Vibration Characteristics of Hydrodynamic Fluid Film Pocket Journal Bearings

Abstract

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.

1. Introduction

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 [1]. In order to minimise the computational effort, particularly in transient response analyses, some further simplifications such as infinitely short or long bearing approximations are frequently used [2]. Improvements to these techniques to achieve better solution accuracy using more complicated yet still manageable approaches have also been developed [3]. Alternatively, for steady-state solutions where it is possible to prepare the bearing data a priori, numerical solutions to the full Reynolds equation can be tabulated or plotted for subsequent use [4]. For different bearing types, the different bearing clearance profiles need to be considered when obtaining the corresponding solutions [5]. Such data is also needed for the different types of pocket bearings, whose respective static and dynamic characteristics are of interest. This paper illustrates the approach used to obtain solutions for these pocket bearings.

2. Theory

2.1. General Theory

The schematic of a simple circular hydrodynamic fluid film bearing is shown in Figure 1. Following the usual assumptions of hydrodynamic lubrication theory, the Reynolds equation can be written as [1]𝜕ℎ𝜕𝑋³𝜕𝑃+𝜕𝜕𝑋ℎ𝜕𝑍³𝜕𝑃𝜕𝑍=−6𝜇𝑈𝜕ℎ𝜕𝑋+12𝜇𝑉, (1) where 𝜋ℎ=𝐶−𝑧sin𝜓−𝑦cos𝜓,𝜓=𝜃+𝜙+2. (2)(3) Upon nondimensionalisation, (1) becomes 𝜕𝜕𝜓ℎ³𝜕𝑃+𝐷𝜕𝜓𝐿²𝜕𝜕𝑍ℎ³𝜕𝑃𝜕𝑍=−2𝑧′−𝑦sin𝜓+2𝑦′+𝑧cos𝜓, (4) in which𝑍𝑍=,𝐿/2𝑃𝑃=6𝜇𝜔(𝑅/𝐶)²,ℎℎ=𝐶=1−𝑧sin𝜓−𝑦cos𝜓,𝑦𝑦=𝐶,andsoforth.ℎ^′=−𝑧^′sin𝜓−𝑦^′cos𝜓. (5) The pressure boundary conditions are that 𝑃=0 at 𝑍=±1 (bearing edges) and at 𝜓₁ and 𝜓₂ (the boundary coordinates at the onset and end of the fluid film, resp.). The Reynolds cavitation boundary condition is used to define the cavitation region [1].

Figure 1 

Schematic of a circular bearing.

Upon integration of the film pressure, the fluid film force components in y and z directions are 𝐹_𝑦𝐹_𝑧=−^𝜓2_𝜓1^𝐿/2_−𝐿/2𝑃0𝑊cos𝜓sin𝜓𝑅𝑑𝑍𝑑𝜓=. (6) The corresponding nondimensional force components are [3]𝑓_𝑦𝑓_𝑧=−3^𝜓2_𝜓1¹₋₁𝑃𝑑cos𝜓sin𝜓0𝑍𝑑𝜓=1𝜋𝑆. (7) For small perturbations Δ𝑦, Δ𝑧, Δ𝑦^′, and Δ𝑧^′ in the 𝑦 and 𝑧 directions about the equilibrium position, the dynamic bearing coefficients are defined as Δ𝑓_𝑧Δ𝑓_𝑦𝐾=−^𝑧𝑧𝐾_𝑧𝑦𝐾_𝑦𝑧𝐾_𝑦𝑦Δ𝑧Δ𝑦−𝐶_𝑧𝑧𝐶_𝑧𝑦𝐶_𝑦𝑧𝐶_𝑦𝑦ΔΔ𝑧′𝑦′. (8) Ignoring higher-order terms, the Taylor series expansion about the equilibrium position (𝑦_𝑜, 𝑧_𝑜, 𝑦′=0 and 𝑧′=0) yields 𝐾_𝑧𝑧=−𝜕𝑓_𝑧/𝜕𝑧,𝐶_𝑧𝑦=−𝜕𝑓_𝑧/𝜕𝑦′, and so forth.

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 𝑦 and 𝑧 directions which equals zero and 𝑊, respectively, in (6); it is the perturbation in these resultant force components which is used to evaluate the dynamic bearing coefficients.

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.

2.2. Pocket Bearing Configurations

Two types of pocket bearings are to be considered. The first type, formed in a similar way to an elliptic bearing (Figure 2), is schematically shown in Figure 3 and is referred to as type 1. The two halves of a bearing block are held together, separated by shims corresponding to the ellipticity 2𝑑₁. A circular hole of diameter 𝐷 is cut, and the shims are then removed to form a normal elliptic bearing with the top pad centre a distance 𝑑₁ below, and the bottom pad centre a distance 𝑑₁ above the bearing centre. Concentric arc portions (the pockets) of a specified angular extent are then machined as shown in Figure 3. Such a pocket bearing has three curvature centres: one above one below the bearing centre for the side arcs just like the elliptic bearings and one coinciding with the bearing centre for the pockets.

Figure 2 

Schematic of an elliptic bearing.

Figure 3 

Schematic of type-1 pocket bearing.

The second type of pocket bearing, referred to as type 2, is shown schematically in Figure 4. It can be conveniently machined by NC machines. It also has three arc centres, but the two for the side arcs are located at a distance 𝑑₂ to the left and right of the bearing centre, the left centre for the left arcs on both the top and bottom pads, and the right one for the right arcs, again on both pads. The angular extent of the side arcs is determined by the arc radii and the centre offset 𝑑₂. The two types of the pocket bearings are expected to have similar vibration characteristics.

Figure 4 

Schematic of type-2 pocket bearing.

Normally, the bearing clearance 𝐶 is used to nondimensionalise the film thickness and the relative journal to bearing displacements. Due to the bearing profile, each pocket bearing has two clearances: 𝐶_𝑝 for the pocket portion and 𝐶_𝑒 for the side arcs. Using different clearances will produce different nondimensional bearing coefficients, but for a given bearing, the dimensional coefficients should not be affected. Following the approach for elliptic bearings [5], 𝐶_𝑒 could be chosen as the nondimensionalising parameter. However, the type-2 pocket bearing does not have such a corresponding clearance, though it also has two clearances: 𝐶_𝑝 for the pocket portion and 𝐶_𝑠 for the side arcs. Hence, in order to have comparable results, the pocket clearance 𝐶_𝑝, common to both types of pocket bearings, is used as the nondimensionalising parameter.

2.3. Type-One Pocket Bearing

From (2), the nondimensional film thickness for the side arcs becomes ℎ_𝑒=𝐶_𝑒𝐶_𝑝−𝑧sin𝜓−𝑦cos𝜓 (9) and for the pocket arcs ℎ_𝑝=1−𝑧sin𝜓−𝑦cos𝜓. (10) Note that in these equations, 𝑦 and 𝑧 are the nondimensional journal centre coordinates from the corresponding arc centres.

For the pocket portion: 𝑦_𝑝=𝜀sin𝜙−𝛽,𝑧_𝑝=−𝜀cos𝜙−𝛽, (11) and for the bottom side arcs,𝑦_𝑏=𝜀_𝑏sin𝜙_𝑏−𝛽,𝑧_𝑏=−𝜀_𝑏cos𝜙_𝑏−𝛽, (12) where, from Figure 5,𝜀_𝑏=𝜀²+𝛿²+2𝜀𝛿cos𝜙,sin𝜙_𝑏=𝜀𝜀_𝑏sin180^∘−𝜙. (13) For the top side arcs,𝑦_𝑡=𝜀_𝑡sin𝜙_𝑡−𝛽,𝑧_𝑡=−𝜀_𝑡cos𝜙_𝑡−𝛽, (14) where, from Figure 5,𝜀_𝑡=𝜀²+𝛿²−2𝜀𝛿cos𝜙,sin180^∘−𝜙_𝑡=𝜀𝜀_𝑡sin𝜙. (15) The clearances 𝐶_𝑒 and 𝐶_𝑝 are related by the pocket arc extent 𝛼 and the ellipticity of the side arcs 𝑑₁. As shown in Figure 6, at the intersection between the side and pocket arcs, one has 𝑅²_𝑒=𝑅²_𝑝+𝑑²₁−2𝑅_𝑝𝑑₁cos180^∘−𝛼2. (16) Solving for 𝑅_𝑝 and omitting higher-order terms𝑅_𝑝=−𝑑₁𝛼cos2+𝑅²_𝑒−𝑑²₁sin²𝛼2≈𝑅_𝑒−𝑑₁𝛼cos2. (17) The journal radius can be expressed as 𝑟=𝑅_𝑒−𝐶_𝑒=𝑅_𝑝−𝐶_𝑝. (18) Substitution of (17) into (18) gives𝐶_𝑒−𝐶_𝑝=𝑅_𝑒−𝑅_𝑝≈𝑑₁𝛼cos2, (19) or 𝐶_𝑒𝐶_𝑝𝑑≈1+¹cos(𝛼/2)𝐶_𝑝=1+𝛿₁𝛼cos2 (20) which is always greater than 1.

Figure 5 

Geometric relationship between the arc centres for elliptic and type-1 pocket bearings.

Figure 6 

Geometric relationship between 𝐶𝑒 and 𝐶𝑝 for type-1 pocket bearing.

2.4. Type-Two Pocket Bearing

Again, following an approach similar to that for the type 1 bearings, one has for the side arcs ℎ_𝑠=𝐶_𝑠𝐶_𝑝−𝑧sin𝜓−𝑦cos𝜓 (21) and for the pocket arcsℎ_𝑝=1−𝑧sin𝜓−𝑦cos𝜓. (22) The clearances 𝐶_𝑠 and 𝐶_𝑝 are related by the pocket arc extent 𝛼 and the offset of the side arc centers 𝑑₂.As shown in Figure 7, at the intersection between the side and pocket arcs, one has 𝑅²_𝑠=𝑅²_𝑝+𝑑²₂−2𝑅_𝑝𝑑₂cos90^∘−𝛼2. (23) Solving for 𝑅_𝑝 and omitting higher-order terms, 𝑅_𝑝=𝑑₂𝛼sin2+𝑅²_𝑠−𝑑²₂cos²𝛼2≈𝑅_𝑠+𝑑₂𝛼sin2. (24) The journal radius can be expressed as 𝑟=𝑅_𝑠−𝐶_𝑠=𝑅_𝑝−𝐶_𝑝. (25) From (24) and (25), this gives𝐶_𝑝−𝐶_𝑠≈𝑑₂𝛼sin2, (26) or 𝐶_𝑠𝐶_𝑝𝑑≈1−²sin(𝛼/2)𝐶_𝑝=1−𝛿₂𝛼sin2. (27) To relate the offset 𝑑₂ to the ellipticity 𝑑₁, it is assumed that, apart from the same pocket arc extent 𝛼, the horizontal widths of the two bearings are the same, that is, 𝑅_𝑠+𝑑₂=𝑅²_𝑒−𝑑²₁≈𝑅_𝑒. (28) Substitution of (24) and (17) into (28) gives𝑅_𝑝−𝑑₂𝛼sin2+𝑑₂=𝑅_𝑝+𝑑₁𝛼cos2, (29) or𝑑₂=𝑑₁cos(𝛼/2)().1−sin𝛼/2 (30)

Figure 7 

Geometric relationship between 𝐶𝑠 and 𝐶𝑝 for type-2 pocket bearings.

The relationships between 𝑦, 𝑧 and 𝜀, 𝜙 are the same as before. However, the calculations for 𝜀_𝑙, 𝜀_𝑟, 𝜙_𝑙, and 𝜙_𝑟 need different expressions. From Figure 8, 𝜀_𝑙=𝜀²+𝛿²₂+2𝜀𝛿₂sin𝜙,sin90^∘−𝜙_𝑙=𝜀𝜀_𝑙sin90^∘,𝜀+𝜙_𝑟=𝜀²+𝛿²₂−2𝜀𝛿₂sin𝜙,sin90^∘+𝜙_𝑟=𝜀𝜀_𝑟sin90^∘−𝜙. (31)

Figure 8 

Geometric relationship between the arc centres for type-2 pocket bearing.

3. Solution Procedures

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 [5], thereby ensuring that the Reynolds condition is used as the cavitation boundary. All results here assume same zero gauge inlet, outlet, and cavitation pressures.

4. Results

Figure 9 compares the nondimensional static and dynamic characteristics of the two types of pocket bearings as a function of load inclination beta (𝛽), using the same base clearance 𝐶_𝑝 and the the same pocket extent of 𝛼=60^∘. Apart from some difference in the attitude angle, the nondimensional characteristics are virtually identical.

(a)

(b)

(c)

Figure 9 

Comparison of the nondimensional bearing characteristics for the two types of pocket bearings ((1) and (2) correspond to type 1 and 2, resp.).

Figures 10 and 11 compare the bearing characteristics of the type-1 pocket bearing to a similar elliptic bearing as functions of the load inclination angle 𝛽 for given eccentricities 𝜀=0.2 and 0.4, respectively. It can be seen that in general there is less variation in the dynamic bearing characteristics of the pocket bearing due to the change of 𝛽.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10 

Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at 𝜀=0.2.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11 

Nondimensional bearing characteristics for elliptic and type-1 pocket bearings at 𝜀=0.4.

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 12. Table 1 lists the corresponding data used to characterise the rotor and the bearings. In house vibration analysis software was used to determine the natural frequencies and mode shapes over a speed range up to 5000 rpm. Table 2 lists these natural frequencies, and Figure 13 shows the Campbell diagrams for the elliptic bearing supports and for the type-1 pocket bearing supports.

Table 1 

Sample rotor bearing data.

Table 2 

Damped natural frequencies of a rotor supported by elliptic or pocket bearings ((1) refers to type 1, etc.).

Figure 12 

Sample rotor bearing system.

(a)

(b)

Figure 13 

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 2 indicate that the two types of the pocket bearings with similar dimensions could have quite different effects on the system vibration characteristics.

5. Conclusions

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

Nomenclature

𝐶:	Radial clearance =𝑅−𝑟
𝐶𝑦𝑦,⋯:	Nondimensional bearing damping coefficients
𝐷:	Bearing diameter
𝑑:	Ellipticity or offset
𝑒:	Eccentricity
𝐹𝑦, 𝐹𝑧:	Film force components
𝑓𝑦, 𝑓𝑧:	Nondimensional force components
ℎ:	Film thickness
𝐾𝑦𝑦,⋯:	Nondimensional bearing stiffness coefficients
𝐿:	Bearing width
𝑂:	Centres
𝑃:	Pressure (gauge)
𝑅:	Bearing radius
𝑟:	Journal radius
𝑆:	Sommerfeld number =𝜇𝜔𝑅𝐿(𝑅/𝐶)2/(𝜋𝑊)
𝑡∶	Time
𝑈, 𝑉:	Velocities at journal surface in 𝑋 and 𝑌 directions, respectively
𝑊:	External load
𝑋, 𝑌, 𝑍:	Cartesian coordinates at bearing surface
𝑥, 𝑦, 𝑧:	Cartesian coordinates at curvature centers
𝛼:	Pocket extent
𝛽:	Load inclination from vertical
Δ:	Perturbation; variation in
𝛿:	Nondimensional ellipticity/offset =𝑑/𝐶
𝜀:	Nondimensional eccentricity =𝑒/𝐶
𝜇:	Absolute viscosity
𝜙:	Attitude angle
𝜃:	Angular coordinate from line of centres =𝑋/𝑅
𝜏:	Nondimensional time =𝜔𝑡
𝜔:	Speed
𝜓:	Angular coordinate from 𝑦 axis

Superscripts (If Not Otherwise Defined):

:	Nondimensional
′:	Differentiation with respect to nondimensional time 𝜏
̇‌:	Differentiation with respect to dimensional time 𝑡

Subscripts (If Not Otherwise Defined):

𝑜:	Steady state
𝑏,𝑗:	Bearing, journal
𝑏, 𝑡:	Bottom lobe, top lobe
𝑒, 𝑝, 𝑠:	Elliptic arc, pocket arc, side arc
𝑙, 𝑟:	Left arc, right arc
1, 2:	Pocket type 1, pocket type 2.