design and test of a pareto optimal flat pad aerostatic bearing

8
Tribology International 41 (2008) 181–188 Design and test of a Pareto optimal flat pad aerostatic bearing Nikhil Bhat, Simon M Barrans School of Computing and Engineering, University of Huddersfield, Huddersfield HD1 3DH, UK Received 15 June 2006; received in revised form 1 February 2007; accepted 24 July 2007 Available online 27 September 2007 Abstract This paper discusses the design and assessment of a Pareto optimal aerostatic flat pad bearing. The design of such bearings is a multi- variable, multi-criteria problem. The criteria used were load capacity, flow rate and stiffness. The design technique does not rely on weighting the criteria at the outset but allows a final design to be selected from the Pareto set. The final design was assessed using an automated experimental rig to obtain load/deflection data, mass flow rates, and film pressure profiles for a range of gap heights. The experimental performance of the Pareto optimal bearing is shown to compare well with the theoretical prediction at lower gap heights. r 2007 Elsevier Ltd. All rights reserved. Keyword: Air bearing 1. Introduction The majority of problems in real life are essentially multi-criteria and in many cases the objectives will conflict with one another. Many classes of search strategies have been used for multi-objective optimization. The usual treatment for multi-objective problems in the tribology field is to convert a multi-objective problem into a single- objective problem or to give preference to one criterion over the other(s). For example, Stout et al. optimized the performance of journal and flat pad aerostatic bearings by maximizing either the load capacity or the stiffness but not both [1–3]. Similarly Tawfik and Stout used minimum power consumption as the basis for the optimization of hybrid journal bearings [4]. A similar approach in the form of weighted criteria approach has been used by Hashimoto and Matsumoto [5], Wang et al. [6] and Bogy and Zhu [7]. In these works the objective function was generated from the weighted sum of normalized criteria values. Although the weighting method is popular, it requires the definition of weights or utility functions. This will not always be simple and can have a dramatic effect on the solution obtained. Also, the application of weights results in solutions which are not necessary Pareto optimal solutions. Recently genetic algorithms (GAs) have been used in many tribological optimization problems because of their high probability of obtaining a global optimum [8–12]. One recent application of GAs is the optimization of porous bearings as demonstrated by Wang et al [13]. In this work the goal of multi-objective optimization was achieved by incorporating the concept of Pareto optimality into the selection of mating groups for the GAs. The Pareto optimum set is defined as a non-inferior set of solutions for which there is no way of improving any criterion without worsening at least one other criterion. The Pareto optimum does not result in a single solution, but rather a set of non-inferior or non-dominated solutions. The minima in the Pareto sense are going to be on the boundary of the feasible set in criteria space, or at the locus of the tangent points of the objective functions. The marrying of the Pareto optimal concept with GAs, as demonstrated by Wang, better maintains the diversity of the population than the traditional Roulette wheel selec- tion method and therefore helps prevent premature convergence. However as with any GA, the final outcome depends heavily on the initial population. To overcome these limitations, Pareto optimization method using a uniformly distributed search sequence (UDS) has been suggested. ARTICLE IN PRESS www.elsevier.com/locate/triboint 0301-679X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2007.07.011 Corresponding author. Tel.: +44 1484 472407; +44 1484 472252. E-mail address: [email protected] (S.M. Barrans).

Upload: nikhil-bhat

Post on 02-Jul-2016

214 views

Category:

Documents


1 download

TRANSCRIPT

ARTICLE IN PRESS

0301-679X/$ - s

doi:10.1016/j.tr

�CorrespondE-mail addr

Tribology International 41 (2008) 181–188

www.elsevier.com/locate/triboint

Design and test of a Pareto optimal flat pad aerostatic bearing

Nikhil Bhat, Simon M Barrans�

School of Computing and Engineering, University of Huddersfield, Huddersfield HD1 3DH, UK

Received 15 June 2006; received in revised form 1 February 2007; accepted 24 July 2007

Available online 27 September 2007

Abstract

This paper discusses the design and assessment of a Pareto optimal aerostatic flat pad bearing. The design of such bearings is a multi-

variable, multi-criteria problem. The criteria used were load capacity, flow rate and stiffness. The design technique does not rely on

weighting the criteria at the outset but allows a final design to be selected from the Pareto set. The final design was assessed using an

automated experimental rig to obtain load/deflection data, mass flow rates, and film pressure profiles for a range of gap heights. The

experimental performance of the Pareto optimal bearing is shown to compare well with the theoretical prediction at lower gap heights.

r 2007 Elsevier Ltd. All rights reserved.

Keyword: Air bearing

1. Introduction

The majority of problems in real life are essentiallymulti-criteria and in many cases the objectives will conflictwith one another. Many classes of search strategies havebeen used for multi-objective optimization. The usualtreatment for multi-objective problems in the tribologyfield is to convert a multi-objective problem into a single-objective problem or to give preference to one criterionover the other(s). For example, Stout et al. optimized theperformance of journal and flat pad aerostatic bearings bymaximizing either the load capacity or the stiffness but notboth [1–3]. Similarly Tawfik and Stout used minimumpower consumption as the basis for the optimization ofhybrid journal bearings [4]. A similar approach in the formof weighted criteria approach has been used by Hashimotoand Matsumoto [5], Wang et al. [6] and Bogy and Zhu [7].In these works the objective function was generated fromthe weighted sum of normalized criteria values.

Although the weighting method is popular, it requiresthe definition of weights or utility functions. This will notalways be simple and can have a dramatic effect on thesolution obtained. Also, the application of weights results

ee front matter r 2007 Elsevier Ltd. All rights reserved.

iboint.2007.07.011

ing author. Tel.: +441484 472407; +44 1484 472252.

ess: [email protected] (S.M. Barrans).

in solutions which are not necessary Pareto optimalsolutions.Recently genetic algorithms (GAs) have been used in

many tribological optimization problems because of theirhigh probability of obtaining a global optimum [8–12]. Onerecent application of GAs is the optimization of porousbearings as demonstrated by Wang et al [13]. In this workthe goal of multi-objective optimization was achieved byincorporating the concept of Pareto optimality into theselection of mating groups for the GAs. The Paretooptimum set is defined as a non-inferior set of solutionsfor which there is no way of improving any criterionwithout worsening at least one other criterion. The Paretooptimum does not result in a single solution, but rather aset of non-inferior or non-dominated solutions. Theminima in the Pareto sense are going to be on the boundaryof the feasible set in criteria space, or at the locus of thetangent points of the objective functions.The marrying of the Pareto optimal concept with GAs,

as demonstrated by Wang, better maintains the diversity ofthe population than the traditional Roulette wheel selec-tion method and therefore helps prevent prematureconvergence. However as with any GA, the final outcomedepends heavily on the initial population.To overcome these limitations, Pareto optimization

method using a uniformly distributed search sequence(UDS) has been suggested.

ARTICLE IN PRESS

Table 1

Range of design variables

S.R No Input variable Range

1 Aspect ratio (L/B) 1–2

2 Breadth of bearing (B) 60–75mm

3 b/B ratio 0.1–0.25

4 Supply pressure value 4–6 bar

5 Number of orifices along length 3–10

6 Spacing ratio for orifices along length(S1/S2) 1–3

7 Diameter of orifice 1.194mm

8 Length of orifice feed pipe 400–1000mm

N. Bhat, S.M. Barrans / Tribology International 41 (2008) 181–188182

2. Multi-criteria optimization code

The multi-criteria optimization code consists of an finite-element analysis (FEA) code for flat pad bearingsdeveloped by Bhat and Barrans [14,15] combined withthe Pareto optimization code. For easier comparison ofdifferent trial points, the objective function values (in thiscase inverse load, inverse stiffness and flow) are minimized.The combined code works through three stages asdescribed below.

2.1. Stage 1

The input parameters for the program are the number ofdesign variables, n, number of trial points, number ofperformance criteria, and the number of functionalconstraints. Upper and lower bounds or constraints areset on the design variables.

A UDS is then used to generate a uniformly distributedset of trial points over the design space. The design variablespace has by definition, as many dimensions as there aredesign variables. This will frequently extend beyond threedimensions and it has been shown that in such cases, even asimple grid of points is not uniformly distributed. Howeverthe UDS point generation algorithm proposed by Soboland Statnikov ensures that points are distributed uniformlythorough an n-dimensional design space irrespective of thenumber of points [16]. Within the program used here, theUDS is first generated within an n-dimensional unit cube.This sequence is then transformed into the n-dimensionaldesign variable space to generate a set of trial points.

The design variable values represented by each trialpoint are then used to define input data for the FEAprogram.

2.2. Stage 2

The performance of the designs defined by trial pointscreated in step 1 is then determined by the FEA program.After evaluating the performance for every trial point,control is passed back on to the optimization program.Here trial point performance is checked against thefunctional and criteria constraints.

Performance criteria constraints ensure that the perfor-mance of a bearing does not drop below some level definedby the designer with respect to one or more criteria. Sincethe code is designed to minimize criteria, these constraintsare upper bounds.

Functional constraints can be included so as to excludethose designs that fall within the design variable andperformance criteria constraints but are not feasible formanufacturing (e.g. the orifice diameter is too small or theorifice length is too large).

Those trial points that satisfy the requirements of thedesign variable constraints, the performance criteria con-straints, and the functional constraints form the feasibleset.

2.3. Stage 3

The final step is to find the Pareto set among all thefeasible points. To achieve this, each criterion of eachfeasible point is compared with the related criterion of allother feasible points. A point is classified as Pareto optimalif at least one criterion is improved without worsening theother criteria when compared with all other feasible points.If it happens that two feasible points have equal values forall of the criteria and they both improve at least onecriterion when compared with other feasible points, thenthey are both Pareto points.

3. Pareto optimal flat pad bearing design

The multi-criteria optimization technique for flat padbearing was demonstrated by Bhat and Barrans using awide range of design variables for a Rank L bearing [15].For the purpose of this paper the range of design variableswas restricted to values that could be tested using theexperimental rig developed, as shown in Table 1. Thedimensional parameters are defined in Fig. 1.For journal aerostatic bearings, the criteria chosen for

optimization were stiffness and flow at zero eccentricity(since stiffness is maximum and flow is minimum at zeroeccentricity) and load at 0.5 eccentricity (since after 0.5eccentricity the bearing can exhibit negative stiffness).However unlike journal aerostatic bearings, flat padbearings do not have a natural position and henceeccentricity cannot be defined. Indeed, they will operateover a range of clearances depending on the load theysupport. It was therefore decided to optimize this bearingtype by investigating performance over a range of gapheights as demonstrated in previous work by the authors[15]. The only difficulty in applying the optimization codeto this problem was in mapping the relevant trial pointcoordinates onto the number of orifices. This range isobviously a set of discrete values rather than a continuousrange. To overcome this problem, points were mappedfrom the unit cube to the nearest number of orifices. Thiswill have disturbed the uniformity of the series slightly.Fig. 2 shows the distribution of trial points within the

criteria space. The X-, Y- and Z-axes represent 1/load area,1/stiffness area, and flow area, respectively. The data has

ARTICLE IN PRESSN. Bhat, S.M. Barrans / Tribology International 41 (2008) 181–188 183

been partially normalized such that the maximum value ineach dimension is 1. Only the Pareto optimal points havebeen shown for the sake of clarity. In order to select a finaldesign for manufacture, the Pareto were ranked by equallyweighting the 3 normalized criteria (weighting the rawcriteria values would not produce sensible results sincethe criteria are non-commensurable). Unlike a normal

B

L

S2S1

b

Fig. 1. Rank L bearing.

P39

P239

P247

P255

X

Y

Z

P39

P239

P247

P255

XY

Z

Fig. 2. Distribution of poin

weighted criteria method which would select a design fromthe entire feasible set, only points from within the Paretoset were considered. Hence, the final design will be Paretooptimal. This ranking process showed that Point 255 givesthe best compromise with respect to the 3 criteria. PointP239 is the point with lowest integrated flow over the gapvariation. P39 gives the maximum stiffness and P247 givesthe maximum load capacity. Table 2 shows the designvariable values for Point 255.

P39

P239

P247P255

X

Y

Z

P39

P239

P247P255

X

Y

Z

ts within criteria space.

Table 2

Values of design variables for point no. 255

S.R No Input Variable Value

1 L/B ratio 2.0

2 Breadth of bearing (B) 72mm

3 b/B 0.2

4 Supply pressure value 5.9 bar

5 Number of orifices along length 5

6 Spacing ratio for orifices along length 1.71

7 Diameter of orifice 1.194mm

8 Length of orifice feed pipe 875.8mm

ARTICLE IN PRESS

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20

Gap Height (microns)

Load C

apacity (

N)

Pareto_Point (255) Max_Loadpoint (247)

Max_Stiffness (39) Min_Flow (239)

Single orifice bearing0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16 18 20

Gap Height (micron)

Stiffness (

N/m

icro

n)

Pareto_Point (255)Point_Loadmax (247)Point_Stiffmax (39)Point_Flowmin (239)Single orifice bearing

0.0E+0

2.0E-4

4.0E-4

6.0E-4

8.0E-4

1.0E-3

1.2E-3

1.4E-3

0 5 10 15 20

Gap Height (micron)

Flo

w r

ate

(K

g/s

)

PARETO_Point (255)Point_Loadmax (247)Point_Stiffmax (39)Point_Flowmin (239)Single Orifice Bearing

Fig. 3. Comparison of performance characteristics of Pareto optimal bearings.

N. Bhat, S.M. Barrans / Tribology International 41 (2008) 181–188184

Fig. 3 shows the comparison of load, stiffness, and flowfor point P255 with other Pareto optimal points P39, P239,and P247 over a range of gap heights. In addition theirperformance was also compared with a single centralorifice bearing previously analyzed by Bhat [17]. Apartfrom the number of orifices and their position, the singleorifice bearing had comparable dimensions to the Paretooptimal bearings. Point P255 can be seen to give goodperformance in terms of load capacity and stiffness. Thelower flow rates shown for points P39 and P239 illustratesthe fact a Pareto set gives the designer a range of potentialdesigns with different merits. The much lower flow rates forthe single orifice bearing is to be expected since a minimumof six orifices were allowed during the optimizationprocedure. Hence, bearing designs with fewer orifices andpotentially lower flow rates were not present when thePareto set was being selected.

4. Experimental verification

In order to validate any theoretical results a number ofmeasurements need to be taken for aerostatic air bearings.These include the pressure both prior to the restrictor andwithin the bearing film, the film thickness, bearing loadcapacity, and air flow rate.

A technique commonly used to measure a flat padbearing pressure profile is to mount a pressure sensor on aplatform which is fixed to a linear stage as demonstrated byFan et al. [18]. However the results were presented onlyalong a line across the bearing surface (for example thecenterline of the bearing) rather than over the entiresurface. An earlier system developed by Kassab et al. [19]used a platform with a row of pressure tappings, each oneconnected to a Piezometer tube.In the test apparatus used by Fan et al., Chen et al. [20],

and Holster et al. [21] the load was applied to the flat padbearing through a loading shaft which was basically ajournal aerostatic bearing with provision to apply a deadload through the top end of bearing. With this systemthe application of load was easy, however a preciseadjustment in gap height was not possible since the gapheight was directly influenced by the amount of loadapplied. In all the cases the load was measured with thehelp of load cells. LVDTs were used to measure the gapheight in the work of Kassab et al. [19] and Holster et al.Non-contact displacement transducers (NCDTs) as usedby Chen et al. are preferable when a moving stage is placedunder the bearing since they offer a more reliablemeasurement of distance between surfaces with tangentialrelative motion [20].

ARTICLE IN PRESSN. Bhat, S.M. Barrans / Tribology International 41 (2008) 181–188 185

It is also important that flat pad bearing testingequipment incorporates a robust data acquisition system.This system needs to be synchronized with the movementof the X–Y stage so that the various sensors can map thedata at different points over the bearing surface.

5. Design of hardware

The test rig arrangement is shown schematically inFig. 4. It consists of an outer frame (1) on which the rest ofthe assembly is mounted. A base plate (2) is fixed onto theframe using screws (11). The X–Y stage (3) is mounted ontothe base plate. The adaptor brackets (4) on which thereaction plate (5) is mounted are screwed on to the X–Y

stage.A thin air film was formed between the bearing (6) and

the reaction plate (5) due to air flowing through thebearing. The ball (7), pillar(8), and the cover plate (10)acted as a thrust bearing which enabled the air bearing toalign itself parallel to the thrust plate due to the pressure inthe clearance (12). The other end of the pillar was fitted tothe hydraulic slave cylinder which could be moved in andout of the frame using the master cylinder for very fineadjustments or the hand pump for course adjustment. Thepillar and hydraulic ram were supported by a cross-beam (15).

The pressure across one-quarter of the bearing wasmapped with the help of the pressure transducer (13) whichwas screwed into reaction plate (5). The combined move-ment of the X and the Y stages of the X–Y stage scannedthe air gap formed between the bearing and the reactionplate.

Multiple NCDTs (9) mounted along the side of thebearing were used to measure the air gap between the

11

9LINE TO

HAND PUMP 13

7

LINE TO

MASTER CYLINDER

1

15

14

Fig. 4. Test rig

bearing and the reaction plate. The thrust load exerted bythe air bearing was measured using a pair of load cells (14)supporting the cross-beam.The test rig instrumentation was based around the

InstruNet general purpose data acquisition system [22]with data being recorded onto an Excel spread-sheet. TheX–Y stage was driven by a UNIDEX 500 programmablecontroller [23]. The digital I/O lines on the UNIDEXcontroller and the InstruNet DAQ system were used tosynchronize these devices. This allowed the mapping of thepressure profile under a bearing to be automated.

6. Comparison of theoretical results with experimental

results

The Pareto optimum bearing to be tested is as shown inFig. 5. Air was supplied to the bearing through capillarytubes which were glued to the bearing plate as shownin Fig. 5. All the experiments were conducted in a cleanroom facility so that the operating variables such assupply pressure, temperature, and air quality remainedconsistent.The Pareto optimal bearing was tested at different gap

heights within its operational range and pressure profilesover a quarter of the bearing plotted for each height. Thesepressure profiles at 5 and 15mm are shown in Figs. 6 and 7,respectively, along with the corresponding theoreticalresults. In both cases it can be seen that the experimentaland the theoretical pressure profiles are in good agreementalthough the theory seems to underestimate the initialpressure drop from the orifice. This is due to the fact thatthe pressure measured at the orifice is the total pressurerather than the static pressure. The low measured pressurefor the central orifice (particularly at the higher flying

1

8

2

12

4

3

5

HYDRAULIC CYLINDER

(SLAVE)0

6

assembly.

ARTICLE IN PRESS

10 holes 2,2 mm Diameter through the Plate

72

53,98

30,59

14,4

72

49

144

20 mm Dia Ball

2 holes M5 10 deep

20

90¡

Capillary tube flushed with

the bottom end of bearing surface

Bearing Section

Glue Cone

Capillary tube

11

Fig. 5. Layout of Pareto optimal bearing.

0

14.4

28.8

43.2

57.6

72

0

10.8

21.6

32.4

1

2

3

4

5

6

Pre

ssure

(B

ar)

Distance along Length (mm)

Distance alongBreadth (mm)

THEORETICAL PRESSURE PROFILE

0

14.4

28.8

43.2

57.6

72

0

14.4

28.8

1

2

3

4

5

6

Pre

ssure

(B

ar)

Distance along Length (mm)

Distance alongBreadth (mm)

EXPERIMENTAL PRESSURE PROFILE

Fig. 6. Comparison of theoretical and experimental pressure profiles (gap height ¼ 5mm).

N. Bhat, S.M. Barrans / Tribology International 41 (2008) 181–188186

height) is simply because the pressure transducer was neverdirectly under this orifice.

A comparison of the experimental and theoretical loadcapacities and flow rates for the Pareto optimal bearing isgiven in Table 3. The discrepancy between the experimental

and the theoretical results, particularly at larger gapheights is due to the inability of the FEA code to simulatethe pressure recovery region of flow. This pressure recoveryis less significant at lower gap heights since the flow ispredominantly viscous.

ARTICLE IN PRESS

0

14.4

28.8

43.2

57.6

72

0

10.8

21.6

32.4

1

2

3

4

5

6

Pre

ssure

(B

ar)

Distance along Length (mm)

Distance alongBreadth (mm)

THEORETICAL PRESSURE PROFILE

0

14.4

28.8

43.2

57.6

72

0

10.8

21.6

32.4

1

2

3

4

5

6

Pre

ssure

(B

ar)

Distance along Length (mm)

Distance alongBreadth (mm)

EXPERIMENTAL PRESSURE PROFILE

Fig. 7. Comparison of theoretical and experimental pressure profile (gap height ¼ 15mm).

Table 3

Comparison between theoretical and experimental results

Gap height (mm) Load capacity (N) Flow rate (kg/s)

Theory Practical % Difference Theory Practical % Difference

5 2447 2424 0.90 1.51E�05 1.50E�05 0.26

10 2445 2348 3.9 1.21E�04 1.42E�04 1.73

15 2437 2266 7.06 4.06E�04 4.26E�04 4.92

0

2

4

6

8

10

12

14

16

0

3

61.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

Absolu

te P

ressure

(b

ar)

Distance along length, L (mm)

Dis

tance

alo

ng

bre

adth

, B

(m

m)

Fig. 8. Air gap pressure profile near feed pipe orifice(gap height ¼ 40 mm).

N. Bhat, S.M. Barrans / Tribology International 41 (2008) 181–188 187

The pressure recovery process is illustrated by Fig. 8which shows the experimental air pressure around anorifice, measured at a much higher resolution than used for

the quarter bearing maps. A large pressure trough isobserved as the air enters into the bearing gap. Thedevelopment of the pressure profile in the air gap isdependant on the entrance conditions characterized by theReynolds and Mach numbers.Bender et al. pointed out that irrespective of the value of

Reynolds number and Mach number at the entrance, theflow is always accelerating [24]. Thus an initial pressuretrough (also known as a vena contracta) is inevitable in thecase of diverging flow. It was shown in [18] that withincreased gap height, the pressure trough increases.

7. Conclusions

A Pareto optimal multi-orifice bearing was designed thatgave a good compromise of load capacity, stiffness, andflow. The example clearly demonstrated the multi-criterianature of air bearing design.The design methodology used ensures that the final

design falls within the Pareto set. The speed of thetechnique is almost entirely due to the speed of the finite-element code used and the accuracy of the final solution is

ARTICLE IN PRESSN. Bhat, S.M. Barrans / Tribology International 41 (2008) 181–188188

principally dependent on the quality of the finite-elementmodel and the number of trial points used.

This technique was demonstrated by experimentaltesting of a Pareto optimal bearing. This showed excellentagreement between theoretical and experimental results atlower gap heights.

Particularly at higher gap heights, the experimentalpressure profile had pressure troughs near the orifice of thebearing which were not observed in the theoretical pressureprofiles. The current FEA code requires development toallow these pressure troughs to be simulated and henceallow accurate modeling of the flow through air bearingswith larger gap heights employing capillary tube restrictors.

References

[1] Pink EJ, Stout KJ. Design procedures for orifice compensated gas

journal bearings based on experimental data. Tribol Int 1978;11:

63–74.

[2] Stout KJ, Sweeney F. Design of aerostatic flat pad bearings using

pocketed orifice restrictors. Tribol Int 1984;17(4):415–22.

[3] Stout KJ. Design of aerostatic flat pad bearings using annular orifice

restrictors. Tribol Int 1985;18(4):209–14.

[4] Stout KJ, Tawfik M. Optimization of slot entry hybrid gas bearings.

Tribol Int 1982;15:31–6.

[5] Hashimoto, Matsumoto K. Improvement of operating characteristics

of high speed hydrodynamic journal bearings by optimum design.

ASME J Tribol 2001:305–12.

[6] Wang N, Ho C-L, Cha K-C. Engineering optimum design of fluid

film lubricated bearings. Tribol Trans 2000;43:377–86.

[7] Hong Z, Bogy DB. Hard disc drive air bearing design: modified

DIRECT algorithm and its application to slider air bearing surface

optimization. Tribol Int 2004;37(2):193–201.

[8] Boedo S, Eshkabilov L. Optimal shape design of steadily loaded journal

bearings using genetic algorithms. Tribol Trans 2003;46:134–43.

[9] Kotera H, Shima S. Tribol trans: shape optimization to perform

prescribed air lubrication using GA. Tribol Trans 2000;43(4):837–41.

[10] Kotera H, Hirasawa T, Senga S, Shimam S. A study on the effect of

air on the dynamic motion of MEMS device and its shape

optimization. Tribol Trans 2000;43(4):842–6.

[11] Wang N, Chang Y-Z. A hybrid search algorithm for porous air

bearing optimisation. Tribol Trans 2002;45(4):471–7.

[12] Kato T, Soutome H. Friction material design for brake pads using

database. Tribol Trans 2001;44:137–41.

[13] Wang N, Chang Y-Z. Application of genetic algorithms to multi-

objective optimization of air bearings. Tribol Lett 2004;17(2):119–25.

[14] Bhat N, Barrans SM. Optimization of journal bearings with the aid of

finite element analysis. In: Proceedings of the NAFEMS world

congress, 2003. Glasgow: NAFEMS; 2003.

[15] Barrans SM, Bhat N, Optimisation of flat pad air bearings with the

aid of finite element analysis. In: Proceedings of the seventh

international lamdamap conference, 2005. Bedford: Euspen; 2005.

p. 392–401.

[16] Statnikov R, Matusov J. Multicriteria optimization and engineering.

New York: Chapman & Hall; 1995.

[17] Bhat N. Pareto optimal design of air bearings. PhD thesis, University

of Huddersfield, 2005. p. 205–7.

[18] Fan K-C, Ho C-C, Mon J-I. Development of multiple-microhole

aerostatic air bearing system. J MicromechMicroeng 2002;12:636–43.

[19] Kassab S, Noureldeen E, Shawky M. Effects of operating conditions

and supply hole diameter on the performance of rectangular

aerostatic bearings. Tribol Int 1997;30(7):533–45.

[20] Chen MF, Lin YT. Static and dynamic stability analysis of grooved

rectangular aerostatic thrust bearings by modified resistance method.

Tribol Int 2002;35:329–38.

[21] Holster P, Jacobs J. The measurement and finite element analysis of

the dynamic stiffness of non-uniform clearance gas bearings, thrust

bearings. Trans ASME 1991;113:768–76.

[22] http://www.instrunet.com/p/i100/specs.html.

[23] http://www.aerotech.com/products/controllers/u500.html.

[24] Al-Bender F, Van Brussel H. Symmetric radial laminar channel flow

with particular reference to aerostatic bearings. J Tribol 1992;114:

630–6.