effect of dam depth and relief track depth on steady-state and dynamic performance ... · 2019. 7....

Research ArticleEffect of Dam Depth and Relief Track Depth onSteady-State and Dynamic Performance Parameters of3-Lobe Pressure Dam Bearing

Ashutosh Kumar and S. K. Kakoty

Indian Institute of Technology Guwahati, Guwahati, India

Correspondence should be addressed to Ashutosh Kumar; [email protected]

Received 18 October 2016; Accepted 31 January 2017; Published 14 March 2017

Academic Editor: GunHee Jang

Copyright © 2017 Ashutosh Kumar and S. K. Kakoty. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The present study analyzes the effect of pressure dam depth and relief track depth on the performance of three-lobe pressure dambearing. Different values of dam depth and relief track depth are taken in nondimensional form in order to analyze their effect.Results are plotted for different parameters against eccentricity ratios and it is shown that the effect of pressure dam depth and relieftrack depth has great significance on stability and other performance parameters. Study of stability and performance characteristicsis undertaken simultaneously.

1. Introduction

In present time, the industry is required to run the turbo-machines for long time at high speeds. The most commonlyused cylindrical bearings are found to be unstable at highspeed. It is found that stability can be increased by usingnoncircular bearing instead of circular bearing. Further, useof pressure dammakes the bearing very stable at high speeds.The stability analysis of finite length bearing was carriedout by Nicholas and Allaire [1]. Nicholas et al. [2] reportedexperimental whirl frequency ratio (WFR) for near optimumpressure dam bearings supporting flexible rotors. The effectof ellipticity ratio on the performance of displaced centrebearing is observed and found that there is an increase invertical stiffness with increase in ellipticity [3]. Performanceanalysis of three-lobe pressure dam bearing was carried outbyMehta and Rattan [4] and found that it has very high valueof minimum threshold speed and zone of infinite stability.The relief track width is one of the important parametersthat affect the performance of pressure dam bearings. Lundand Thomson [5] had done the analysis of steady-state anddynamic characteristics of grooved, two-lobe, and three-lobebearings and presented the result in tabular form. In thesame line, Soni et al. [6] studied two-lobe journal bearing

using linearized turbulent lubrication theory by FEM andGalerkin’s technique.

Kumar et al. [7] had done the detail study and analysisof two-lobe pressure bearing and shown the superiority ofthese bearings over conventional circular bearing. Static anddynamic characteristics of orthogonally displaced pressuredambearing had been studied byMehta [8].Mehta and Singh[9] studied the dynamic behavior of offset half pressure dambearing. Effect of micropolar lubrication couple stress fluidson two-lobe andmultilobe pressure dam bearing was studiedby many researchers. In this line, Sharma and Ratan [10]studied the performance of two-lobe pressure dam bearingsunder micropolar fluid lubrication and found an increase inload carrying capacity.

From the available literature, it has been proven thatincorporation of pressure dam and relief track is usefulin improving the stability of a multilobe pressure dambearing. So, the pressure dam depth, relief track depth,pressure dam axial width, relief track axial width, pressuredam circumferential length, and relief track circumferentiallength are supposed to have great influence on static anddynamic behaviors of two-lobe and multilobe pressure dambearings. Therefore, the present study has been undertakento investigate the effect of pressure dam depth and relief track

HindawiAdvances in TribologyVolume 2017, Article ID 1380367, 11 pageshttps://doi.org/10.1155/2017/1380367

https://doi.org/10.1155/2017/1380367

2 Advances in Tribology

R

Step

Lobe 3

Lobe 2

XStep

Z

Oil supply

Relief trackLoad

R

O1

O2

O3

C − Cm

Cm

R + C

Lobe 1

Figure 1: Three-lobe pressure dam bearing.

Oil grooves Oil groovesLobe 2 and lobe 3Step

Lobe 1

L

L

Dw

Rw

Dw = Dw/L

Rw = Rw/L

Figure 2: Individual lobes of three-lobe pressure dam bearing.

depth on the steady-state and dynamic characteristics of two-lobe and three-lobe pressure dam bearings.

2. Theory

Figure 1 shows the schematic diagram of three-lobe pressuredam bearing, whereas Figure 2 shows different lobes withpressure dam and relief track. O1, O2, and O3 are the centerof lobes 1, 2, and 3, respectively. Each lobe of the bearing iscircular but the geometric configuration of the bearing is not.Lobe 2 and lobe 3 are provided with a rectangular pressuredam of depth𝐷𝑑 and width (axial length)𝐷𝑤. Similarly, lobe1 is provided with relief track of depth 𝑅𝑑 and width 𝑅𝑤. Eachdam subtends an angle of 90∘ at the center starting from thebeginning of the individual lobe. Bearing is provided withthree oil supply holes of 10∘ at 0∘, 120∘, and 240∘. Two differentclearances are shown: aminor clearance (𝐶𝑚) for the centeredshaft and a major clearance (𝐶) for the circle circumscribedby the lobe radius. So, the center of each lobe is shifted by a

distance 𝑒𝑝, which is given by 𝑒𝑝 = 𝐶 − 𝐶𝑚, and this 𝑒𝑝 isnondimensionalized by dividing it by major radial clearance(𝐶) which is known as ellipticity ratio; that is, 𝛿 = 𝑒𝑝/𝐶 =1 − 𝐶𝑚/𝐶.

From the geometry, eccentricity ratios of individual lobesare given as

𝜀1 = √𝜀2 + 𝛿2 + 2𝜀𝛿 cos (𝜙);

𝜀2 = √𝜀2 + 𝛿2 − 2𝜀𝛿 cos(𝜋3 + 𝜙);

𝜀3 = √𝜀2 + 𝛿2 − 2𝜀𝛿 cos(𝜋3 − 𝜙).

(1)

Similarly, attitude angles of individual lobes are given as

𝜙1 = tan−1 ( 𝜀 sin𝜙𝛿 + 𝜀 cos𝜙) ;

𝜙2 = 2𝜋3 − tan−1 (𝜀 sin (𝜋/3 + 𝜙)

𝛿 − 𝜀 cos (𝜋/3 + 𝜙)) ,

𝜙3 = −(2𝜋3 − tan−1 (𝜀 sin (𝜋/3 − 𝜙)

𝛿 − 𝜀 cos (𝜋/3 − 𝜙))) .

(2)

Reynolds equation for incompressible, nonviscous lami-nar flow for hydrodynamic lubrication is given as

𝜕𝜕𝑥 (ℎ3

𝜕𝑝𝜕𝑥) +

𝜕𝜕𝑧 (ℎ3

𝜕𝑝𝜕𝑧) = 6𝜂𝑈

𝜕ℎ𝜕𝑥 + 12𝜂

𝜕ℎ𝜕𝑡 . (3)

This equation in nondimensionalized (by using substitu-tions) form is given as

𝜕𝜕𝜃 (ℎ

3 𝜕𝑝𝜕𝜃) + (

𝐷𝐿 )2 𝜕𝜕𝑧 (ℎ

3 𝜕𝑝𝜕𝑧) =

𝑑ℎ𝑑𝜃 + 2𝜆

𝜕ℎ𝜕𝜏 . (4)

Equation (2) is the nondimensional Reynolds equationfor dynamic state condition. Let 𝜀0 and 𝜙0 be the steady-stateeccentricity ratio and attitude angle, respectively. Now, con-sidering that the journal whirls with small amplitude aboutits mean steady-state position and considering only first-

Advances in Tribology 3

order perturbation (neglecting higher order term), nondi-mensional pressure and film thickness can be expressed as[11]

𝑝 = 𝑝0 + 𝜀1𝑒𝑖𝜔𝜏𝑝1 + 𝜀0𝜙1𝑒𝑖𝜔𝜏𝑝2;ℎ = ℎ0 + 𝜀1𝑒𝑖𝜔𝜏 cos 𝜃 + 𝜀0𝜙1𝑒𝑖𝜔𝜏 sin 𝜃.

(5)

On substituting 𝑝 and ℎ in (4) and equating coefficient of𝜀0, 𝜀1𝑒𝑖𝜔𝜏, and 𝜀0𝜙1𝑒𝑖𝜔𝜏, the following set of three equations isobtained when higher-order terms are neglected:

𝜕𝜕𝜃 (ℎ0

3 𝜕𝑝0𝜕𝜃 ) + (𝐷𝐿 )2 𝜕𝜕𝑧 (ℎ0

3 𝜕𝑝0𝜕𝑧 ) =𝜕ℎ0𝜕𝜃 , (6)

𝜕𝜕𝜃 (ℎ0

3 𝜕𝑝1𝜕𝜃 ) + (𝐷𝐿 )2 𝜕𝜕𝑧 (ℎ0

3 𝜕𝑝1𝜕𝑧 )

+ 3 𝜕𝜕𝜃 (ℎ02 𝜕𝑝0𝜕𝜃 cos 𝜃)

+ 3 (𝐷𝐿 )2 𝜕𝜕𝑧 (ℎ0

2 𝜕𝑝0𝜕𝑧 cos 𝜃) = − sin 𝜃+ 2𝑖𝜔 cos 𝜃,

(7)

𝜕𝜕𝜃 (ℎ0

3 𝜕𝑝2𝜕𝜃 ) + (𝐷𝐿 )2 𝜕𝜕𝑧 (ℎ0

3 𝜕𝑝2𝜕𝑧 )

+ 3 𝜕𝜕𝜃 (ℎ02 𝜕𝑝0𝜕𝜃 sin 𝜃)

+ 3 (𝐷𝐿 )2 𝜕𝜕𝑧 (ℎ0

2 𝜕𝑝0𝜕𝑧 sin 𝜃) = cos 𝜃 + 2𝑖𝜔 sin 𝜃.

(8)

Equation (6) gives the steady-state pressure distributionand (7) and (8) give the dynamic pressure distribution in eachindividual lobe.

On solving these equations, the following steady-statepressure and dynamic pressure are obtained:

𝑝0𝑖,𝑗 =[(𝑝0𝑖+1,𝑗 + 𝑝0𝑖−1,𝑗) + 𝐶0 (𝑝0𝑖,𝑗+1 + 𝑝0𝑖,𝑗−1) − 𝐶1 (𝑝0𝑖+1,𝑗 − 𝑝0𝑖−1,𝑗) + (𝜀 sin 𝜃/ℎ03) (Δ𝜃)2]

2 [1 + 𝐶0] ,𝑝1𝑖,𝑗= [(𝑝1𝑖+1,𝑗 + 𝑝1𝑖−1,𝑗) + 𝐶0 (𝑝1𝑖,𝑗+1 + 𝑝1𝑖,𝑗−1) − 𝐶1 (𝑝1𝑖+1,𝑗 − 𝑝1𝑖−1,𝑗) − 𝐶2 (𝑝0𝑖+1,𝑗 − 𝑝0𝑖−1,𝑗) + 𝐶3 (𝑝0𝑖+1,𝑗 − 2𝑝0𝑖,𝑗 + 2𝑝0𝑖−1,𝑗) + 𝐶0 × 𝐶3 (𝑝0𝑖,𝑗+1 − 2𝑝0𝑖,𝑗 + 2𝑝0𝑖,𝑗−1) + 𝐶4]2 [1 + 𝐶0] ,𝑝2𝑖,𝑗= [(𝑝2𝑖+1,𝑗 + 𝑝2𝑖−1,𝑗) + 𝐶0 (𝑝2𝑖,𝑗+1 + 𝑝2𝑖,𝑗−1) − 𝐶1 (𝑝2𝑖+1,𝑗 − 𝑝2𝑖−1,𝑗) + 𝐶5 (𝑝0𝑖+1,𝑗 − 𝑝0𝑖−1,𝑗) + 𝐶6 (𝑝0𝑖+1,𝑗 − 2𝑝0𝑖,𝑗 + 2𝑝0𝑖−1,𝑗) + 𝐶0 × 𝐶6 (𝑝0𝑖,𝑗+1 − 2𝑝0𝑖,𝑗 + 2𝑝0𝑖,𝑗−1) − 𝐶7]2 [1 + 𝐶0] ,

(9)

where

𝐶0 = (𝐷𝐿 )2 (Δ𝜃Δ𝑧)

2 ;𝐶1 = 3𝜀2ℎ0 sin 𝜃 (Δ𝜃) ;𝐶2 = 32ℎ22

(2𝜀 cos 𝜃 + ℎ0) sin 𝜃 (Δ𝜃) ,

𝐶3 = 3ℎ0 cos 𝜃;

𝐶4 = (sin 𝜃 − 2𝑖𝜔 cos 𝜃) (Δ𝜃)2ℎ03;

𝐶5 = 3 (Δ𝜃)2ℎ02(ℎ0 cos 𝜃 − 2𝜀 sin2 𝜃)

𝐶6 = 3ℎ0 sin 𝜃;

𝐶7 = (Δ𝜃)2ℎ03(cos 𝜃 + 2𝑖𝜔 sin 𝜃) .

(10)

Boundary conditions used for the steady-state pressureand dynamic pressure distribution are as follows:

𝜕𝑝𝑖𝜕𝜃 = 0, 𝑝𝑖 = 0 at 𝜃 = 𝜃𝑟,𝑝𝑖 (𝜃, 𝑧) = 0 when 𝜃𝑠 ≤ 𝜃 ≤ 𝜃𝑒,

(11)

where 𝑝𝑖 = 𝑝0, 𝑝1, 𝑝2, 𝜃𝑠 is starting angle of groove withrespect to the vertical axis, 𝜃𝑒 is angle at which the grooveends with respect to the vertical axis, and 𝜃𝑟 is angle at whichthe film cavitates with respect to vertical axis.

Since the pressure distribution is symmetrical about thecenter line, only half of the bearing has been analyzed.The nondimensionalized pressure distribution equations aresolved by using finite difference method (FDM). Each halfof the bearing is divided into 88 and 16 elements alongcircumferential length and axial length, respectively. Gauss-Seidel method with successive overrelaxation technique sat-isfying boundary conditions has been used for the numericalintegration.

(𝑝𝑖,𝑗)new = (𝑝𝑖,𝑗)old + (Error)𝑖,𝑗 × 𝑜𝑟𝑓. (12)


The convergence criterion has been taken as

(∑𝑝𝑖,𝑗)𝑁−1 − (∑𝑝𝑖,𝑗)𝑁(∑𝑝𝑖,𝑗)𝑁

≤ 10−6. (13)

Since the load is acting in vertical direction (𝑊𝑥), the atti-tude angle keeps on changing till the horizontal componentbecomes zero for each eccentricity ratio.

The present analysis has been carried out for the bearingwith the following parameters:

𝐷𝑑 = 0.75;𝐷𝑤 = 0.5;𝑅𝑑 = 2.0;𝑅𝑤 = 0.5.

(14)

The present study considers the ellipticity ratio (𝛿) = 0.5and 𝐿/𝐷 = 1. However, dam depth ratio and relief trackratio are not constant. At a time, one of these two variablesis kept constant and the other one is varied in a certain rangeand the effect of the variable parameter over the performanceparameter is analyzed.

Direct and cross-coupled nondimensional dynamic coef-ficients (stiffness and damping coefficients) are determinedby separating real and imaginary parts of horizontal andvertical dynamic loads.

Horizontal and vertical components of dynamic load dueto dynamic pressures 𝑝1 and 𝑝2 are given, respectively, as

𝑊𝑧1 = ∫𝜃𝑒

𝜃𝑠

∫10𝑝1 cos 𝜃 𝑑𝜃 𝑑𝑧;

𝑊𝑧2 = ∫𝜃𝑒

𝜃𝑠


𝑊𝑥1 = ∫𝜃𝑒

𝜃𝑠

∫10𝑝1 sin 𝜃 𝑑𝜃 𝑑𝑧;

𝑊𝑥2 = ∫𝜃𝑒

𝜃𝑠


𝐾𝑥𝑥 = −Re (𝑊𝑥1) ;𝐾𝑥𝑧 = −Re (𝑊𝑥2) ,𝐾𝑧𝑥 = −Re (𝑊𝑧1) ;𝐾𝑧𝑧 = −Re (𝑊𝑧2)𝐶𝑥𝑥 = −Im (𝑊𝑥1) ;𝐶𝑥𝑧 = −Im (𝑊𝑥2) ,𝐶𝑧𝑥 = −Im (𝑊𝑧1) ;𝐶𝑧𝑧 = −Im (𝑊𝑧2) .

(15)

Nondimensional mass parameter, flow coefficient, fric-tion variable, and Sommerfeld number are given as

𝑀 = 𝑚𝑐𝜔2𝑊 =𝐾0𝜔2 ;

where, 𝐾0 = (𝐾𝑋𝑋𝐶𝑍𝑍 + 𝐾𝑍𝑍𝐶𝑋𝑋) − (𝐾𝑋𝑍𝐶𝑍𝑋 + 𝐾𝑍𝑋𝐶𝑋𝑍)𝐶𝑋𝑋 + 𝐶𝑍𝑍 , 𝜔2 = (𝐾𝑋𝑋 − 𝐾0) (𝐾𝑍𝑍 − 𝐾0) − 𝐾𝑋𝑍𝐶𝑍𝑋𝐶𝑋𝑋𝐶𝑍𝑍 + 𝐶𝑋𝑍𝐶𝑍𝑋 ,

𝑞𝑧 = 12 (𝐷𝐿 )2 ∫2𝜋0

ℎ30 𝜕𝑝0𝜕𝑧 𝑑𝜃;

𝜇 = 𝜇 (𝑅𝐶) =∫2𝜋0

(3ℎ (𝜕𝑝0/𝜕𝑧) + 1/ℎ0) 𝑑𝜃6𝑊 ;

𝑆 = 16𝜋𝑊.

(16)

The nondimensional film thickness in the entire rectan-gular pressure dam region is given as

ℎ = ℎ0 + 𝐷𝑑. (17)

Similarly, for the relief track region,

ℎ = ℎ0 + 𝑅𝑑, (18)where ℎ0 = 1 + 𝜀𝑖 cos(𝜃 − 𝜙); 𝑖 = 1, 2, and 3 for lobes 1, 2, and3, respectively.


0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

Eccentricity ratio (𝜀)

Load

carr

ying

capa

city

(W)

Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Figure 3: Effect of relief track depth ratio on load carrying capacity.

48505254565860626466

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Eccentricity ratio (𝜀)

Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

Attit

ude a

ngle(𝜙)

Figure 4: Effect of relief track depth ratio on attitude angle.

Since 𝑀 is closely related to the journal rotational speedand journal mass is normally constant, it is more convenientto define a speed parameter 𝜔𝑆 which is equal to square rootof𝑀.3. Results and Discussion

Various performance parameters have been plotted againsteccentricity ratio (𝜀) for different pressure dam depth ratio(𝐷𝑑) and relief track depth ratio (𝑅𝑑). In the first section, theeffect of the relief track depth ratio has been shownby keepingother parameters constant (𝐷𝑑 = 0.75; 𝐷𝑤 = 0.5; 𝑅𝑤 =0.75). Relief track depth ratio has been taken as (𝑅𝑑 = 0.2, 0.4,0.6, 0.8, and 1.6). In the second section, the effect of thepressure dam depth ratio has been shown by keeping otherparameters constant (𝑅𝑑 = 1.6; 𝐷𝑤 = 0.5; 𝑅𝑤 = 0.75). Damdepth ratio has been taken as (𝐷𝑑 = 0.2, 0.3, 0.8, and 1.4).

Throughout the study, 𝐿/𝐷 ratio has been taken as 1.0 andellipticity ratio (𝛿) has been taken as 0.5.

Figures 3–7 show the effect of relief track depth ratio onthe steady-state parameters. Load carrying capacity decreases

0.10.110.120.130.140.150.160.170.18


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

Flow

coeffi

cien

t(Q)

Figure 5: Effect of relief track depth ratio on flow coefficient.

00.5

11.5

22.5

33.5

4

Som

mer

feld

num

ber (S)


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

Figure 6: Effect of relief track depth ratio on Sommerfeld number.

with increase in 𝑅𝑑. As shown in Figure 3, the variation inrelief track depth ratio between 0.2 and 0.6 has a significanteffect on load carrying capacity but as the relief track depthratio goes beyond 0.6, the variation in load carrying capacitybecomes very minor. Presence of relief track increases thefilm thickness and hence pressure at the bottom of theshaft decreases; consequently the load carrying capacity alsodecreases.

Attitude angle of bearing decreases with increase in relieftrack depth ratio as seen in Figure 4. For higher value ofrelief track depth ratio, the attitude angle first increases witheccentricity ratio and then decreases.

The behavior of flow coefficient is more or less similar tothe load carrying capacity. Figure 5 shows that flow coefficientalso decreases with increase in relief track depth ratio and fora given relief track depth ratio flow coefficient increases withincrease in eccentricity ratio.

The Sommerfeld number and friction variable behave insimilar manner for different relief track depth ratio as shownin Figures 6 and 7, respectively. In both cases, as the relieftrack ratio increases, the Sommerfeld number and frictionvariable also increase, but, for a given relief track depth ratio,


0

50

100

150


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

Fric

tion

varia

ble(

F)

Figure 7: Effect of relief track depth ratio on friction variable.

02468

10121416


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

KxxKzz

Dire

ct st

iffne

ssco

effici

ent(K

xx,K

zz)

Figure 8: Effect of relief track depth ratio on direct stiffnesscoefficient.

these keep on decreasing with increase in eccentricity ratio.For all relief track depth ratio, Sommerfeld number and flowcoefficient converge at eccentricity ratio of 0.441.

Figures 8–11 show the variation in dynamic coefficientswith eccentricity ratio for different relief track depth ratio.Figure 8 depicts that 𝐾𝑥𝑥 and 𝐾𝑧𝑧 increase with increasein 𝑅𝑑 but with increase in eccentricity ratio for given 𝑅𝑑,𝐾𝑥𝑥 first decreases and then increases, whereas 𝐾𝑧𝑧 keepson decreasing. For all the value of 𝑅𝑑, 𝐾𝑧𝑧 is found to beconverging at an eccentricity ratio of 0.441.

Figure 9 shows the variation in cross-coupled stiffnesscoefficient with eccentricity ratio for different value of 𝑅𝑑.𝐾𝑥𝑧 increaseswith increase in𝑅𝑑, whereas𝐾𝑧𝑥 decreaseswithincrease in 𝑅𝑑. For given 𝑅𝑑, there is a decrease in𝐾𝑥𝑧 but anincrease in𝐾𝑧𝑥 can be seen.

Figures 8 and 9 also depict that both direct and cross-coupled stiffness coefficients increase with increase in relieftrack depth ratio. As a result of this, the bearing gets moreand more stiff.

−25−20−15−10

−505

101520


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

KxzKzx

Cros

s-co

uple

d sti

ffnes

sco

effici

ent(K

xz,K

zx)

Figure 9: Effect of relief track depth ratio on cross-coupled stiffnesscoefficient.

05

101520253035404550


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

CxxCzz

Dire

ct d

ampi

ngco

effici

ent(Cxx,C

zz)

Figure 10: Effect of relief track depth ratio on direct dampingcoefficient.

The behavior of direct damping coefficient and cross-coupled damping coefficient can be seen from Figures 10and 11. Both direct and cross-coupled damping coefficientsincrease with increase in relief track depth ratio. The directand cross-coupled damping coefficients converge with eccen-tricity ratio for all the given relief track depth ratios and thesebecome almost equal for eccentricity ratio of 0.441.

Figure 12 shows the effect on mass parameter whichdetermines the stability of the bearing. As the relief trackdepth ratio increases, themass parameter increases andhencethe stability also increases. Beyond eccentricity ratio of 0.4,mass parameter shows negative value for all relief trackdepth ratio. Negative value of mass parameter shows thatthe bearing is stable for all speeds [12]. The bearing becomesinfinitely stable beyond 𝜀 = 0.35 for 𝑅𝑑 = 1.6.

Figure 13 shows that, with increase in relief track depthratio, the whirling tendency of shaft reduces. This reductionin whirling tendency improves the stability of the shaft. Forthe relief track depth ratio up to 0.8, the plot shows the whirlratio up to 0.389 eccentricity ratio but for 𝑅𝑑 = 1.6 it shows


−1

0

1

2

3

4

5

6


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

CxzCzx

Cros

s-co

uple

d da

mpi

ngco

effici

ent(Cxz,C

zx)

Figure 11: Effect of relief track depth ratio on cross-coupleddamping coefficient.

0

20

40

60

80

100

120

140

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45Eccentricity ratio (𝜀)

Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

Mas

s par

amet

er(M

)

Figure 12: Effect of relief track depth ratio on mass parameter.

only up to 0.351 eccentricity ratio.This is because of the reasonthat in case of𝑅𝑑 = 1.6 the bearing goes to the infinite stabilityzone after 𝜀 = 0.351, whereas in case of𝑅𝑑 = 0.2 to 0.8 bearingattains the infinite stability after 𝜀 = 0.389.

The dam depth ratio shows different behavior comparedto the relief track depth ratio. Pressure dam is provided inthe upper half of the bearing by providing a step. Whenthe lubricating oil passes through this passage due to stepprovided, a high pressure zone is created which pushes thebearing downward. This high pressure results in increase inapparent load on the bearing. Consequently, the whirlingtendency of shaft is reduced and bearing shows better stabilityeven at low load and high speed condition.

Figure 14 shows that there is no significant change inload carrying capacity with dam depth ratio. In general, loadcarrying capacity is observed to increase, with increase ineccentricity ratio for all the value of dam depth ratio.

With increase in dam depth ratio, attitude angle increasesfor a given eccentricity ratio. It can also be observed fromFigure 15 that, for a given dam depth ratio, as the eccentricityratio increases, the attitude angle first increases and attains its

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.2

0.25

0.3

0.35

0.4

0.5

0.45


Rd = 0.2

Rd = 0.4Rd = 0.6

Rd = 0.8

Rd = 1.6

Whi

rl ra

tio(𝜔

)

Figure 13: Effect of relief track depth ratio on whirl ratio.

00.05

0.10.15

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.50.45

0.20.25

0.30.35

0.4

0.50.45


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Load

carr

ying

capa

city(W

)

Figure 14: Effect of dam depth ratio on load carrying capacity.

maximum value near about 0.289 eccentricity ratio and thendecreases.

The flow coefficient decreases as the dam depth ratioincreases for a given eccentricity ratio; however, flow coeffi-cient increases with increase in eccentricity ratio for a givendamdepth ratio, as seen in Figure 16. Keeping the eccentricityratio constant, the damdepth ratio shows significant decreasein flow coefficient up to 𝑅𝑑 = 0.8, but beyond that the changein flow coefficient becomes negligible.

Sommerfeld number and friction variable show the samebehavior with dam depth ratio. As shown in Figures 17 and18, the Sommerfeld number and friction variable decreasewith increase in dam depth ratio. The effect of dam depthratio is dominant for lower eccentricity ratio but, beyond theeccentricity ratio of 0.2, Sommerfeld number and frictionvariable become almost constant for all the value of damdepth ratio.

Figures 19–22 show the variation in direct and cross-coupled stiffness and damping coefficients. For the lowereccentricity ratios, reduction in stiffness and damping coef-ficient is shown with increase in dam depth ratio. But, forany given eccentricity ratio beyond 0.2, 𝐾𝑥𝑥 becomes more


464748495051525354

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.50.45Eccentricity ratio (𝜀)

Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Attit

ude a

ngle(𝜙)

Figure 15: Effect of dam depth ratio on attitude angle.

0.10.110.120.130.140.150.160.170.18


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Flow

coeffi

cien

t(Q)

Figure 16: Effect of dam depth ratio on flow coefficient.

or less the same for all the values of dam depth ratio as seenin Figure 19. For different dam depth ratio, both𝐾𝑥𝑥 and𝐾𝑧𝑧converge separately at higher eccentricity ratio. Figure 19 alsoshows that 𝐾𝑧𝑧 has higher value than 𝐾𝑥𝑥 for a given damdepth ratio and eccentricity ratio but as the eccentricity ratioincreases,𝐾𝑥𝑥 becomes higher than𝐾𝑧𝑧.

Figure 20 shows that 𝐾𝑥𝑧 decreases with increase in damdepth ratio for a given eccentricity ratio but as the eccentricityratio increases both𝐾𝑥𝑧 and𝐾𝑧𝑥 become constant separatelyfor any given dam depth ratio. For higher eccentricity ratios,𝐾𝑥𝑧 becomes the same for all the dam depth ratios and thesame can be seen for𝐾𝑧𝑥.

In case of direct damping coefficient, as the dam depthratio increases, both 𝐶𝑥𝑥 and 𝐶𝑧𝑧 decrease. For any givenvalue of dam depth ratio and eccentricity ratio less than 0.2,𝐶𝑧𝑧 is higher than𝐶𝑥𝑥, but, for the same dam depth ratio andeccentricity ratio more than 0.2, 𝐶𝑥𝑥 has higher value than𝐶𝑧𝑧. At 𝜀 = 0.2, both 𝐶𝑥𝑥 and 𝐶𝑧𝑧 have the same value forgiven dam depth ratio as seen in Figure 21. For all dam depthratios, 𝐶𝑥𝑥 converges at 𝜀 = 0.441 and the similar situationhas been observed for 𝐶𝑧𝑧.

0

1

2

3

4

5

6

7


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Som

mer

feld

num

ber (S)

Figure 17: Effect of dam depth ratio on Sommerfeld number.

0

50

100

150

200

250


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Fric

tion

varia

ble(

F)

Figure 18: Effect of dam depth ratio on friction variable.

Figure 22 shows that cross-coupled damping coefficientdecreases as the dam depth ratio of three-lobe pressure dambearing increases. For a given dam depth ratio, the differencebetween 𝐶𝑥𝑧 and 𝐶𝑧𝑥 is very less. For the lower eccentricityratio (up to 0.25), 𝐶𝑧𝑥 is higher than 𝐶𝑥𝑧, but, for highereccentricity ratio (beyond 0.25),𝐶𝑥𝑧 exceeds𝐶𝑧𝑥. Lower damdepth ratio shows more influence on 𝐶𝑥𝑧 and 𝐶𝑧𝑥 than thehigher dam depth ratio.

For the lower eccentricity ratio (up to 0.25), massparameter decreases with increase in dam depth ratio butas the eccentricity ratio increases and goes beyond 0.25, thepattern changes and the bearing with higher dam depth ratioshows higher stability as the mass parameter increases asseen in Figure 23. Bearing becomes infinitely stable beyondeccentricity ratio of 0.35 for dam depth ratio of 0.8 and1.4, whereas bearing attains infinite stability after eccentricityratio of 0.4 in case of dam depth ratio of 0.2 and 0.3.

Figure 24 shows that, with increase in dam depth ratio,the whirling tendency of shaft reduces. This reduction inwhirling tendency improves the stability of the journal. Forthe dam depth ratio of 0.2 and 0.4, the plot shows the whirlratio up to 0.389 eccentricity ratio, but, for 𝐷𝑑 = 0.8 and 1.4,it shows only up to 0.351 eccentricity ratio. This is because of


0

5

10

15

20

25

30

35


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

KxxKzz

Dire

ct st

iffne

ssco

effici

ent(K

xx,K

zz)

Figure 19: Effect of dam depth ratio on direct stiffness coefficient.

−50−40−30−20−10

010203040


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

KxzKzx

Cros

s-co

uple

d sti

ffnes

sco

effici

ent(K

xz,K

zx)

Figure 20: Effect of dam depth ratio on cross-coupled stiffnesscoefficient.

0102030405060708090

100


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Dire

ct d

ampi

ngco

effici

ent(Cxx,C

zz)

CxxCzz

Figure 21: Effect of dam depth ratio on direct damping coefficient.

the reason that in case of 𝐷𝑑 = 0.8 and 1.4 the bearing goesto the infinite stability zone after 𝜀 = 0.351, whereas in caseof 𝑅𝑑 = 0.2 to 0.3 bearing attains the infinite stability after𝜀 = 0.389.

2

4

6

8

10

12

14

00.1 0.15 0.2 0.25 0.3 0.35 0.4 0.50.45


Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

CxzCzx

Cros

s-co

uple

d da

mpi

ngco

effici

ent(Cxz,C

zx)

Figure 22: Effect of dam depth ratio on cross-coupled dampingcoefficient.

0

50

100

150

200

250

300

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45Eccentricity ratio (𝜀)

Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Mas

s par

amet

er(M

)

Figure 23: Effect of dam depth ratio on mass parameter.

4. Conclusion

From the study carried out here, the following conclusions aremade:

(1) As the relief track depth ratio increases, the stiffness ofthe bearing increases. As a result of that, the thresholdspeed and zone of infinite stability increase.

(2) With increase in relief track ratio, load carryingcapacity, attitude angle, and flow coefficient decrease,whereas Sommerfeld number and friction variableincrease.

(3) Relief track ratio between 0.2 and 0.6 has muchsignificant effect on load carrying capacity and flowcoefficient for a given eccentricity ratio but, for higherrelief track ratio, this effect becomes insignificant.

(4) For a given eccentricity ratio, the effect of pressuredam depth ratio on load carrying capacity is almostnegligible.

(5) Pressure dam depth ratio does not have significanteffect on performance parameters of journal bearingif eccentricity ratio goes beyond 0.2.


0.10.1

0.15 0.2 0.25 0.3 0.35 0.4 0.45Eccentricity ratio (𝜀)

0.20.15

0.250.3

0.350.4

0.50.45

Whi

rl ra

tio(𝜔

)

Dd = 0.2

Dd = 0.3

Dd = 0.8

Dd = 1.4

Figure 24: Effect of dam depth ratio on whirl ratio.

(6) The incorporation of pressure dam and relief track ina three-lobe journal bearing increases the stability andthreshold speed but the depth of pressure dam andrelief rack has to be set very carefully because it hasgreat significance on overall performance.

Nomenclature

𝐶: Radial clearance, m𝐶𝑚: Minimum film thickness for acentered shaft, m𝐶𝑥𝑥, 𝐶𝑥𝑧, 𝐶𝑧𝑥, 𝐶𝑧𝑧: Oil-film damping coefficients, Ns/m𝐶𝑥𝑥, 𝐶𝑥𝑧, 𝐶𝑧𝑥, 𝐶𝑧𝑧: Nondimensional oil-film dampingcoefficients 𝐶𝑥𝑥 = 𝐶𝑥𝑥(𝜔𝐶/𝑊)𝑒: Eccentricity, m𝜀: Eccentricity ratio 𝜀 = 𝑒/𝐶𝑒1, 𝑒2, 𝑒3: Eccentricity for each lobe, mℎ: Oil-film thickness, m

ℎ: Nondimensional oil-film thicknessℎ = ℎ/𝐶𝐾𝑥𝑥, 𝐾𝑥𝑧, 𝐾𝑧𝑥, 𝐾𝑧𝑧: Oil-film stiffness coefficients, N/m𝐾𝑥𝑥, 𝐾𝑥𝑧, 𝐾𝑧𝑥, 𝐾𝑧𝑧: Nondimensional oil-film stiffnesscoefficients𝐾𝑥𝑥 = 𝐾𝑥𝑥(𝐶/𝑊)𝐿: Bearing length, m𝐷𝑑: Pressure dam depth, m𝐷𝑑: Nondimensional dam depth𝐷𝑑 = 𝐷𝑑/𝐶𝐷𝑤: Pressure dam width, m𝐷𝑤: Nondimensional dam depth𝐷𝑤 = 𝐷𝑤/𝐿𝑅𝑑: Relief track depth, m𝑅𝑑: Nondimensional relief track depth𝑅𝑑 = 𝑅𝑑/𝐶𝑅𝑤: Relief track width, m𝑅𝑤: Nondimensional relief track depth𝑅𝑤 = 𝑅𝑤/𝐿𝑚: Mass of the rotor per bearing, kg𝜔: Angular velocity of the journal, rad/s

𝑀: Nondimensional mass parameter𝑀 = 𝑚𝐶𝜔2/𝑊𝑁: Speed of the journal, rps𝐷: Diameter of journal, m𝑝: Load per unit bearing area 𝑝 = (𝑊/𝐿𝐷),N/m2𝑝: Nondimensional film pressure𝑝 = 𝑝𝐶2/6𝜂𝑈𝑅𝑅: Bearing radius, m𝑈: Sliding speed, m/s𝜂: Coefficient of viscosity, Pa-s𝜙: Beating attitude angle, rad𝑆: Sommerfeld number 𝑆 = 𝜂𝑁/𝑝(𝑅/𝐶)2𝜇: Nondimensional friction variable𝜇 = 𝜇(𝑅/𝐶)𝑊: Load carrying capacity, N𝑊: Nondimensional load carrying capacity𝑊 = 𝑊𝐶2/6𝜂𝑈𝑅2𝐿𝑄: Nondimensional flow coefficient𝑝1, 𝑝2: Nondimensional perturbed pressure𝛿: Ellipticity ratio 𝛿 = 𝑑/𝐶𝑑: Distance of lobe center from bearinggeometry center, m𝑡: time, s𝜏: Nondimensional time 𝜏 = 𝜔𝑝𝑡𝜔𝑝: Angular velocity of whirl, rad/s𝜔𝑗: Nondimensional speed parameter𝑊𝑥: Vertical component of the resultant load,N𝑊𝑧: Horizontal component of the resultantload, N.

Competing Interests

The authors declare that they have no competing interests.

References

[1] J. C. Nicholas and P. E. Allaire, “Analysis of step journalbearings—finite length, stability,” ASLE Transactions, vol. 23,no. 2, pp. 197–207, 1980.

[2] J. C. Nicholas, P. E. Allaire, and D. W. Lewis, “Stiffness anddamping coefficients for finite length step journal bearings,”ASLE Transactions, vol. 23, no. 4, pp. 353–362, 1980.

[3] A. Singh and B. K. Gupta, “Static and dynamic properties ofoil films in displaced centres elliptical bearings,” Proceedingsof the Institution of Mechanical Engineers, Part C: Journal ofMechanical Engineering Science, vol. 197, no. 3, pp. 159–165, 1983.

[4] N. P. Mehta and S. S. Rattan, “Stability analysis of three-lobebearings with pressure dams,”Wear, vol. 167, no. 2, pp. 181–185,1993.

[5] J. W. Lund and K. K.Thomson, “A calculation method and datafor the dynamic coefficient of oil-lubricated journal bearings,”in Proceedings of the ASME Design and Engineering Conference,pp. 1–28, New York, NY, USA, 1978.

[6] S. C. Soni, R. Sinhasan, and D. V. Singh, “Performance charac-teristics of noncircular bearings in laminar and turbulent flowregimes,” ASLE Transactions, vol. 24, no. 1, pp. 29–41, 1981.


[7] A. Kumar, R. Sinhasan, and D. V. Singh, “Performance charac-teristics of two-lobe hydrodynamic journal bearings,” Journal ofLubrication Technology, vol. 102, no. 4, pp. 425–429, 1980.

[8] N. P. Mehta, “Static and dynamic characteristics of orthogo-nally-displaced pressure dam bearings,” Tribology Transactions,vol. 36, no. 2, pp. 201–206, 1993.

[9] N. P. Mehta and A. Singh, “Stability analysis of finite offset-halves pressure dam bearing,” Journal of Tribology, vol. 108, no.2, pp. 270–274, 1986.

[10] S. Sharma and S. S. Ratan, “Micropolar lubricant effects onthe performance of a two-lobe bearing with pressure dam,”International Journal of Engineering Science and Technology, vol.2, no. 10, pp. 5637–5646, 2010.

[11] S. Das, S. K. Guha, and A. K. Chattopadhyay, “Linear stabilityanalysis of hydrodynamic journal bearings under micropolarlubrication,” Tribology International, vol. 38, no. 5, pp. 500–507,2005.

[12] D. F. Li, K. C. Choy, and P. E. Allaire, “Stability and transientcharacteristics of four multilobe journal bearing configura-tions,” Journal of lubrication technology, vol. 102, no. 3, pp. 291–299, 1980.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttps://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


effect of dam depth and relief track depth on steady-state and dynamic performance ... · 2019. 7....

Documents