This paper describes a non-linear power take-off system within a buoyant heaving wave energy converter. The power take-off uses the relative velocity between the main structure of the wave energy converter and an inertial referenced mass contained within its outer envelope. The power take-off consists of a linear spring and a quadratic damper. In this paper the mathematical principles behind a working time domain simulation of a non-linear power take-off as applied to a relative motion inertial referenced wave energy converter are presented. The principles could, however, be applied to any differential motion system. The resulting numerical simulation is used to investigate the performance of a representative system in irregular waves, and the methodology necessary to determine optimal power extraction is discussed and applied. ‘Optimisation’ of the power take-off parameters is discussed with respect to limiting the overall relative motion and large accelerations, as indicators of the expected lifetime and the maintenance requirements of the wave energy converter.