This study presents a nonlinear non-stationary analysis of a rotor-disk-bearing system comprising a flexible shaft supported by two flexible bearings. The model incorporates both the geometric nonlinearity of the shaft and the linear and nonlinear stiffness characteristics of the bearings. The governing equations are derived using Hamilton's extended principle and discretized via the Galerkin method, employing the mode shapes of a rotating beam on elastic supports. This process yields a set of nonlinear ordinary differential equations. Asymptotic analysis is applied to these reduced equations to systematically investigate the influence of key parameters including bearing stiffness, damping coefficients, and unbalanced masses on the dynamic response. Particular attention is given to resonance crossing and the potential occurrence of the Sommerfeld effect. The analytical solutions are verified against numerical integration using the Runge-Kutta method, showing excellent agreement. The results demonstrate that increasing the system's acceleration reduces the peak resonance amplitude but causes it to occur earlier. Furthermore, under low acceleration conditions, increased nonlinear stiffness or unbalance mass can induce the Sommerfeld effect, preventing the rotor from traversing the critical speed.