Non-linear damping coefficients from an asymmetric roll decay time series: An analytical method

Abstract

A method to extract non-linear ship roll damping coefficients from an asymmetric roll decay record is developed by following an analytical route. Customarily, the existence of bias in the roll equation is dealt with either through a constant term in the external forcing or through an asymmetric restoring. With suitable variable transformations and time scaling, the conditions of equivalence between these two representations are established. The main body of the paper begins with the derivation of the law of (asymmetric and non-linear) roll motion decay, initiated from an angle near to the vanishing angle. This involves the following three steps: (a) finding the solution (‘orbit’) of the corresponding Hamiltonian system, (b) using the velocity along the Hamiltonian orbit for approximating the amount of energy dissipated in a half-cycle of the real system, through exact analytical calculation of a number of elliptic integrals and (c) extracting the decay law by balancing the dissipated energy against the lost potential one. The use of this law for deriving non-linear roll damping coefficients is subsequently described. The applicability of the proposed methodology is demonstrated with an example.