By using magnetorheological (MR) fluid as the lubricating oil in a traditional squeeze film damper (SFD), one can build a variable-damping SFD, thereby controlling the vibration of a rotor by controlling the magnetic field. This study aims to control the vibration of a flexible rotor system using a magnetorheological squeeze film damper (MR-SFD). In order to evaluate the performance of the damper, the Bingham plastic model is used for the MR fluid and the hydrodynamic equation of MR-SFD is presented. Usually, the numerical methods are necessary for solving this equation. These methods are too costly and time consuming, especially in the simulation of complex rotors and the implementation of model-based controllers. To fix this issue, an innovative estimated equation for pressure distribution in MR-SFD is presented in this paper. By integration of this explicit expression, the hydrodynamic forces of MR-SFD are easily calculated as an algebraic equation. It is shown that the pressure and forces, which are calculated from the introduced expression, are consistent with the corresponding results of the original equations. Furthermore, considering the structural and parametric uncertainties of the system, proportional-integral-furthermore controller (PID) and sliding mode controllers are chosen for reducing the vibration level of the flexible rotor system, which is modeled by the finite element method. The time and frequency responses of a flexible rotor in the presence of these controllers show a good performance in reducing vibration of the shaft's midpoint, although near the rotor's critical speed the results of the sliding mode controller (SMC) are better than the corresponding results of the PID controller. The last part of this article is devoted to an analysis of the system's uncertainties. The results of the open loop system indicate that changes in the stiffness coefficient of the elastic foundation and the temperature of the MR fluid (two uncertainties of the system) strongly affects the outputs while using the controllers well increases the robustness of the system. The obtained results indicate that both the PID and sliding mode controllers have good performance against the uncertainty of the stiffness coefficient, but for changes in the MR fluid's temperature, the SMC presents better outputs compared to the PID controller, especially for high rotational speeds.