Chaos of rub-impact rotor supported by bearings with nonlinear suspension

The nonlinear dynamic analysis of the rotor-bearing system is studied in this paper and is supported by oil-film short bearings with nonlinear suspension. An observation of a nonlinearly supported model and the rub-impact between rotor and stator is needed. Therefore for more precise analysis of rotor-bearing systems, the rub-impact between rotor and stator is also proposed in this paper. The displacements in the horizontal and the vertical directions charactering the theoretical model of the system are considered with non-dimensional speed ratio. Inclusive of the analysis methods of the dynamic trajectory, the power spectrum, the Poincaré maps and the bifurcation diagrams are used to analyze the behavior of the rotor centre and bearing centre in the horizontal and vertical directions under different operating conditions. The maximum Lyapunov exponent analysis is also used in this study to identify the chaotic motion. It is concluded that the trajectory of rotor centre and bearing centre have undesirable vibrations. Especially at s=5.6, the rotor centre is at chaotic motion but the bearing centre is still at quasi-periodic motion. With the analysis of the dynamic behavior of these operating conditions, the theoretical and practical idea for controlling rotor-bearing systems and optimizing their operation can be more precise.     

Non-linear stability analysis of a rigid rotor on tilting pad journal bearings

This paper describes a method for conducting non-linear stability analysis on the synchronous motion of the journal axis and tilting pads in a system consisting of a symmetrical rigid rotor mounted on tilting pad journal bearings with two pads. The method makes it possible to obtain the limit curves separating the stable operating condition, characterized by the above motion, from the so-called unstable one characterized by motions of the pads and journal with a fundamental frequency equal to one half the rotor angular speed. A variational method is used to carry out stability analysis on the synchronous solution of the non-linear equations of motion, obtained in an approximated analytical form using a particularized harmonic balance method.     

High-speed stability of a rigid rotor supported by aerostatic journal bearings with compound restrictors

Demands for higher rotational speed and accuracy for effective manufacture of small holes on printed circuit boards and very small precise parts have been increasing remarkably. Aerostatic journal bearings with compound restrictors have greater stiffness than those with conventional inherently compensated restrictors and are one of the most effective candidates to satisfy these demands. In this work, the instability of a rigid rotor supported by aerostatic journal bearings with compound restrictors was investigated numerically and experimentally. It was found that this type of aerostatic bearings showed a much higher threshold speed for instability compared with bearings with inherently compensated restrictors.     

Rotor dynamic instability analysis on hybrid air journal bearings

Hybrid air journal bearings with multi-array of 1, 2, 3, 4, or 5-row orifice feedings are analyzed for the problem of rotor dynamic instability. The bearing stiffness and damping coefficients are calculated numerically to determine threshold rotor mass under various operating conditions. The hybrid porous air journal bearings are also analyzed for comparison to investigate the similarities in dynamic characteristics between the multi-array of orifice feeding bearings and the porous bearings. The results show that the porous bearing is more stable than the orifice feeding bearing at lower rotation speeds (Λ<0.1) or at higher rotation speeds (Λ>1) with lower feeding parameters (λ_P<10⁻⁸). The 5-row orifice feeding bearing is more stable than the porous bearing at moderate speeds (0.3<Λ<0.6) with lower feeding parameters (λ₀<10⁻⁴).     

Bifurcation analysis of flexible rotor supported by couple-stress fluid film bearings with non-linear suspension systems

This study analyzes the dynamic behavior of a flexible rotor supported by two couple-stress fluid film journal bearings with non-linear suspension systems. The analysis employs a short journal bearing assumption and considers the respective effects on the dynamic response of the system of the lubricant type (i.e. Newtonian or non-Newtonian), the bearing housing suspension system (i.e. linear or non-linear), the rotor speed and the degree of rotor unbalance. The behavior of the system is analyzed by reference to bifurcation diagrams, dynamic trajectory diagrams and Poincaré maps, respectively. In general, the results show that the dynamics of the system are significantly dependent on the rotor speed and the degree of rotor unbalance. Furthermore, it is found that the use of a couple-stress fluid lubricant enhances the dynamic stability of the rotor-bearing system considerably compared to that obtained when using a traditional Newtonian lubricant. Finally, the results show that the common assumption of a linear journal housing suspension system results in a significant underestimation of the vibrational amplitudes of both the rotor and the bearing and should therefore be replaced by a non-linear assumption.     

On the non-linear stability of self-acting gas journal bearings

The non-linear stability of finite length self-acting gas journal bearings is studied by solving a time-dependent Reynolds equation using finite difference method. Two threshold values are discovered instead of one through which the self-acting gas journal bearings are changed from stable to unstable state. Keeping other parameters unchanged, when the mass of rotor exceeds the upper threshold value of mass, the system gets unstable; when the mass is less than the lower threshold value of mass, the system is stable; when the mass is between these two threshold values, the stability of bearings is determined by the amplitude of disturbance. Similar facts of two threshold values are also found when the stability parameter is chosen as external force or aspect ratio.     

Nonlinear dynamic and bifurcation analysis of short aerodynamic journal bearings

A numerical analysis of a rigid rotor supported by relative short aerodynamic journal bearings is presented for nonlinear dynamic behaviors and bifurcation. The compressible Reynolds' equation is solved by the finite differences method and the successive over relation method and a time-dependent mathematical model for aerodynamic journal bearings is studied. A comparison of the results for the system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams is made and dynamic behavior of the rotor center in the horizontal and vertical directions under different operating conditions is analyzed. The analysis shows how the existence of a complex dynamic behavior comprises periodic and subharmonic response of the rotor center. This paper shows how the dynamic behavior of this type of system varies with changes in bearing number and squeeze number.     

Stability of tri-taper journal bearings under dynamic load using a non-linear transient method

Stability of a tri-taper journal bearing which is subjected to steady, periodic and variable rotating loads is studied theoretically using a non-linear transient approach. The hydrodynamic forces are initially obtained by solving the Reynolds equation, satisfying the Reynolds boundary conditions. Further, the transient behavior of rotor bearing system is predicted by substituting these hydrodynamic forces in the equations of motion and then solving them by fourth order Runge Kutta method. Stability of the rotor bearing system is determined from the journal locus. Comparative studies predict that the dynamic performance is superior for a bearing with high ramp size and aspect ratio.     

Stability of a rigid rotor in turbulent hybrid porous journal bearings

Stability characteristics of hybrid porous journal bearings with a turbulent fluid film have been investigated theoretically following Constantinescu's turbulent lubrication theory. The stability curves have been drawn for different Re, eccentricity ratios, slenderness ratios and bearing speed parameters. In the absence of any experimental data, laminar flow results obtained by this analysis have been compared and found to be in excellent agreement with the previous results. It is observed that turbulence deteriorates the stability of the rotor and for better performance the value of the bearing feeding parameter, β, should be kept small.     

Theoretical analysis of the non-linear behavior of a flexible rotor supported by herringbone grooved gas journal bearings

This paper studies the behavior of a flexible rotor supported by a herringbone-grooved gas journal-bearing system. A hybrid method is employed to develop a time-dependent mathematical model of the bearing system. The finite difference method is employed with the successive over relaxation technique to solve the Reynolds equation. The system state trajectories, Poincaré maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and the journal center in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and quasi-periodic responses of the rotor and the journal center. The present numerical study illustrates the relationship between the dynamic behavior of this type of system and the rotor mass and bearing number. As such, the present results provide a deeper understanding of the non-linear dynamics of gas film rotor-bearing systems.     

Thermo-hydrodynamic analysis of herringbone grooved journal bearings

Steady-state and stability characteristics of herringbone grooved journal bearings (HGJBs) are found considering thermal effect. The temperature of the fluid film rises significantly due to the frictional heat, thereby the viscosity of the fluid and the load carrying capacity decrease. A thermodynamic analysis requires the simultaneous solution of Reynolds equation along with energy equation of the fluid and heat conduction equations in the bush and the shaft. The linearized first-order perturbation technique is employed for the prediction of stiffness and damping coefficients of the oil film. Thereafter mass parameter and whirl ratio are found from the stability analysis. It is difficult to obtain the solution due to the numerical instability when the bearing is operated at high eccentricity ratios.     

Nonlinear behavior analysis of a rotor on two-lobe wave journal bearings

The dynamics of an unbalanced rigid rotor on two-lobe journal bearings with wave profile, symmetrically placed at its ends, is studied in numerical way. The radial dissymmetry of the shape given to the bearing bore makes the angular orientation of the same bearing one of the main parameters in the study of the system dynamics. Further parameters adopted in the investigations are the amplitude of the wave component which characterizes the profile of the bearing bore, the static unbalance and the speed of the rotor. Brute-force integration of system equations has been carried out in order to obtain qualitative portraits of the system dynamics in terms of bifurcation diagrams and trajectories of the journal centre. The study also illustrates the way numerical continuation method can highlight the dependence of the stable, synchronous response on the adopted parameters and allow a tracking of the same motion in a simulated run-up of the rotor.     

Performance analysis of micropolar lubricated journal bearings using GDQ method

In this work generalized differential quadrature (GDQ) method as a simple, efficient and high order numerical technique is used for the solution of modified Reynolds equation to obtain the performance of micropolar lubricated hydrodynamic circular and noncircular lobed journal bearings. Effects of micropolarity of the lubricant as well as noncircularity parameter (preload) of the bearings are investigated. Comparing the GDQ method results with the results of FDM/FEM, good agreement between them is observed. The results also show that though the micropolarity of the lubricants improves the performance of the bearings, however, the rate of enhancement is affected by noncircularity of the bearings.     

Non-linear analysis of finite hydrodynamic journal bearings in laminar and turbulent flow regimes

The Navier-Stokes equations were modified by adopting the non-linear turbulence theory proposed by Constantinescu to take into account the turbulent stresses. The modified equations were solved by the finite element method using Galerkin's technique and a suitable iterative procedure.The performance characteristics of a finite hydrodynamic journal bearing (aspect ratio b = 1) were studied in terms of the load-carrying capacity, non-dimensional stiffness and damping coefficients and critical journal mass at various eccentricities for Reynolds' numbers up to 13 300. The computed results are compared with those obtained using the twodimensional triangular element formulation and the linearized turbulence theory.     

Dynamic performance of plain gas journal bearings

This paper presents dynamic performance characteristics of plain gas journal bearings. The perturbation formulation suggested by Lund has been modified to obtain stiffness and damping properties. Since rotor bearing axes are never perfectly parallel, the effect of skew has also been considered. Stability studies have been carried out for selected compressibility parameters     

Transient response of a journal supported on elastic bearings

By considering the inherent flexibility of the bearing liner, the non-linear dynamic response of a journal bearing system is determined in this study. The transient analysis is presented and the journal centre motion trajectories are plotted for rigid and flexible bearings. Effects of the different parameters such as the eccentricity ratio, and the deformation coefficient etc on non-linear trajectories are reported in this paper. The transient response of the system is also obtained for different values of the normalized journal mass which shows that a rotating system supported in a flexible bearing may have a better performance under dynamic conditions than one supported in rigid bearings.     

An approximate solution of oil film forces of turbulent finite length journal bearing

An approximate analytical method is proposed for calculating oil film forces of turbulent finite length journal bearings. The dynamic ‘π’ oil film assumption is usually taken to determine oil film forces in the dynamic analysis of hydrodynamic journal bearing-rotor systems. However oil film field is not in the ‘π’ zone, i.e. the start position of the oil film is not 0, and the termination position of the oil film is not ‘π’. Based on the variational principle, separation of variables is employed to obtain the pressure distribution in this paper. The pressure distribution of the infinitely long journal bearing model is taken as a circumferential separable function of the pressure distribution. The start and termination positions of oil film in the circumferential direction are determined by using the continuity condition. The axial separable function of the pressure distribution is obtained by the variational principle and the circumferential separable function. The results calculated by the proposed method are in good agreement with the oil film forces by the finite element method, and the computing cost is reduced greatly. Meanwhile, the influence of the parameters on the oil film forces is analyzed.     

Rapid performance evaluation of journal bearings

This paper describes a rapid method for evaluating the significant design parameters such as load capacity, maximum pressure, flow, power loss, and maximum temperature in the oil film. The proposed analytical pressure expression is a modification of that given by Reason and Narang. An analytical expression for maximum pressure is presented. The accuracy of the proposed modification is validated up to an eccentricity ratio of 0.99. The effective temperature rise, which depends on the fraction of heat generation carried away by lubricant, is chosen to be a function of the eccentricity ratio. An expression for maximum temperature, based on existing experimental findings, is given. A journal bearing design table is provided to help the designer without the involvement of numerical and mathematical complexities.     

Stability of journal bearing systems using non-Newtonian lubricants: a non-linear transient analysis

The effect of the non-Newtonian behaviour of lubricants, resulting from the addition of polymers, on the performance of hydrodynamic journal bearings is investigated. The model of non-Newtonian lubricant developed by Dien and Elrod is taken into consideration. An attempt has been made to evaluate the mass parameter (a measure of stability) besides finding out the steady-state characteristics of finite journal bearings with non-Newtonian lubricants. A non-linear time transient analysis is carried out for the stability analysis.     

A parametric study on bubbly lubrication of high-speed journal bearings

The influence of aerated oil on high-speed journal bearings is examined by classical thermohydrodynamic lubrication theory coupled with analytical models for viscosity and density of the air–oil mixture in the fluid-film bearing. Convection to the walls, mixing with supply oil and recirculating oil, and some degree of journal misalignment are considered. The parameters considered for the study of bubbly lubrication are oil aeration level, air bubble size, shaft misalignment and shaft speed. The results show that air bubbles can more clearly bring on increasing load capacity under high-speed operation of a plain journal bearing than under previous normal speed operation. Moreover, the load capacity may be increased more by oil aeration under the conditions of shaft misalignment and increasing speed.