In this work a formulation for the analysis of shaft-rotor-bearing type rotating systems is extendend to accommodate the effects of hydrodynamic bearings in its dynamic response. These effects, which are associated to the nonlinear force on the shaft at the bearings, are dependent of the transverse displacements, transverse linear velocities an the angular veolicty of the shaft. The structure behavior is modeled by employing the finite element method. The shaft is represented by the two node timoshenko model for bearns, with four desgrees-of-freedom per node and Hermite interpolation functions to represent the displacement fields along the bearn axis. Rotors are modeled by using concentrated inertia elements associated to the shaft degrees-of-freedom at the bearing nodal point. In the numerical analysis considering the time integration of the system differential equation, a step-by-step procedure was employed with the newmark technique in this unconditionally stable form. Due to the nonlearities associated with the hydrodynamic bearings, the solution of the system of equations is obtained using a modified Newton-Raphson precedure at each time step for solution convergence. In the evaluation of the proposed computacional system, comparison with solutions obtained from analytical/numerical results available in the literature are used. Also, a numeric represemtation of tilting-pad bearings is presented using the theory for plain journal bearings, under the same simplified conditions. In this case an evaluation of the numerical procedure is given by comparing calculated solutions with experimental results obtained from the evaluation of a hydrogenaration plant provided by CEPEL-Brazilian Research Center For Eletrobras. In both plain an tilting-pad journal bearing numerical procedures, the idealized Jeffcott rotor is employed as a case study for different operating conditions. As a result, it is shown that the solutions associated to the main oil whirl and oil whip effects and afterwards dynamic stabilization are represented by the proposed numerical procedures employed.