several segments. The traditional Hybrid-frequency-time-domain solution is obtained computing the Fourier transform for the entire duration to account for the initial conditions for each segment. An alternative approach developed in this work is to account for the initial conditions for each segment into the equation of motion instead of computing the Fourier transform for the entire duration. Thus, the duration of the FFT computed in the segment-by-segment approach is equal to the duration of a segment. Using this approach, the efficiency of the Hybrid-frequency-time-domain procedure is improved. However, the segment-by-segment approach is less stable than the traditional approach because the nonzero initial conditions lead to larger unbalanced forces in the iterative process and care must be exercised in the selection of various parameters to ensure stability and accuracy.
The major objectives of this study are:
- Assessment of the feasibility of using the Hybrid-frequency-time-domain method to solve flexible structures subjected to Morison’s equation type forces applied in the original or the deformed position at each time.
- Evaluation of the segment-by-segment approach used in the Hybrid-frequency-time-domain procedure by comparing its solutions with the ones obtained with the direct integration in time.
- Investigation of the effect of adding a fictitious damping to both sides of the linearized equation of motion.