This paper is a study of application of Mathieu instability evaluations for Floating Production System (FPS) design in deepwater. The instability of the Mathieu’s equation with damping is investigated based on the infinite determinant and harmonic balance method. General stability charts are generated in a relevant parametric plane to detect the unstable conditions. It has been found that high-order regions of instability are more sensitive to the damping than loworder ones. The regions of the principal (1st) and 2nd unstable zones are the major concerns for FPS design. It is our intention to study damping effects on suppressing Mathieu instability.
DDCV is one of the FPS types with deep draft. In the light damped system (damping less than 1% of critical damping), the DDCV may excite the Mathieu instability due to long period waves if the ratio of the incident wave period/pitch natural period is equal to or close to 0.5, or if the ratio of incident difference period / pitch natural period is equal to or close to 1.0. If the instability occurs at the principal (1st) unstable zone, it requires relatively large damping in the system to suppress the Mathieu instability. If the instability occurs at the second unstable zone, it needs only relatively small damping in the system to suppress the Mathieu instability. Thus the instability in the second unstable zone may vanish with the presence of damping due to mooring lines and risers/riser supports coupling inside the moonpool.
The Tension Leg Platform (TLP) is a FPS with tendons connecting the platform to the seabed. The parametric responses excited by the interactions of tendon tension variation and lateral dynamics of tendons are studied by using Mathieu’s equation. It may induce the Mathieu instability at a certain mode of tendon lateral motion dependent on the magnitude of fluctuation of the tendon tension and damping of tendon transverse motions.
Mathieu’s instability could be triggered in the light damped system by the frequency components, which are not necessarily regular waves. It is worth checking Mathieu’s equation in the design in order to exclude any possibility of instability problems.
In this paper, Mathieu instability evaluations for DDCV and TLP tendon design are based on a stability chart (including damping effects) which assumes incident waves being regular waves. It is recommended to develop a Mathieu stability diagram, which considers damping effects in both bi-frequency and irregular waves with known spectrum in the future.