Atsuo Maki, Naoya Umeda, Martin Renilson, Tetsushi Ueta, Analytical formulae for predicting the surf-riding threshold for a ship in following seas, Springer Japan, Springer-Verlag, Journal of Marine Science and Technology, Vol.15, No.3, pp.218-229, Aug. 2010.
Abstract: Making use of Melnikov's method, a generalized formula for predicting the surf-riding threshold is developed as an extension to the applications of Kan and Spyrou. A new analytical formula for calculating the surfriding threshold of a ship in following seas is also proposed in light of nonlinear dynamical system theory. By applying a continuous piecewise linear approximation to the waveinduced surge force, a heteroclinic bifurcation point is obtained analytically with an uncoupled surge equation. Results calculated using these formulae are presented, and they show good agreement with those obtained utilizing numerical bifurcation analysis. Further, it was confirmed that the surf-riding threshold obtained using the proposed formulae agrees reasonably well with that obtained experimentally for an unconventional vessel.