Tilting pad journal bearings (TPJBs) are widely applied in the high-speed rotor system whose working speed is higher than its critical speed due to excellent hydrodynamic lubrication and stability. Pivot stiffness is one of the key design parameters of TPJBs compared to other journal bearings and has become particularly important for optimizing the performance of TPJB-rotor systems. In order to improve the vibration and critical characteristics of rotor systems, the transient dynamic characteristic of a TPJB-rotor system passing through the critical speed is investigated considering different pivot stiffness ratios. A time-varying dynamics model of a symmetrical single-disc rotor supported by four-pad TPJBs is established considering constant acceleration conditions and nonlinear hydrodynamic bearing force. The disc vibration characteristic, journal vibration characteristic, pad vibration characteristic, and hydrodynamic bearing force are analyzed by using Bode plot, shaft center orbit, pad phase orbit, waterfall plot, and time history. The results show that the pivot stiffness plays a major role in the suppression of resonance amplitude and working amplitude of a TPJB-rotor system, without changing the frequency characteristic of the system. This study provides a theoretical basis for the pivot stiffness design of TPJBs and the vibration suppression of rotor systems.