Theoretical analyses of hydrodynamic fluid film bearings with different bearing profiles rely on solutions of the Reynolds equation. This paper presents an approach used for analysing the so-called pocket bearings formed from a combination of offset circular bearing profiles. The results show that the variation of the dynamic bearing characteristics with different load inclinations for the pocket bearings is less than that for the elliptic bearing counterpart. It is shown that the natural frequencies as well as the critical speeds, and hence the vibrational behaviour, can also be significantly different for an industrial rotor supported by the different bearings.
In order to increase productivity and reduce machine downtime, hydrodynamic bearings are frequently used to support high-speed rotating machinery where reliability, long running life, and minimum vibration levels are of primary concern. Simple circular bore journal bearings sometimes cause instability, which may result in catastrophic failure of the machinery. To improve resistance to such failure, different bearing types with different bearing clearance profiles have been developed and used in practice. Typical examples are elliptic bearings and tilting pad bearings, the former providing stabilizing preload and the latter minimizing the troublesome cross-coupled bearing forces. Another bearing type, formed by a combination of offset circular bearing profiles and referred to as a pocket bearing, is also sometimes used in machinery such as turbogenerators.
The theoretical analysis of hydrodynamic bearings relies on the solutions of Reynolds equation, a partial differential equation derived from the Navier-Stokes and continuity equations under certain simplifying assumptions [
The schematic of a simple circular hydrodynamic fluid film bearing is shown in Figure
Schematic of a circular bearing.
Upon integration of the film pressure, the fluid film force components in
Note that for more than one pad or clearance profile, it is the resultant of the forces on all the pads or clearance profiles in the
The above is the summary of the general theory for determining the static and dynamic bearing characteristics (i.e., Sommerfeld number, attitude angle, stiffness coefficients, and damping coefficients). Different bearing configurations impose different geometric relationships for the position of the journal in the bearing.
Two types of pocket bearings are to be considered. The first type, formed in a similar way to an elliptic bearing (Figure
Schematic of an elliptic bearing.
Schematic of type-1 pocket bearing.
The second type of pocket bearing, referred to as type 2, is shown schematically in Figure
Schematic of type-2 pocket bearing.
Normally, the bearing clearance
From (
For the pocket portion:
Geometric relationship between the arc centres for elliptic and type-1 pocket bearings.
Geometric relationship between
Again, following an approach similar to that for the type 1 bearings, one has for the side arcs
Geometric relationship between
The relationships between
Geometric relationship between the arc centres for type-2 pocket bearing.
To calculate the static and dynamic characteristics of the pocket bearings, Gauss-Seidel iteration with successive over relaxation is used to solve the finite difference formulation of the Reynolds equation [
Figure
Comparison of the nondimensional bearing characteristics for the two types of pocket bearings ((1) and (2) correspond to type 1 and 2, resp.).
Figures
Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at
Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at
A sample vibration analysis is also performed for an industrial rotor subjected to gravity loading and supported on either type of pocket bearings or similar elliptic bearings. The rotor is shown schematically in Figure
Sample rotor bearing data.
Rotor: Young’s modulus = 210 GPa, density = 7850 kg/m3 | |||||
No. | Length (m) | Diameter (m) | Disk mass (kg) | Disk Ip (kg-m2) | Disk Id (kg-m2) |
1 | 0.14 | 0.135 | 0 | 0.0358 | 0.0436 |
2 | 0.16 | 0.18 | 0 | 0.1294 | 0.1329 |
3 | 0.148 | 0.18 | 0 | 0.1193 | 0.1131 |
4 | 0.148 | 0.18 | 0 | 0.1193 | 0.1131 |
5 | 0.115 | 0.258 | 8.392 | 0.5447 | 0.3336 |
6 | 0.115 | 0.277 | 1.185 | 0.5447 | 0.3336 |
7 | 0.115 | 0.277 | 1.185 | 0.5447 | 0.3336 |
8 | 0.106 | 0.328 | 11.87 | 1.291 | 0.7217 |
9 | 0.106 | 0.353 | 0.921 | 1.291 | 0.7217 |
10 | 0.049 | 0.345 | 0 | 0.535 | 0.2747 |
11 | 0.076 | 0.44 | 89.29 | 8.643 | 4.408 |
12 | 0.058 | 0.5 | 47.73 | 6.559 | 3.317 |
13 | 0.15 | 0.556 | 93.86 | 20.57 | 8.875 |
14 | 0.224 | 0.556 | 139.5 | 30.48 | 14.42 |
15 | 0.2 | 0.556 | 111 | 25.38 | 12.52 |
16 | 0.219 | 0.556 | 128.4 | 28.57 | 14 |
17 | 0.217 | 0.556 | 126.6 | 28.19 | 13.88 |
18 | 0.216 | 0.556 | 122.8 | 27.57 | 13.77 |
19 | 0.215 | 0.556 | 119.1 | 27.05 | 13.66 |
20 | 0.214 | 0.556 | 119.1 | 26.87 | 13.59 |
21 | 0.221 | 0.547 | 126.7 | 26.83 | 13.6 |
22 | 0.198 | 0.55 | 117.2 | 24.64 | 12.55 |
23 | 0.159 | 0.704 | 4.15 | 30.62 | 16.34 |
24 | 0.159 | 0.704 | 4.15 | 30.62 | 16.34 |
25 | 0.16 | 0.497 | 249.4 | 30.81 | 16.46 |
26 | 0.015 | 0.345 | 0 | 0.1638 | 0.0821 |
27 | 0.168 | 0.353 | 1.467 | 2.056 | 1.335 |
28 | 0.168 | 0.353 | 1.467 | 2.056 | 1.335 |
29 | 0.17 | 0.353 | 1.484 | 2.081 | 1.359 |
30 | 0.187 | 0.315 | 0 | 1.419 | 1.043 |
31 | 0.07 | 0.385 | 38.39 | 3.034 | 1.559 |
32 | 0.238 | 0.315 | 0 | 1.806 | 1.59 |
33 | 0.238 | 0.315 | 0 | 1.806 | 1.59 |
34 | 0.067 | 0.42 | 25.1 | 2.904 | 1.489 |
35 | 0.065 | 0.58 | 67.21 | 12.73 | 6.436 |
36 | 0.07 | 0.58 | 131 | 22.1 | 11.16 |
Bearings: | |||||
No. | Length (m) | Diameter (m) | Clearance (mm) | Viscosity (Ns/m2) | Location |
1 | 0.092 | 0.18 | 0.1327 | 0.014 | 2 |
2 | 0.204 | 0.315 | 0.2322 | 0.014 | 32 |
Damped natural frequencies of a rotor supported by elliptic or pocket bearings ((1) refers to type 1, etc.).
Speed (rpm) | 1000 | 2000 | 3000 | 4000 | 5000 |
Elliptic | Real, Imag (rad/s) | ||||
Mode 1 | |||||
Mode 2 | |||||
Mode 3 | |||||
Real, Imag (rad/s) | |||||
Mode 1 | |||||
(1) | |||||
(2) | |||||
Mode 2 | |||||
(1) | |||||
(2) | |||||
Mode 3 | |||||
(1) | |||||
Mode 4 | |||||
(1) | |||||
(2) |
Sample rotor bearing system.
Campbell diagrams for a rotor supported by (a) elliptic bearings or (b) type-1 pocket bearings.
It can be seen that the system behaves quite differently with the different types of bearing supports. The rotor with the elliptic bearing has three natural frequencies but two critical speeds in the speed range; while the rotor with the type-1 pocket bearing has four natural frequencies but only one critical speed. The second mode in both cases is a backward whirl and is therefore unlikely to be excited. Using equivalent clearances for the different pocket bearing models, the natural frequencies in Table
An approach to evaluate the static and dynamic bearing characteristics of pocket type bearings, suited for subsequent steady-state vibration analysis of rotating machinery involving such bearings, is presented.
Compared to similar elliptic bearings, the pocket bearings tend to provide less fluctuation in the dynamic bearing coefficients for different load inclinations and may produce significantly different vibration behaviour in a given rotor system.
The two types of pocket bearings investigated here (produced by different machining procedures), apart from some difference in attitude angle, have virtually identical static and dynamic bearing characteristics
Radial clearance
Nondimensional bearing damping coefficients
Bearing diameter
Ellipticity or offset
Eccentricity
Film force components
Nondimensional force components
Film thickness
Nondimensional bearing stiffness coefficients
Bearing width
Centres
Pressure (gauge)
Bearing radius
Journal radius
Sommerfeld number
Time
Velocities at journal surface in
External load
Cartesian coordinates at bearing surface
Cartesian coordinates at curvature centers
Pocket extent
Load inclination from vertical
Perturbation; variation in
Nondimensional ellipticity/offset
Nondimensional eccentricity
Absolute viscosity
Attitude angle
Angular coordinate from line of centres
Nondimensional time
Speed
Angular coordinate from
Nondimensional
Differentiation with respect to nondimensional time
Differentiation with respect to dimensional time
Steady state
Bearing, journal
Bottom lobe, top lobe
Elliptic arc, pocket arc, side arc
Left arc, right arc
Pocket type 1, pocket type 2.