Analytical models were developed for the dynamic analysis of an articulated loading platform (ALP) and a moored tanker system under regular wave excitations. Three different mathematical models were developed: two single-degree-of-freedom (SDOF) systems, one with a continuous and the other with a piecewise-nonlinear restoring function, and a two-degree-of-freedom (2-DOF) system with a piecewise- nonlinear restoring function.
The dynamic response and stability of the SDOF system were investigated by using the harmonic balance method and direct numerical integrations. A simple and efficient algorithm was developed to analyze the piecewise-nonlinear system by the harmonic balance method and fast Fourier transformation technique. The highly nonlinear characteristics, including subharmonics, instabilities, period-doubling bifurcations, Lyapunov exponents, and domains of attraction were examined. Possible occurrence of chaotic motion was investigated by means of direct numerical integration and the application of the classical stability theory with the aid of Hill’s type equation. It was discovered that chaotic motion developed either through period-doubling bifurcations or occurred suddenly at extremely high excitations in an unsymmetric system with continuous restoring function. Such motions will only take place under the action of a train of extremely large waves. However, the behavior of a nonlinear system is very sensitive to any changes in system parameters. The threshold of onset of chaos will change as the system parameter varies. Chaotic motions in a piecewise-nonlinear system are more likely to occur in a less severe ocean environment as compared to the continuous system.
Time domain numerical simulations were used to study the 2-DOF system. Only low order subharmonic motions were observed to co-exist with harmonic motions. Due to the nonlinearities involved in the system, subharmonic oscillations with slowly varying amplitudes have occurred at certain range of parameters even though the excitations were monochromatic. Chaotic motions were not observed in the 2-DOF system, but all the possible types of bifurcations were discovered.