During the starting of the vehicle, the friction clutch engagement sometimes generates judder. Judder prevents vehicles from starting smoothly, harms the ride comfort, and may produce damage to the drivetrain components. This unpleasant phenomenon, which often manifests in the form of noisy torsional vibrations of the drivetrain or a violent surging of starting vehicles, is an example of the many annoying problems that automotive engineers have been experiencing since the car was invented.Engineers and scientists have identified some causes of transient torsional oscillations connected to judder. Vibrations generated by the clutch facings when a special type of relationship between the friction coefficient and sliding speed occurs account for the most important source of judder. Other potential judder sources are: misalignments in the drivetrain that may induce fluctuating pressure between sliding components, some thermoelastic phenomena on contact surfaces, and/or torsional excitations from the engine and universal joint in the presence of special resonant conditions.The present paper concentrates on a research of selfexcited vibrations that are generated by a special type of friction force identified by an increase of the coefficient of friction on clutch facings when the slip speed on friction surfaces decreases. The self-excited vibrations are very interesting nonlinear phenomena. The amplitude of selfexcited vibration depends on the system parameters alone. It does not depend on the initial energy or initial conditions.The ultimate goal of the present research was to develop a reliable mathematical model for the dry clutch engagement to allow the easy engineering analysis and design of engagement for various facing materials and drivetrain configurations. The reliability of the model was to be assured by a strict correlation of the simulation with experimental results.The first basic task of the present paper was to model experimental engagements performed on a facing dynamometer that is commonly used in the automotive industry. The mathematical description, both qualitative and quantitative, was corroborated and repeatedly adjusted by extensive testing. The final model was used to study the behavior of various existing and hypothetical facing materials in dyno testing. Some selected results of this study are presented.