A nonlinear model of a pinned-free long shaft rotor system with a stabilized support at arbitrary position is established by employing the generalized Hamilton principle. The lower-order polynomials are introduced to construct Ritz basis to analyze the critical rotational velocity of the system and the optimal position of the stabilized support. The result of the optimal position and the bifurcation shape are verified by applying the finite element method. Furthermore,the bifurcation modes and post-buckling solutions of the rotor system are investigated while the rotational velocity exceeding the critical value. The results show that the optimal position of the stabilized support for the pinned-free long shaft rotor system is at 73% length of the beam from the pinned end. While the rotational velocity exceeds the critical value,the trivial deflection loses its stability through the pitchfork bifurcation and the axial displacement loses its stability by the transcritical bifurcation. The investigation provides a theoretic basis for installing the stabilized support for the long shaft rotor system and for understanding the geometrical nonlinear effect of the system working at supercritical state.