The torsional vibrations that occur in the motorcycle steering system pose a significant problem for motorcyclist’s safety. These vibrations are particularly conspicuous in the motorcycles operated with high dynamic loads, where the impact of freeplay and friction emerges due to mechanical wear. The vibrations generated in the motorcycle steering system in the presence of freeplay and friction have peculiar nonlinear nature typical of stick-slip processes. Due to the threshold-like phenomena reflected in the nonlinearities related to freeplay and friction, it is difficult to simulate the behaviour of such systems and comprehensive preliminary research must be first carried out for this purpose. In this paper, simulation tests of the torsional vibrations that occur in a simplified equivalent model of the steering system (torsional pendulum) have been described. A mathematical model of the system has been given, inclusive of an original method of generating external inputs. Computer software developed in the Matlab-Simulink environment has been presented. The impact of computational procedures on simulation results, including impact of the method used to implement the slick-slip model in the simulation program, impact of the type of the algorithms used to integrate differential equations, and impact of the preset numerical parameters, has been analysed.