Study On Nonlinear Motion Behavior Of Coupled Heave-Pitch For The Spar Platform

A Spar platform is one kind of new platforms designed for water depth from 500m to 3 000m. The typical heave natural frequency of the Spar is approximately double the pitch natural frequency due to its configuration characteristic, while, the natural frequency of surge is much lower than that of heave or pitch. Hence, for the safety of the platform and overall recognition and grasping the motion characters under complicated environment loads, it is important to study the nonlinear dynamic characters of the platform considering the coupled relation between heave motion and pitch motion. In this paper, the theory and method of nonlinear dynamics are adopted to study the nonlinear dynamic behavior and motion stability of the Spar platform under complicated environment loads. In addition, the coupled heave-pitch motion characters are analyzed with the effect of mooring tension. The main task and the conclusions are as follow:The domestic and overseas development of the research for deep water platforms is summarized synthetically, especially, the development of the Spar platform. The main problem of dynamic characters for the Spar platform is analyzed and reviewed, and the research direction is determined based on the above analysis.The main body of the Spar is simplified as an isolated, truncated and vertical floating circular cylinder. Surge wave excitation force, heave wave excitation force and pitch wave excitation moment are obtained respectively by using first order diffraction theory. According to first order radiation theory, added mass coefficients and added damping coefficients of surge and heave, added rotational moment coefficient of pitch and added rotational damping moment coefficient of pitch are obtained respectively.In order to solve the linear strong parametrical excitation equation, the method for solving strong parametrical excitation Mathieu equation is brought out. With the introduction of a new conversion, the strong parametrical excitation problem is converted to weak parametrical excitation problem. The second order approximate periodic solutions are obtained and the transition curves are gained by using a modified strained parametrical method. The instability domain is determined by the transition curves. The stability of periodic solution is investigated and the parametric value which induces double-period bifurcation is got. Without considering the effect of mooring tension and considering the effect of coupled heave and pitch, the nonlinear coupled heave-pitch parametric equations of Spar platform main body are established in which heave motion contains the effect of wave excitation force and pitch motion does not contain the effect of wave excitation moment. The bifurcation equations are obtained in the presence of 2:1 internal resonance of this system. The local bifurcation characters and stability of the periodic solutions are investigated. Osmosis phenomenon and jumping phenomenon of the Spar platform can be seen under the condition of internal resonance. Hopf bifurcations of zero solution are found when damping is ignored and the frequency of wave, frequency of heave and pitch satisfy certain relation. According to Routh- Hurwitz criterion, the stability domain of non-zero solution of heave motion is obtained and increasing pitch damping can reduce the instability domain.With and without mooring tension respectively, considering wave frequency equal to sum of heave frequency and pitch frequency, the motion characters of the platform are investigated in the presence of harmonic resonance. Applying method of multiple scales, the bifurcation equations are obtained in the presence of 2:1 internal resonance. The local bifurcation characters and stability of the periodic solution are investigated. According to Routh-Hurwitz criterion, the stability domain of zero solution is obtained and increasing heave or pitch damping can reduce the instability domain. With and without considering the effect of mooring tension respectively, numerical method is applied to obtain the characters of coupled heave-pitch motion and to analyze the influence of the mooring tension on the platform motion.