This paper introduces mathematical modelling techniques that are designed to analyse the dynamic behaviour of a class of rotary vane engines. These engines employ non-circular gears for the transmission of primary torque and the modelling explores ways to ensure that torque fluctuations in the flywheel are small. The basic physical principles of rigid-body dynamics are used to formulate a system of non-linear ordinary differential equations describing the rolling motion of two-dimensional rigid laminae. It is argued that such a system offers a valuable first approximation for the study of a realistic engine consisting of two connected combustion units.