## 1 Introduction  
Hydrodynamic bearings play an important role in the dynamics of turbomachines and are widely used in industry because they allow high loads at low and high rotating speeds. The bearings properties highly influence the dynamic characteristics of a rotor, and instability phenomena tend to especially appear at high rotating speeds. The latest advances in the metal-mechanical industry have overcome most bearing machining problems. New materials and designs have brought increased bearing life, which are no longer the main reason for fault or malfunction in certain applications. Thus, the interactions of bearings with the surrounding mechanical system are drawing attention. The dynamics of the bearing is mainly governed by support zones between its fixed and movable parts since these transmit all vibration between the fixed parts of the bearing and the moving parts of the system. Therefore, the oil film in these interfaces, mainly provided to increase the bearings' life, completely separating the contact surfaces, has a primordial role in the dynamic interactions. This thin film will, therefore, absorb part of the transmitted energy. Although the equation governing the behavior of the oil film inside the bearing, Reynolds' equation, dates from the late nineteenth century, it was only after mid-twentiethcentury that its complete solution became viable, mainly due to the advent of computer science. The first use of computers for the solution of the Reynolds Equation, with the correct boundary conditions, was achieved in 1956 by [PERSON]. [PERSON] analyzed the pressure distribution in elliptical bearings [1], partial arc [2] and three-lobe bearings [3], experimentally investigating the instability of these geometries. From then, with very little time, a wide spectrum of solutions were developed for numerous types of bearings, using both gas and oil as a lubricant [4]. Regarding the oil film dynamics, [PERSON] [5; 6] published a method for linearizing the bearing forces through the calculation of coefficients of stiffness and damping, to be introduced in the equation of motion of the rotating system, which has improved the studies in rotor dynamics. From then on, the study of linear coefficients behavior and their representativeness in the dynamic response of machines spread, becoming the most widely used way for representing the influence of hydrodynamic bearings on the behavior of rotating systems, as can be seen in [7; 8; 9; 10]. Despite the conveniences introduced by using linear coefficients of stiffness and damping for bearings, these bearings can present, in some cases, nonlinear nature that influences the equation of motion for the entire rotating system. The instability behavior of rotors supported by bearings is widely known as a nonlinear phenomenon and continues to attract the attention of researchers [11; 12]. However, nonlinear vibrations can occur in many other, and more realistic, operational conditions. Therefore, nonlinear rotor models were developed and analyzed by several authors, as can be seen in [13; 14; 15; 16; 17; 18]. In [PERSON] et al. [17], cylindrical hydrodynamic bearings were subjected to several operating conditions in order to evaluate different nonlinear behaviors. The authors tested the influence of structural damping, the gyroscopic effect, journal eccentricity and external excitation force. Simulation of the journal orbits inside the bearings permitted the observation of differences between the linear and nonlinear models, validated through experimental procedures. It was concluded that the greatest nonlinear impact factor on hydrodynamic forces was external excitation. The results, using linear and nonlinear approaches, diverged with the increase of the unbalance mass, indicating that the linear model was unsatisfactory for system representation in these conditions.