Derivation of Hydrodynamic Bearing Coefficients Using the Minimum Square Method

1997 ◽  
Vol 119 (4) ◽  
pp. 802-807 ◽  
Author(s):  
Carmen M. Mu¨ller-Karger ◽  
Andre´s L. Granados
Keyword(s):  
Transient Response ◽  
Equilibrium Position ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Linear Analysis ◽  
Frequency Domain Method ◽  
Transient Solution ◽  
Nonlinear Simulation ◽  
Hydrodynamic Bearing ◽  
Dynamic Coefficients

A linear analysis of the parameters for the orbital transient response of journal-bearing systems is made with the purpose of computing the bearing dynamic coefficients using the minimum square method. The journal-bearing response is obtained from a nonlinear simulation that includes a transient solution of the Reynolds equation. The minimum square method permits the adjustment of coefficients with only one orbit and does not need prior linearization of the response. Therefore it was found to be advantageous compared with the more traditional experimental method of using a frequency domain method with two orbital responses. Three different Sommerfeld numbers were analyzed. Comparisons between the eight adjusted coefficients and the linear coefficients obtained from perturbations of the Reynolds equation about the equilibrium position permit the establishment of the ranges where the bearings behave linearly.

Theoretical Condition for the Cxy=Cyx Relation in Fluid Film Journal Bearings

1989 ◽  
Vol 111 (3) ◽  
pp. 426-429 ◽  
Author(s):  
T. Kato ◽  
Y. Hori
Keyword(s):  
Reynolds Equation ◽  
Journal Bearing ◽  
Journal Bearings ◽  
Fluid Film ◽  
Finite Width ◽  
Damping Coefficients ◽  
Dynamic Coefficients ◽  
Cross Terms ◽  
Linear Damping ◽  
Theoretical Condition

A computer program for calculating dynamic coefficients of journal bearings is necessary in designing fluid film journal bearings and an accuracy of the program is sometimes checked by the relation that the cross terms of linear damping coefficients of journal bearings are equal to each other, namely “Cxy = Cyx”. However, the condition for this relation has not been clear. This paper shows that the relation “Cxy = Cyx” holds in any type of finite width journal bearing when these are calculated under the following condition: (I) The governing Reynolds equation is linear in pressure or regarded as linear in numerical calculations; (II) Film thickness is given by h = c (1 + κcosθ); and (III) Boundary condition is homogeneous such as p=0 or dp/dn=0, where n denotes a normal to the boundary.

A Numerical Solution for the Incompressible Hybrid Journal Bearing With Cavitation

1969 ◽  
Vol 91 (3) ◽  
pp. 508-515 ◽  
Author(s):  
Stanley Heller ◽  
Wilbur Shapiro
Keyword(s):  
Vapor Pressure ◽  
Pressure Gradient ◽  
Numerical Solution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Journal Bearings ◽  
Supply Circuit ◽  
Hydrodynamic Bearing

A numerical solution is presented for determining performance for hybrid journal bearings with arbitrary clearance distribution and cavitation. Regions of cavitation are determined by solution of the incompressible Reynolds’ equation. The pressures in the cavitated regions are immediately adjusted to a specified vapor pressure with zero pressure gradient. The continuity of mass equation permits coupling the influence of the external supply circuit and the methods of recess compensation to the Reynolds’ equation. Results are presented for geometrically similar hydrodynamic, hydrostatic, and hybrid bearings. Favorable comparisons are made with previously published results for the hydrodynamic bearing.

Nonlinear Effects in a Plain Journal Bearing: Part 1—Analytical Study

1991 ◽  
Vol 113 (3) ◽  
pp. 555-561 ◽  
Author(s):  
F. K. Choy ◽  
M. J. Braun ◽  
Y. Hu
Keyword(s):  
Equilibrium Position ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Fluid Film ◽  
Liquid Oxygen ◽  
Solution Technique ◽  
Space Applications ◽  
Linear Solution ◽  
Film Pressure ◽  
Linear And Nonlinear

Hydrodynamic/hydrostatic journal bearings have been widely used in various types of high speed rotating machinery. For space applications, the issue of using cryogenic fluids as working lubricants has steadily gained in significance. The primary goal of this paper is to model the nonlinearities that occur in a hydrodynamic journal bearing with both cryogenic and oil lubricants. Results will be examined through bearing fluid film pressure distribution and bearing linear and nonlinear stiffness characteristics. The numerical model that couples a variable property Reynolds equation with the dynamics of the rotor is solved by means of a finite difference solution technique. The procedure for the fluid film pressure solution involves an iterative scheme that solves the Reynolds equation coupled with the equations of state for liquid oxygen (LO2). The pressure curve is then integrated to calculate bearing supporting forces. A two-dimensional Newton-Raphson iteration method is used to locate the journal equilibrium position from which both linear and nonlinear bearing stiffness are evaluated by means of the small perturbation technique. The effects of load on the linear/nonlinear plain journal bearing characteristics are analyzed and presented in a parametric form. The relationship between the accuracy of the linear solution and the various orders (3rd, 5th, and 7th power for ΔX) of the nonlinear approximation are also discussed. The validity of both linear and nonlinear solutions at various distances from the journal equilibrium position is also examined. A complete parametric study on the effects of load, temperature, operating speed, and shaft misalignment will be given in Part 2 of this paper.

Nonlinear Dynamic Oil-Film Forces in Infinite-Short Journal Bearings

2011 ◽  
Author(s):  
Lihua Yang ◽  
Weimin Wang ◽  
Lie Yu
Keyword(s):  
Nonlinear Dynamic ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Analytic Solutions ◽  
Journal Bearings ◽  
Oil Film ◽  
Dynamic Coefficients ◽  
Representation Model ◽  
Linear And Nonlinear

In this paper, the analytic solutions of oil-film forces in infinite-short cylindrical journal bearing are calculated by solving its corresponding Reynolds equation. On this base, the linear and nonlinear dynamic coefficients of the bearing are predicted. By comparing the dynamic oil-film forces approximately represented by dynamic coefficients with the analytic solutions, the accuracy of this representation model is investigated. The results show that more orders of dynamic coefficients are included in representation model, the obtained approximate oil-film forces are more close to their analytic solutions. This can be a reference to illustrate the feasibility and applicability of representing oil-film forces by applying the dynamic coefficients of bearings.

Determination of Dynamic Coefficients in a Hydrodynamic Journal Bearing Based on the 3-D Navier-Stokes Equations and Considering Cavitation Effects

2012 ◽  
Author(s):  
Changhu Xing ◽  
Minel J. Braun
Keyword(s):  
Pressure Distribution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Maximum Magnitude ◽  
Navier Stokes Equations ◽  
Dynamic Coefficients ◽  
Finite Perturbation ◽  
Hydrodynamic Journal Bearing

Dynamic coefficients are very important for the stability of a hydrodynamic journal bearing and therefore for its design. In order to determine the stiffness, damping and added mass coefficients of the hydrodynamic bearing, the finite perturbation method around its stabilization position was employed. Based on the Reynolds equation with Gumbel cavitation algorithm, the maximum magnitude of the perturbation was judged by comparing results from finite perturbation (numerical way) to those from infinitesimal perturbation (additional analytical equations need to be derived based on order analysis), as well as theoretical analysis. Using the determined perturbation amplitude, the full three-dimensional Navier-Stokes equations in CFD-ACE+ were used to evaluate coefficients from an actual lubricant and compare to those obtained with Reynolds equation. Finally, a homogeneous gaseous cavitation algorithm is coupled with the Navier-Stokes equation to establish the pressure distribution in the bearing. When gas concentration was varied, the pressure distribution as well as the dynamic coefficients changed significantly.

scholarly journals A superlinear iteration method for calculation of finite length journal bearing's static equilibrium position

2017 ◽  
Vol 4 (5) ◽  
pp. 161059 ◽  
Author(s):  
Wenjie Zhou ◽  
Xuesong Wei ◽  
Leqin Wang ◽  
Guangkuan Wu
Keyword(s):  
Finite Length ◽  
Carrying Capacity ◽  
Equilibrium Position ◽  
Journal Bearing ◽  
Secant Method ◽  
Static Equilibrium ◽  
Computational Time ◽  
Load Carrying Capacity ◽  
Dynamic Coefficients ◽  
Load Carrying

Solving the static equilibrium position is one of the most important parts of dynamic coefficients calculation and further coupled calculation of rotor system. The main contribution of this study is testing the superlinear iteration convergence method—twofold secant method, for the determination of the static equilibrium position of journal bearing with finite length. Essentially, the Reynolds equation for stable motion is solved by the finite difference method and the inner pressure is obtained by the successive over-relaxation iterative method reinforced by the compound Simpson quadrature formula. The accuracy and efficiency of the twofold secant method are higher in comparison with the secant method and dichotomy. The total number of iterative steps required for the twofold secant method are about one-third of the secant method and less than one-eighth of dichotomy for the same equilibrium position. The calculations for equilibrium position and pressure distribution for different bearing length, clearance and rotating speed were done. In the results, the eccentricity presents linear inverse proportional relationship to the attitude angle. The influence of the bearing length, clearance and bearing radius on the load-carrying capacity was also investigated. The results illustrate that larger bearing length, larger radius and smaller clearance are good for the load-carrying capacity of journal bearing. The application of the twofold secant method can greatly reduce the computational time for calculation of the dynamic coefficients and dynamic characteristics of rotor-bearing system with a journal bearing of finite length.

scholarly journals A Nonlinear Model for Prediction of Dynamic Coefficients in a Hydrodynamic Journal Bearing

2004 ◽  
Vol 10 (6) ◽  
pp. 507-513 ◽  
Author(s):  
Jerzy T. Sawicki ◽  
T. V. V. L. N. Rao
Keyword(s):  
Equilibrium Position ◽  
Nonlinear Dynamic ◽  
Static Pressure ◽  
Journal Bearing ◽  
Dynamic Pressure ◽  
Higher Order ◽  
Pressure Gradients ◽  
Damping Coefficients ◽  
Dynamic Coefficients ◽  
Stiffness And Damping

This paper investigates the variation of nonlinear stiffness and damping coefficients in a journal orbit with respect to equilibrium position. The journal orbit is obtained by the combined solution of equations of motion and Reynolds equation. In the linearized dynamic analysis, dynamic pressure is written as a perturbation of static pressure and pressure gradients at equilibrium position. However, in order to obtain nonlinear dynamic coefficients about equilibrium position, the dynamic pressure gradients in the orbit are also written as the first order perturbation of static pressure gradients and higher order pressure gradients for displacement and velocity perturbations. The dynamic coefficients are functions of bearing displacement and velocity perturbations. The higher order pressure gradients at equilibrium position are evaluated at various eccentricity ratios and L/D ratios of 0.5 and 1.0. The variation of nonlinear dynamic coefficients is analyzed for three Sommerfeld numbers of a two-axial groove journal bearing under the action of an external synchronous load along and perpendicular to the radial journal load. Results indicate that the oil film nonlinearities affect the journal motion at lower eccentricity ratios (higher Sommerfeld numbers) with wide variation in stiffness and damping coefficients.

Journal Motion Simulation of Hybrid Journal Bearing Considering Viscosity Variation Due to Temperature Change

Tribology ◽  
2005 ◽  
Author(s):  
Vijay Kumar ◽  
S. C. Sharma ◽  
S. C. Jain
Keyword(s):  
Dynamic Response ◽  
Equilibrium Position ◽  
Journal Bearing ◽  
Critical Mass ◽  
Nonlinear Dynamic Analysis ◽  
Stability Margin ◽  
Linear Analysis ◽  
Constant Flow ◽  
Bearing System ◽  
Flow Valve

A journal bearing system, if journal is disturbed from its equilibrium position, experiences change in the hydrodynamic forces acting on it. This disturbs the equilibrium of the journal and makes its center to whirl around the static equilibrium position. The dynamic response of a journal bearing system under these conditions can be obtained using either linear or non-linear equation of journal motion. The present work is aimed to determine realistic dynamic response of hole-entry hybrid journal bearing system compensated with constant flow valve restrictor. In this paper, the nonlinearized dynamic response of the journal bearing system is studied by considering two cases of journal mass (MJ) with respect to critical mass (Mlc obtained from linear analysis. i.e MJ = Mlc and MJ > Mlc. The deviation in stability margins is established by comparing the results obtained from the linearized and nonlinearized stability analysis for each case namely, isothermal, elastohyrdostatic, thermohydroststic and thermoelastohydrostatic. The coupled solution of Reynold’s, energy, conduction and elasticity equations is obtained using finite element method and the equation of motion is computed using fourth order Runga-Kutta method. The results obtained in the present work for nonlinear dynamic analysis of a constant flow valve compensated hole-entry hybrid journal bearing shows a increase in stability margin as compared to linear analysis for a case when isothermal conditions are assumed and bearing is considered rigid. When variation of viscosity with temperature is considered i.e. THS case, the stability margin is found to be about 20% higher than that estimated by linear analysis.

Calculation of the Dynamic Coefficients of a Journal Bearing, Using a Variational Approach

1986 ◽  
Vol 108 (3) ◽  
pp. 421-424 ◽  
Author(s):  
P. Klit ◽  
J. W. Lund
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variational Approach ◽  
Numerical Evaluation ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Temperature Dependency ◽  
The Finite Element Method ◽  
Dynamic Coefficients ◽  
The Individual

The dynamic bearing coefficients are obtained from a solution to the variational equivalent of Reynolds equation. A perturbation method is applied to find the individual dynamic coefficients. The Finite Element Method is used in the numerical evaluation of the equations. The flow is assumed to be laminar, the lubricant is Newtonian. Allowance is made for viscosity-temperature dependency in circumferential and axial directions.

A Study of the Stability of an Externally-Pressurized Gas-Lubricated Thrust Bearing With a Flexible Damped Support

1978 ◽  
Vol 100 (3) ◽  
pp. 364-368 ◽  
Author(s):  
D. A. Boffey
Keyword(s):  
Feasibility Study ◽  
Equilibrium Position ◽  
Reynolds Equation ◽  
Thrust Bearing ◽  
Dynamic Instability ◽  
Pneumatic Hammer ◽  
Gas Bearings ◽  
Dynamic Coefficients ◽  
The Stability ◽  
Improve Stability

Externally-pressurized gas bearings are prone to a dynamic instability known as pneumatic hammer. This paper examines the possibility of using a flexible damped bearing support to suppress the instability. A circular thrust bearing having a central feed hole and pocket is employed in the feasibility study. The linearized gas film dynamic coefficients are derived using an adaptation of an existing solution to Reynolds equation for a long rectangular bearing. Only stability of the equilibrium position is considered. Results obtained for a support having a stiffness comparable to the stiffness of the gas film show that damping in the support can substantially improve stability.

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