In the transmission systems of vehicles, unforced vibrations can be observed during the sliding phase of clutch engagement. These vibrations are due to frictional forces and may generate noise. Several studies have shown that the stability of such friction systems is highly sensitive to parameters (friction law, damping) which lead to significant dispersions. Therefore, uncertain parameters must be considered in the stability analysis of a clutch system. This paper investigates the ability of the multi-element generalized polynomial chaos to take an increasing number of uncertain parameters into account in the stability analysis of a clutch system: it focuses on accuracy, on the criterion for the choice of the order of truncation, on the criterion for the choice of the number of elements and on computation time. The objective is to propose a low-cost model of high accuracy, compared with the Monte Carlo method.