Calculation of RAO Motion Response Factors for Ship Motion in Waves

The motion response of ships in waves represents a critical research topic in ship dynamics. The Response Amplitude Operator (RAO) characterizes the amplitude response of ship motions under wave excitations at different frequencies, serving as a fundamental parameter in frequency-domain analysis.

RAO calculation typically involves three key stages: First, establishing a hydrodynamic model incorporating hull geometry, mass distribution, and wave frequency/direction parameters. Second, solving fluid forces using potential flow theory or numerical methods to compute six-degree-of-freedom motion responses. Third, normalizing response amplitudes against wave amplitudes to derive RAO values. In MATLAB implementation, this translates to defining hydrodynamic coefficient matrices and solving complex linear systems using matrix inversion methods.

MATLAB's computational advantages for RAO analysis stem from its robust matrix operations and specialized toolboxes like PDE Toolbox for boundary element methods. Key programming considerations include wave frequency discretization strategies, efficient handling of hydrodynamic data files (such as .mat files containing added mass/damping coefficients), and optimizing frequency-domain equation solutions through vectorized operations. A standard workflow entails: configuring wave spectrum parameters, loading hydrodynamic coefficients, solving motion equations using matrix division operators (e.g., backslash operator for linear systems), and visualizing RAO curves with plot customization functions.

Interpreting RAO curves enables ship design optimization—for instance, modifying hull forms to mitigate heave or pitch responses, thereby enhancing seakeeping performance. These analyses also support operational decisions by predicting vessel behavior in specific sea states, providing theoretical basis for route planning and motion-limiting criteria. MATLAB's visualization tools further facilitate comparative analysis through multi-line plotting and interactive curve inspection features.