The calculated dynamic response of an excited rotating system can be significantly affected by uncertainties in its mechanical properties, such as mass, stiffness, geometrical imperfections, or loadings. For this reason, it is essential to understand and quantify the influence of uncertain parameters on the predicted rotor response. This paper aims to optimize the propagation of random input uncertainties for a rotordynamic problem and assess their influence on the dynamic behaviour of an unbalanced rotor. The Harmonic balance method (HBM) and a non-intrusive Polynomial Chaos Expansion (PCE) are used to evaluate the stochastic response of a finite element rotor. The proposed stochastic approach is based on a numerical quadrature calculation of integrals for finding the coefficients of the PCE. The method is initially applied to evaluate the stochastic response of a linear rotodynamic system, leading to the original concept of stochastic Campbell diagram and further extended to nonlinear rotordynamic problems, using the Asymptotic Numerical Method (ANM).