Mechanical systems are typically described with finite element models resulting in high-dimensional dynamical systems. The high-dimensional space excludes the application of certain investigation methods like numerical continuation and bifurcation analysis to investigate the dynamical behaviour and its parameter dependence. Nevertheless, the dynamical behaviour usually lives on a low-dimensional manifold but typically no closed equations are available for the macroscopic quantities of interest. Therefore, an equation-free approach is suggested here to analyse and investigate the vibration behaviour of nonlinear rotating machinery. This allows then in the next step to optimize the rotor design specifications to reduce unbalance vibrations of a rotor-bearing system with nonlinear factors like the oil film dynamics. As an example we provide a simple model of a passenger car turbocharger where we investigate how the maximal vibration amplitude of the rotor depends on the viscosity of the oil used in the bearings.Copyright © 2009 by ASME