The tool RD-07 computes the dynamic coefficients of journal bearings with axial grooves, including dynamic stiffness and damping coefficients.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

## Tutorial

Example: A Journal Bearing with Two Axial Grooves

The input in the tool is

After clicking "Run", the results will be shown as follows in another page.

Kxx=  27591.51796414  N/m

Kyy=  15149.4992524  N/m

Kxy=  117098.76307453  N/m

Kyx=  -64606.7648857  N/m

Cxx=  1230.36686844  N-s/m

Cyy=  708.10322867  N-s/m

Cxy=  40.30104742  N-s/m

Cyx=  40.34992044  N-s/m

## Subject Review

### Introduction

Journal bearing, as a type of hydrodynamic bearings, are commonly used in rotor systems. The fundamental purpose of journal bearings is to provide radial support to a rotating shaft. The relative motion between the rotating shaft and the inside journal bearing surface results in a hydrodynamic pressure film, which generates force to support the rotating shaft. Fig. 1 shows a typical pressure distribution in circumferential direction of a journal bearing with two axial grooves.

Fig. 1 A typical pressure distribution of a journal bearing with two axial grooves

### Governing Equation

The governing equation for pressure distribution in fluid film bearings is Reynolds equation. Reynolds equation is derived from the Navier-Stoke equation and the continuity equation with the following assumptions: (i) the fluid is Newtonian fluid with constant viscosity; (ii) inertial and body force terms are negligible compared to the viscous term; (iii) the variation of pressure across the film thickness is negligibly small; (iv) the flow is laminar.

The Reynolds equation for a journal bearing with incompressible fluid is

where p is the distribution of the fluid pressure; x=Rθ is the circumferential coordinate in the direction of rotation as shown in Fig. 2; ω and μ are the shaft rotating speed and fluid viscosity; h=C+ecos(θ) is the fluid film thickness, with C being the radial clearance and e the eccentricity.

The dimensionless representation of Reynolds equation is

where

Boundary conditions implemented in this software is p=0 at z=0 and z=1; p=0 in grooves; p=0, dp/dθ=0 at cavitation zone (Reynolds boundary conditions).

Fig. 2 Coordinate in hydrodynamic lubrication

### Solution Method

There are many methods to solve the Reynolds equation, such as finite element method, finite perturbation method, infinitesimal perturbation method. Infinitesimal perturbation method will be briefly introduced.

For small amplitude whirl motions of frequency about the center, the pressure and film thickness can be written as:

where |ε1|<<ε0, |Φ1|<<Φ0; h0=1+εcos(θ)

Substitute above equations to the original equation and retaining up to first linear terms, to obtain the following equations

Above equations are solved by a finite difference method that uses the successive over relaxation (SOR) scheme satisfying the boundary conditions.

In rotordynamics, bearings are commonly modeled as pointwise springs and dampers. The linearized bearing forces in XY coordinate are

With the perturbed pressure p1 and p2, the bearing stiffness and damping in r-Φ coordinate are then given by

The bearing stiffness and damping coefficients in the xy coordinate are given by

### References

1. D. Childs, Turbomachinery Rotordynamics: Phenomena, Modeling, & Analysis, 1993, John Wiley & Sons Inc, New York.

2. W.J. Chen and E.J. Gunter, Introduction to Dynamics of Rotor-Bearing Systems, 2007, Trafford Publishing.