Turbocharger (TC) rotor-floating ring bearing (FRB) system is characterised by high speed as well as high non-linearity. Using the run-up and run-down simulation method, this paper systematically investigates the influence of unbalance on the rotordynamic characteristics of a real TC-FRB system over the speed range from 0 Hz to 3500 Hz. The rotor is discretized by the finite element method, and the desired oil film forces at each simulation step are calculated by an efficient analytical method. The imposed unbalance amount and distribution are the variables considered in the performed non-stationary simulations. The newly obtained results evidently show the distinct phenomena brought about by the variations of the unbalance offset, which confirms that the unbalance level is a critical parameter for the system response. In the meantime, the variations of unbalance distribution, i.e. out-of-phase and in-phase unbalance, can lead to entirely different simulation results as well, which proves the distribution of unbalance is not negligible during the dynamic analysis of the rotor-FRB system. Additionally, considerable effort has been placed on the description as well as discussion of a unique phenomenon termed Critical Limit Cycle Oscillation (CLC Oscillation), which is of great importance and interest to the TC research and development.