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.     

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.     

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).     

Non-linear dynamic analysis of rub-impact rotor supported by turbulent journal bearings with non-linear suspension

This study presents a dynamic analysis of a rub-impact rotor supported by two turbulent model journal bearings with non-linear suspension. The dynamics of the rotor center and bearing center are studied. The analysis of the rotor-bearing system is investigated under the assumptions of turbulent lubricant flow and a short bearing approximation. The spatial displacements in the horizontal and vertical directions are considered for various non-dimensional speed ratios and dimensionless unbalance parameter. The numerical results show that the stability of the system varies with the non-dimensional speed ratios and the non-dimensional unbalance parameters. The contribution of this study is to provide a further understanding of the characteristics and dynamic behaviors of the rotor-bearing system.     

Static and dynamic characteristics of short wavy journal bearings

A general analysis has been done to investigate the static and dynamic performance of short wavy journal bearing. Reynolds equation is solved for both steady‐ and unsteady‐state operations. The steady‐state analysis has been done and compared with circular bearing with respect to their load capacity, flow rate, pressure distribution and friction factor. The unsteady‐state analysis has been used to determine the rotor dynamic coefficients for various eccentricity ratios. These coefficients are used to determine the stability parameters, such as critical mass and whirl ratio, and prove that wave bearing has not only better load carrying capacity but also provides better stability at high speed than circular bearing. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Linear stability of short journal bearings with consideration of flow rheology and surface roughness

The combined effects of surface roughness and flow rheology on the linear stability of a rigid rotor supported on short-length journal bearings are analyzed. The modified Reynolds equation and the rotor motion equation are linearized about the equilibrium position and the fluid film is modeled as spring and damping elements. From the characteristic equation, the instability threshold is then obtained with various surface roughness parameters (standard derivation, σ, and Peklenik number, γ), flow rheology (power-law index, n) and eccentricity ratios (). The results show that the size of the stability regions of different roughness patterns has the following characteristics: longitudinal (γ>1)>isotropic (γ=1)>smooth>transverse (γ<1). The stability of the bearing system deteriorates with decreasing power-law index. Moreover, there are crossing points in the vicinity of =0.45 among the curves of dimensionless speed parameter ( ) associated with various power-law indices and surface roughness parameters.     

Investigations on the thermal effects in non-circular journal bearings

A comparative study based on the thermal performance of elliptical and offset-halves journal bearings has been carried out by solving energy equation while assuming Parabolic Temperature Profile Approximation across the fluid film for faster computation of temperatures. Investigation for the rise in oil film temperatures, thermal pressures, load capacity, and power loss for three commercially available grade oils have been carried out for bearing configurations under study. It has been found that the offset-halves journal bearing runs cooler when compared with elliptical journal bearing profile with minimum power loss and good load capacity using Oil 2 as lubricant for which minimum thermal degradation has been observed.     

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.     

Load performance analysis of three‐pad fixing pad aerodynamic journal bearings with parabolic grooves

Three‐pad fixing pad aerodynamic journal bearings (TPFPAJBs) have been widely used in precision instruments due to their low friction, high stability and non‐pollution. In order to improve the load performance of TPFPAJBs, parabolic grooves are opened in the bearing pad surfaces. By opening parabolic grooves in various bearing pad surfaces, the effects of the orientation angle, distance, width and depth of the grooves on the load performance of micro‐grooved TPFPAJBs can be investigated. The numerical results show that the location of the micro‐grooved bearing pads can greatly affect the load performance of micro‐grooved TPFPAJBs. When the given bearing pad surface is grooved, the effects of the bearing number and width‐to‐diameter ratio on the load performance of micro‐grooved TPFPAJBs can be studied. The bearing number and width‐to‐diameter ratio are observed to have significant influences on the load performance of micro‐grooved TPFPAJBs. Copyright © 2015 John Wiley & Sons, Ltd.     

Rotation effects on hybrid air journal bearings

Rotation effects of hybrid air journal bearings with multi-arrays of orifice feedings are investigated numerically. Porous air bearings are also solved for comparison. The results show that bearing load capacity W increases faster with eccentricity ratio than with rotation speed, i.e. bearing number Be. There are optimum orifice diameters, i.e. optimum feeding parameters λ_o, which give maximum load capacity W for orifice feeding; but for porous feeding, load capacity W increases with feeding parameters λ_p. It was found that the load capacity increases with feeding arrays of orifices and five rows of orifice feedings can approximate the operations of porous bearings very well. It was also found that load capacity W does not increase further when air supply pressure exceeds 5 atm because there is a critical pressure ratio through orifice (e.g. (P_o/P_s)<0.53).     

Dynamic stiffness and damping coefficients of aerodynamic tilting-pad journal bearings

The dynamic gas–film forces of aerodynamic bearing often can be characterized by eight linear stiffness and damping coefficients. How to theoretically predict these coefficients is a very difficult issue for tilting-pad gas bearing design because of its structural complexity. The current study presents a novel and universal theoretical analysis method for calculating the dynamic stiffness and damping coefficients of aerodynamic tilting-pad bearing. The gas–film pressure within the bearing is expressed in the form of dimensionless compressible gas-lubricated Reynolds equation, which is solved by means of the finite element method. With the assumption that the journal and the pads are disturbed with the same frequency, the dynamic coefficients of tilting-pad gas bearing are computed by using the partial derivative method and the equivalent coefficient method. Finally, the investigations are conducted about the effects of bearing number, perturbation frequency of the journal and the pads, eccentricity ratios, preload and length-to-diameter ratio of the bearing on the dynamic coefficients of aerodynamic tilting-pad journal bearing. The numerical results indicate that the dynamic stiffness and damping coefficients of tilting-pad gas bearing are closely related with these factors. The proposed analytical method provides a valuable means of predicting dynamic performances of tilting-pad gas bearing. The solution can be used for the purpose of prediction of dynamic behavior of the rotor systems supported by aerodynamic tilting-pad bearings.     

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.     

An elastohydrodynamic study on two-axial-groove journal bearings

An elastohydrodynamic study of two-axial-groove journal bearings operating in laminar and superlaminar flow regimes is presented. The effect of transition is included in turbulence analysis. Static and dynamic performance characteristics are presented for several values of deformation coefficient and Reynolds number. Illustrative examples are cited to demonstrate the significance of the present investigation.     

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     

气体动压径向短轴承动力学和分岔分析

径向气体润滑轴承是精确定位以及微旋转机械中的重要部件,其非线性动力学特性对整个器件的稳定性和可靠性具有重要的影响。文中以气体动压径向短轴承—转子系统为研究对象,建立气体轴承—转子系统耦合非线性动力学方程,以轴承数为分岔参数,分析长径比为0.75时在不同初始偏心率条件下,系统动力学特性随转子转速变化的影响。     

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.     

Measurement of wear in radial journal bearings

In this article, the measurement of wear in radial journal bearings is discussed, where a distinction is made between stationary and non-stationary contact conditions. Starting with Holm/Archard's wear law, equations are derived for the calculation of the specific wear rate k of the bearing material as a function of the wear depth d, measured after an experiment or a period of use in practice. It is also possible to calculate a value for the maximum allowable apparent pressure p_a at given values of k, required lifetime and maximum allowable wear depth d, or a value of d at a given value of p_a. In order to facilitate the use of the equations mentioned above, non-dimensional diagrams are presented. Two examples are given to explain the use of these diagrams.     

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.