A nonlinear mathematical model for coupled heave, pitch, and roll motions of a high-speed planing hull

A mathematical method is presented for time‐domain simulation of coupled heave, pitch, and roll motions of a planing hull. This method is introduced by using 2D+t theory and employs potential field related to hydrodynamic impact of an asymmetric wedge with roll speed to solve four relations for the involved added masses in the motion. Momentum variation of the derived added mass terms is used to compute 2D normal force and roll moment. Two‐dimensional hydrostatic moment and moment due to hydrodynamic forces are taken into consideration and time derivative of the wetted half‐beam is determined by using the roll speed. Three‐dimensional forces are computed by integrating the 2D forces over the length of the boat and new added mass terms are derived for the coupled heave, pitch, and roll motions. Ultimately, a nonlinear system of equations for the motion is presented by putting the forces together. Validity of the proposed method is assessed using an extensive set of test cases, which are conducted in four steps. Predicting coupled heave and pitch motions without roll motion, dynamic response of a 2D wedge due to asymmetric impact, computing the hydrodynamic coefficients in coupled heave and roll motions, and predicting roll motion of a planing boat due to a forced pitch motion are involved in the validation steps, respectively. Results of the first three steps are compared against experimental results, while the results of the last step are compared against the results of previously published simulations. Favorable agreement has been displayed between the obtained results and the available data in all of these steps. Finally, the proposed method is used to investigate the effects of the roll motion on the heave and pitch motions of planing hulls. Based on the obtained results, amplitudes of the heave and pitch motions exhibit an increase when the boat is free of roll which is due to an increase in the exciting force, accelerations, and reduction of heave and pitch added masses in this situation.