Research ArticleMathematical Model and Analysis ofthe Water-Lubricated Hydrostatic Journal Bearingsconsidering the Translational and Tilting Motions

Hui-Hui Feng1 Chun-Dong Xu1 and Jie Wan2

1 School of Mechanical Engineering Southeast University Nanjing 211189 China2 CSR Qishuyan Institute Co Ltd Changzhou 213011 China

Correspondence should be addressed to Hui-Hui Feng fhhwjmail163com

Received 14 April 2014 Revised 13 June 2014 Accepted 17 June 2014 Published 17 July 2014

Academic Editor Nam-Il Kim

Copyright copy 2014 Hui-Hui Feng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The water-lubricated bearings have been paid attention for their advantages to reduce the power loss and temperature rise andincrease load capacity at high speed To fully study the complete dynamic coefficients of two water-lubricated hydrostatic journalbearings used to support a rigid rotor a four-degree-of-freedommodel considering the translational and tiltingmotion is presentedThe effects of tilting ratio rotary speed and eccentricity ratio on the static and dynamic performances of the bearings areinvestigated The bulk turbulent Reynolds equation is adopted The finite difference method and a linear perturbation methodare used to calculate the zeroth- and first-order pressure fields to obtain the static and dynamic coefficients The results suggestthat when the tilting ratio is smaller than 04 or the eccentricity ratio is smaller than 01 the static and dynamic characteristics arerelatively insensitive to the tilting and eccentricity ratios however for larger tilting or eccentricity ratios the tilting and eccentriceffects should be fully considered Meanwhile the rotary speed significantly affects the performance of the hydrostatic water-lubricated bearings

1 Introduction

Hydrostatic journal bearings are applied widely in spindle-bearing systems owning to their favorable performancecharacteristics However with the requirement of highermachining speed the limitations of the conventional oil filmbearings are apparently due to their remarkable power loss aswell as the temperature rise Therefore the water-lubricatedbearings were developed and have been studied to fulfillthe targets of lower power loss lower temperature rise andheavier load capacity at high speed

Many studies related to water-lubricated bearings havebeen reported in the literatures in the past few years Liu et alcompared the oil-lubricated andwater-lubricated hybrid slid-ing bearings and the results show that the latter benefitsmorefrom improved processing precision and efficiency [1] Yuanet al study the static and dynamic characteristics of water-lubricated hybrid journal bearings compensated by short

capillaries [2] Yoshimoto et al investigated the static charac-teristics of water-lubricated hydrostatic conical bearings withspiral grooves for high speed spindles [3] Gao et al analyzedthe effects of eccentricity ration on pressure distributionof water-lubricated plain journal bearings by computationalfluid dynamics (CFD) [4] In summary extensive researcheshave been conducted in the area of water lubricated bearingsin various aspects numerical methods [1] performance ofthe bearings with various geometries [2 3 5 6] effects ofvarious kinds of restrictors upon performance of a bearing[7] et al However their studies were restricted to the static ordynamic characteristics of plain or grooved journal bearingsconsidering only translational motion of the journal

In actual practice the bearings and the journals maynot be properly aligned as a result of improper assemblyor noncentral loading As a consequence not only shouldthe translational motion of the rotor be studied but alsothe tilting motion of the rotor should be investigated

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 353769 15 pageshttpdxdoiorg1011552014353769

2 Mathematical Problems in Engineering

Front bearing Rear bearingMass center

3

Z

Y

120579y

120579x

X

52 ∘

R26

l1 l2

lm1 lm2

Figure 1 The arrangement and coordinate system of the bearings [6]

Displacement of mass center

120579

Figure 2 The rigid rotor-bearing model

As a result the complete stiffness and damping coefficientsof a journal bearing which is important to the vibration ofa rotor should be taken into consideration in four degreesof freedom including the translation in 119909 119910 direction andtilting about the 119909- and 119910-axis Numerous studies concerningthe tilting motion of journal are available in the literatures[8ndash10] Recently Jang et al [11ndash13] thoroughly investigatedthe dynamic characteristics of the journal and groove thrustbearings used to support a HDD spindle considering bothof the translational motion and tilting motion of the journalResults show that the tilting motions have an important rolein the dynamic characteristics of the proposed bearings

Unlike the conventional oil film bearings water-lubricated bearings utilized in spindle are different inworking conditions and characteristics However forhydrostatic water-lubricated journal bearings we are notaware of any previous investigations to study their dynamiccharacteristics considering the translational and titlingmotions Therefore in this work we aim to fully studythe complete dynamic coefficients for two water-lubricatedhydrostatic journal bearings used to support a rigid rotorThedynamic characteristics will be categorized into four groupscoefficients of force to displacement coefficients of force toangle coefficients of moment to angle and coefficients ofmoment to displacement In the present study in order tofully study the variations of the complete static and dynamic

characteristics of the proposed water-lubricated bearings theinfluences of the tilting ratio rotary speed and eccentricityratio on the bearings have been studied

2 Mathematical Models

Figure 1 shows the schematic representations for a rigid rotorsupported by a pair of identical water-lubricated journalbearings as well as the geometry of a hydrostatic water-lubricated journal bearing Pressed water enters the bearingacross an orifice restrictor flows into the film lands and thenexits the bearing The recess pressure is regarded as uniformAs shown in Figure 1 the rigid rotor moved in an inertialreference frame 119862mdash119883119884119885 the rotor tilts about 119883 by an angle120579119909and tilts about 119884 by an angle 120579

119910 The rigid rotor-bearing

model is shown in Figure 2

21 Reynolds Equation For an isoviscous incompressiblefluid the Reynolds equation governing the turbulent bulkflow in nondimensional form is given as

120597

120597120593

(

1198661199091198673

120583

120597119875

120597120593

) +

1199032

1198972

120597

120597120582

(

1198661199111198673

120583

120597119875

120597120582

) = Γ1

120597119867

120597120593

+ Γ2

120597119867

120597120591

(1)

Mathematical Problems in Engineering 3

where Γ1

= 1205830Ω21198751199041205992 Γ2

= 1205830Ω1199011199041205992 120599 = 119888119903 and the

details of the turbulent coefficients 119866119909 119866119911can be obtained as

follows [6 14ndash16]

119881119909= minus119866119909

ℎ2

120583

120597119901

120597119909

+ 119866119869

119880

2

119881119885= minus119866119885

ℎ2

120583

120597119901

120597119911

119877119861=

120588ℎ

120583

[1198812

119909+ 1198812

119885]

12

119877119869=

120588ℎ

120583

[(119881119909minus 119880)2

+ 1198812

119885]

12

119891119869= 0066119877

minus025

119869

119891119861= 0066119877

minus025

119861

119896119869= 119891119869119877119869

119896119861= 119891119861119877119861

119866119909= 119866119911=

2

(119896119869+ 119896119861)

(2)

With the increase of rotary speed or film depth the flow islikely to become turbulent from laminar state The turbulentcoefficients 119866

119909 119866119910 and 119866

119911dependent on the fluid velocity

field are obtained as follows [15]

119866119909= min

1

12

119866119909

119866119910= min

1

12

119866119910

119866119911= min

1

12

119866119911

(3)

At the bearing exit plane the pressure takes a constantvalue equal to the ambient pressure

22 Continuity Equation The dimensionless continuityequation at the recess is defined by the global balancebetween the flow through the orifice restrictor and the recessoutflow into the film lands

120582radic(1 minus 119875119903) = int

1198781+1198783

(

61205830119871119903Ω

1198751199041198882

119867 minus

12119871

119903

1198661199091198673

120583

120597119875

120597120593

)119889120582

+ int

1198782+1198784

12119903

119871

1198661199111198673

120583

120597119875

120597120582

119889120593

(4)

where 120582 = 3radic21205871205721198892

0120583(1198883

radic120588119875119904) 119875119903is the recess pressure

11987813

is the circumferential recess boundary and 11987824

is the axialrecess boundary

23 Perturbation Analysis The journal center rotates aboutits steady-state position (119909

0 1199100 1205791199090 1205791199100) with a small whirl

which is generated from the variations due to the translations

of the rotormass center and variations due to the tilting angles[11ndash13] For small amplitude motions the dimensionless filmthickness and pressure fields are expressed as the sum of azeroth-order field and first-order field describing the steady-state condition and perturbed motion respectively

The dimensionless perturbed film expression consideringthe tilting angles is [12]

119867119894119895

= 1198670119894119895

+ Δ120576119909sdot sin 120579

119894119895+ Δ120576119910sdot cos 120579

119894119895+ 120593119869Δ120579119909+ 120595119869Δ120579119910

1198670119894119895

= 1 + 1205760cos120593119894119895

+ 120593119869120579119909+ 120595119869120579119910

(5)

where

120593119869=

120574119897119894119895cos 120579119909cos 120579119894119895

119888

120595119869= minus

120574119897119894119895cos 120579119910sin 120579119894119895

119888

(6)

where 119897119894119895

is the distance between the grid node of eachbearing and the mass center of the rotor supported on thebearings 120574 = 1 for the front journal bearing and 120574 = minus1 forthe rear journal bearing

The dimensionless perturbed pressure expression is

119875 = 1198750+

120597119875

120597Δ120576119909

Δ120576119909 +

120597119875

120597Δ120576119910

Δ120576119910 +

120597119875

120597Δ120579119909

Δ120579119909+

120597119875

120597Δ120579119910

Δ120579119910

+

120597119875

120597Δ120576

Δ120576 +

120597119875

120597Δ120576 119910

Δ120576 119910 +

120597119875

120597Δ

120579119909

Δ

120579119909+

120597119875

120597Δ

120579119910

Δ

120579119910

(7)

Substitution of the perturbed equations (5)ndash(7) intothe Reynolds equation yields the zeroth- and first-orderexpressions

120597

120597120593

(

1198661199091198673

120583

120597119875120585

120597120593

) +

1199032

1198972(

1198661199111198673

120583

120597119875120585

120597120582

) = 119865120585

(120585 = 0 119909 119910 120579119909 120579119910 119910

120579119909

120579119910)

1198650= Γ1

1205971198670

120597120593

119865120576119909

= minus

120597

120597120593

[

11986611990931198672

0

120583

sin 120579

1205971198750

120597120593

]

minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

sin 120579

1205971198750

120597120582

]

+ Γ1(sum

120597 (sin 120579)

120597120593

) + Γ2(sum

120597 (sin 120579)

120597120591

)

119865120576119910

= minus

120597

120597120593

[

11986611990931198672

0

120583

cos 1205791205971198750

120597120593

]

minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

cos 1205791205971198750

120597120582

]

4 Mathematical Problems in Engineering

+ Γ1(sum

120597 (cos 120579)120597120593

) + Γ2(sum

120597 (cos 120579)120597120591

)

119865120579119909

= minus

120597

120597120593

[

11986611990931198672

0

120583

120593119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120593119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120593119869)

120597120593

) + Γ2(sum

120597 (120593119869)

120597120591

)

119865120579119910

= minus

120597

120597120593

[

11986611990931198672

0

120583

120595119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120595119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120595119869)

120597120593

) + Γ2(sum

120597 (120595119869)

120597120591

)

119865120576 119909

= Γ2sin 120579

119865120576 119910

= Γ2cos 120579

119865120576120579119909

= Γ2120593119869

119865120576120579119910

= Γ2120595119869

(8)

The perturbed quality into the orifice diameter can beobtained by Taylor expansion

119876119894119899

= 120582(1 minus 1198751199030)12

minus

120582

2

(1 minus 1198751199030)minus12

(119875119903minus 1198751199030) (9)

Substitution of the perturbed equations (5)ndash(7) intothe continuity equation yields the zeroth- and first-orderexpressions

120582radic(1 minus 1198751199030) = int

1198781+1198783

(

61205830119871119903Ω

1198751199041198882

1198670119894119895

minus

12119871

119903

1198661199091198673

0119894119895

120583

1205971198750

120597120593

)119889120582

+ int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

1205971198750

120597120582

119889120593

minus

120582

2

(1 minus 1198751199030)minus12

119875120585

=

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

sdot sin 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

11986611991131198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

cos 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119910

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120593119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120593119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120593119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120595119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120595119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120595119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119910

120597120582

119889120593

(10)

Mathematical Problems in Engineering 5

The finite difference method (FDM) and a successiveover-relaxation (SOR) scheme are implemented to solve (1)ndash(10) to find the pressure distribution When the steady andperturbed pressure distributions are obtained the static anddynamic coefficients can be solved

24 Static Characteristics The quality frictional power lossand pump power are calculated by integration of the pressurefield on the bearing surfaces

119876119894119899

= 120582radic1 minus 119875119903 (11)

ℎ119891=

120583(119903Ω)2

ℎ0

∬119903119889120593119889119911 +

120583Ω2

ℎ0

∬1199033

119889119903 119889120593 (12)

ℎ119901= 119875119904119876 (13)

San Andres et al carried out a systematic research on thewater-lubricated hydrostatic journal bearings both theoreti-cally and experimentally [17] Numerical and experimentalresults show that predictions of the bearing performancecharacteristics like flow rate load capacity and rotor dynamicforce coefficients are not affected by the small temperaturevariations (Δ119879 lt 10

∘C) in the water hydrostatic journalbearings As a result an adiabatic isothermal assumption ismade in this study All the heat produced in the bearings byfriction is considered absorbed by water film The averageelevated temperature is given by

Δ119879 =

ℎ119891+ ℎ119901

119876120588119862V (14)

where 120588 is the density of water and 119862V is the specific heatcapacity of water

25 Dynamic Characteristics The dynamic coefficients ofthe journal bearings can be calculated by integrating theperturbed pressure across the fluid filmThere are 16 stiffnessand damping coefficients and the dynamic coefficients can begrouped into four categories coefficients of force to displace-ment coefficients of force to angle coefficients of moment toangle and coefficients of moment to displacement

[119870119869] =

[

[

[

[

[

[

[

119870119909119909

119870119909119910

119870119909120579119909

119870119909120579119910

119870119910119909

119870119910119910

119870119910120579119909

119870119910120579119910

119870119872119909119909

119870119872119909119910

119870119872119909120579119909

119870119872119909120579119909

119870119872119910119909

119870119872119910119910

119870119872119909120579119909

119870119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875119909119875119910

119875120579119909

119875120579119910] 119889120593 119889120582

Table 1 Operating condition of the test bearing [17]

Orifice diameter (mm) 249 Supply temperature (∘C) 55Length (mm) 762 Supply pressure (MPa) 4Film thickness (um) 127 Rotary speed (rpm) 24600

41∘

R381

Figure 3 Geometry of the test water-lubricated hydrostatic journalbearing [17]

[119861119869] =

[

[

[

[

[

[

[

119887119909119909

119887119909119910

119887119909120579119909

119887119909120579119910

119887119910119909

119887119910119910

119887119910120579119909

119887119910120579119910

119887119872119909119909

119887119872119909119910

119887119872119909120579119909

119887119872119909120579119909

119887119872119910119909

119887119872119910119910

119887119872119909120579119909

119887119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875 119909119875119910119875 120579119909

119875 120579119910] 119889120593 119889120582

(15)

26 The Numerical Solution Procedure The finite differencemethod (FDM) and a successive over-relaxation (SOR)scheme are implemented to solve (1)ndash(10) governing theflow in the film to find the pressure distribution The over-relaxation factor always lies between (1sim2) By trial and errorone can determine an optimum value of the relaxation factorfor the fastest convergence Normally 17 is a good startingpoint for determining the relaxation factor [18] In thenumerical procedure implemented any negative pressurescalculated in the cavitation zone are arbitrarily set equalto zero (or ambient) pressure [18] The pressure iterationsare continued until the following convergence criterion issatisfied

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895minus 119875(119896minus1)

119894119895

10038161003816100381610038161003816

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895

10038161003816100381610038161003816

le 120575 (16)

where 120575 is the convergence criteria which is set as 10minus3

Once the pressure field is established for the water filmother performance parameters follow from the pressuredistributionThe steady and perturbed pressure distributions

6 Mathematical Problems in Engineering

00 01 02 03 04 05

0

1000

2000

3000

4000

5000

6000

7000

8000Lo

ad ca

paci

ty (N

)

Eccentricity ratio

Experimental resultsNumerical predictions

minus1000

(a) Comparison of load capacity

00 01 02 03 04 0500

04

08

12

16

20

Mas

s flow

rate

(kg

s)

Eccentricity ratio

Experimental resultsNumerical predictions

(b) Comparison of mass flow rate

Figure 4 Comparison of static performance of experimental results[17] and numerical predictions in respect to eccentricity ratio

obtained are subsequently integrated to yield the desiredstatic and dynamic coefficients

The fluid film was discretized by rectangular grid withunequal intervals which are 0033 0088 and 01154 forthe film land recess and return groove respectively in thecircumferential direction and 002 and 004 for the film landand recess area in the axial direction The total number ofthe grid is 31 times 102 It takes about 35 seconds for the staticperformance to achieve convergence A few validation testsweremade with a coarser grid of 31 times 86 and finer grid of 37 times

158 with different intervals and in no case did the predictedresults of the static characteristics differ by more than 01percent from those obtained by the initial grid

3 Results and Discussion

31 Comparisons of Present Solution with ExperimentalResults The present numerical solution has been correlated

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10218

220

222

224

226

228

230

232

234

236

Qua

lity

(Lm

in)

Tilting ratiominus01

(a) The effect of tilting angle on the quality

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10240242244246248250252254256258260262264

Gro

ss p

ower

loss

(W)

Tilting ratiominus01

(b) The effect of tilting angle on the power loss

00 01 02 03 04 05 06 07 08 09 10148

152

156

160

164

168

172

Tilting ratio

Tem

pera

ture

rise

(∘C)

minus01

Front bearingRear bearing

(c) The effect of tilting angle on the temperature rise

Figure 5 The effect of tilting angle on the static performance of thewater-lubricated bearing

Mathematical Problems in Engineering 7

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

Stiff

ness

of f

orce

to d

ispla

cem

ent (

Nm

)

Tilting ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

Eccentricity ratio = 0

minus05

minus10

minus15

minus01

times108

(a) The stiffness of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Tilting ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus tilting ratio

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

40

Tilting ratio

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

Eccentricity ratio = 0times10

6

minus05

minus10

minus15

minus01

(c) The stiffness of moment to angle versus tilting ratio

Tilting ratio00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

Eccentricity ratio = 0times10

7

minus1

minus2

minus3

minus4

minus01

(d) The stiffness of moment to displacement versus tilting ratio

Figure 6 The effect of tilting ratio on the stiffness

and validated with the experimental results available inthe literature A five-recess turbulent-flow water-lubricatedhydrostatic bearing operating at a high rotational speed istested by San Andres et al [17] Table 1 shows the bearingdescription and operating conditions and Figure 3 shows theschematic of the bearing Figure 4 shows a comparison ofmeasured load capacity and flow rate and the numerical pre-dictions in respect to eccentricity ratio As shown in Figure 4a good agreement is observed between the present numericalpredictions and the experimental results available in the

reference The calculated load capacity results correlate wellwith the experimental results with a maximum differenceof 115 The average of the flow rate for the experimentalresults is about 12 kgs while that for the numerical resultsis approximately 145 kgs

32 Effects of Tilting Ratio on the Static Performances of theBearings The geometric parameters for the bearings havebeen presented in Figure 1 the supply pressure is 15MPa the

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Front bearing Rear bearingMass center

3

Z

Y

120579y

120579x

X

52 ∘

R26

l1 l2

lm1 lm2

Figure 1 The arrangement and coordinate system of the bearings [6]

Displacement of mass center

120579

Figure 2 The rigid rotor-bearing model

As a result the complete stiffness and damping coefficientsof a journal bearing which is important to the vibration ofa rotor should be taken into consideration in four degreesof freedom including the translation in 119909 119910 direction andtilting about the 119909- and 119910-axis Numerous studies concerningthe tilting motion of journal are available in the literatures[8ndash10] Recently Jang et al [11ndash13] thoroughly investigatedthe dynamic characteristics of the journal and groove thrustbearings used to support a HDD spindle considering bothof the translational motion and tilting motion of the journalResults show that the tilting motions have an important rolein the dynamic characteristics of the proposed bearings

Unlike the conventional oil film bearings water-lubricated bearings utilized in spindle are different inworking conditions and characteristics However forhydrostatic water-lubricated journal bearings we are notaware of any previous investigations to study their dynamiccharacteristics considering the translational and titlingmotions Therefore in this work we aim to fully studythe complete dynamic coefficients for two water-lubricatedhydrostatic journal bearings used to support a rigid rotorThedynamic characteristics will be categorized into four groupscoefficients of force to displacement coefficients of force toangle coefficients of moment to angle and coefficients ofmoment to displacement In the present study in order tofully study the variations of the complete static and dynamic

characteristics of the proposed water-lubricated bearings theinfluences of the tilting ratio rotary speed and eccentricityratio on the bearings have been studied

2 Mathematical Models

Figure 1 shows the schematic representations for a rigid rotorsupported by a pair of identical water-lubricated journalbearings as well as the geometry of a hydrostatic water-lubricated journal bearing Pressed water enters the bearingacross an orifice restrictor flows into the film lands and thenexits the bearing The recess pressure is regarded as uniformAs shown in Figure 1 the rigid rotor moved in an inertialreference frame 119862mdash119883119884119885 the rotor tilts about 119883 by an angle120579119909and tilts about 119884 by an angle 120579

119910 The rigid rotor-bearing

model is shown in Figure 2

21 Reynolds Equation For an isoviscous incompressiblefluid the Reynolds equation governing the turbulent bulkflow in nondimensional form is given as

120597

120597120593

(

1198661199091198673

120583

120597119875

120597120593

) +

1199032

1198972

120597

120597120582

(

1198661199111198673

120583

120597119875

120597120582

) = Γ1

120597119867

120597120593

+ Γ2

120597119867

120597120591

(1)

Mathematical Problems in Engineering 3

where Γ1

= 1205830Ω21198751199041205992 Γ2

= 1205830Ω1199011199041205992 120599 = 119888119903 and the

details of the turbulent coefficients 119866119909 119866119911can be obtained as

follows [6 14ndash16]

119881119909= minus119866119909

ℎ2

120583

120597119901

120597119909

+ 119866119869

119880

2

119881119885= minus119866119885

ℎ2

120583

120597119901

120597119911

119877119861=

120588ℎ

120583

[1198812

119909+ 1198812

119885]

12

119877119869=

120588ℎ

120583

[(119881119909minus 119880)2

+ 1198812

119885]

12

119891119869= 0066119877

minus025

119869

119891119861= 0066119877

minus025

119861

119896119869= 119891119869119877119869

119896119861= 119891119861119877119861

119866119909= 119866119911=

2

(119896119869+ 119896119861)

(2)

With the increase of rotary speed or film depth the flow islikely to become turbulent from laminar state The turbulentcoefficients 119866

119909 119866119910 and 119866

119911dependent on the fluid velocity

field are obtained as follows [15]

119866119909= min

1

12

119866119909

119866119910= min

1

12

119866119910

119866119911= min

1

12

119866119911

(3)

At the bearing exit plane the pressure takes a constantvalue equal to the ambient pressure

22 Continuity Equation The dimensionless continuityequation at the recess is defined by the global balancebetween the flow through the orifice restrictor and the recessoutflow into the film lands

120582radic(1 minus 119875119903) = int

1198781+1198783

(

61205830119871119903Ω

1198751199041198882

119867 minus

12119871

119903

1198661199091198673

120583

120597119875

120597120593

)119889120582

+ int

1198782+1198784

12119903

119871

1198661199111198673

120583

120597119875

120597120582

119889120593

(4)

where 120582 = 3radic21205871205721198892

0120583(1198883

radic120588119875119904) 119875119903is the recess pressure

11987813

is the circumferential recess boundary and 11987824

is the axialrecess boundary

23 Perturbation Analysis The journal center rotates aboutits steady-state position (119909

0 1199100 1205791199090 1205791199100) with a small whirl

which is generated from the variations due to the translations

of the rotormass center and variations due to the tilting angles[11ndash13] For small amplitude motions the dimensionless filmthickness and pressure fields are expressed as the sum of azeroth-order field and first-order field describing the steady-state condition and perturbed motion respectively

The dimensionless perturbed film expression consideringthe tilting angles is [12]

119867119894119895

= 1198670119894119895

+ Δ120576119909sdot sin 120579

119894119895+ Δ120576119910sdot cos 120579

119894119895+ 120593119869Δ120579119909+ 120595119869Δ120579119910

1198670119894119895

= 1 + 1205760cos120593119894119895

+ 120593119869120579119909+ 120595119869120579119910

(5)

where

120593119869=

120574119897119894119895cos 120579119909cos 120579119894119895

119888

120595119869= minus

120574119897119894119895cos 120579119910sin 120579119894119895

119888

(6)

where 119897119894119895

is the distance between the grid node of eachbearing and the mass center of the rotor supported on thebearings 120574 = 1 for the front journal bearing and 120574 = minus1 forthe rear journal bearing

The dimensionless perturbed pressure expression is

119875 = 1198750+

120597119875

120597Δ120576119909

Δ120576119909 +

120597119875

120597Δ120576119910

Δ120576119910 +

120597119875

120597Δ120579119909

Δ120579119909+

120597119875

120597Δ120579119910

Δ120579119910

+

120597119875

120597Δ120576

Δ120576 +

120597119875

120597Δ120576 119910

Δ120576 119910 +

120597119875

120597Δ

120579119909

Δ

120579119909+

120597119875

120597Δ

120579119910

Δ

120579119910

(7)

Substitution of the perturbed equations (5)ndash(7) intothe Reynolds equation yields the zeroth- and first-orderexpressions

120597

120597120593

(

1198661199091198673

120583

120597119875120585

120597120593

) +

1199032

1198972(

1198661199111198673

120583

120597119875120585

120597120582

) = 119865120585

(120585 = 0 119909 119910 120579119909 120579119910 119910

120579119909

120579119910)

1198650= Γ1

1205971198670

120597120593

119865120576119909

= minus

120597

120597120593

[

11986611990931198672

0

120583

sin 120579

1205971198750

120597120593

]

minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

sin 120579

1205971198750

120597120582

]

+ Γ1(sum

120597 (sin 120579)

120597120593

) + Γ2(sum

120597 (sin 120579)

120597120591

)

119865120576119910

= minus

120597

120597120593

[

11986611990931198672

0

120583

cos 1205791205971198750

120597120593

]

minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

cos 1205791205971198750

120597120582

]

4 Mathematical Problems in Engineering

+ Γ1(sum

120597 (cos 120579)120597120593

) + Γ2(sum

120597 (cos 120579)120597120591

)

119865120579119909

= minus

120597

120597120593

[

11986611990931198672

0

120583

120593119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120593119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120593119869)

120597120593

) + Γ2(sum

120597 (120593119869)

120597120591

)

119865120579119910

= minus

120597

120597120593

[

11986611990931198672

0

120583

120595119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120595119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120595119869)

120597120593

) + Γ2(sum

120597 (120595119869)

120597120591

)

119865120576 119909

= Γ2sin 120579

119865120576 119910

= Γ2cos 120579

119865120576120579119909

= Γ2120593119869

119865120576120579119910

= Γ2120595119869

(8)

The perturbed quality into the orifice diameter can beobtained by Taylor expansion

119876119894119899

= 120582(1 minus 1198751199030)12

minus

120582

2

(1 minus 1198751199030)minus12

(119875119903minus 1198751199030) (9)

Substitution of the perturbed equations (5)ndash(7) intothe continuity equation yields the zeroth- and first-orderexpressions

120582radic(1 minus 1198751199030) = int

1198781+1198783

(

61205830119871119903Ω

1198751199041198882

1198670119894119895

minus

12119871

119903

1198661199091198673

0119894119895

120583

1205971198750

120597120593

)119889120582

+ int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

1205971198750

120597120582

119889120593

minus

120582

2

(1 minus 1198751199030)minus12

119875120585

=

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

sdot sin 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

11986611991131198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

cos 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119910

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120593119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120593119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120593119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120595119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120595119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120595119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119910

120597120582

119889120593

(10)

Mathematical Problems in Engineering 5

The finite difference method (FDM) and a successiveover-relaxation (SOR) scheme are implemented to solve (1)ndash(10) to find the pressure distribution When the steady andperturbed pressure distributions are obtained the static anddynamic coefficients can be solved

24 Static Characteristics The quality frictional power lossand pump power are calculated by integration of the pressurefield on the bearing surfaces

119876119894119899

= 120582radic1 minus 119875119903 (11)

ℎ119891=

120583(119903Ω)2

ℎ0

∬119903119889120593119889119911 +

120583Ω2

ℎ0

∬1199033

119889119903 119889120593 (12)

ℎ119901= 119875119904119876 (13)

San Andres et al carried out a systematic research on thewater-lubricated hydrostatic journal bearings both theoreti-cally and experimentally [17] Numerical and experimentalresults show that predictions of the bearing performancecharacteristics like flow rate load capacity and rotor dynamicforce coefficients are not affected by the small temperaturevariations (Δ119879 lt 10

∘C) in the water hydrostatic journalbearings As a result an adiabatic isothermal assumption ismade in this study All the heat produced in the bearings byfriction is considered absorbed by water film The averageelevated temperature is given by

Δ119879 =

ℎ119891+ ℎ119901

119876120588119862V (14)

where 120588 is the density of water and 119862V is the specific heatcapacity of water

25 Dynamic Characteristics The dynamic coefficients ofthe journal bearings can be calculated by integrating theperturbed pressure across the fluid filmThere are 16 stiffnessand damping coefficients and the dynamic coefficients can begrouped into four categories coefficients of force to displace-ment coefficients of force to angle coefficients of moment toangle and coefficients of moment to displacement

[119870119869] =

[

[

[

[

[

[

[

119870119909119909

119870119909119910

119870119909120579119909

119870119909120579119910

119870119910119909

119870119910119910

119870119910120579119909

119870119910120579119910

119870119872119909119909

119870119872119909119910

119870119872119909120579119909

119870119872119909120579119909

119870119872119910119909

119870119872119910119910

119870119872119909120579119909

119870119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875119909119875119910

119875120579119909

119875120579119910] 119889120593 119889120582

Table 1 Operating condition of the test bearing [17]

Orifice diameter (mm) 249 Supply temperature (∘C) 55Length (mm) 762 Supply pressure (MPa) 4Film thickness (um) 127 Rotary speed (rpm) 24600

41∘

R381

Figure 3 Geometry of the test water-lubricated hydrostatic journalbearing [17]

[119861119869] =

[

[

[

[

[

[

[

119887119909119909

119887119909119910

119887119909120579119909

119887119909120579119910

119887119910119909

119887119910119910

119887119910120579119909

119887119910120579119910

119887119872119909119909

119887119872119909119910

119887119872119909120579119909

119887119872119909120579119909

119887119872119910119909

119887119872119910119910

119887119872119909120579119909

119887119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875 119909119875119910119875 120579119909

119875 120579119910] 119889120593 119889120582

(15)

26 The Numerical Solution Procedure The finite differencemethod (FDM) and a successive over-relaxation (SOR)scheme are implemented to solve (1)ndash(10) governing theflow in the film to find the pressure distribution The over-relaxation factor always lies between (1sim2) By trial and errorone can determine an optimum value of the relaxation factorfor the fastest convergence Normally 17 is a good startingpoint for determining the relaxation factor [18] In thenumerical procedure implemented any negative pressurescalculated in the cavitation zone are arbitrarily set equalto zero (or ambient) pressure [18] The pressure iterationsare continued until the following convergence criterion issatisfied

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895minus 119875(119896minus1)

119894119895

10038161003816100381610038161003816

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895

10038161003816100381610038161003816

le 120575 (16)

where 120575 is the convergence criteria which is set as 10minus3

Once the pressure field is established for the water filmother performance parameters follow from the pressuredistributionThe steady and perturbed pressure distributions

6 Mathematical Problems in Engineering

00 01 02 03 04 05

0

1000

2000

3000

4000

5000

6000

7000

8000Lo

ad ca

paci

ty (N

)

Eccentricity ratio

Experimental resultsNumerical predictions

minus1000

(a) Comparison of load capacity

00 01 02 03 04 0500

04

08

12

16

20

Mas

s flow

rate

(kg

s)

Eccentricity ratio

Experimental resultsNumerical predictions

(b) Comparison of mass flow rate

Figure 4 Comparison of static performance of experimental results[17] and numerical predictions in respect to eccentricity ratio

obtained are subsequently integrated to yield the desiredstatic and dynamic coefficients

The fluid film was discretized by rectangular grid withunequal intervals which are 0033 0088 and 01154 forthe film land recess and return groove respectively in thecircumferential direction and 002 and 004 for the film landand recess area in the axial direction The total number ofthe grid is 31 times 102 It takes about 35 seconds for the staticperformance to achieve convergence A few validation testsweremade with a coarser grid of 31 times 86 and finer grid of 37 times

158 with different intervals and in no case did the predictedresults of the static characteristics differ by more than 01percent from those obtained by the initial grid

3 Results and Discussion

31 Comparisons of Present Solution with ExperimentalResults The present numerical solution has been correlated

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10218

220

222

224

226

228

230

232

234

236

Qua

lity

(Lm

in)

Tilting ratiominus01

(a) The effect of tilting angle on the quality

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10240242244246248250252254256258260262264

Gro

ss p

ower

loss

(W)

Tilting ratiominus01

(b) The effect of tilting angle on the power loss

00 01 02 03 04 05 06 07 08 09 10148

152

156

160

164

168

172

Tilting ratio

Tem

pera

ture

rise

(∘C)

minus01

Front bearingRear bearing

(c) The effect of tilting angle on the temperature rise

Figure 5 The effect of tilting angle on the static performance of thewater-lubricated bearing

Mathematical Problems in Engineering 7

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

Stiff

ness

of f

orce

to d

ispla

cem

ent (

Nm

)

Tilting ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

Eccentricity ratio = 0

minus05

minus10

minus15

minus01

times108

(a) The stiffness of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Tilting ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus tilting ratio

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

40

Tilting ratio

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

Eccentricity ratio = 0times10

6

minus05

minus10

minus15

minus01

(c) The stiffness of moment to angle versus tilting ratio

Tilting ratio00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

Eccentricity ratio = 0times10

7

minus1

minus2

minus3

minus4

minus01

(d) The stiffness of moment to displacement versus tilting ratio

Figure 6 The effect of tilting ratio on the stiffness

and validated with the experimental results available inthe literature A five-recess turbulent-flow water-lubricatedhydrostatic bearing operating at a high rotational speed istested by San Andres et al [17] Table 1 shows the bearingdescription and operating conditions and Figure 3 shows theschematic of the bearing Figure 4 shows a comparison ofmeasured load capacity and flow rate and the numerical pre-dictions in respect to eccentricity ratio As shown in Figure 4a good agreement is observed between the present numericalpredictions and the experimental results available in the

reference The calculated load capacity results correlate wellwith the experimental results with a maximum differenceof 115 The average of the flow rate for the experimentalresults is about 12 kgs while that for the numerical resultsis approximately 145 kgs

32 Effects of Tilting Ratio on the Static Performances of theBearings The geometric parameters for the bearings havebeen presented in Figure 1 the supply pressure is 15MPa the

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering 3

where Γ1

= 1205830Ω21198751199041205992 Γ2

= 1205830Ω1199011199041205992 120599 = 119888119903 and the

details of the turbulent coefficients 119866119909 119866119911can be obtained as

follows [6 14ndash16]

119881119909= minus119866119909

ℎ2

120583

120597119901

120597119909

+ 119866119869

119880

2

119881119885= minus119866119885

ℎ2

120583

120597119901

120597119911

119877119861=

120588ℎ

120583

[1198812

119909+ 1198812

119885]

12

119877119869=

120588ℎ

120583

[(119881119909minus 119880)2

+ 1198812

119885]

12

119891119869= 0066119877

minus025

119869

119891119861= 0066119877

minus025

119861

119896119869= 119891119869119877119869

119896119861= 119891119861119877119861

119866119909= 119866119911=

2

(119896119869+ 119896119861)

(2)

With the increase of rotary speed or film depth the flow islikely to become turbulent from laminar state The turbulentcoefficients 119866

119909 119866119910 and 119866

119911dependent on the fluid velocity

field are obtained as follows [15]

119866119909= min

1

12

119866119909

119866119910= min

1

12

119866119910

119866119911= min

1

12

119866119911

(3)

At the bearing exit plane the pressure takes a constantvalue equal to the ambient pressure

22 Continuity Equation The dimensionless continuityequation at the recess is defined by the global balancebetween the flow through the orifice restrictor and the recessoutflow into the film lands

120582radic(1 minus 119875119903) = int

1198781+1198783

(

61205830119871119903Ω

1198751199041198882

119867 minus

12119871

119903

1198661199091198673

120583

120597119875

120597120593

)119889120582

+ int

1198782+1198784

12119903

119871

1198661199111198673

120583

120597119875

120597120582

119889120593

(4)

where 120582 = 3radic21205871205721198892

0120583(1198883

radic120588119875119904) 119875119903is the recess pressure

11987813

is the circumferential recess boundary and 11987824

is the axialrecess boundary

23 Perturbation Analysis The journal center rotates aboutits steady-state position (119909

0 1199100 1205791199090 1205791199100) with a small whirl

which is generated from the variations due to the translations

of the rotormass center and variations due to the tilting angles[11ndash13] For small amplitude motions the dimensionless filmthickness and pressure fields are expressed as the sum of azeroth-order field and first-order field describing the steady-state condition and perturbed motion respectively

The dimensionless perturbed film expression consideringthe tilting angles is [12]

119867119894119895

= 1198670119894119895

+ Δ120576119909sdot sin 120579

119894119895+ Δ120576119910sdot cos 120579

119894119895+ 120593119869Δ120579119909+ 120595119869Δ120579119910

1198670119894119895

= 1 + 1205760cos120593119894119895

+ 120593119869120579119909+ 120595119869120579119910

(5)

where

120593119869=

120574119897119894119895cos 120579119909cos 120579119894119895

119888

120595119869= minus

120574119897119894119895cos 120579119910sin 120579119894119895

119888

(6)

where 119897119894119895

is the distance between the grid node of eachbearing and the mass center of the rotor supported on thebearings 120574 = 1 for the front journal bearing and 120574 = minus1 forthe rear journal bearing

The dimensionless perturbed pressure expression is

119875 = 1198750+

120597119875

120597Δ120576119909

Δ120576119909 +

120597119875

120597Δ120576119910

Δ120576119910 +

120597119875

120597Δ120579119909

Δ120579119909+

120597119875

120597Δ120579119910

Δ120579119910

+

120597119875

120597Δ120576

Δ120576 +

120597119875

120597Δ120576 119910

Δ120576 119910 +

120597119875

120597Δ

120579119909

Δ

120579119909+

120597119875

120597Δ

120579119910

Δ

120579119910

(7)

Substitution of the perturbed equations (5)ndash(7) intothe Reynolds equation yields the zeroth- and first-orderexpressions

120597

120597120593

(

1198661199091198673

120583

120597119875120585

120597120593

) +

1199032

1198972(

1198661199111198673

120583

120597119875120585

120597120582

) = 119865120585

(120585 = 0 119909 119910 120579119909 120579119910 119910

120579119909

120579119910)

1198650= Γ1

1205971198670

120597120593

119865120576119909

= minus

120597

120597120593

[

11986611990931198672

0

120583

sin 120579

1205971198750

120597120593

]

minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

sin 120579

1205971198750

120597120582

]

+ Γ1(sum

120597 (sin 120579)

120597120593

) + Γ2(sum

120597 (sin 120579)

120597120591

)

119865120576119910

= minus

120597

120597120593

[

11986611990931198672

0

120583

cos 1205791205971198750

120597120593

]

minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

cos 1205791205971198750

120597120582

]

4 Mathematical Problems in Engineering

+ Γ1(sum

120597 (cos 120579)120597120593

) + Γ2(sum

120597 (cos 120579)120597120591

)

119865120579119909

= minus

120597

120597120593

[

11986611990931198672

0

120583

120593119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120593119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120593119869)

120597120593

) + Γ2(sum

120597 (120593119869)

120597120591

)

119865120579119910

= minus

120597

120597120593

[

11986611990931198672

0

120583

120595119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120595119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120595119869)

120597120593

) + Γ2(sum

120597 (120595119869)

120597120591

)

119865120576 119909

= Γ2sin 120579

119865120576 119910

= Γ2cos 120579

119865120576120579119909

= Γ2120593119869

119865120576120579119910

= Γ2120595119869

(8)

The perturbed quality into the orifice diameter can beobtained by Taylor expansion

119876119894119899

= 120582(1 minus 1198751199030)12

minus

120582

2

(1 minus 1198751199030)minus12

(119875119903minus 1198751199030) (9)

Substitution of the perturbed equations (5)ndash(7) intothe continuity equation yields the zeroth- and first-orderexpressions

120582radic(1 minus 1198751199030) = int

1198781+1198783

(

61205830119871119903Ω

1198751199041198882

1198670119894119895

minus

12119871

119903

1198661199091198673

0119894119895

120583

1205971198750

120597120593

)119889120582

+ int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

1205971198750

120597120582

119889120593

minus

120582

2

(1 minus 1198751199030)minus12

119875120585

=

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

sdot sin 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

11986611991131198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

cos 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119910

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120593119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120593119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120593119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120595119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120595119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120595119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119910

120597120582

119889120593

(10)

Mathematical Problems in Engineering 5

The finite difference method (FDM) and a successiveover-relaxation (SOR) scheme are implemented to solve (1)ndash(10) to find the pressure distribution When the steady andperturbed pressure distributions are obtained the static anddynamic coefficients can be solved

24 Static Characteristics The quality frictional power lossand pump power are calculated by integration of the pressurefield on the bearing surfaces

119876119894119899

= 120582radic1 minus 119875119903 (11)

ℎ119891=

120583(119903Ω)2

ℎ0

∬119903119889120593119889119911 +

120583Ω2

ℎ0

∬1199033

119889119903 119889120593 (12)

ℎ119901= 119875119904119876 (13)

San Andres et al carried out a systematic research on thewater-lubricated hydrostatic journal bearings both theoreti-cally and experimentally [17] Numerical and experimentalresults show that predictions of the bearing performancecharacteristics like flow rate load capacity and rotor dynamicforce coefficients are not affected by the small temperaturevariations (Δ119879 lt 10

∘C) in the water hydrostatic journalbearings As a result an adiabatic isothermal assumption ismade in this study All the heat produced in the bearings byfriction is considered absorbed by water film The averageelevated temperature is given by

Δ119879 =

ℎ119891+ ℎ119901

119876120588119862V (14)

where 120588 is the density of water and 119862V is the specific heatcapacity of water

25 Dynamic Characteristics The dynamic coefficients ofthe journal bearings can be calculated by integrating theperturbed pressure across the fluid filmThere are 16 stiffnessand damping coefficients and the dynamic coefficients can begrouped into four categories coefficients of force to displace-ment coefficients of force to angle coefficients of moment toangle and coefficients of moment to displacement

[119870119869] =

[

[

[

[

[

[

[

119870119909119909

119870119909119910

119870119909120579119909

119870119909120579119910

119870119910119909

119870119910119910

119870119910120579119909

119870119910120579119910

119870119872119909119909

119870119872119909119910

119870119872119909120579119909

119870119872119909120579119909

119870119872119910119909

119870119872119910119910

119870119872119909120579119909

119870119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875119909119875119910

119875120579119909

119875120579119910] 119889120593 119889120582

Table 1 Operating condition of the test bearing [17]

Orifice diameter (mm) 249 Supply temperature (∘C) 55Length (mm) 762 Supply pressure (MPa) 4Film thickness (um) 127 Rotary speed (rpm) 24600

41∘

R381

Figure 3 Geometry of the test water-lubricated hydrostatic journalbearing [17]

[119861119869] =

[

[

[

[

[

[

[

119887119909119909

119887119909119910

119887119909120579119909

119887119909120579119910

119887119910119909

119887119910119910

119887119910120579119909

119887119910120579119910

119887119872119909119909

119887119872119909119910

119887119872119909120579119909

119887119872119909120579119909

119887119872119910119909

119887119872119910119910

119887119872119909120579119909

119887119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875 119909119875119910119875 120579119909

119875 120579119910] 119889120593 119889120582

(15)

26 The Numerical Solution Procedure The finite differencemethod (FDM) and a successive over-relaxation (SOR)scheme are implemented to solve (1)ndash(10) governing theflow in the film to find the pressure distribution The over-relaxation factor always lies between (1sim2) By trial and errorone can determine an optimum value of the relaxation factorfor the fastest convergence Normally 17 is a good startingpoint for determining the relaxation factor [18] In thenumerical procedure implemented any negative pressurescalculated in the cavitation zone are arbitrarily set equalto zero (or ambient) pressure [18] The pressure iterationsare continued until the following convergence criterion issatisfied

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895minus 119875(119896minus1)

119894119895

10038161003816100381610038161003816

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895

10038161003816100381610038161003816

le 120575 (16)

where 120575 is the convergence criteria which is set as 10minus3

Once the pressure field is established for the water filmother performance parameters follow from the pressuredistributionThe steady and perturbed pressure distributions

6 Mathematical Problems in Engineering

00 01 02 03 04 05

0

1000

2000

3000

4000

5000

6000

7000

8000Lo

ad ca

paci

ty (N

)

Eccentricity ratio

Experimental resultsNumerical predictions

minus1000

(a) Comparison of load capacity

00 01 02 03 04 0500

04

08

12

16

20

Mas

s flow

rate

(kg

s)

Eccentricity ratio

Experimental resultsNumerical predictions

(b) Comparison of mass flow rate

Figure 4 Comparison of static performance of experimental results[17] and numerical predictions in respect to eccentricity ratio

obtained are subsequently integrated to yield the desiredstatic and dynamic coefficients

The fluid film was discretized by rectangular grid withunequal intervals which are 0033 0088 and 01154 forthe film land recess and return groove respectively in thecircumferential direction and 002 and 004 for the film landand recess area in the axial direction The total number ofthe grid is 31 times 102 It takes about 35 seconds for the staticperformance to achieve convergence A few validation testsweremade with a coarser grid of 31 times 86 and finer grid of 37 times

158 with different intervals and in no case did the predictedresults of the static characteristics differ by more than 01percent from those obtained by the initial grid

3 Results and Discussion

31 Comparisons of Present Solution with ExperimentalResults The present numerical solution has been correlated

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10218

220

222

224

226

228

230

232

234

236

Qua

lity

(Lm

in)

Tilting ratiominus01

(a) The effect of tilting angle on the quality

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10240242244246248250252254256258260262264

Gro

ss p

ower

loss

(W)

Tilting ratiominus01

(b) The effect of tilting angle on the power loss

00 01 02 03 04 05 06 07 08 09 10148

152

156

160

164

168

172

Tilting ratio

Tem

pera

ture

rise

(∘C)

minus01

Front bearingRear bearing

(c) The effect of tilting angle on the temperature rise

Figure 5 The effect of tilting angle on the static performance of thewater-lubricated bearing

Mathematical Problems in Engineering 7

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

Stiff

ness

of f

orce

to d

ispla

cem

ent (

Nm

)

Tilting ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

Eccentricity ratio = 0

minus05

minus10

minus15

minus01

times108

(a) The stiffness of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Tilting ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus tilting ratio

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

40

Tilting ratio

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

Eccentricity ratio = 0times10

6

minus05

minus10

minus15

minus01

(c) The stiffness of moment to angle versus tilting ratio

Tilting ratio00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

Eccentricity ratio = 0times10

7

minus1

minus2

minus3

minus4

minus01

(d) The stiffness of moment to displacement versus tilting ratio

Figure 6 The effect of tilting ratio on the stiffness

and validated with the experimental results available inthe literature A five-recess turbulent-flow water-lubricatedhydrostatic bearing operating at a high rotational speed istested by San Andres et al [17] Table 1 shows the bearingdescription and operating conditions and Figure 3 shows theschematic of the bearing Figure 4 shows a comparison ofmeasured load capacity and flow rate and the numerical pre-dictions in respect to eccentricity ratio As shown in Figure 4a good agreement is observed between the present numericalpredictions and the experimental results available in the

reference The calculated load capacity results correlate wellwith the experimental results with a maximum differenceof 115 The average of the flow rate for the experimentalresults is about 12 kgs while that for the numerical resultsis approximately 145 kgs

32 Effects of Tilting Ratio on the Static Performances of theBearings The geometric parameters for the bearings havebeen presented in Figure 1 the supply pressure is 15MPa the

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

+ Γ1(sum

120597 (cos 120579)120597120593

) + Γ2(sum

120597 (cos 120579)120597120591

)

119865120579119909

= minus

120597

120597120593

[

11986611990931198672

0

120583

120593119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120593119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120593119869)

120597120593

) + Γ2(sum

120597 (120593119869)

120597120591

)

119865120579119910

= minus

120597

120597120593

[

11986611990931198672

0

120583

120595119869

1205971198750

120597120593

] minus

1199032

1198972

120597

120597120582

[

11986611991131198672

0

120583

120595119869

1205971198750

120597120582

]

+ Γ1(sum

120597 (120595119869)

120597120593

) + Γ2(sum

120597 (120595119869)

120597120591

)

119865120576 119909

= Γ2sin 120579

119865120576 119910

= Γ2cos 120579

119865120576120579119909

= Γ2120593119869

119865120576120579119910

= Γ2120595119869

(8)

The perturbed quality into the orifice diameter can beobtained by Taylor expansion

119876119894119899

= 120582(1 minus 1198751199030)12

minus

120582

2

(1 minus 1198751199030)minus12

(119875119903minus 1198751199030) (9)

Substitution of the perturbed equations (5)ndash(7) intothe continuity equation yields the zeroth- and first-orderexpressions

120582radic(1 minus 1198751199030) = int

1198781+1198783

(

61205830119871119903Ω

1198751199041198882

1198670119894119895

minus

12119871

119903

1198661199091198673

0119894119895

120583

1205971198750

120597120593

)119889120582

+ int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

1205971198750

120597120582

119889120593

minus

120582

2

(1 minus 1198751199030)minus12

119875120585

=

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

sdot sin 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

11986611991131198672

0119894119895sin 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

cos 120579119894119895119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895cos 120579119894119895

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576119910

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120593119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120593119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119909

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120593119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119909

120597120582

119889120593

int

1198781+1198783

61205830119871119903Ω

1198751199041198882

120595119869119889120582 minus int

1198781+1198783

12119871

119903

31198661199091198672

0119894119895120595119869

120583

1205971198750

120597120593

119889120582 minus int

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579119910

120597120593

119889120582

+int

1198782+1198784

12119903

119871

31198661199111198672

0119894119895120595119869

120583

1205971198750

120597120582

119889120593 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576 119910

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119909

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119909

120597120582

119889120593

minusint

1198781+1198783

12119871

119903

1198661199091198673

0119894119895

120583

120597119875120576120579 119910

120597120593

119889120582 + int

1198782+1198784

12119903

119871

1198661199111198673

0119894119895

120583

120597119875120576120579 119910

120597120582

119889120593

(10)

Mathematical Problems in Engineering 5

The finite difference method (FDM) and a successiveover-relaxation (SOR) scheme are implemented to solve (1)ndash(10) to find the pressure distribution When the steady andperturbed pressure distributions are obtained the static anddynamic coefficients can be solved

24 Static Characteristics The quality frictional power lossand pump power are calculated by integration of the pressurefield on the bearing surfaces

119876119894119899

= 120582radic1 minus 119875119903 (11)

ℎ119891=

120583(119903Ω)2

ℎ0

∬119903119889120593119889119911 +

120583Ω2

ℎ0

∬1199033

119889119903 119889120593 (12)

ℎ119901= 119875119904119876 (13)

San Andres et al carried out a systematic research on thewater-lubricated hydrostatic journal bearings both theoreti-cally and experimentally [17] Numerical and experimentalresults show that predictions of the bearing performancecharacteristics like flow rate load capacity and rotor dynamicforce coefficients are not affected by the small temperaturevariations (Δ119879 lt 10

∘C) in the water hydrostatic journalbearings As a result an adiabatic isothermal assumption ismade in this study All the heat produced in the bearings byfriction is considered absorbed by water film The averageelevated temperature is given by

Δ119879 =

ℎ119891+ ℎ119901

119876120588119862V (14)

where 120588 is the density of water and 119862V is the specific heatcapacity of water

25 Dynamic Characteristics The dynamic coefficients ofthe journal bearings can be calculated by integrating theperturbed pressure across the fluid filmThere are 16 stiffnessand damping coefficients and the dynamic coefficients can begrouped into four categories coefficients of force to displace-ment coefficients of force to angle coefficients of moment toangle and coefficients of moment to displacement

[119870119869] =

[

[

[

[

[

[

[

119870119909119909

119870119909119910

119870119909120579119909

119870119909120579119910

119870119910119909

119870119910119910

119870119910120579119909

119870119910120579119910

119870119872119909119909

119870119872119909119910

119870119872119909120579119909

119870119872119909120579119909

119870119872119910119909

119870119872119910119910

119870119872119909120579119909

119870119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875119909119875119910

119875120579119909

119875120579119910] 119889120593 119889120582

Table 1 Operating condition of the test bearing [17]

Orifice diameter (mm) 249 Supply temperature (∘C) 55Length (mm) 762 Supply pressure (MPa) 4Film thickness (um) 127 Rotary speed (rpm) 24600

41∘

R381

Figure 3 Geometry of the test water-lubricated hydrostatic journalbearing [17]

[119861119869] =

[

[

[

[

[

[

[

119887119909119909

119887119909119910

119887119909120579119909

119887119909120579119910

119887119910119909

119887119910119910

119887119910120579119909

119887119910120579119910

119887119872119909119909

119887119872119909119910

119887119872119909120579119909

119887119872119909120579119909

119887119872119910119909

119887119872119910119910

119887119872119909120579119909

119887119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875 119909119875119910119875 120579119909

119875 120579119910] 119889120593 119889120582

(15)

26 The Numerical Solution Procedure The finite differencemethod (FDM) and a successive over-relaxation (SOR)scheme are implemented to solve (1)ndash(10) governing theflow in the film to find the pressure distribution The over-relaxation factor always lies between (1sim2) By trial and errorone can determine an optimum value of the relaxation factorfor the fastest convergence Normally 17 is a good startingpoint for determining the relaxation factor [18] In thenumerical procedure implemented any negative pressurescalculated in the cavitation zone are arbitrarily set equalto zero (or ambient) pressure [18] The pressure iterationsare continued until the following convergence criterion issatisfied

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895minus 119875(119896minus1)

119894119895

10038161003816100381610038161003816

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895

10038161003816100381610038161003816

le 120575 (16)

where 120575 is the convergence criteria which is set as 10minus3

Once the pressure field is established for the water filmother performance parameters follow from the pressuredistributionThe steady and perturbed pressure distributions

6 Mathematical Problems in Engineering

00 01 02 03 04 05

0

1000

2000

3000

4000

5000

6000

7000

8000Lo

ad ca

paci

ty (N

)

Eccentricity ratio

Experimental resultsNumerical predictions

minus1000

(a) Comparison of load capacity

00 01 02 03 04 0500

04

08

12

16

20

Mas

s flow

rate

(kg

s)

Eccentricity ratio

Experimental resultsNumerical predictions

(b) Comparison of mass flow rate

Figure 4 Comparison of static performance of experimental results[17] and numerical predictions in respect to eccentricity ratio

obtained are subsequently integrated to yield the desiredstatic and dynamic coefficients

The fluid film was discretized by rectangular grid withunequal intervals which are 0033 0088 and 01154 forthe film land recess and return groove respectively in thecircumferential direction and 002 and 004 for the film landand recess area in the axial direction The total number ofthe grid is 31 times 102 It takes about 35 seconds for the staticperformance to achieve convergence A few validation testsweremade with a coarser grid of 31 times 86 and finer grid of 37 times

158 with different intervals and in no case did the predictedresults of the static characteristics differ by more than 01percent from those obtained by the initial grid

3 Results and Discussion

31 Comparisons of Present Solution with ExperimentalResults The present numerical solution has been correlated

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10218

220

222

224

226

228

230

232

234

236

Qua

lity

(Lm

in)

Tilting ratiominus01

(a) The effect of tilting angle on the quality

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10240242244246248250252254256258260262264

Gro

ss p

ower

loss

(W)

Tilting ratiominus01

(b) The effect of tilting angle on the power loss

00 01 02 03 04 05 06 07 08 09 10148

152

156

160

164

168

172

Tilting ratio

Tem

pera

ture

rise

(∘C)

minus01

Front bearingRear bearing

(c) The effect of tilting angle on the temperature rise

Figure 5 The effect of tilting angle on the static performance of thewater-lubricated bearing

Mathematical Problems in Engineering 7

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

Stiff

ness

of f

orce

to d

ispla

cem

ent (

Nm

)

Tilting ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

Eccentricity ratio = 0

minus05

minus10

minus15

minus01

times108

(a) The stiffness of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Tilting ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus tilting ratio

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

40

Tilting ratio

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

Eccentricity ratio = 0times10

6

minus05

minus10

minus15

minus01

(c) The stiffness of moment to angle versus tilting ratio

Tilting ratio00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

Eccentricity ratio = 0times10

7

minus1

minus2

minus3

minus4

minus01

(d) The stiffness of moment to displacement versus tilting ratio

Figure 6 The effect of tilting ratio on the stiffness

and validated with the experimental results available inthe literature A five-recess turbulent-flow water-lubricatedhydrostatic bearing operating at a high rotational speed istested by San Andres et al [17] Table 1 shows the bearingdescription and operating conditions and Figure 3 shows theschematic of the bearing Figure 4 shows a comparison ofmeasured load capacity and flow rate and the numerical pre-dictions in respect to eccentricity ratio As shown in Figure 4a good agreement is observed between the present numericalpredictions and the experimental results available in the

reference The calculated load capacity results correlate wellwith the experimental results with a maximum differenceof 115 The average of the flow rate for the experimentalresults is about 12 kgs while that for the numerical resultsis approximately 145 kgs

32 Effects of Tilting Ratio on the Static Performances of theBearings The geometric parameters for the bearings havebeen presented in Figure 1 the supply pressure is 15MPa the

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

The finite difference method (FDM) and a successiveover-relaxation (SOR) scheme are implemented to solve (1)ndash(10) to find the pressure distribution When the steady andperturbed pressure distributions are obtained the static anddynamic coefficients can be solved

24 Static Characteristics The quality frictional power lossand pump power are calculated by integration of the pressurefield on the bearing surfaces

119876119894119899

= 120582radic1 minus 119875119903 (11)

ℎ119891=

120583(119903Ω)2

ℎ0

∬119903119889120593119889119911 +

120583Ω2

ℎ0

∬1199033

119889119903 119889120593 (12)

ℎ119901= 119875119904119876 (13)

San Andres et al carried out a systematic research on thewater-lubricated hydrostatic journal bearings both theoreti-cally and experimentally [17] Numerical and experimentalresults show that predictions of the bearing performancecharacteristics like flow rate load capacity and rotor dynamicforce coefficients are not affected by the small temperaturevariations (Δ119879 lt 10

∘C) in the water hydrostatic journalbearings As a result an adiabatic isothermal assumption ismade in this study All the heat produced in the bearings byfriction is considered absorbed by water film The averageelevated temperature is given by

Δ119879 =

ℎ119891+ ℎ119901

119876120588119862V (14)

where 120588 is the density of water and 119862V is the specific heatcapacity of water

25 Dynamic Characteristics The dynamic coefficients ofthe journal bearings can be calculated by integrating theperturbed pressure across the fluid filmThere are 16 stiffnessand damping coefficients and the dynamic coefficients can begrouped into four categories coefficients of force to displace-ment coefficients of force to angle coefficients of moment toangle and coefficients of moment to displacement

[119870119869] =

[

[

[

[

[

[

[

119870119909119909

119870119909119910

119870119909120579119909

119870119909120579119910

119870119910119909

119870119910119910

119870119910120579119909

119870119910120579119910

119870119872119909119909

119870119872119909119910

119870119872119909120579119909

119870119872119909120579119909

119870119872119910119909

119870119872119910119910

119870119872119909120579119909

119870119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875119909119875119910

119875120579119909

119875120579119910] 119889120593 119889120582

Table 1 Operating condition of the test bearing [17]

Orifice diameter (mm) 249 Supply temperature (∘C) 55Length (mm) 762 Supply pressure (MPa) 4Film thickness (um) 127 Rotary speed (rpm) 24600

41∘

R381

Figure 3 Geometry of the test water-lubricated hydrostatic journalbearing [17]

[119861119869] =

[

[

[

[

[

[

[

119887119909119909

119887119909119910

119887119909120579119909

119887119909120579119910

119887119910119909

119887119910119910

119887119910120579119909

119887119910120579119910

119887119872119909119909

119887119872119909119910

119887119872119909120579119909

119887119872119909120579119909

119887119872119910119909

119887119872119910119910

119887119872119909120579119909

119887119872119909120579119909

]

]

]

]

]

]

]119869

= minus∬

[

[

[

[

[

[

sin 120579

cos 120579minus120574119897119894119895cos 120579

120574119897119894119895sin 120579

]

]

]

]

]

]

[119875 119909119875119910119875 120579119909

119875 120579119910] 119889120593 119889120582

(15)

26 The Numerical Solution Procedure The finite differencemethod (FDM) and a successive over-relaxation (SOR)scheme are implemented to solve (1)ndash(10) governing theflow in the film to find the pressure distribution The over-relaxation factor always lies between (1sim2) By trial and errorone can determine an optimum value of the relaxation factorfor the fastest convergence Normally 17 is a good startingpoint for determining the relaxation factor [18] In thenumerical procedure implemented any negative pressurescalculated in the cavitation zone are arbitrarily set equalto zero (or ambient) pressure [18] The pressure iterationsare continued until the following convergence criterion issatisfied

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895minus 119875(119896minus1)

119894119895

10038161003816100381610038161003816

sum119898

119895=1sum119899

119894=1

10038161003816100381610038161003816119875(119896)

119894119895

10038161003816100381610038161003816

le 120575 (16)

where 120575 is the convergence criteria which is set as 10minus3

Once the pressure field is established for the water filmother performance parameters follow from the pressuredistributionThe steady and perturbed pressure distributions

6 Mathematical Problems in Engineering

00 01 02 03 04 05

0

1000

2000

3000

4000

5000

6000

7000

8000Lo

ad ca

paci

ty (N

)

Eccentricity ratio

Experimental resultsNumerical predictions

minus1000

(a) Comparison of load capacity

00 01 02 03 04 0500

04

08

12

16

20

Mas

s flow

rate

(kg

s)

Eccentricity ratio

Experimental resultsNumerical predictions

(b) Comparison of mass flow rate

Figure 4 Comparison of static performance of experimental results[17] and numerical predictions in respect to eccentricity ratio

obtained are subsequently integrated to yield the desiredstatic and dynamic coefficients

The fluid film was discretized by rectangular grid withunequal intervals which are 0033 0088 and 01154 forthe film land recess and return groove respectively in thecircumferential direction and 002 and 004 for the film landand recess area in the axial direction The total number ofthe grid is 31 times 102 It takes about 35 seconds for the staticperformance to achieve convergence A few validation testsweremade with a coarser grid of 31 times 86 and finer grid of 37 times

158 with different intervals and in no case did the predictedresults of the static characteristics differ by more than 01percent from those obtained by the initial grid

3 Results and Discussion

31 Comparisons of Present Solution with ExperimentalResults The present numerical solution has been correlated

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10218

220

222

224

226

228

230

232

234

236

Qua

lity

(Lm

in)

Tilting ratiominus01

(a) The effect of tilting angle on the quality

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10240242244246248250252254256258260262264

Gro

ss p

ower

loss

(W)

Tilting ratiominus01

(b) The effect of tilting angle on the power loss

00 01 02 03 04 05 06 07 08 09 10148

152

156

160

164

168

172

Tilting ratio

Tem

pera

ture

rise

(∘C)

minus01

Front bearingRear bearing

(c) The effect of tilting angle on the temperature rise

Figure 5 The effect of tilting angle on the static performance of thewater-lubricated bearing

Mathematical Problems in Engineering 7

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

Stiff

ness

of f

orce

to d

ispla

cem

ent (

Nm

)

Tilting ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

Eccentricity ratio = 0

minus05

minus10

minus15

minus01

times108

(a) The stiffness of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Tilting ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus tilting ratio

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

40

Tilting ratio

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

Eccentricity ratio = 0times10

6

minus05

minus10

minus15

minus01

(c) The stiffness of moment to angle versus tilting ratio

Tilting ratio00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

Eccentricity ratio = 0times10

7

minus1

minus2

minus3

minus4

minus01

(d) The stiffness of moment to displacement versus tilting ratio

Figure 6 The effect of tilting ratio on the stiffness

and validated with the experimental results available inthe literature A five-recess turbulent-flow water-lubricatedhydrostatic bearing operating at a high rotational speed istested by San Andres et al [17] Table 1 shows the bearingdescription and operating conditions and Figure 3 shows theschematic of the bearing Figure 4 shows a comparison ofmeasured load capacity and flow rate and the numerical pre-dictions in respect to eccentricity ratio As shown in Figure 4a good agreement is observed between the present numericalpredictions and the experimental results available in the

reference The calculated load capacity results correlate wellwith the experimental results with a maximum differenceof 115 The average of the flow rate for the experimentalresults is about 12 kgs while that for the numerical resultsis approximately 145 kgs

32 Effects of Tilting Ratio on the Static Performances of theBearings The geometric parameters for the bearings havebeen presented in Figure 1 the supply pressure is 15MPa the

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

00 01 02 03 04 05

0

1000

2000

3000

4000

5000

6000

7000

8000Lo

ad ca

paci

ty (N

)

Eccentricity ratio

Experimental resultsNumerical predictions

minus1000

(a) Comparison of load capacity

00 01 02 03 04 0500

04

08

12

16

20

Mas

s flow

rate

(kg

s)

Eccentricity ratio

Experimental resultsNumerical predictions

(b) Comparison of mass flow rate

Figure 4 Comparison of static performance of experimental results[17] and numerical predictions in respect to eccentricity ratio

obtained are subsequently integrated to yield the desiredstatic and dynamic coefficients

The fluid film was discretized by rectangular grid withunequal intervals which are 0033 0088 and 01154 forthe film land recess and return groove respectively in thecircumferential direction and 002 and 004 for the film landand recess area in the axial direction The total number ofthe grid is 31 times 102 It takes about 35 seconds for the staticperformance to achieve convergence A few validation testsweremade with a coarser grid of 31 times 86 and finer grid of 37 times

158 with different intervals and in no case did the predictedresults of the static characteristics differ by more than 01percent from those obtained by the initial grid

3 Results and Discussion

31 Comparisons of Present Solution with ExperimentalResults The present numerical solution has been correlated

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10218

220

222

224

226

228

230

232

234

236

Qua

lity

(Lm

in)

Tilting ratiominus01

(a) The effect of tilting angle on the quality

Front bearingRear bearing

00 01 02 03 04 05 06 07 08 09 10240242244246248250252254256258260262264

Gro

ss p

ower

loss

(W)

Tilting ratiominus01

(b) The effect of tilting angle on the power loss

00 01 02 03 04 05 06 07 08 09 10148

152

156

160

164

168

172

Tilting ratio

Tem

pera

ture

rise

(∘C)

minus01

Front bearingRear bearing

(c) The effect of tilting angle on the temperature rise

Figure 5 The effect of tilting angle on the static performance of thewater-lubricated bearing

Mathematical Problems in Engineering 7

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

Stiff

ness

of f

orce

to d

ispla

cem

ent (

Nm

)

Tilting ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

Eccentricity ratio = 0

minus05

minus10

minus15

minus01

times108

(a) The stiffness of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Tilting ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus tilting ratio

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

40

Tilting ratio

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

Eccentricity ratio = 0times10

6

minus05

minus10

minus15

minus01

(c) The stiffness of moment to angle versus tilting ratio

Tilting ratio00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

Eccentricity ratio = 0times10

7

minus1

minus2

minus3

minus4

minus01

(d) The stiffness of moment to displacement versus tilting ratio

Figure 6 The effect of tilting ratio on the stiffness

and validated with the experimental results available inthe literature A five-recess turbulent-flow water-lubricatedhydrostatic bearing operating at a high rotational speed istested by San Andres et al [17] Table 1 shows the bearingdescription and operating conditions and Figure 3 shows theschematic of the bearing Figure 4 shows a comparison ofmeasured load capacity and flow rate and the numerical pre-dictions in respect to eccentricity ratio As shown in Figure 4a good agreement is observed between the present numericalpredictions and the experimental results available in the

reference The calculated load capacity results correlate wellwith the experimental results with a maximum differenceof 115 The average of the flow rate for the experimentalresults is about 12 kgs while that for the numerical resultsis approximately 145 kgs

32 Effects of Tilting Ratio on the Static Performances of theBearings The geometric parameters for the bearings havebeen presented in Figure 1 the supply pressure is 15MPa the

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

Stiff

ness

of f

orce

to d

ispla

cem

ent (

Nm

)

Tilting ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

Eccentricity ratio = 0

minus05

minus10

minus15

minus01

times108

(a) The stiffness of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Tilting ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus tilting ratio

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

00 01 02 03 04 05 06 07 08

00

05

10

15

20

25

30

35

40

Tilting ratio

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

Eccentricity ratio = 0times10

6

minus05

minus10

minus15

minus01

(c) The stiffness of moment to angle versus tilting ratio

Tilting ratio00 01 02 03 04 05 06 07 08

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

Eccentricity ratio = 0times10

7

minus1

minus2

minus3

minus4

minus01

(d) The stiffness of moment to displacement versus tilting ratio

Figure 6 The effect of tilting ratio on the stiffness

and validated with the experimental results available inthe literature A five-recess turbulent-flow water-lubricatedhydrostatic bearing operating at a high rotational speed istested by San Andres et al [17] Table 1 shows the bearingdescription and operating conditions and Figure 3 shows theschematic of the bearing Figure 4 shows a comparison ofmeasured load capacity and flow rate and the numerical pre-dictions in respect to eccentricity ratio As shown in Figure 4a good agreement is observed between the present numericalpredictions and the experimental results available in the

reference The calculated load capacity results correlate wellwith the experimental results with a maximum differenceof 115 The average of the flow rate for the experimentalresults is about 12 kgs while that for the numerical resultsis approximately 145 kgs

32 Effects of Tilting Ratio on the Static Performances of theBearings The geometric parameters for the bearings havebeen presented in Figure 1 the supply pressure is 15MPa the

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

Tilting ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

Eccentricity ratio = 0

minus04

minus01

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus tilting ratio

00 01 02 03 04 05 06 07 08

000408121620242832

Tilting ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus12

minus16

minus20

minus24

minus28

minus32

minus01

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

00

04

08

12

16

20

24

28

32

Tilting ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

Eccentricity ratio = 0

minus04

minus08

minus01

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus tilting ratio

00 01 02 03 04 05 06 07 08

0

1

2

3

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Tilting ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

Eccentricity ratio = 0

minus01

minus1

minus2

minus3

times104

(d) The damping of moment to displacement versus tilting ratio

Figure 7 The effect of tilting ratio on the damping coefficients

orifice diameter is 06mm the pressurized water is suppliedat a temperature of 30∘C and the viscosity is 000087 Pa sdot s

Figure 1 shows the coordinate system for the tiltingmotion of the rotor As the film variation of each bearingvaries with different values for the tilting angles and bearingspan the tilting ratio is proposed here to investigate the effectof misalignment The tilting ratio is defined as follows

120581 =

119897119898radic1205792

119909+ 1205792

119910

119888

(17)

where 119897119898is the distance between the titling center and the left

edge of the front bearing or the right edge of the rear bearingwhich is shown in Figure 1

In order to obtain a better physical insight into theeffect of misalignment the static characteristics have beenpresented with eccentricity ratio equal to zero Figures 5(a)ndash5(c) show the calculated quality power loss and temperaturerise for different values of tilting ratios for each journalbearing operating at 10000 rpm It is observed that the qualityof each bearing undergoes a reduction by only 63 when

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

5000 10000 15000 20000 25000 3000020

21

22

23

24

25G

ross

qua

lity

(Lm

in)

Rotary speed (rpm)

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

5000 10000 15000 20000 25000 300000

200

400

600

800

1000

1200

1400

Rotary speed (rpm)

Gro

ss p

ower

loss

(W)

(b) The effect of rotary speed on the power loss

5000 10000 15000 20000 25000 300000

1

2

3

4

5

6

7

8

9

10

Rotary speed (rpm)

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 8 The effect of rotary speed on the static performance of the water-lubricated bearing

the tilting ratio increases from zero to 075 The total powerloss of each bearing keeps almost unchanged when the tiltingratio is not greater than 04 However when the tiltingratio continues to increase the power loss gets a slightincrease by approximately 5 Meanwhile the temperaturerise increases with increased tilting ratio The reason for thisis that increasing tilting angles will increase the film variationof each bearing which promote the hydrodynamic andturbulent effect It is noticed that the maximum temperaturerise for the investigated water-lubricated bearing is 17∘C forthe rotor-bearing system investigated here which is far lowerthan the conventional oil film bearings This in turn verifiesthe validation of the adiabatic assumption In conclusionwhen the tilting ratio is smaller than 04 the influence oftilting ratio on the static performance of a water-lubricatedhydrostatic journal bearing can be ignored

33 Effects of Tilting Ratio on the Dynamic Characteristicsof the Bearings Figures 6 and 7 show the variation of thedynamic coefficients of the two identical bearings in respectto the tilting ratio with a rotary speed of 10000 rpm Theeccentricity ratio is assumed to be zero to exclude the effectof the clearance change due to the eccentricity ratioThe solidlines represent the dynamic coefficients of the front journalbearing while the dotted lines represent those of the rearbearing

According to the results the dynamic coefficients of forceto displacement andmoment to angle for the front bearing areapproximately equal to those for the rear bearing due to thefact that the tilting center almost coincides with the bearingspan center However the coupled dynamic coefficients offorce to angle and moment to displacement for the twobearings have close proximitymagnitudes but in the opposite

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

5000 10000 15000 20000 25000 30000

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Rotary speed (rpm)

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

minus3

minus4

times108

(a) The stiffness of force to displacement versys rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Rotary speed (rpm)

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(b) The stiffness of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

minus1

minus2

minus3

minus4

minus5

times106

(c) The stiffness of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

0

1

2

3

4

5

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

minus1

minus2

minus3

minus4

minus5

times107

(d) The stiffness ofmoment to displacement versus rotary speed

Figure 9 The effect of rotary speed on the stiffness coefficients

direction due to the fact that the two bearings are arranged atboth sides of the mass center which is the origin of the tiltingmotions It is observed that with the increase of the tiltingratio the stiffness and damping coefficients of the bearingskeep almost independent of the tilting ratio when it is notlarger than 04 however when the tilting ratio continuesto increase the effect on dynamic coefficients is significantwith a maximum variation rate of 167 for the stiffnessand 492 for the damping coefficients Furthermore thelarger the tilting ratio is the greater the differences amongthe coefficients are This can be ascribed to the variations

of the film thickness induced by the tilting angles andbearing span which can be calculated according to (5) Theresults indicate that for a small tilting ratio the effect ofthe misalignment on the performance of hydrostatic water-lubricated journal bearings can be ignored but for a relativelygreater tilting ratio the effect of misalignment should betaken into consideration

34 Effects of Rotary Speed on the Static Performances ofthe Bearings Figure 8 depicts the variation of the staticcharacteristics for each journal bearing in respect to

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

5000 10000 15000 20000 25000 30000

00

02

04

06

08

10

12

14

16

Rotary speed (rpm)

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus02

minus04

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus04

minus08

minus12

minus16

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Rotary speed (rpm)

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

minus04

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

(c) The damping of moment to angle versus rotary speed

5000 10000 15000 20000 25000 30000

00

04

08

12

16

20

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Rotary speed (rpm)

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

minus04

minus08

minus12

minus16

minus20

times104

(d) The damping of moment to displacement versus rotary speed

Figure 10 The effect of rotary speed on the damping coefficients

the rotary speed In the case of a water-lubricated hydrostaticbearing operating with eccentricity ratio and tilting anglesequal to zero it may be observed that the static characteristicsof the front bearing are the same as those of the rearbearing As shown in Figure 8(a) the quality of each bearingapparently remains unchanged with increased rotary speedAs shown in Figures 8(b) and 8(c) the gross power loss foreach bearing increases dramatically from 123W to 1232Wwhen the rotary speed increases from 5000 rpm to 30000 rpmbecause the frictional power loss is closely related to therotary speed according to (12) Furthermore the temperature

rise undergoes a sharp increase from07∘C to 8∘C It should bepointed out that most of the predicted temperature rises arehigher than the actual values presumably due to the adiabaticassumption imposed on the analysis Considering the valuesof the temperature rise the adiabatic flow assumption is fullyjustified for the bearing studied However when the rotaryspeed continues to increase the energy equation should beincluded to predict the temperature rise precisely

35 Effects of Rotary Speed on the Dynamic Characteristics ofthe Bearings Figures 9 and 10 show the dynamic coefficients

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

00 01 02 03 04 05220

225

230

235

240G

ross

qua

lity

(Lm

in)

Eccentricity ratio

Front bearingRear bearing

(a) The effect of rotary speed on the quality

Front bearingRear bearing

00 01 02 03 04 05240

241

242

243

244

245

246

247

248

249

250

Gro

ss p

ower

loss

(W)

Eccentricity ratio

(b) The effect of rotary speed on the power loss

00 01 02 03 04 05148

150

152

154

156

158

160

Eccentricity ratio

Front bearingRear bearing

Tem

pera

ture

rise

(∘C)

(c) The effect of rotary speed on the temperature rise

Figure 11 The effect of eccentricity ratio on the static performance of the water-lubricated bearing

of each water-lubricated bearing in respect to the rotaryspeedThe eccentricity ratio and tilting angles are assumed aszero to exclude their influences on the film thickness Accord-ing to the results a higher rotary speed generates largercoupled stiffness of force to displacement and moment toangle but the influence of rotary speed on the relative directcoefficients is smallThe direct stiffness coefficients of force toangle and moment to displacement are relatively insensitiveto the variation of the rotary speed The magnitudes of thecross-coupled stiffness of force to displacement and momentto angle are comparable to those of the direct stiffnesswhich demonstrates the importance of hydrodynamic effectsUnlike the stiffness the damping coefficients are independentof the rotary speed in the aligned condition The reasonfor this is that the perturbed pressure due to the perturbedvelocity is not related to the rotary speed of the rotor

36 Effects of Eccentricity Ratio on the Static Performances ofthe Bearings The influence of eccentricity ratio in alignedcondition on static performances of each bearing is as shownin Figure 11 The tilting ratio is assumed as zero to excludethe influence of tilting effect Figure 11(a) indicates that thevalue of quality for each bearing decreases slowly with theeccentricity ratio As is shown in Figure 11(b) the powerloss is almost constant at first and then increases withincreased eccentricity ratio The maximum temperature riseacross the bearing length is about 16∘C at the eccentricityratio equal to 05 This is expected since a smaller clearanceproduces a larger frictional power loss and a smaller flowrate However it should be pointed out that in the alignedcondition the static characteristics of the hydrostatic water-lubricated journal bearings vary slightly with eccentricityratio

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

00 01 02 03 04 05

0

1

2

3

4St

iffne

ss o

f for

ce to

disp

lace

men

t (N

m)

Eccentricity ratio

kxx front bearingkxy front bearingkyx front bearingkyy front bearing

kxx rear bearingkxy rear bearingkyx rear bearingkyy rear bearing

minus1

minus2

times108

(a) The stiffness of force to displacement versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of f

orce

to an

gle (

Nr

ad)

Eccentricity ratio

kx120579x front bearingkx120579y front bearingky120579x front bearingky120579y front bearing

kx120579x rear bearingkx120579y rear bearingky120579x rear bearingky120579y rear bearing

minus1

minus2

minus3

minus4

times107

(b) The stiffness of force to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mr

ad)

Rotary speed (rpm)

kmx120579x front bearingkmx120579y front bearingkmy120579x front bearingkmy120579y front bearing

kmx120579x rear bearingkmx120579y rear bearingkmy120579x rear bearingkmy120579y rear bearing

times106

minus1

minus2

(c) The stiffness of moment to angle versus eccentricity ratio

00 01 02 03 04 05

0

1

2

3

4

Stiff

ness

of m

omen

t to

angl

e (N

mm

)

Rotary speed (rpm)

kmxx front bearingkmxy front bearingkmyx front bearingkmyy front bearing

kmxx rear bearingkmxy rear bearingkmyx rear bearingkmyy rear bearing

times107

minus1

minus2

minus3

minus4

(d) The stiffness of moment to displacement versus eccentricityratio

Figure 12 The effect of eccentricity ratio on the stiffness coefficients

37 Effects of Eccentricity Ratio on the Dynamic Characteristicsof the Bearings Figures 12 and 13 show a variation ofstiffness and damping coefficients of each bearing versusthe eccentricity ratio It may be noticed that the dynamiccoefficients are almost constant as the eccentricity ratioincreases from 0 to 01 The direct stiffness of force todisplacement and moment to angle decreases gradually withincreased eccentricity ratio Generally the coupled stiffnessvaries significantly with the eccentricity ratio The coupledcoefficients of force to angle and moment to displacement

for the front bearing have the same magnitude as thosefor the rear bearing but in the opposite direction Thedamping coefficients are also relatively insensitive to theeccentricity ratio when it is not larger than 01 However inlarger eccentric condition the damping coefficients vary witheccentricity ratio and the larger the eccentricity ratio is thegreater the coefficients change In summary for the smalleccentric condition (le01) the influence of eccentricity ratioon the full dynamic coefficients for the hydrostatic water-lubricated journal bearing operating in aligned condition can

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

00 01 02 03 04 05

00

04

08

12

16

20

Eccentricity ratio

bxx front bearingbxy front bearingbyx front bearingbyy front bearing

bxx rear bearingbxy rear bearingbyx rear bearingbyy rear bearing

minus04

minus08

times105

Dam

ping

of f

orce

to d

ispla

cem

ent (

Ns

m)

(a) The damping of force to displacement versus eccentricity ratio

00 01 02 03 04 05

00

05

10

15

20

Eccentricity ratio

bx120579x front bearingbx120579y front bearingby120579x front bearingby120579y front bearing

bx120579x rear bearingbx120579y rear bearingby120579x rear bearingby120579y rear bearing

minus05

minus10

minus15

minus20

times104

Dam

ping

of f

orce

to an

gle (

Ns

rad)

(b) The damping of force to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Eccentricity ratio

bmx120579x front bearingbmx120579y front bearingbmy120579x front bearingbmy120579y front bearing

bmx120579x rear bearingbmx120579y rear bearingbmy120579x rear bearingbmy120579y rear bearing

times103

Dam

ping

of m

omen

t to

angl

e (N

ms

rad)

minus04

minus08

(c) The damping of moment to angle versus eccentricity ratio

00 01 02 03 04 05

00

04

08

12

16

20

24

Dam

ping

of m

omen

t to

angl

e (N

ms

m)

Eccentricity ratio

bmxx front bearingbmxy front bearingbmyx front bearingbmyy front bearing

bmxx rear bearingbmxy rear bearingbmyx rear bearingbmyy rear bearing

times104

minus04

minus08

minus12

minus16

minus20

(d) The damping of moment to displacement versus eccentricityratio

Figure 13 The effect of eccentricity ratio on the damping coefficients

be ignored however for a larger eccentric condition theinfluence should be fully discussed

4 Conclusion

This paper investigated the complete dynamic coefficientsfor two hydrostatic water-lubricated journal bearings usedto support a rigid rotor considering the translational andtilting motion The bulk turbulent flow model and FDMmethod is used to numerically predict the performance of the

bearingsThe results show that the proposedwater-lubricatedhydrostatic bearing has the potential to fulfill the target oflower power loss temperature rise and larger load capacityat high speed On the basis of the results presented thefollowing conclusions can be drawn

(1) For a small tilting ratio (lt04) the influence of tiltingratio on the static and dynamic characteristics of awater-lubricated hydrostatic journal bearing is rela-tively small however when the tilting ratio continues

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

to increase the power loss and temperature riseincrease gradually while the quality decreases andthe effect of tilting ratio on the dynamic coefficientsshould be taken into consideration

(2) The quality of the bearings is relatively insensitiveto the rotary speed however the power loss andtemperature rise increase sharply with the rotaryspeed in an aligned condition The direct stiffnesscoefficients vary significantly with the rotary speeddue to the hydrodynamic effect while the dampingcoefficients are almost constant

(3) For a relatively smaller eccentric condition (le01) thestatic and dynamic characteristics of the hydrostaticwater-lubricated journal bearings vary slightly witheccentricity ratio However for a larger eccentriccondition the dynamic characteristics increase ordecrease significantly with the eccentricity ratio

Nomenclature

119888 Design film thickness1198890 Orifice diameter

ℎ Film thickness1198971 Distance between the mass center and

front journal bearing center1198972 Distance between the mass center and rear

journal bearing center1198971198981 Distance between the mass center and theleft edge of front journal bearing

1198971198982 Distance between the mass center and theright edge of rear journal bearing

119903 Radius of the bearing119863 Diameter of a journal bearing119865 The bearing force119871 The length of a journal bearing119875119904 Supply pressure

119882 The external loadΩ Rotary speed120572 Flow coefficient120588 Density120583 Viscosity120582 Orifice design coefficient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Liu B Lin and X F Zhang ldquoNumerical design method forwater-lubricated hybrid sliding bearingsrdquo International Journalof Precision Engineering andManufacturing vol 9 no 1 pp 47ndash50 2008

[2] X Y Yuan G Y Zhang B Li and X Miao ldquoTheoreticaland experimental results of water-lubricated high-speed short-capillary-compensated hybrid journal bearingsrdquo in Proceedingsof IJTC STLEASME International Joint Tribology Conferencepp 391ndash398 San Antonio Tex USA October 2006

[3] S Yoshimoto T Kume and T Shitara ldquoAxial load capacityof water-lubricated hydrostatic conical bearings with spiralgrooves for high speed spindlesrdquoTribology International vol 31no 6 pp 331ndash338 1998

[4] G Y Gao ZW Yin D Jiang et al ldquoNumerical analysis of plainjournal bearing under hydrodynamic lubrication by waterrdquoTribology International vol 75 pp 31ndash38 2014

[5] S Nakano T Kishibe T Inoue and H Shiraiwa ldquoAn advancedmicroturbine system with water-lubricated bearingsrdquo Interna-tional Journal of Rotating Machinery vol 2009 Article ID718107 12 pages 2009

[6] H H Feng ldquoDynamic characteristics of a rigid spindle sup-ported by water-lubricated bearingsrdquo Applied Mechanics andMaterials vol 401ndash403 pp 121ndash124 2013

[7] J Corbett R J Almond D J Stephenson and Y B P KwanldquoPorous ceramic water hydrostatic bearings for improved foraccuracy performancerdquo CIRP Annals Manufacturing Technol-ogy vol 47 no 1 pp 467ndash470 1998

[8] O Ebrat Z P Mourelatos N Vlahopoulos and KVaidyanathan ldquoCalculation of journal bearing dynamiccharacteristics including journal misalignment and bearingstructural deformationrdquo Tribology Transactions vol 47 no 1pp 94ndash102 2004

[9] S C Jain S C Sharma and T Nagaraju ldquoMisaligned journaleffects in liquid hydrostatic non-recessed journal bearingsrdquoWear vol 210 no 1-2 pp 67ndash75 1997

[10] S C Sharma V M Phalle and S C Jain ldquoCombined influenceof wear and misalignment of journal on the performanceanalysis of three-lobe three-pocket hybrid journal bearingcompensated with capillary restrictorrdquo Journal of Tribology vol134 no 1 Article ID 011703 2012

[11] GH Jang and JWYoon ldquoDynamic characteristics of a coupledjournal and thrust hydrodynamic bearing in a HDD spindlesystem due to its groove locationrdquo Microsystem Technologiesvol 8 no 4-5 pp 261ndash270 2002

[12] G H Jang and S H Lee ldquoDetermination of the dynamiccoefficients of the coupled journal and thrust bearings by theperturbation methodrdquo Tribology Letters vol 22 no 3 pp 239ndash246 2006

[13] HW Kim G H Jang and S H Lee ldquoComplete determinationof the dynamic coefficients of coupled journal and thrustbearings considering five degrees of freedom for a general rotor-bearing systemrdquo Microsystem Technologies vol 17 no 5ndash7 pp749ndash759 2011

[14] G G Hirs ldquoA bulk-flow theory for turbulence in lubricantfilmsrdquo Journal of Lubrication Technology vol 95 no 2 pp 137ndash146 1973

[15] J Frene M Arghir and V Constantinescu ldquoCombined thin-film and Navier-Stokes analysis in high Reynolds numberlubricationrdquo Tribology International vol 39 no 8 pp 734ndash7472006

[16] R Bassani E Ciulli B Piccigallo M Pirozzi and U StaffilanoldquoHydrostatic lubrication with cryogenic fluidsrdquo Tribology Inter-national vol 39 no 8 pp 827ndash832 2006

[17] L San Andres D Childs and Z Yang ldquoTurbulent-flow hydro-static bearings analysis and experimental resultsrdquo InternationalJournal of Mechanical Sciences vol 37 no 8 pp 815ndash829 1995

[18] M M Khonsari and E R Booser Applied Tribology Wiley-Interscience London UK 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of