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/m^{3} | |||||

No. | Length (m) | Diameter (m) | Disk mass (kg) | Disk Ip (kg-m^{2}) | Disk Id (kg-m^{2}) |

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/m^{2}) | 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.