A new analytical approach is applied to the riser's equations of motion, aimed at the real-time processing of solutions under hard boundary conditions. Based on this, a mathematical framework is devised, with the objective of supplying practical assistance for challenging riser positioning cases, leading to both live visualizations of the riser displacement during its maneuvering and on-the-fly planning of its desired top-end movement.
The new analytical solution is based on a non-causal analysis using the two-sided Laplace Transform. By transforming a set of approximate PDEs that closely resemble those of the complete riser model, we're able to devise improper transfer functions between any two arbitrary points across the riser. Then, an inverse transformation applied to these transfer functions yields convolutions that may be quickly processed. Nonlinear boundary conditions are dealt with via a proposed iterative method.
Quickly computable simulations for the horizontal displacement along the riser's length are presented, allowing for live, simultaneous estimation of the structure's motion at various cross-sections. Likewise, stress/tension estimates at these points are presented in real time.
Furthermore, various top-end trajectories and motion strategies are derived for scenarios previously unsolved in the literature, involving non-static initial conditions, disturbances and time-variable tensions acting on the bottom-end of the riser. This is done to demonstrate the theory's applicability in dire weather conditions.
Finally, the proposed solution is validated and compared to previous results in modeling and trajectory planning of risers. The speed of the presented approach is attested and properly analyzed.