With small-sized high-speed rotors, such as exhaust turbochargers in cars and trucks, floating bush bearings are used. The multi-body simulation of the run-up of exhaust gas turbocharger rotors with nonlinearly modelled floating bush bearings represents a great challenge for hydrodynamics and rotor dynamics. The development of computer time-efficient methods and modelling techniques as well as the explanation of the various instability phenomena were the major objectives of the FVV project “Run-up Simulation” (No. 912). The task was given by Forschungsvereinigung Verbrennungskraftmaschinen e. V. (FVV) and was solved at Karlsruhe Institute of Technology (KIT) and University of Kassel.