This paper investigates the accelerating up transient vibrations of a rotor system under both the random and uncertain-but-bounded parameters. The Polynomial Chaos Expansion (PCE) coupled with the Chebyshev Surrogate Method (CSM) is used to analyses the propagations of the two categorizes of uncertainties. The output responses will possess the characteristics of both bounded quantities and statistical moments. As a hybrid non-intrusive uncertainty quantification (UQ) procedure, the deterministic rotor model is taken as a black box and will only be executed at some parameter points. A number of uncertain physical parameters are studied and the corresponding transient responses are presented. The accuracy and efficiency are verified by the Monte Carlo simulations (MCS) in combination with the scanning scheme and also other hybrid analysis framework. It will provide guidance for the accurate transient dynamic analysis of engineering problems with hybrid uncertainties.