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

Authors: J. Sharana Basavaraja; Satish C. Sharma; S. C. Jain
Journal: Advances in Tribology
 (2008)

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## 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
(W¯0).
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.


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## Body

## 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] (1)∂∂α(h¯36∂p¯∂α)+∂∂β(h¯36∂p¯∂β)=Ω∂h¯∂α+2∂h¯∂t¯.


## 3. Fluid Film Thickness

The geometry of the symmetric hole-entry circular
as well as multilobe hybrid journal bearing has been shown schematically in Figures1(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](2)h¯=1δ−(X¯j+x¯−X¯Li)cos⁡α−(Z¯j+z¯−Z¯Li)sin⁡α, 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¯Li and Z¯Li are the lobe center coordinates of ith  lobe.(a) Hole-entry type journal bearing
system (symmetric configuration). (b) Two-lobe hole-entry type journal bearing system (symmetric
configuration).
(a)(b)


## 4. Restrictor Flow Equation

The flow rate of the lubricant through the
orifice restrictor is defined as [17](3)Q¯R=C¯S2(1−p¯c)1/2,
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](4)p¯=∑j=14Njp¯j,where Nj is elemental shape function,
and using the approximate value of p¯ (1) can be expressed as(5)∂∂α[h¯36∂∂α(∑j=14Njp¯j)]+∂∂β[h¯36∂∂β(∑j=14Njp¯j)]−Ω∂h¯∂α−2∂h¯∂t¯=Re,where Re 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,(6)∬ΩeNiRedα⁢dβ=0.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: (7)[F¯]n×n{p¯}n×1={Q¯}n×1+Ω{R¯H}n×1+x˙¯j{R¯xj}n×1+z˙¯j{R¯zj}n×1,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 aneth element, the elements of the above
matrices are defined as follows:
(7a)F¯ije=∬Aeh¯3F¯2[∂Ni∂α∂Nj∂α+∂Ni∂β∂Nj∂β]dα⁢dβ,⁢(7b)Q¯ie=∫Γe{(h¯3F¯2∂p¯∂α−Ω(1−F¯1F¯0)h¯)l+(h¯3F¯2∂p¯∂β)m}NidΓ¯e,(7c)R¯Hie=∬Ae{(1−F¯1F¯0)h¯}∂Ni∂αdα⁢dβ,(7d)R¯XJie=∬AeNicos⁢α⁢dα⁢dβ,(7e)R¯ZJie=∬AeNisin⁡α⁢dα⁢dβ,
where l and m are the directions cosines and i, j=1,2,…, nle (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,(8)p¯|β=∓1.0=0.0.(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 (X˙¯j,Z˙¯j=0), 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 (X¯j,Z¯j) 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:(9)F¯x=0,F¯z−W¯0=0.The
fluid-film reaction terms F¯x and F¯z in (9) are expanded by Taylor series about the ith journal center position, and the increments (ΔX¯ji,ΔZ¯ji) on the journal coordinates are obtained.
Iterations are continued until the following convergence criterion is not
satisfied: (10)[((ΔX¯ji)2+(ΔZ¯ji)2)1/2((X¯ji)2+(Z¯ji)2)1/2]×100<0.001,where X¯ji,Z¯ji are the coordinates of the ith 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:(11)S¯ij=−∂F¯i∂q¯J,(i=x,z),wherei = direction of force or moment.q¯J = direction of journal center displacement (q¯J=X¯J,Z¯J).Fluid-film damping coefficients:(12)C¯ij=−∂F¯i∂q˙¯J,(i=x,z),here, q˙¯J represents the velocity component of journal
center (q˙¯J=X˙¯J,Z˙¯J).


## 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 (δ) and
restrictor design parameter (C¯S2). The chosen values of bearing operating parameter are most generally
used values as given in Table 1.Table 1
Bearing
operating and geometric parameters.
Bearing aspect ratio (λ)1Land width ratio(a¯b)0.25Restrictor design parameter(C¯S2)0.02–0.1Offset factor (δ)0.5, 0.75, 1.0, 1.25, 1.5External load(W¯0)1.25Number of holes per row12Number of rows2Concentric pressure ratio (β*)0.5Type of restrictororificeIn 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(F¯0) 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,L/D = 1, Ω = 1, δ=0.5. W¯0 = 5.0.
CharacteristicsPresentRef. [19]ε0.34260.340ϕ80.99782.100S¯xx/W¯01.2031.110S¯xz/W¯01.8121.700S¯zx/W¯0−3.5004−3.610S¯zz/W¯05.15445.130C¯xx/W¯02.8012.620−(C¯xz=C¯zx)/W¯00.25680.238C¯zz/W¯09.33489.430Figure 3
Variation of external load carrying capacity(F¯0) with eccentricity (ε).The static
and dynamic performance characteristics which include maximum pressure(p¯),
minimum fluid film thickness (h¯min),
bearing flow (Q¯),
stiffness coefficients (S¯ij),
damping coefficients (C¯ij),
and threshold speed (ω¯th) have been presented for various values of
offset factors (δ),
restrictor design parameter (C¯S2),
and external load (W¯0). The results are presented through Figures
(4–14) and discussed in the following paragraph.Figure 4
Variation of maximum pressure(p¯).Figure 5
Variation of fluid film thickness(h¯).Figure 6
Variation of bearing flow(Q¯).Figure 7
Variation of direct stiffness(S¯xx).Figure 8
Variation of direct stiffness(S¯zz).Figure 9
Variation of direct stiffness(S¯zz).Figure 10
Variation of cross coupled stiffness(S¯zx).Figure 11
Variation of direct damping(C¯xx).Figure 12
Variation of direct damping(C¯zz).Figure 13
Variation of direct damping(C¯zz).Figure 14
Variation of threshold speed(ϖ¯th).Figure4 shows
the variation of maximum pressure (p¯).
It has been observed that the value of maximum pressure (p¯) for a specified external load increases with
restrictor design parameter (C¯S2).
From the graph, it may be observed for the given value of external load (W¯0) and the restrictor design parameter (C¯S2) 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 (p¯) increases in case of two-lobe hole-entry
hybrid journal bearing with offset factor (δ) more than one compared to circular hole-entry
hybrid journal bearing. The decrease in the value of fluid film thickness (h¯) owing to the change of geometry of the bearing
is responsible for the increase in the value of maximum pressure (p¯).Figure5 shows the variation of minimum fluid-film thickness (h¯min).
It is observed that the value of minimum fluid film thickness (h¯min) for a specified external load reduces with an
increase in restrictor design parameter (C¯S2) for all the cases of  offset factor (δ) 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 (δ) 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 δ = 0.75 operates with higher value of minimum fluid film thickness (h¯min) compared to other cases of offset factor (δ).
The
increase or decrease in the value of minimum fluid film thickness (h¯min) has been found of the order of 0.5 < .75
>1.0 >1.25 >1.3. The maximum value of minimum fluid film thickness (h¯min) 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 h¯min may be obtained in a two-lobe hole-entry journal
bearing by carefully selecting the value of restrictor design parameter (C¯S2) and offset factor (δ).At a constant value of external load (W¯0), the bearing flow (Q¯) in Figure 6 is found to be lower for the case of two-lobe hole-entry
hybrid journal bearing with offset factor (δ) 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 (δ). 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 (S¯xx,S¯zz) 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 (C¯S2) at which the value of direct stiffness coefficient (S¯xx) is the maximum for a
particular case of offset  factor (δ).
As seen from Figure 7, at a value of restrictor design parameter (C¯S2=.04),
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 (C¯S2=.08),
the hole-entry hybrid journal bearing with an offset factor δ = 1.0 has the
highest value of direct stiffness coefficient (S¯xx).
The results from Figure 7 indicate that the restrictor design parameter (C¯S2) becomes an important parameter when the
designer aims to maximize the value of (S¯xx) at constant external load (W¯0). Figure 8 shows, at a constant external load
W¯0,
fluid film stiffness
coefficients (S¯zz) which in general show an initial increasing
trend and then decrease with an increase in restrictor
design  parameter (C¯S2) 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 (δ) greater than one for a chosen load.Figure9 shows the effect of direct fluid-film stiffness
coefficients (S¯zz) with an increase in offset
factor (δ).
The value of direct fluid-film stiffness coefficient (S¯zz) gets decreased with an increase in the value
of external load (W¯0) for all the cases of offset factor (δ). Further, it may be observed that the value of direct fluid-film stiffness coefficient (S¯zz) is maximum in case of two-lobe hole entry
hybrid journal bearing with an offset factor δ = 1.3.Figure10
shows the effect of cross coupled fluid-film stiffness coefficients (S¯zx) with an increase in offset factor (δ).
It can be observed from the figure that the values of S¯zx in case of two-lobe hole-entry journal bearing
follow the similar trend for the different values of offset factor (δ).
The two-lobe hole-entry hybrid journal bearing with offset factor of 0.5 has
the least value of S¯zx.The
variation of direct damping coefficients(C¯xx,C¯zz) with offset factor (δ)
is shown in Figures 11
and 12.  At constant load (W¯0), damping coefficients (C¯xx,C¯zz) are increased with an increase of offset
factor (δ).
It is observed from Figures 11 and 12 that the two-lobe hole-entry hybrid
journal bearing with offset factor (δ) greater than one exhibits higher value of
direct damping coefficients (C¯xx,C¯zz) compared to that of circular hole-entry hybrid
journal bearing.Figure13 shows the variation of direct fluid-film coefficient (C¯zz) against restrictor design parameter (C¯S2).
The value of direct fluid-film damping coefficient (C¯zz) gets increased with an increase in offset
factor (δ) for the chosen value of external load (W¯0). Further, it may be observed that the value
of direct fluid-film damping coefficient (C¯zz) decreases with an increase in the value
of restrictor design parameter (C¯S2) for all the cases of offset factor (δ) except for the case of offset factor δ=0.5. For the case of offset
factor δ=0.5, the value of direct fluid-film damping coefficient (C¯zz) nearly
remains a constant with an increase in restrictor design parameter (C¯S2).Figure14 shows variation with stability threshold speed  (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 (ωJ) is less than stability threshold  (ω¯th), that is, ωJ<ωth.
From Figure 7, it may be observed that there exists a particular value of
restrictor design parameter (C¯S2) at which the value of stability threshold
speed  (ω¯th) is the maximum for a particular case of offset
factor (δ).
As seen from the Figure 7, at a value of restrictor design parameter (C¯S2=.06),
the two-lobe hole-entry hybrid journal bearing with an offset factor δ=1.25
has the highest value of stability threshold speed  (ω¯th) and at a value of restrictor design parameter (C¯S2=.08),
the hole-entry hybrid journal bearing with an offset factor δ=1.0 has the
highest value of stability threshold speed  (ω¯th).
The results from Figure 7 indicate that the restrictor design parameter (C¯S2) becomes an important parameter when the
designer aims to maximize the value of  (ω¯th) at constant external load (W¯0).


## 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(δ) 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 (S¯zz) and
damping (C¯xx) 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 (W¯0) and for the same
bearing geometric parameters.(3)
A two-lobe hole-entry journal bearing system with offset factor(δ) 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.



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