Performance of an Orifice Compensated Two-Lobe Hole-Entry Hybrid Journal Bearing

Abstract

The work presented in this paper aims to study the performance of a two-lobe hole-entry hybrid journal bearing system compensated by orifice restrictors. The Reynolds equation governing the flow of lubricant in the clearance space between the journal and bearing together with the equation of flow through an orifice restrictor has been solved using FEM and Galerkin's method. The bearing performance characteristics results have been simulated for an orifice compensated nonrecessed two-lobe hole-entry hybrid journal bearing symmetric configuration for the various values of offset factor (𝛿), restrictor design parameter (𝐶_𝑆2), and the value of external load (𝑊₀). Further, a comparative study of the performance of a two-lobe hole-entry hybrid journal bearing system with a circular hole-entry symmetric hybrid journal bearing system has also been carried out so that a designer has a better flexibility in choosing a suitable bearing configuration. The simulated numerical results indicate that for the two-lobe symmetric hole-entry hybrid journal bearing system with an offset factor (𝛿) greater than one provides 30 to 50 percent larger values of direct stiffness and direct damping coefficients as compared to a circular symmetric hole-entry hybrid journal bearing system.

1. Introduction

Hybrid journal bearings are extensively used in machine tools under conditions of heavy loads and higher speeds. The stability and unsteady behavior of the hybrid journal bearings is greatly influenced by bearing geometry, and accordingly various designs have been used by designers to achieve the desired objective. To ensure better dynamic stability than a circular plain journal bearing, multilobe journal bearings are used. Multilobe hydrodynamic journal bearings have been investigated for their antiwhril characteristics by many researchers [1–5]. Pinkus [1] studied the performance of an elliptical journal bearing in terms of steady-state load carrying capacity and power loss using the FDM. Malik [2] theoretically studied an elliptical hydrodynamic journal bearing and compared its performance over a wide range of load conditions and provided the comprehensive design data including the static and dynamic characteristics for the two-lobed journal bearing for different aspect ratios. The study dealing with the effects of surface ellipticity on the dynamically loaded cylindrical bearing was carried out by Goenka and Booker [3] for an optimum bearing shape on the basis of maximizing the minimum film thickness. The notable observation about these studies is that they are all concerned with hydrodynamic journal bearing systems. Few studies dealing with the noncircular multirecess hydrostatic/hybrid journal bearing systems have also been reported in literature recently [6–8]. Ghosh and Satish [7], using a small amplitude perturbation analysis, determined the rotordynamic coefficients of a multilobe hybrid journal bearing system having large recesses. They extended their work further [8] to study the stability of a rigid rotor supported on a multilobe hybrid journal bearing system with short sills following a linear vibration theory for small amplitude oscillations of the journal center about its steady-state position. It was observed that a multilobe recessed hybrid journal bearing system with an offset factor more than one exhibits better dynamic performance than the circular hybrid journal bearings. The recessed journal bearings are unable to generate a substantial hydrodynamic action because recess constitutes a large bearing area, thus leaving very less area for lands. Thus, recessed bearings when operating in hybrid mode at higher speeds are not suitable for heavily loaded applications. Hence, nonrecessed journal bearings, that is, hole-entry hybrid journal bearings were developed to gain the advantage of available large land area to generate substantial hydrodynamic action. Such bearings give better performance than the recessed bearings along with the ease of manufacture and reduced cost of machining [9–12].

Recent developments towards the use of hybrid bearings in high-speed turbopumps and advanced machine tools have further necessitated to focus research activities in the area of hybrid bearings. Most of the available nonrecessed hydrostatic/hybrid bearing studies are generally confined to circular bearings [13–15]. However, a thorough review of literature reveals that the performance of nonrecessed multilobe hole-entry hybrid journal bearing has not yet been investigated. Therefore, a study of a nonrecessed multilobe hole-entry hybrid journal bearing is planned to bridge the gap in literature. In this paper, nonrecessed orifice compensated two-lobe hole-entry hybrid journal bearing is investigated for its static and dynamic characteristics. The performance characteristics of two-lobe hole-entry hybrid journal bearing have been compared with that of a circular hole-entry orifice compensated hybrid journal bearing for the same bearing operating and geometric parameters. Results are presented for various values of offset factor (𝛿), restrictor design parameter (𝐶_𝑆2), and for the value of external load for symmetric hole configuration. The computed results presented in this work are expected to be quite useful to the bearing designers as well as for the academic community.

2. Analysis

The Reynolds equation governing the laminar flow of an incompressible lubricant in the clearance space of journal and bearing in nondimensional form is expressed as [16, 17] 𝜕𝜕𝛼(ℎ³6𝜕𝑝𝜕𝛼)+𝜕𝜕𝛽(ℎ³6𝜕𝑝𝜕𝛽)=Ω𝜕ℎ𝜕𝛼+2𝜕ℎ𝜕𝑡.(1)

3. Fluid Film Thickness

The geometry of the symmetric hole-entry circular as well as multilobe hybrid journal bearing has been shown schematically in Figures 1(a) and 1(b). The expression for nondimensional fluid film thickness for a multilobe rigid journal bearing with reference to fixed coordinate axis is given as [6]ℎ=1𝛿−𝑋_𝑗+𝑥−𝑋^𝑖𝐿cos𝛼−𝑍_𝑗+𝑧−𝑍^𝑖𝐿sin𝛼,(2) where 𝑋_𝑗 and 𝑍_𝑗 are the equilibrium coordinates of the journal center, and 𝑥 and 𝑧 are time dependent perturbation coordinates of the journal center measured from their equilibrium position (𝑥,𝑧=0forsteadycase). 𝑋^𝑖𝐿 and 𝑍^𝑖𝐿 are the lobe center coordinates of 𝑖th lobe.

(a)

(b)

Figure 1 

(a) Hole-entry type journal bearing system (symmetric configuration). (b) Two-lobe hole-entry type journal bearing system (symmetric configuration).

4. Restrictor Flow Equation

The flow rate of the lubricant through the orifice restrictor is defined as [17]𝑄_𝑅=𝐶_𝑆21−𝑝_𝑐^1/2,(3) where the term 𝑝_𝑐 represents the pressure at the hole.

5. Finite Element Formulation

The lubricant flow field in the clearance space of a circular bearing has been discretized into four-noded isoparametric elements and using the Lagrangian interpolation function, the pressure at a point in the element is bilinearly distributed and expressed approximately as [18]𝑝=⁴_𝑗=1𝑁_𝑗𝑝_𝑗,(4)where 𝑁_𝑗 is elemental shape function, and using the approximate value of 𝑝 (1) can be expressed as𝜕𝜕𝛼[ℎ³6𝜕𝜕𝛼(⁴_𝑗∑₌₁𝑁_𝑗𝑝_𝑗)]+𝜕𝜕𝛽[ℎ³6𝜕𝜕𝛽(⁴−Ω𝜕∑_𝑗=1𝑁_𝑗𝑝_𝑗)]ℎ𝜕𝛼−2𝜕ℎ𝜕𝑡=𝑅^𝑒,(5)where 𝑅^𝑒 is known as residue. The element equations are obtained by applying Galerkin’s technique. As per this technique, minimization of residue is obtained by orthogonalizing the residue with interpolation functions, that is,_Ω𝑒𝑁_𝑖𝑅^𝑒𝑑𝛼𝑑𝛽=0.(6)By integrating the second-order term by parts to obtain 𝐶⁰ continuity and differentiating (4) with respect to 𝑡, the resulting equation for a typical element is obtained in matrix form as follows: 𝐹_𝑛×𝑛𝑝_𝑛×1=𝑄_𝑛×1+Ω𝑅_𝐻_𝑛×1+𝑥̇_𝑗𝑅_𝑥𝑗_𝑛×1+𝑧_𝑗̇𝑅_𝑧𝑗_𝑛×1,(7)where 𝑛 = total number of nodes in lubricant flow field.

After modification for continuity of flow between bearing and restrictor and incorporating appropriate boundary condition, the system (7) is solved for nodal pressure and nodal flows.

For an 𝑒th element, the elements of the above matrices are defined as follows: 𝐹^𝑒𝑖𝑗=_𝐴𝑒ℎ³𝐹₂𝜕𝑁^𝑖𝜕𝛼𝜕𝜕𝑁𝛼^𝑗+𝜕𝜕𝑁𝛽^𝑖𝜕𝜕𝑁𝛽^𝑗𝑑𝛼𝑑𝛽,(7a)𝑄^𝑒𝑖=_Γ𝑒ℎ³𝐹₂𝜕𝑝𝜕𝛼−Ω1−𝐹₁𝐹₀ℎ𝑙+ℎ³𝐹₂𝜕𝑝𝜕𝛽𝑚𝑁^𝑖𝑑Γ^𝑒,(7b)𝑅^𝑒𝐻_𝑖=_𝐴𝑒1−𝐹₁𝐹₀ℎ𝜕𝜕𝑁𝛼^𝑖𝑑𝛼𝑑𝛽,(7c)𝑅^𝑒𝑋_𝐽𝑖=_𝐴𝑒𝑁_𝑖cos𝛼𝑑𝛼𝑑𝛽,(7d)𝑅^𝑒𝑍_𝐽𝑖=_𝐴𝑒𝑁_𝑖sin𝛼𝑑𝛼𝑑𝛽,(7e)𝑙𝑚𝑖𝑗=1,2,… where 𝑛^𝑒𝑙 and 𝑝||_𝛽=∓1.0=0.0.(8) are the directions cosines and 𝑋̇_𝑗,𝑍̇_𝑗=0, (𝑋_𝑗,𝑍_𝑗), 𝐹_𝑥=0,𝐹_𝑧−𝑊₀=0.(9) (number of nodes per element).

6. Boundary Conditions

The boundary conditions for the lubricant flow field are as follows.

(1)Nodes situated on the external boundary of the bearing have zero pressure, 𝐹_𝑥(2)The nodes situated on a hole have equal pressure.(3)Flow of lubricant through the restrictor is equal to the bearing input flow.(4)The nodal flows are zero at internal nodes except those situated on holes.

7. Solution Procedure

The study of an orifice compensated multilobe hole-entry hybrid journal bearing system needs an iterative solution scheme to establish solutions of the flow field system (5) with the restrictor flow (3) as constraint with boundary conditions. Assuming constant viscosity and the steady-state case (𝐹_𝑧), the lubricant flow field system (5), after adjustment for flow through orifice restrictor (3) and modification for boundary conditions, is solved for a specified journal center position 𝑖th using Gauss elimination technique. If the solution is to be obtained for a specified vertical external load, one additional iterative loop is needed to establish the equilibrium journal center position using the following equations:(Δ𝑋^𝑖𝑗,Δ𝑍^𝑖𝑗)The fluid-film reaction terms [((Δ𝑋^𝑖𝑗)²+(Δ𝑍^𝑖𝑗)²)^1/2((𝑋^𝑖𝑗)²+(𝑍^𝑖𝑗)²)^1/2]×100<0.001,(10) and 𝑋^𝑖𝑗,𝑍^𝑖𝑗 in (9) are expanded by Taylor series about the 𝑖th journal center position, and the increments 𝑆_𝑖𝑗=−𝜕𝐹_𝑖𝜕𝑞_𝐽,𝑖=𝑥,𝑧,(11) on the journal coordinates are obtained. Iterations are continued until the following convergence criterion is not satisfied: 𝑖where 𝑞_𝐽 are the coordinates of the 𝑞_𝐽=𝑋_𝐽,𝑍_J journal center position. The overall iterative solution scheme is presented in Figure 2.

Figure 2 

Overall iterative scheme.

8. Fluid Film Stiffness and Damping Coefficients

Fluid-film stiffness and damping coefficients are computed using the expressions given below.

Fluid-film stiffness coefficients:𝐶_𝑖𝑗=−𝜕𝐹_𝑖𝜕𝑞̇_𝐽,𝑖=𝑥,𝑧,(12)where

𝑞̇_𝐽 = direction of force or moment.

𝑞̇_𝐽=𝑋̇_𝐽,𝑍̇_J = direction of journal center displacement (𝛿).

Fluid-film damping coefficients: (𝐶_𝑆2)here, 𝜆 represents the velocity component of journal center ((𝑎_𝑏)).

9. Results and Discussion

The performance characteristics for circular and two-lobe symmetric hole-entry hybrid journal bearing have been computed using the solution scheme as discussed earlier. The results have been computed for various values of offset factor ((𝐶_𝑆2)) and restrictor design parameter 𝛿. The chosen values of bearing operating parameter are most generally used values as given in Table 1.

Table 1 

Bearing operating and geometric parameters.

In order to check the validity of the computed results obtained from the developed program, the computed results have been compared with those already available in the literature. To the best of author’s knowledge, no results are available in the literature for the performance characteristics of nonrecessed multilobe hole-entry hybrid journal bearing. Hence, to validate the results, first the values of fluid film reaction 𝜀 for orifice compensated hole-entry hydrostatic journal bearing are computed and compared with that of Rowe et al. [10]. The results compare well for wide range of eccentricity ratio (ε) as shown in Figure (3). Further, using Reynolds cavitation boundary condition, the performance characteristics of two-lobe hydrodynamic journal bearing are compared with the data of Lund and Thomsen [19]. The results match very well as shown in Table 2.

Table 2 

Comparison of performance characteristics of two-lobe hydrodynamic journal bearing, 𝜙 = 1, Ω = 1, 𝑆_𝑥𝑥/𝑊₀. 𝑆_𝑥𝑧/𝑊₀ = 5.0.

Figure 3 

Variation of external load carrying capacity (𝑄) with eccentricity ((𝑆𝑖𝑗)).

The static and dynamic performance characteristics which include maximum pressure (𝐶_𝑖𝑗), minimum fluid film thickness (𝜔_th), bearing flow (𝛿), stiffness coefficients (𝐶_𝑆2), damping coefficients (𝑊₀), and threshold speed (𝑝) have been presented for various values of offset factors (ℎ), restrictor design parameter (𝑄), and external load (𝑆_𝑥𝑥). The results are presented through Figures (4–14) and discussed in the following paragraph.

Figure 4 

Variation of maximum pressure (𝑆𝑧𝑧).

Figure 5 

Variation of fluid film thickness (𝑆𝑧𝑧).

Figure 6 

Variation of bearing flow (𝑆𝑧𝑥).

Figure 7 

Variation of direct stiffness (𝐶𝑥𝑥).

Figure 8 

Variation of direct stiffness (𝐶𝑧𝑧).

Figure 9 

Variation of direct stiffness (𝐶𝑧𝑧).

Figure 10 

Variation of cross coupled stiffness (𝜛th).

Figure 11 

Variation of direct damping (𝑝).

Figure 12 

Variation of direct damping (𝑝).

Figure 13 

Variation of direct damping (𝐶𝑆2).

Figure 14 

Variation of threshold speed 𝑊0.

Figure 4 shows the variation of maximum pressure (𝐶_𝑆2). It has been observed that the value of maximum pressure (𝑝) for a specified external load increases with restrictor design parameter (𝛿). From the graph, it may be observed for the given value of external load ((ℎ)) and the restrictor design parameter (𝑝) that the value of maximum pressure is higher for two-lobe hole-entry hybrid journal bearing. It is also observed that the value of maximum pressure (ℎ_min) increases in case of two-lobe hole-entry hybrid journal bearing with offset factor (ℎ_min) more than one compared to circular hole-entry hybrid journal bearing. The decrease in the value of fluid film thickness (𝐶_𝑆2) owing to the change of geometry of the bearing is responsible for the increase in the value of maximum pressure (𝛿).

Figure 5 shows the variation of minimum fluid-film thickness 𝛿. It is observed that the value of minimum fluid film thickness (𝛿) for a specified external load reduces with an increase in restrictor design parameter 𝛿 for all the cases of offset factor (ℎ_min) except for the case of offset factor (𝛿) = 0.5. At a constant external load, the two-lobe hole-entry hybrid journal bearing with offset factor (ℎ_min) more than one operates at higher eccentricities compared to circular hole-entry hybrid journal bearing. Also Figure 5 indicates that the two-lobe hole-entry hybrid journal bearing with offset factor (ℎ_min) = 0.75 operates with higher value of minimum fluid film thickness ℎ_min compared to other cases of offset factor (𝐶_𝑆2). The increase or decrease in the value of minimum fluid film thickness (𝛿) has been found of the order of 0.5 < .75 >1.0 >1.25 >1.3. The maximum value of minimum fluid film thickness 𝑊₀ is not exactly for δ = 1.0, but rather it is observed to be at the values between δ = 0.75 and δ = 0.9. Similar tend has been reported by Goenka and Booker [3]. This trend may be attributed due to the clearance change in the bearing for the values of offset factor (δ). It may be observed that a desired value of (𝑄) may be obtained in a two-lobe hole-entry journal bearing by carefully selecting the value of restrictor design parameter (𝛿) and offset factor (𝛿).

At a constant value of external load ((𝛿)), the bearing flow (𝑆_𝑥𝑥,𝑆_𝑧𝑧) in Figure 6 is found to be lower for the case of two-lobe hole-entry hybrid journal bearing with offset factor (𝐶_𝑆2) greater than one compared to bearing flow in circular hole-entry hybrid journal bearing. But the bearing flow is higher for the case of two-lobe hole-entry journal bearing with offset factor less than one. This change in bearing flow may be accounted for the change in profile of the bearing due to change in offset factor (𝑆_𝑥𝑥). The decrease in the flow rate in case of two-lobe hole-entry hybrid journal bearing with offset greater than one minimizes the pumping power.

The effects of offset factor (𝛿) on the value of direct stiffness coefficients (𝐶_𝑆2=.04) have been presented in Figures 7 and 8. From Figure 7, it may be observed that there exists a particular value of restrictor design parameter (𝑆_𝑥𝑥) at which the value of direct stiffness coefficient (𝐶_𝑆2=.08) is the maximum for a particular case of offset factor (𝑆_𝑥𝑥). As seen from Figure 7, at a value of restrictor design parameter (𝐶_𝑆2), the two-lobe hole-entry hybrid journal bearing with an offset factor δ = 1.25 has the highest value of direct stiffness coefficient (𝑆_𝑥𝑥) and at a value of restrictor design parameter 𝑊₀, the hole-entry hybrid journal bearing with an offset factor δ = 1.0 has the highest value of direct stiffness coefficient 𝑊₀. The results from Figure 7 indicate that the restrictor design parameter (𝑆_𝑧𝑧) becomes an important parameter when the designer aims to maximize the value of (𝐶_𝑆2) at constant external load ((𝑆_𝑧𝑧)). Figure 8 shows, at a constant external load (𝛿), fluid film stiffness coefficients (𝑆_𝑧𝑧) which in general show an initial increasing trend and then decrease with an increase in restrictor design parameter (𝛿) except for the case of offset factor δ = 0.5. The two-lobe hole-entry hybrid journal bearing indicates a higher value of direct fluid-film stiffness coefficients (𝑆_𝑧𝑧) compared with circular hole-entry hybrid journal bearing for a value of offset factor 𝑊₀ greater than one for a chosen load.

Figure 9 shows the effect of direct fluid-film stiffness coefficients (𝛿) with an increase in offset factor (𝑆_𝑧𝑧). The value of direct fluid-film stiffness coefficient 𝑆_𝑧𝑥 gets decreased with an increase in the value of external load ((𝛿)) for all the cases of offset factor 𝑆_𝑧𝑥. Further, it may be observed that the value of direct fluid-film stiffness coefficient (𝛿) is maximum in case of two-lobe hole entry hybrid journal bearing with an offset factor δ = 1.3.

Figure 10 shows the effect of cross coupled fluid-film stiffness coefficients (𝑆_𝑧𝑥) with an increase in offset factor (𝐶_𝑥𝑥,𝐶_𝑧𝑧). It can be observed from the figure that the values of (𝛿) in case of two-lobe hole-entry journal bearing follow the similar trend for the different values of offset factor 𝑊₀. The two-lobe hole-entry hybrid journal bearing with offset factor of 0.5 has the least value of (𝐶_𝑥𝑥,𝐶_𝑧𝑧).

The variation of direct damping coefficients (𝛿) with offset factor (𝛿) is shown in Figures 11 and 12. At constant load ((𝐶_𝑥𝑥,𝐶_𝑧𝑧)), damping coefficients (𝐶_𝑧𝑧) are increased with an increase of offset factor (𝐶_𝑆2). It is observed from Figures 11 and 12 that the two-lobe hole-entry hybrid journal bearing with offset factor (𝐶_𝑧𝑧) greater than one exhibits higher value of direct damping coefficients (𝛿) compared to that of circular hole-entry hybrid journal bearing.

Figure 13 shows the variation of direct fluid-film coefficient 𝑊₀ against restrictor design parameter (𝐶_𝑧𝑧). The value of direct fluid-film damping coefficient (𝐶_𝑆2) gets increased with an increase in offset factor (𝛿) for the chosen value of external load (𝛿=0.5). Further, it may be observed that the value of direct fluid-film damping coefficient 𝛿=0.5 decreases with an increase in the value of restrictor design parameter (𝐶_𝑧𝑧) for all the cases of offset factor (𝐶_𝑆2). except for the case of offset factor (𝜔_th). For the case of offset factor (𝐶_𝑆2), the value of direct fluid-film damping coefficient (𝜔_𝐽) nearly remains a constant with an increase in restrictor design parameter (𝜔_th)

Figure 14 shows variation with stability threshold speed 𝜔_𝐽<𝜔_th against the restrictor design parameter (𝐶_𝑆2). For a chosen value of load at a constant speed, the system is asymptotically stable when the operating speed of the journal (𝜔_th) is less than stability threshold (𝛿), that is, (𝐶_𝑆2=.06). From Figure 7, it may be observed that there exists a particular value of restrictor design parameter 𝛿=1.25 at which the value of stability threshold speed (𝜔_th) is the maximum for a particular case of offset factor (𝐶_𝑆2=.08). As seen from the Figure 7, at a value of restrictor design parameter 𝛿=1.0, the two-lobe hole-entry hybrid journal bearing with an offset factor (𝜔_th) has the highest value of stability threshold speed (𝐶_𝑆2) and at a value of restrictor design parameter (𝜔_th), the hole-entry hybrid journal bearing with an offset factor 𝑊₀ has the highest value of stability threshold speed (𝛿). The results from Figure 7 indicate that the restrictor design parameter (𝛿) becomes an important parameter when the designer aims to maximize the value of (𝑆_𝑧𝑧) at constant external load ((𝐶_𝑥𝑥)).

10. Conclusions

On the basis of the theoretical results presented in the paper, the following conclusions have been drawn.

(1) Two-lobe hole-entry hybrid journal bearing with an offset factor 𝑊₀ greater than one requires a reduced quantity of lubricant as compared to circular hybrid hole-entry journal bearing, hence reduced pumping power is needed.(2) A two-lobe hole-entry hybrid journal bearing system with an offset factor (𝛿) greater than one provides a larger value of stiffness 𝑎_𝑏 and damping 𝑐 coefficients. These values are found to be of the order of 15.9% and 36.2%, respectively more vis-a-vis a circular hole-entry journal bearing system for the same value of external load (𝐶₁=𝑐) and for the same bearing geometric parameters.(3)A two-lobe hole-entry journal bearing system with offset factor 𝐶₂ greater than one provides enhanced value of stability threshold speed margin of the order of 14.4% to that of circular hole-entry hybrid journal bearing system.

Nomenclature

𝐶𝑖𝑗:	Bearing land width, mm
(𝑖,𝑗=𝑥,𝑧):	Radial clearance, mm
𝑑0, 𝑒:	Clearance due to circumscribed, inscribed circle on the bearing, mm
ℎ:	Damping coefficients 𝐿, N.sec.mm−1
𝑡:	Orifice diameter, mm
𝐹:	Journal eccentricity, mm
(𝜕ℎ/𝜕𝑡≠0), 𝐹0:	Fluid-film thickness, bearing length mm
(𝜕ℎ/𝜕𝑡=0):	Time, second
𝑔:	Fluid film reaction 𝑅𝐽, N
𝑅𝑏:	Fluid film reaction 𝑅𝐿, N
𝑝:	Acceleration due to gravity, m.sec−2
𝑝∗, 𝑄:	Radius of journal and bearing, respectively, mm
𝑆𝑖𝑗:	Radius of lobe, mm
(𝑖,𝑗=𝑥,𝑧), 𝑊0:	Pressure, concentric design pressure N.mm−2
𝑋,𝑌,𝑍:	Bearing flow, mm3.sec−1
𝑋𝐽,𝑍𝐽:	Stiffness coefficients, 𝑋𝑖𝐿,𝑍𝑖𝐿, N.mm−1
𝑖th:	External load, N
𝑥,z:	Cartesian coordinates
𝜇:	Coordinates of steady-state equilibrium journal center from geometric center of bearing, coordinates, mm
𝜇𝑟:	Lobe center coordinates of the 𝜔𝐽 lobe
𝜔th:	Horizontal and vertical co-ordinate measured from steady-state equilibrium position of journal center, mm
𝜔𝐼:	Dynamic viscosity of lubricant, N.sec.m−2
(𝑔/𝑐)1/2:	Dynamic viscosity of lubricant at reference temperature and pressure, N.sec.m−2
𝜙, 𝜌:	Journal rotational speed, threshold speed rad.sec−1
𝜓𝑑:	𝑎𝑏=𝑎𝑏/𝐿, rad.sec−1
:	Attitude angle, rad
𝐶𝑖𝑗=𝐶𝑖𝑗(𝑐3/𝜇𝑅4𝐽):	Density of lubricant, kg.mm−3
:	Coefficient of discharge for orifice

Nondimensional Parameters

𝐶𝑆2=1/12(3𝜋𝑑20𝜇𝑟𝜓𝑑/𝑐3)(2/𝜌𝑝𝑠)1/2∶: Land width ratio
(𝐹,𝐹0)=(𝐹,𝐹0)/𝑝𝑠𝑅2𝐽
(ℎ)=(ℎ)/𝑐 Orifice design parameter
(𝑝,𝑝𝑠,𝑝𝑐,𝑝max)=(𝑝,𝑝𝑠,𝑝𝑐,𝑝max)/𝑝𝑠
𝑄=𝑄(𝜇/𝑐3𝑝𝑠)
𝑆𝑖𝑗=𝑆𝑖𝑗(𝑐/𝑝𝑠𝑅2𝐽)
𝑊0=𝑊0/𝑝𝑠𝑅2𝐽
(𝑋𝐽,𝑥,𝑍𝐽,𝑧)=(𝑋𝐽,𝑥,𝑍𝐽,z)/𝑐
(𝑋𝑖𝐿,𝑍𝑖𝐿)=(𝑋𝑖𝐿,𝑍𝑖𝐿)/𝑐
(𝛼,𝛽)=(𝑋,𝑌)/𝑅𝐽
𝛽∗	𝑝∗/𝑝𝑠
: Circumferential axial coordinates	𝜀
𝑒/𝑐 = Concentric design pressure ratio,	𝜆=𝐿/𝐷
= Ω=𝜔𝐽(𝜇𝑅2𝐽/𝑐2𝑝𝑠): Eccentricity ratio
𝜔th=𝜔th/𝜔𝐼: Aspect ratio
𝑡=𝑡(𝑐2𝑝𝑠/𝜇𝑅2𝐽): Speed parameter
𝛿=𝐶1/𝐶2
𝑂𝐽
𝑂𝐿𝑖: Offset factor
[𝐹]: Journal center	{𝑝}
{𝑄}: Lobe center	{𝑅𝐻}

Matrices

{𝑅𝑋𝑗},{𝑅𝑍𝑗}∶:	Fluidity matrix
𝐽:	Nodal pressure vector
𝑅:	Nodal flow vector
𝑠:	Vector due to hydrodynamic terms
−	Right-hand side vectors due to journal center velocities

Subscripts and Superscripts

0:	Steady-state solution
𝑙:	Journal
𝑖:	Restrictor
:	Supply
:	Corresponding nondimensional parameter
:	Lobe
:	Lobe number