# Effect ofL

Authors: Sanyam Sharma; C. M. Krishna
Journal: Advances in Tribology
 (2015)

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## Abstract

Two-lobe pressure dam bearings are commonly used in high speed rotating machineries. Their use is considered more stable than plain cylindrical bearings. Such bearings have a dam in the upper half whereas the lower half is provided with a relief track. Performance of two-lobe pressure dam bearings under micropolar fluid is evaluated. Finite element method is used to solve the modified Reynolds equation. Fluid film pressures are obtained by solving modified Reynolds equation. Thus pressure obtained is used to find performance characteristics of this bearing. The dynamic performance characteristics are studied at variousL
/
D ratios. Three L
/
D ratios are considered 1.6, 2.0, and 2.4 for the analysis purpose. Results obtained are presented for various micropolar parameters. Results show that stability of two-lobe pressure dam bearings increases with decrease in aspect ratio (L
/
D).


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## Body

## 1. Introduction

Pressure dam bearing is widely acceptable in high speed rotating machineries due to its good stability characteristics over plain two-lobe bearing. Lobed bearing came into existence, to overcome the limitations of circular bearing such as low load carrying capacity, stiffness, and stability. Effect ofL
/
D ratios on the stability of circular bearing was discussed by [1–3]. They found that L
/
D ratio affects the stability up to some extent. Therefore, researchers showed interest to examine the noncircular (lobed) bearing. The analysis of lobed bearing was given by [4–7]. These researchers analyzed the performances of such bearings and found that lobed bearings were much better in stability, which was the main problem of high speed rotating machineries. The problem was minimized by changing the bearing geometry from circular to lobed but could not be eliminated completely. Further, researchers continued work on the stability criterion and investigated circular and noncircular bearings by cutting a step (pressure dam) and relief rack in bottom half. Circular bearing with pressure dam was examined by [8, 9] and different variants of lobed bearing providing pressure dam were analyzed such as two-lobe by [10, 11] and three-lobe by [12, 13]. All the researchers engaged in analysis of lobed bearing with pressure dam concluded that the performance of pressure dam bearing was better than plain lobed bearing (without pressure dam).Due to the contamination of dirt or foreign particles in the commercial lubricant, classical fluid theory does not work well. Therefore to solve the lubrication problem [14] develops micropolar fluid theory. Micropolar fluids are fluids with microstructure. They contain rigid, randomly oriented, or spherical particles suspended in viscous medium. To increase effectiveness of the Newtonian lubricants, polymer additives are mixed to the Newtonian lubricant in order to improve their performance characteristics and then such fluid behaves as micropolar fluid. Increase in the viscosity of such lubricant is also observed as the additives are introduced in Newtonian lubricant. Load carrying capacity increases as the concentration of additives and contaminants increases [15]. There are many physical examples of micropolar fluids as ferrofluids, blood flows, bubbly liquids, and liquid crystals [16] in nuclear power plant where heat transfer agent sodium is used as a micropolar lubricant [17].As per the Eringens [14] theory, micropolar fluid was superior in comparison to Newtonian fluid. Higher threshold of stability was achieved [18] in circular bearing with micropolar lubrication. Lobed bearings were examined by [19, 20]. Rahmatabadi et al. [19] extended the work of Das et al. [18] for the analysis of lobed bearing under micropolar lubrication and investigated the static performance of lobed bearing and observed that micropolar lubricants could produce significant increase in the static performance characteristics. Rahmatabadi et al. [20] considered the effects of bearing orientation on the performance of lobed bearing and reported that the performance of the two-lobe bearing was significantly affected by the bearing orientation, as the number of lobes in a bearing was increased.During the literature survey, author found that few researchers [21] worked on two-lobe bearing with pressure dam. Since micropolar fluid is found to have considerable effect on the performance characteristics of two-lobe pressure dam bearing, it is expected that the performance of two-lobe pressure dam bearing system will also be affected. This present paper deals with the effect of L
/
D ratio on the dynamic performance of two-lobe pressure dam bearing.


## 2. Bearing Geometry

Performance analysis of such bearing is obtained by adding the individual lobe performance evaluated separately. The upper lobe is provided with pressure dam and lower lobe with relief track is shown in Figures1 and 2. Lengths l
1 and l
2 are circumferential lengths before and after dam. Consider
(1)
ε
1
2
=
ε
2
+
δ
v
2
+
2
ε
δ
v
cos
⁡
ϕ
,
ε
2
2
=
ε
2
+
δ
v
2
-
2
ε
δ
v
cos
⁡
ϕ
,
ϕ
1
=
ta
n
-
1
ε
sin
ϕ
ε
cos
⁡
ϕ
+
δ
v
180
π
,
ϕ
2
=
ta
n
-
1
ε
sin
ϕ
ε
cos
⁡
ϕ
-
δ
v
180
π
.Figure 1
Two-lobe pressure dam bearing.Figure 2
Pressure dam geometry.


## 3. Analysis

### 3.1. Modified Reynolds Equation

Based on [18] work the generalized form of the Reynolds equation for micropolar fluids is given below:
(2)
∂
∂
θ
ψ
(
Λ
,
N
,
h
-
)
μ
∂
p
-
∂
θ
+
R
L
·
∂
2
∂
z
ψ
Λ
,
N
,
h
-
μ
∂
p
-
∂
z
=
∂
h
-
∂
θ
+
12
∂
h
¯
∂
t
,
where
(3)
ϕ
Λ
,
N
,
h
-
=
h
-
3
+
12
h
-
l
m
2
-
6
N
h
-
2
l
m
coth
⁡
N
h
-
l
m
2
.N
2 and Λ are the two important parameters of a micropolar fluid. Linear and angular momentum equations are coupled by a parameter N
2, called the coupling number; the second nondimensional parameter is the characteristic length (l
m) which represents the interaction between the bearing geometry and fluid film. l
m depend on the size of the lubricant molecules. At smaller l
m the effect of microstructure is very pronounced. Lower values of l
m mean the larger values of the characteristic length (l
m
=
C
/
Λ). Thus, small l
m indicates that characteristic length of the substructure is larger compared to that of clearance dimension. At high l
m values, the effect of substructure becomes less significant and in the limiting case when l
m
→
∞, the effect of individuality of the substructure is lost and the lubricant reduces to Newtonian one. The following dimensionless quantities are used:
(4)
l
m
=
c
Λ
,
θ
=
x
R
,
Z
-
=
Z
L
,
h
-
=
h
c
m
,
p
-
=
p
c
m
2
μ
U
R
.
Pressure boundary conditions are applied:
(5)
P
¯
θ
,
Z
¯
=
0
at
θ
=
0
,
θ
2
P
¯
θ
,
Z
¯
=
0
at
z
¯
=
±
1
∂
p
¯
∂
θ
θ
,
z
=
0
at
θ
=
θ
2
.


### 3.2. Solution Procedure

Equation (2) is solved by finite element method, using Galerkin’s approach. The domain is divided into four noded isoparametric elements, such as 100 elements in circumferential direction and 120 elements in the axial direction. The elements matrices are generated for all elements and assembled to get global coefficient matrix. Thus finite element equations are obtained. Fluid film pressures are computed at nodal points by applying Reynolds’ boundary conditions as given in (5).


## 3.1. Modified Reynolds Equation

Based on [18] work the generalized form of the Reynolds equation for micropolar fluids is given below:
(2)
∂
∂
θ
ψ
(
Λ
,
N
,
h
-
)
μ
∂
p
-
∂
θ
+
R
L
·
∂
2
∂
z
ψ
Λ
,
N
,
h
-
μ
∂
p
-
∂
z
=
∂
h
-
∂
θ
+
12
∂
h
¯
∂
t
,
where
(3)
ϕ
Λ
,
N
,
h
-
=
h
-
3
+
12
h
-
l
m
2
-
6
N
h
-
2
l
m
coth
⁡
N
h
-
l
m
2
.N
2 and Λ are the two important parameters of a micropolar fluid. Linear and angular momentum equations are coupled by a parameter N
2, called the coupling number; the second nondimensional parameter is the characteristic length (l
m) which represents the interaction between the bearing geometry and fluid film. l
m depend on the size of the lubricant molecules. At smaller l
m the effect of microstructure is very pronounced. Lower values of l
m mean the larger values of the characteristic length (l
m
=
C
/
Λ). Thus, small l
m indicates that characteristic length of the substructure is larger compared to that of clearance dimension. At high l
m values, the effect of substructure becomes less significant and in the limiting case when l
m
→
∞, the effect of individuality of the substructure is lost and the lubricant reduces to Newtonian one. The following dimensionless quantities are used:
(4)
l
m
=
c
Λ
,
θ
=
x
R
,
Z
-
=
Z
L
,
h
-
=
h
c
m
,
p
-
=
p
c
m
2
μ
U
R
.
Pressure boundary conditions are applied:
(5)
P
¯
θ
,
Z
¯
=
0
at
θ
=
0
,
θ
2
P
¯
θ
,
Z
¯
=
0
at
z
¯
=
±
1
∂
p
¯
∂
θ
θ
,
z
=
0
at
θ
=
θ
2
.


## 3.2. Solution Procedure

Equation (2) is solved by finite element method, using Galerkin’s approach. The domain is divided into four noded isoparametric elements, such as 100 elements in circumferential direction and 120 elements in the axial direction. The elements matrices are generated for all elements and assembled to get global coefficient matrix. Thus finite element equations are obtained. Fluid film pressures are computed at nodal points by applying Reynolds’ boundary conditions as given in (5).


## 4. Results and Discussion

Various dynamic characteristics are presented in the form of plots for three differentL
/
D ratios as shown in Table 1 while other parameters are held constant like eccentricity ratio (
ε
), ellipticity ratio (ζ), coupling number (N
2), dam depth, and relief track length. The values of characteristics length l
m are chosen (1–50), as beyond the l
m
=
50, all the performance characteristics were found to be independent of “l
m” beyond the value l
m
=
50. This is the typical range taken by all researchers.Table 1
Parameters for two-lobe pressure dam bearing.
Bearing geometry parameters
Aspect ratioL/D
1.6, 2.0, 2.4
Operating parameters
Eccentricity ratioε
0.20
Ellipticity ratioζ
0.3
Dam parameters
Dam width ratioD
-
w
0.4, 0.2
Dam depth ratioD
-
d
0.25
Relief  track width ratioL
-
t
0.25
Micropolar parameters
Coupling numberN
2
0.9
Characteristics lengthl
m
1, 3, 5, 10, 20, 40, 50Figure3 depicts that load carrying capacity increases as there is increase in L
/
D ratio and is found to be maximum at aspect ratio 2.4. It means at higher L
/
D ratio the bearing can sustain higher load. For a particular value of L
/
D ratio, it is observed that load is higher at lower value of characteristics length (l
m) and decreases as l
m increases. Load capacity is higher at higher value of D
w. Figure 4 shows that attitude angle increases as there is increase in L
/
D ratio and it is found to be maximum at aspect ratio 2.4. It means at higher L
/
D ratio of the bearing, for a particular value of L
/
D ratio, it is observed that attitude angle is higher at lower value of characteristics length (l
m) and decreases as characteristics length (l
m) decreases. It is also observed that there is a minimum attitude angle for each L
/
D ratio. Attitude angle is observed higher at larger dam width ratio. Figure 5 depicts that K
x
x first increases for lower values of l
m and then becomes constant for the higher values of characteristics lengths. Higher material length does not produce much effect on the stiffness K
x
x. Figure 6 depicts that K
x
y is found to be maximum at smaller L
/
D ratio 1.6. It is observed that K
x
y is higher at lower value of characteristics length (l
m) and decreases as l
m decreases. Figures 7, 8, and 9 show the behavior of K
y
y, C
x
x, and C
x
y, respectively, and show that all the three are on the same trend. These stiffness coefficients are found on higher side, as L
/
D ratio increases. For all values of L
/
D ratio these coefficients decrease as l
m increases. Figures 10 and 11 show the behavior of cross coupling stiffness C
x
y and C
y
x, respectively, and depict that these coefficients decrease as L
/
D ratio increases. These coefficients are of negative values and near about the same. Figure 12 depicts that C
y
y increases as there is increase in L
/
D ratio and is found to be maximum at a ratio 2.4. It is also observed that for a particular L
/
D ratio C
y
y is higher at lower value of characteristics length (l
m) and decreases as l
m decreases. Damped frequency of whirl (ω
d) is shown in Figure 13 and is found minimum at aspect ratio 1.6. It is also observed that for a particular L
/
D ratio damped frequency of whirl is higher at lower value of characteristics length (l
m) and decreases as l
m decreases. Damped frequency is higher at lower dam width ratio. Figure 14 represents the behavior of critical mass with characteristics length. Critical mass is higher at lower values of l
m, irrespective of dam width ratios and decreases as l
m increases. Critical mass is higher at higher value of dam width ratio for a particular L
/
D ratio. Figure 15 shows the threshold speed variation with l
m which shows that threshold speed ω
s increases up to a particular material length and then decreases with increase in l
m. Threshold speed is higher at higher dam width ratio for the same value of L
/
D ratio.Figure 3
Load variation as a function of characteristics length “l
m.”Figure 4
Attitude angle variation as a function of characteristics length “l
m.”Figure 5
Variation ofK
x
x as a function of characteristics length “l
m.”Figure 6
Variation ofK
x
y as a function of characteristics length “l
m.”Figure 7
Variation ofK
y
x as a function of characteristics length “l
m.”Figure 8
Variation ofK
y
y as a function of characteristics length “l
m.”Figure 9
Variation ofC
x
x as a function of characteristics length “l
m.”Figure 10
Variation ofC
x
y as a function of characteristics length “l
m.”Figure 11
Variation ofC
y
x as a function of characteristics length “l
m.”Figure 12
Variation ofC
y
y as a function of characteristics length “l
m.”Figure 13
Variation of damped frequency of whirlω
d as a function of characteristics length “l
m.”Figure 14
Variation of critical massM
c as a function of characteristics length “l
m.”Figure 15
Variation of threshold speedω
s as a function of characteristics length “l
m.”


## 5. Conclusions

(i)
By incorporating of pressure dam, relief track and modification of lubricant to micropolar fluid result in increase of load capacity with increase in the aspect ratios.
(ii)
There is reduction in attitude angle for low value of characteristics length “l
m,” but it increases with increase in characteristics length “l
m.” Higher aspect ratios result in larger attitude angle. Lower attitude angle is observed at smaller dam width ratio.
(iii)
Damped frequency of whirl is found to be lower at higher dam width ratio, while critical mass and threshold speed increase as dam width ratio increases.
(iv)
Stability of two-lobe pressure dam increases by incorporating pressure dam with micropolar lubrication.



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*Source: 182713-2015-02-25.xml*