research article design and parameter study of a self...

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2013, Article ID 638193, 10 pageshttp://dx.doi.org/10.1155/2013/638193

Research ArticleDesign and Parameter Study of a Self-CompensatingHydrostatic Rotary Bearing

Xiaobo Zuo, Shengyi Li, Ziqiang Yin, and Jianmin Wang

College of Mechatronics Engineering & Automation, National University of Defense Technology, Changsha 410073, China

Correspondence should be addressed to Shengyi Li; [email protected]

Received 7 April 2013; Accepted 20 September 2013

Academic Editor: Shigenao Maruyama

Copyright © 2013 Xiaobo Zuo et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The influence of design parameters on the static performance of a newly designed self-compensating hydrostatic rotary bearingwas investigated. The bearing was designed by incorporating the main attributes of angled-surface self-compensating bearing andopposed-pad self-compensating bearing. A governing model based on flow conservation was built to theoretically study the staticperformance, and the methodology was validated by experiments. It is pointed out that the influence factors on the bearing staticperformance are the designed resistance ratio of the restricting land to the bearing land, the inner resistance ratio of the land betweenpockets to that between the pocket and the drain groove, the initial clearance ratio of the restricting gap to the bearing gap, and thesemiconical angle. Their effects on the load carrying capacity and stiffness were investigated by simulation. Results show that theoptimum designed resistance ratio is 1; the initial clearance ratio should be small, and the inner resistance ratio should be large.

1. Introduction

Hydrostatic bearings are widely used in high precisionmechanisms and machines for the advantage of low friction,high stiffness, high accuracy, and long life. The pressure inthe hydrostatic bearings is generated by external pumps, andcompensation devices are necessary to regulate flow into thepockets. Garg et al. [1] presented a review on the designand development of hybrid/hydrostatic bearings, in whichresearches on the compensation devices are reviewed.Martin[2] summarized the operational principle of constant-flowand constant pressure restrictors. Malanoski and Loeb [3]theoretically investigated the compensation effects of capil-lary, orifice, and constant-flow restrictor on the static stiffnessof hydrostatic thrust bearing. Sharma et al. [4] studied theslot-entry hydrostatic/hybrid journal bearing using finiteelement method and revealed that asymmetric slot-entryjournal bearings provide an improved stability thresholdspeed margin compared with asymmetric hole-entry journalbearings compensated by capillary, orifice, and constant-flowvalve restrictors. Singh et al. [5] studied the performanceof membrane compensated multirecess hydrostatic/hybridjournal bearing and found that selection of recess shapealong with a suitable compensation device is needed to

get an improved performance of the bearing. Kumar et al.[6] investigated the restrictor design parameter of capillarycompensated journal bearings. Gao et al. [7, 8] analyzed thestatic and dynamic characteristics of hydrostatic guides withprogressive mengen flow controller, in which the flow ratedoes not decrease with the increase of the pocket pressure sothat the oil gap keeps constant and the dynamic stiffness isgreatly improved. The influence of the controller parameterson the performance was also discussed. Kang et al. [9, 10] uti-lize a flow equilibriummethod to study the effect of constantcompensation and single-action variable compensation onopen- and closed-type flat bearing. Hydrostatic bearings withthese types of restrictors arewidely used in industry; however,the external compensation devices add to the complexity ofdesign and are sensitive to manufacturing errors.

Self-compensating bearings offer another design scheme.It does not need any external devices, which are substitutedby some internal features, to achieve load carrying capacity.Hoffer [11] proposed an automatic pressure balancing bear-ing, using opposed gap as a means of regulating flow to thepockets on the opposite side of a bearing. In view of themain features, bearings of this type are called opposed-padself-compensating bearings in this paper. Slocum et al. [12–14] developed Hoffer’s ideas and designed self-compensating

2 International Journal of Rotating Machinery

hydrostatic linear motion bearing and journal bearing, usingfluid lines to connect the annulus supply hole and the pocketon the opposite side.The linear motion bearing was designedas modular and can be lubricated by water, which is veryfavorable in machine tool applications. Wasson and Slocum[15] designed another self-compensating hydrostatic journalbearing in which the compensation feature is connected tothe corresponding pocket via grooves on the surface of thebearing so that the external sleeve is nonessential. Kotilainenand Slocum [16] introduced a new manufacturing method ofthe bearing proposed by Wasson and Slocum [15], which isvery cost effective because the cast processing is introduced.Kane et al. [17, 18] proposed a hydrostatic bearing withangled-surface self-compensation. The bearing consists ofseveral round parts which can be easily manufactured andassembled and achieves high stiffness and accuracy. Butthe restricting gap is a whole circular cylinder region, inwhich the lubricate diffusion is inevitable, and the acuteedge between the restricting region and the bearing regionis prone to be damaged. Once it is broken, the relativetwo bearing pockets may be connected at the entrance.These disadvantages are harmful to the performance of thehydrostatic bearing.

For both the opposed-pad and the angled-surface self-compensating bearings, performance is not much affectedby the manufacturing errors. As the restriction is achievedby the gap which varies with the external load, the stiffnessof self-compensating bearings is higher than that of thebearings with fixed restrictors. Another advantage of self-compensating bearings is that they are far less sensitive to dirtbecause they have no small diameter passages. The angled-surface self-compensating design lends the bearing to lowprofile rotary tables, while the previous opposed-pad self-compensating designs do not. In this study the angled-surfaceand the opposed-pad self-compensations are unified to forma hydrostatic rotary bearing. Analysis is based on flow equi-librium method. Design parameters are analyzed, and theireffect on the load capacity and static stiffness is discussed.

2. Bearing Design

The key feature of the angled-surface self-compensatinghydrostatic bearing is a restricting gap region that makesan acute angle relative to a bearing gap region. And themain feature of the opposed-pad self-compensating bearingis the restricting pocket on the opposite side of the bearingpocket. The restoring force in radial direction is reinforcedfor self-compensating bearings because the restricting gapadversely varies with the corresponding bearing gap. Thedesign present in this paper incorporates the main attributesof angled-surface self-compensating bearings and opposed-pad self-compensating bearings. The innovative feature isthe restrictor units on the restricting surface which makesan acute angel relative to the bearing surface. Every bearingpocket has its respective restrictor unit, so the diffusioneffect, which is inevitable in previous angled-surface self-compensating bearing, is avoided. Meanwhile sharp acuteedge which is prone to be broken is not necessary.

Stator

Restrictorring

Circularchannel

Rotorhalf

Spacer

Bearinggap

Restrictinggap

Figure 1: Framework of the designed bearing.

A sectional view of a rotary bearing which incorporatesthis concept is shown in Figure 1. The hydrostatic bearingincludes a rotor assembly and a stator assembly. The rotorassembly is composed of two rotor halves and a spacer;the stator assembly is composed of a stator and a restrictorring. The sectional view of the stator assembly is shown inFigure 2. Framework of the restrictor unit is illustrated bya broken view. The restrictor unit is composed of a supplygroove, a restricting land, and a restrictor pocket. Everyrestrictor pocket is connected to a bearing pocket by drillingholes. Though the fabrication process must be a little morecomplicated, the improvement of the bearing performancemay be considerable. Figure 3 displays the entity pictures ofthe bearing and its parts.

Then, fluid flow path through the bearing is described indetail according to Figures 1 and 2. After entering the bearingthrough the supply hole on the stator, the pressurized fluidfills the circular channel formed by the restrictor ring and therotor assembly. Then, it flows into the restricting gap. There,it first fills the supply groove, and most fluid flows throughthe restricting land, where its pressure drops, and then intothe restrictor pocket; the rest flows through the leakage landinto the drain groove.The fluid in the restrictor pocket entersthe corresponding bearing pocket via the fluid passage andthen through the bearing gap into the drain groove, whereits pressure drops to atmospheric. The atmospheric fluid iscollected into the drain passage and finally flows out of thebearing through the drain hole. Depth of the pockets andthe grooves should be greatly larger than the clearance of therestricting gap and the bearing gap to ensure that the pressureloss across them is negligible.

International Journal of Rotating Machinery 3

Supplygroove

Restrictingland

Restrictorpocket

Collectorhole

Leakageland

Bearingpocket

Drainpassage Fluid

passage

Draingroove

Supplyhole

Drainhole

Restrictorring

Restrictorunit

Stator

Figure 2: Sectional view of the stator assembly.

3. Analysis

3.1. Mathematical Model. The governing equation of thelubricant flow in the hydrostatic bearing is the Reynoldsequation. Since the restricting surface is cylinderical andthe bearing surface is conical, both forms in Cartesiancoordinates and polar coordinates are needed which areshown as follows:

𝜕

𝜕𝑥(

ℎ3

12𝜂

𝜕𝑝

𝜕𝑥) +

𝜕

𝜕𝑧(

ℎ3

12𝜂

𝜕𝑝

𝜕𝑧) =

𝑈

2

𝜕ℎ

𝜕𝑧+

𝜕ℎ

𝜕𝑡,

1

𝑟

𝜕

𝜕𝑟(

𝑟ℎ3

12𝜂

𝜕𝑝

𝜕𝑟) +

1

𝑟2sin2𝛼𝜕

𝜕𝜑(

ℎ3

12𝜂

𝜕𝑝

𝜕𝜑) =

Ω

2

𝜕ℎ

𝜕𝜑+

𝜕ℎ

𝜕𝑡,

(1)

where𝑥,𝑦, 𝑟, and𝜑 are the coordinates in Cartesian and polarcoordinate systems, respectively, ℎ is the fluid film thickness,𝜂 is the fluid dynamic viscosity, 𝑝 is the fluid pressure, 𝑈 andΩ are the line and angular velocity, respectively, and 𝑡 is thetime.

Lumped parameter method is adopted to model thebearing. The bearing is divided into regions where the flowcan be approximated by fully developed laminar flowbetweentwo parallel plates. Using the analogy of electrical networks,the bearing area can be segmented into some discrete zones as

shown in Figure 4, where 𝑅𝑟is the restrictor resistance, 𝑅

𝑏is

the resistance of the bearing land between the pocket and thedrain groove, 𝑅

𝑙is the resistance of the bearing land between

adjacent pockets, and subscript 𝑖 is the pocket number.By using the pressure boundary conditions, the pressure

in these two coordinates is expressed, respectively, as

𝑝𝐶= (

∫𝑎

𝑥(1/ℎ3) 𝑑𝑥

∫𝑎

0(1/ℎ3) 𝑑𝑥

) (𝑝1− 𝑝2) + 𝑝2,

𝑝𝑝= (

∫𝑟

𝑟2

(1/ℎ3𝑟) 𝑑𝑟

∫𝑟1

𝑟2

(1/ℎ3𝑟) 𝑑𝑟) (𝑝1− 𝑝2) + 𝑝2,

(2)

where 𝑎, 𝑟1, and 𝑟

2are coordinates of the boundaries and 𝑝

1

and 𝑝2are the boundary pressure, as shown in Figure 5.

Equation (2) can be simplified by treating the gap heightℎ as a constant in every divided zone so that integration ofthe cube of ℎ is not necessary. Thus, the resistance of a singlezone is defined by the profile, expressed, respectively, as

𝑅𝐶=

12𝜂𝑎

𝑤ℎ3,

𝑅𝑝= −

12𝜂

𝜑𝑤ℎ3

ln 𝑟2

𝑟1

.

(3)

Fluid film thickness varies with rotor motion. As shownin Figure 6, the fluid film thickness can be calculated by

ℎ = ℎ0− 𝛿𝑟cos𝜙𝑖cos 𝜃 − 𝛿

𝑧sin 𝜃, (4)

where ℎ0is the initial gap clearance, 𝛿

𝑟is the radial displace-

ment, 𝛿𝑧is the axial displacement, 𝜙

𝑖is the position angle

from the direction of 𝛿𝑟, and 𝜃 is the semicone angle of the

bearing surface. For the restricting gap region, 𝛿𝑧= 0, 𝜃 = 0.

The nondimensional fluid film thickness is given as

ℎ =ℎ

ℎ0

, (5)

where ℎ0= ℎ𝑏0is the designed clearance of bearing gap.

So the nondimensional fluid film thickness in the bearinggap and the restricting gap is, respectively, written as

ℎ𝑏= 1 − 𝜀 cos𝜙

𝑖cos 𝜃 − 𝜁 sin 𝜃, (6a)

ℎ𝑟= ℎ𝑟0

+ 𝜀 cos𝜙𝑖, (6b)

where ℎ𝑟0

= ℎ𝑟0/ℎ0is the nondimensional initial restricting

clearance, 𝜀 = 𝛿𝑟/ℎ0is the nondimensional radial dis-

placement, that is, eccentricity ratio, and 𝜁 = 𝛿𝑧/ℎ0is the

nondimensional axial displacement.It is seen from the formula of flow resistance as in (3) that

the gap flow resistance changes with the fluid film thickness.When the rotor moves relative to the stator, the fluid filmthickness is as in (4), so the gap flow resistance can becalculated by

𝑅 =𝑅0ℎ3

0

(ℎ0− 𝛿𝑟cos𝜙𝑖cos 𝜃 − 𝛿

𝑧sin 𝜃)3, (7)


(a) (b)

Figure 3: Manufactured parts and the assembled bearing.

Bearing

Restrictorunit

Rri

Rbi

Pb(i−1) Pb(i+1)Pbi

Rl(i−1)iRli(i+1)

Figure 4: Sketch of the lumped parameter model.

where 𝑅0is the initial flow resistance of gap when no rotor

displacement occurs.The continuity condition is expressed as

𝑄𝑟𝑖

= 𝑄𝑏𝑖

+ 𝑄𝑙𝑖(𝑖+1)

− 𝑄𝑙(𝑖−1)𝑖

, (8)

where𝑄𝑟is the flow rate getting into the restrictor pocket,𝑄

𝑏

is the flow rate getting out of the bearing pocket, and 𝑄𝑙is

the flow rate getting through the land between two adjacentpockets.

As the lumped parametermodel is applied, the continuitycondition can be written as

𝑝𝑠− 𝑝𝑏𝑖

𝑅𝑟𝑖

=𝑝𝑏𝑖

𝑅𝑏𝑖

+𝑝𝑏𝑖

− 𝑝𝑏(𝑖−1)

𝑅𝑙(𝑖−1)𝑖

+𝑝𝑏𝑖

− 𝑝𝑏(𝑖+1)

𝑅𝑙𝑖(𝑖+1)

, (9)

where 𝑝𝑠is the supply pressure and 𝑝

𝑏is the pocket pressure.

Then,𝑝𝑏𝑖can be calculated by solving the equation set defined

by (9).

3.2. Load Capacity and Stiffness. The load capacity of a singlediscrete bearing pocket region is calculated by

𝐹𝑖= 𝑝𝑏𝑖𝐴𝑒= 𝐹𝑖𝑝𝑠𝐴𝑒, (10)

where 𝐴𝑒is the effective area of a discrete pocket region. The

nondimensional load capacity is

𝐹𝑖=

𝑝𝑏𝑖

𝑝𝑠

, (11)

and the nondimensional stiffness of the discrete region is

𝑗𝑖= −

𝜕𝐹𝑖

𝜕ℎ. (12)

The load capacity 𝑊 and the stiffness 𝑆 of the bearingequal the vector sum of 𝐹

𝑖and 𝑗𝑖of all the divided regions

in the corresponding direction, respectively.

3.3. Key Parameters. Next, the key parameters affecting thebearing static performance are discussed. By transposition,(9) can be rewritten as

𝑝𝑠= (1 +

𝑅𝑟𝑖

𝑅𝑏𝑖

+𝑅𝑟𝑖

𝑅𝑙𝑖(𝑖−1)

+𝑅𝑟𝑖

𝑅𝑙(𝑖+1)𝑖

)𝑝𝑏𝑖

−𝑅𝑟𝑖

𝑅𝑙𝑖(𝑖−1)

𝑝𝑏(𝑖−1)

−𝑅𝑟𝑖

𝑅𝑙(𝑖+1)𝑖

𝑝𝑏(𝑖+1)

.

(13)

It indicates that the pocket pressure is affected by theresistance ratios. They are analyzed as follows:

𝑅𝑟𝑖

𝑅𝑏𝑖

=𝑅𝑟0ℎ3

𝑟0(ℎ𝑏0

− 𝛿𝑟cos𝜙𝑖cos 𝜃 − 𝛿

𝑧sin 𝜃)3

𝑅𝑏0ℎ3𝑏0(ℎ𝑟0

− 𝛿𝑟cos𝜙𝑖)3

= 𝜆(1 − 𝜀 cos𝜙

𝑖cos 𝜃 − 𝜍 sin 𝜃)

3

(1 − 𝜀 cos𝜙𝑖/ℎ𝑟0)3

,


x

a

O p1 w y

p2

(a) Cartesian coordinate

O

r1

r2

𝜑

r

p2

p1

𝜑w

(b) Polar coordinate

Figure 5: Sketch of boundary conditions.

Oy

hr0

𝛿r

r

𝜙i

𝛿rO

z

𝛽

r

hb0

𝛿z

Figure 6: Schematic diagram of the fluid film thickness.

𝑅𝑟𝑖


=𝑅𝑟0ℎ3

𝑟0

𝑅𝑏0(ℎ𝑟0

− 𝛿𝑟cos𝜙𝑖cos 𝜃)3

𝑅𝑏0

𝑅𝑙0

𝑅𝑙0


=𝜆

𝜅

(1 − 𝜀 cos𝜙𝑖(𝑖+1)

cos 𝜃 − 𝜍 sin 𝜃)3

(1 − 𝜀 cos𝜙𝑖cos 𝜃/ℎ

𝑟0)3

.

(14)

𝜆 = 𝑅𝑟0/𝑅𝑏0

is the designed resistance ratio, which isa very important parameter of the bearing. 𝜅 = 𝑅

𝑙0/𝑅𝑏0

is the inner resistance ratio, which influences the lubricantdiffusion between adjacent pockets. The larger the 𝜅 is, theless the diffusion effect is.ℎ

𝑟0= ℎ𝑟0/ℎ𝑏0is the nondimensional

initial restricting clearance, that is, the initial clearance ratioof the restricting gap to the bearing gap. 𝜆 and 𝜅 are totallydefined by the geometry. It can be summarized from (13) and(14) that the key parameters influencing the pocket pressureare the designed resistance ratio 𝜆, the inner resistance ratio𝜅, the initial clearance ratio ℎ

𝑟0, and the semiconical angle 𝜃.

4. Results and Discussion

The influence of themost important design parameters on thebearing performance was simulated. Results indicate that thepocket number 𝑛 does not obviously affect the performance.It is because 𝑛 does not obviously change the total effective


n = 16, hr0 = 1, 𝜆 = 1, 𝜅 = 0.5

𝜃 = 30∘

𝜃 = 40∘

𝜃 = 50∘

𝜃 = 60∘

𝜃 = 70∘

Wa

0 0.1 0.2 0.3 0.4 0.5

𝜁

0

0.1

0.2

0.3

0.4

0.5

0.6

(a) Axial load versus axial displacement

n = 16, hr0 = 1, 𝜆 = 1, 𝜅 = 0.5

𝜃 = 30∘

𝜃 = 40∘

𝜃 = 50∘

𝜃 = 60∘

𝜃 = 70∘

Wr

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5𝜀

(b) Radial load versus radial displacement

n = 16, hr0 = 1, 𝜆 = 1, 𝜅 = 0.5

S a

𝜃 = 30∘

𝜃 = 40∘

𝜃 = 50∘

𝜃 = 60∘

𝜃 = 70∘

𝜁

0 0.1 0.2 0.3 0.4 0.50.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(c) Axial stiffness versus axial displacement

𝜃 = 30∘

𝜃 = 40∘

𝜃 = 50∘

𝜃 = 60∘

𝜃 = 70∘

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5𝜀

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1n = 16, hr0 = 1, 𝜆 = 1, 𝜅 = 0.5

S r

(d) Radial stiffness versus radial displacement

Figure 7: Load capacity and static stiffness of bearings with different 𝜃.

area of the bearing. However, a large number of pockets areuseful to create excellent averaging effects. For the designedbearing has a large ratio of diameter to length, 𝑛 = 16

is adopted in the designed research. The influences of theabove-mentioned parameters on the load carrying capacityand stiffness are discussed in this section.

4.1. Load Capacity and Stiffness at Different Semicone Angles 𝜃.Figure 7 shows the load capacity and the stiffness at variousdimensionless displacements in the axial and the radial

direction, respectively. It is noticed that, as the semiconeangle increases (𝜃 = 30∘, 40∘, 50∘, 60∘, and 70∘), there is asignificant reduction in radial load capacity and a consider-able increment in axial load capacity, and the stiffness variesin the same trend. The stiffness gets the peak value whenthe rotor is at the initial place; that is, 𝜁 = 0 and 𝜀 = 0.Designers can select an appropriate semicone angle accordingto the practice requirement. If both the axial and the radialload capacity are important, a semicone angle around 40∘ to50∘ is advisable. In the following sections, simulations wereperformed at 𝜃 = 50

∘.


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1S a

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2𝜆

(a) Axial initial stiffness

hr0 = 0.5

hr0 = 1.0

hr0 = 2.0

S r0

𝜅 = 0.5

𝜅 = 1.0

𝜅 = 2.0

1.2 1.4 1.6 1.8 2

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.8 10𝜆

(b) Radial initial stiffness

Figure 8: Stiffness versus designed resistance ratio.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.55

0.6

0.65

0.7

0.75

0.8

0.85

𝜆 = 2

𝜆 = 1.2𝜆 = 1.5

𝜆 = 1

𝜆 = 0.8𝜆 = 0.5

𝜁

n = 16, hr0 = 1, 𝜃 = 50∘, 𝜅 = 0.5

S a

(a) Axial stiffness versus axial displacement

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

𝜆 = 2

𝜆 = 1.2𝜆 = 1.5

𝜆 = 1

𝜆 = 0.8𝜆 = 0.5

𝜀

n = 16, 𝜅 = 0.5, hr0 = 1, 𝜃 = 50∘

S r

(b) Radial stiffness versus radial displacement

Figure 9: Static stiffness of bearings with different 𝜆.

4.2. Influence of Designed Resistance Ratio 𝜆. The designedresistance ratio 𝜆 is always regarded as the most importantfactor. Figure 8 depicts the variation of the initial stiffnesscoefficients 𝑆

𝑎0with𝜆.When the rotor is displaced in the axial

direction, the clearance ratio ℎ𝑟0and the inner resistance ratio

𝜅 do not affect the fluid film stiffness 𝑆𝑎, and 𝑆

𝑎0achieves the

maximum value when 𝜆 = 1. When the rotor is displaced inthe radial direction, the fluid film stiffness 𝑆

𝑟is affected by all

the design parameters. As shown in Figure 8(b), 𝑆𝑟0achieves

the optimal value when 𝜆 is in the interval of 0.8∼1.0. Thismeans that there exists an optimal designed value for𝜆, whichis around 1. A little value of ℎ

𝑟0and a large value of 𝜅 are

favorable to the stiffness.Figure 9 shows the static stiffness versus dimensionless

displacement curves under different values of 𝜆 and fixedvalue of ℎ

𝑟0and 𝜅. The more the deviation from the optimal

value of 𝜆 is, the worse the initial stiffness is. For the bearingwith 𝜆 = 0.5 and 𝜆 = 2, the percentage of reduction of the


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5𝜀

n = 16, 𝜆 = 1, 𝜅 = 0.5, 𝜃 = 50∘

S r

hr0 = 0.5hr0 = 0.8hr0 = 1

hr0 = 1.2hr0 = 1.5

hr0 = 2

Figure 10: Radial static stiffness of bearings with different ℎ𝑟0.

initial stiffness in the axial direction is found to be 11.11% andthat in the radial direction is 8.83% and 13.07%, respectively.However, for the bearing with 𝜆 = 1, the stiffness decreasesrapidly. Its radial stiffness is lower than the others when 𝜀 islarger than 0.4. In most cases, the peak value of the stiffnessis at the initial location, that is, when the rotor displacementis zero. But when 𝜆 = 2, the peak axial stiffness is at thenondimensional axial displacement 𝜍 = 0.4. This unusualevolution can be explained by differentiating the stiffnesswithrespect to the axial displacement.

4.3. Influence of Initial Clearance Ratio ℎ𝑟0. Figure 10 depicts

the variation of radial stiffness with eccentricity in thecondition that the designed resistance ratio 𝜆 = 1 and theinner resistance ratio 𝜅 = 0.5. Compared with a bearinghaving ℎ

𝑟0= 1, the bearing with ℎ

𝑟0= 0.5 can achieve an

initial radial stiffness of 60.0% larger. However, the stiffnessdecreases much more rapidly when ℎ

𝑟0= 0.5. Furthermore,

a very little value of ℎ𝑟0is not advisable in practice because

the designed resistance ratio 𝜆 has a three-power relationshipwith it.

4.4. Influence of Inner Resistance Ratio 𝜅. Figure 11 depicts theinfluence of 𝜅 on the radial stiffness. To avoid the diffusioneffect, drain grooves are sometimes fabricated between theadjacent bearing pockets to make a multipad bearing, inwhich 𝜅 is infinity. The curves indicate that large value of 𝜅is favorable to the bearing stiffness, which is accordant withthe knowledge that the lubricant flow between the pocketsdecreases the bearing stiffness [19]. As 𝜅 = 𝑅

𝑙0/𝑅𝑏0, a larger

value of 𝜅means that the flow resistance between the pocketsis larger; thus, the flow across the inner land is less, which isfavorable to the bearings stiffness. Compared with that of abearing with 𝜅 = 1, the initial radial stiffness of a multipad

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5𝜀

0.3

0.4

0.5

0.6

0.7

0.8

S r

𝜅 = 0.5

𝜅 = 0.8𝜅 = 1

𝜅 = 1.5𝜅 = 2𝜅 = ∞

n = 16, 𝜆 = 1, hr0, and 𝜃 = 50∘

Figure 11: Radial static stiffness of bearings with different 𝜅.

Table 1: Main parameters of the manufactured bearing.

Parameter/Unit ValueAverage diameter/mm 236Length diameter ratio 0.14Semicone angle/∘ 50Pocket number 16Bearing clearance/mm 0.04Design resistance ratio 1.0Initial clearance ratio 1.0Inner resistance ratio 12.7

bearing is 7.96% larger. It means that there exists a limit of𝑆𝑟, so a very large value of 𝜅 is not necessary. In design

practice, a big value of 𝜅 means a large width/length ratio ofthe land region between pockets, which reduces the effectiveload carrying area of the bearing pockets.

5. Stiffness Test

Experimental test of the stiffness has been performed on aprototype bearing, the main geometric parameters of whichare listed inTable 1. Figure 12 is the photo of the test setup.Thestator is fixed on a 100mm thick steel base. Load is applied onthe rotor by a rod, which is pushed by a gas cylinder. In theaxial test, a block is used to avoid loading at a single point;in the radial test, a plate is fixed on the top surface of therotor to aid loading. The force is determined by using an S-shaped load cell with a maximum load capacity of 8,000Nand a resolution of 10N. The displacement of the rotor ismeasured using two inductance sensors with a resolution of0.1 𝜇m. The experiment and simulation results under supplypressure of 1.0MPa are plotted in Figure 13. The predictedvalue of load carrying capacity is larger than that of the tested


(a) (b)

Figure 12: Photos of the test setup.

0 2 4 6 8 10 12 14 16 18 200

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Axial simulationAxial experiment

Radial simulationRadial experiment

Displacement (𝜇m)

Load

(N)

Figure 13: Load versus displacement plot of the prototype.

one. The computed initial stiffness is 459N/𝜇m in the radialdirection and 515N/𝜇m in the axial direction. The testedvalue of initial stiffness is degraded by a percentage of 18%and 16% in the radial and the axial direction, respectively.Thedifference is caused by the complex manufacturing errors,which are not accounted for in the simulation model.

6. Conclusions

A self-compensating hydrostatic rotary bearing is designed,and the mathematical model is built based on flow equilib-rium. The methodology is validated by experiment, and theinfluence of the designed parameters on the static perfor-mance is discussed. On the basis of the numerically simulatedresults presented in this study, the following general conclu-sions may be drawn.

(1) The influencing factors on the bearing static perfor-mance are the designed resistance ratio 𝜆, the innerresistance ratio 𝜅, the initial clearance ratio ℎ

𝑏0, and

the semiconical angle 𝜃.(2) The optimum value of the designed resistance ratio

𝜆 is around 1, which can be yielded by adjustingthe geometry parameters. The bearing with the opti-mum 𝜆 has the maximum initial stiffness, but itsperformance gets worse when the nondimensionaldisplacement is large.

(3) A little value of the clearance rate ℎ𝑟0is beneficial to

the static performance in the radial direction. But it islimited in practice by the need to create a reasonableinitial resistance ratio 𝜆.

(4) The inner resistance ratio 𝜅 affects the radial perfor-mance. A large value of 𝜅 is suggested to reduce thediffusion effect between bearing pockets, but there isa limit for the improvement of the stiffness.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

The authors would like to acknowledge the National NatureScience Foundation of China (no. 50805143) and theNationalMajor Project of Science andTechnology (2010ZX04001-151).

References

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[3] S. B. Malanoski and A. M. Loeb, “The effect of method ofcompensation on hydrostatic bearing stiffness,” Journal of BasicEngineering, Transaction of ASME, vol. 83, pp. 179–187, 1961.

[4] S. C. Sharma, V. Kumar, S. C. Jain, R. Sinhasan, and M. Subra-manian, “Study of slot-entry hydrostatic/hybrid journal bearingusing the finite element method,” Tribology International, vol.32, no. 4, pp. 185–196, 1999.

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[7] D. Gao, D. Zheng, and Z. Zhang, “Theoretical analysis andnumerical simulation of the static and dynamic characteristicsof hydrostatic guides based on progressive mengen flow con-troller,” Chinese Journal of Mechanical Engineering, vol. 23, no.6, pp. 709–716, 2010.

[8] D. Gao, J. Zhao, Z. Zhang, and D. Zheng, “Research on theinfluence of PM controller parameters on the performance ofhydrostatic slide for NC machine tool,” Journal of MechanicalEngineering, vol. 47, no. 18, pp. 186–194, 2011 (Chinese).

[9] Y. Kang, J.-L. Lee, H.-C. Huang et al., “Design for staticstiffness of hydrostatic plain bearings: constant compensations,”Industrial Lubrication and Tribology, vol. 63, no. 3, pp. 178–191,2011.

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