Elastic considerations in the hydrostatic lubrication of capillary-compensated thrust bearings of different configurations

Wear, 111 (1986)

41

41- 62

ELASTIC CONSIDERATIONS IN THE HYDROSTATIC LUBRICATION OF CAPILLARY-COMPENSATED THRUST BEARINGS OF DIFFERENT CONFIGURATIONS R. SINHASAN, S. C. JAIN and S. C. SHARMA Department of Mechanical and Industrial 247667, Uttar Pradesh (India) (Received February 8,1984;

Engineering,

University

of Roorkee,

Roorkee

accepted January 2,1986)

Summary Using the finite element method and a suitable iteration scheme, the effects of interaction between the three parameters, the deformation coefficient cd, Poisson’s ratio v and the restrictor design parameter csz, on the performance characteristics of the capillary-compensated flexible thrust pad bearings have been studied. The results of bearing characteristics for singlepocket and multipocket rectangular thrust bearing confiirations are presented for various values of &, Y and es92.

1. Introduction The performance characteristics of compensated hydrostatic thrust bearings which are widely used in machine tools are generally evaluated by assuming the thrust pad material to be absolutely rigid [l, 21. However, in actual practice the thrust pad deforms because of pressure in the fluid film and the performance characteristics of the bearing are significantly affected. The effects of elastic deformation of the thrust pad on the performance ch~c~~stics of the uncompen~~d circular thrustpad bearing have already been studied by several investigators [3 - 51, but generally in practice the externally pressurized bearings have some kind of compensation (fixed or variable restrictor) in the lubricant supply line (Fig. 1) [6]. So the performance of capillary-compensated flexible circular thrust pad bearings has been studied by the present authors [ 71. The present paper is an extension of the previous work [7]. In the present paper, thrust bearing configurations which are commonly used in hydrostatically lubricated slideways (Fig. 2) [6] are studied. The static and dynamic performance characteristics of the bearing are studied in terms of load capacity, lubricant flow, fluid film stiffness and damping coefficients for various values of the deformation coefficient cd, Poisson’s ratio v and restrictor design parameter &, The study indicates that to obtain the optimum 0043-1643/36/$3.50

0 Elsevier Sequoia/Printed

in The Netherlands

42 THRUST

RIGID

LOAD

F.

BODY SURFACE RECESS

I

h f

ON/OFF

VALVE

LINE

LEXIBLE EARING

FILTER

PAD

-7 4

Fig. 1. Lubricant supply and thrust bearing system: electric motor; STR, strainer; FLT, line filter.

j~jLCAPILLARY

RESTRICTOR

PF, fixed-displacement

pump; M,

performance of a capillary-compensated hydrostatic thrust bearing a proper selection of the deformation coefficient cd, Poisson’s ratio v and restrictor design parameter cSszmust be made and proper consideration of their interactions is essential.

2. Analysis The study of the elastohydrostatic lubrication problem for compensated hydrostatic thrust bearings requires simultaneous solution of the equations governing the flow of the lubricant in the fluid film with the flow of lubricant through the restrictor as a constraint and the elastic deformation of the thrust pad. Using the finite element method, the global system equations are derived for the fluid flow and elastic deformations domains. 2.1. Fluid flow field The flow of incompressible lubricant through the fluid fihn of an externally pressurized thrust bearing, neglecting the hydrodynamic effect, is governed by the following nondimensional Reynolds equation [ 8) :

(1)

<% @‘ -

43

Y

Y

X

X LZ

'1

L-I-

(a)

(d)

(cl

Fig. 2. Hydrostatic thrust pad configurations: (a) configuration I, r2/rl = 2.0; (b) configuration II, square thrust pad, LJL2 = 1.0, l/L, ation III, rectangular thrust pad, two pockets, LJL2 = 4.0,1= 0.3L1, figuration IV, rectangular thrust pad, four pockets, L1/L2 = 2.666, O.O83L2,1= 0.3L1, (12= 0.208L2.

circular thrust pad, = 0.5; (c) configural = O.lLz;(d) conb = 0.5416L2, a1 =

If the elastic deformation of the bearing pad is also taken into account, the fluid film thickness !i is defined as h=ii,+Ali+ti

(2)

where ii, is the fluid film thickness for the rigid bearing, A/i is the perturbation on the film thickness and ii, (= w/h,) is the non-dimensional displacement of the fluid film-thrust pad interface in the 2 direction. The flow field is discretized using eightrnode isoparametric elements [9] (Fig. 3(b)). Using the orthogonality condition of Galerkin’s technique [9] and eqn. (l), the following equation is derived: (3) where -e G

ij =

--

+&*-

aNi

aNj

ap

ap

-

dcudfi

(i, j= 1, . . . . 8)

71

1 ; -*-1’

lb)

@ (cl

(d)

(e)

3

P’

5

Fig. 3. Bearing pad discretization: (a) flexible thrust pad; (b) eight-node two-dimensional isoparametric element; (c) 20-node three-dimensional isoparametric element; (d) pocket zone included; (e) pocket zone excluded.

-e

R 71 =-2

SS%&drudp AC? a7

(i = 1, . . . . 8)

(i= 1, . . . . 8) whereNfaretheshapefunctions(i=l, . ...8) [9]. Following the usual method of assembly [9] for all elements of the lubricant flow field (eqn. (3)) the global system equation is obtained as

El “fX”ft~.)“fX1={8}“fX1+{~7}“fX1

(4)

where nf is the total degrees of freedom in the discretized flow field. {ii} and (0) are the nodal pressures and nodal flows for the entire flow field. For the general dynamic case, the nodal pressures {ij} are expressed as {PI = G%JI+ {&I

(5)

45

where & are the nodal pressures when &/a~ = 0 and j& are the nodal pressures due to the squeeze velocity (a&/& # 0). For the fluid fihn stiffness computation, eqn. (4) is differentiated with respect to fi and the system equation for the nodal pressure derivatives ap,/aT; is obtained 171. 2.1.1. Restrictor flow The equation for the flow of lubricant through the capillary restrictor is expressed as [7,10, 111 s,

= G(1

(6)

-P,)

where

2.1.2. Boundary conditions The boundary conditions pertinent to the lubrication problem are as follows. (a) At the external boundary the nodal pressures are zero. (b) The nodal pressures for nodes on the pocket boundary are equal. (c) The flow of lubricant through the restrictor is equal to the bearing input flow at the pocket. (d) The net nodal flow at each internal node (excluding the pocket and external boundary nodes) is zero. 2.2. Elastic deformation domain The flexible thrust pad represents the elastic deformation domain (Fig. 3). The elastic domain is discretized using 20-node isoparametric hexahedral elements [7,9]. The deformation at any point in the domain is represented by its three components, u, u and w in the X, Y and 2 directions respectively. Using the virtual work principle [ 91 and the parameters x=zt,

Y= Bt,

.z=zt,

u = h,ii

u = h,b

w = h,ii,

[D] = E[b]

P=PSP

(7)

the system equation in the non-dimensional form for the complete discretized elastic deformation domain is derived as [ 71 [i1]{6}

= C,(R)

(3)

where the deformation coefficient is given by c d

=PS -E

th k

The deformation coefficient cd represents a single non-dimensional parameter to account for the flexibility of the thrust pad. The global nodal force

46

matrix {R) is evaluated on the fluid film-thrust pad interface and only the 2 components of the nodal force vector have non-zero values. The displacement components (El, Cl, ti,) for the node (number 1) lying on the interface between the flexible thrust pad and the rigid support are taken to be zero:

3. Performance characteristics Once the matched steady state solutions are established, the performance characteristics in terms of load-carrying capacity F,,, lubricant flow 8, fluid film stiffness s and damping coefficients c are obtained [ 71. The closed form expressions of the pocket pressure PC, resultant film reaction F, fluid film stiffness s and damping coefficients 6 for a dynamically loaded (a&/& f 0) capillary-compensated rigid circular thrust pad bearing have been derived and the final expressions are presented in the Appendix A.

4. Solution procedure The g-node two-dimensional and 20-node three-dimensional isoparametric elements [ 7,9] were used to discretize the two-dimensional lubricant flow field and three-dimensional elastic deformation domain (thrust pad) respectively (Fig. 3). The 2 component iir of the nodal displacements (for the nodes at the fluid film-thrust pad interface) modifies the nodal film thickness and this in turn redefines the fluid film profile. The global system equations (eqns. (4)) for the lubricant flow field are adjusted such that for each pocket there remains one nodal pressure and one nodal flow as unknowns in the adjusted system equations. The unknown nodal flow for each pocket is replaced by the corresponding restrictor flow expression (eqn. (6)), keeping in view the non-dimensional parameters. This reduces the adjusted global equations to a form which has only one nodal pressure for each pocket as unknown. The matched steady state solutions of the nodal pressure and nodal flows for the lubricant flow field and nodal displacements for the elastic deformation domain are obtained using an iterative technique [ 71. The iterations are continued until all the nodal pressures of the current iteration are within 0.1% of the corresponding nodal pressures of the previous iteration (Fig. 4).

PROGRAM

ST&m

0

INCV’l. ,T.NO, CX,fO*

hOT

COEFF.

ACWIEVEO

program; (b) convergence

CONVERGENCE ACHIEVE0 COHVEROENCE OF lTERA?lON OEFORMATtON

0, L”BA,CA,,ON PROI)LEtd STRUCTURAL PROULEY ELAsTO HYOAOS~ATIC PROBLEM

,NCVNO,

a-,, ,L-2,

Fig. 4. (a) Block diagram for computer

MASIER

test.

I

1

,r-nit

I_

48

5. Results

and discussion

On the basis of the analysis and solution procedure presented in this paper and in ref. 7 a computer program has been developed. Using this program, the performance characteristics for a capillary-compensated hydrostatic circular thrust bearing having a rigid thrust pad (cid = 0) were computed. These results are compared with the closed form solutions (Appendix A) in Table 1. The comparison between the two results is very good. The stress analysis portion of the program was checked separately.

TABLE 1 Comparison between the closed form and finite element solutions (circular pocket; ??d = 0.0; hr = 1; r2/rI = 2.0; LR = 1.0)

5 5 5 5 10 10 10 10

Performance characteristics

Closed form solution

Finite element solution

EOC

0.76796

0.76790

Fo

1.30524 0.90862 1.06965

1.30247 0.90847 1.06896

0.86875 1.47656 0.58136 0.86368

0.86871 1.47599 0.58133 0.86184

0.92977 1.58026 0.33293 0.73817

0.92974 1.57969 0.33295 0.73645

0.96360 1.63778 0.17882 0.66856

0.96369 1.63720 0.17882 0.66692

S c YOC

Fo s C?

20 20 20 20

YOC

40 40 40 40

“C

Fo S C Fo S E

To obtain a better practical feeling for the problem, the representative values of the bearing material properties, E (N mP2) and V, [12,13] and the corresponding computed values of the deformation coefficient cd are presented in Table 2. In this paper, the performance characteristics of four capillary-compensated thrust bearing configurations (Fig. 2) have been presented. The bearing pads are elastic and have been discretized using 20-node isoparametric elements (Fig. 3(c)). For the displacement analysis, two cases for each bearing pad have been considered as represented in Figs. 3(d) and 3(e). When the pocket zone is included (Fig. 3(d)), the complete bearing pad has been discretized using two layers of elements in the 2 direction but, when the pocket

49

TABLE 2 Capillary-compensated t,,/hr = 350)

flexible

thrust

pad

bearing

(ps = 20.6843 N mm-‘;

?h = 0.1;

Specimen

Material thrust pad

E (N me2)

Poisson’s ratio v

Deformation coefficient Cd

1 2 3 4 5

Rigid pad Steel Cast iron Bronze (80-10-10) Texotolite (2001)

00 2.00 x 10” 0.90 x 10” 0.76 x 10” 0.04826 x 10”

0.30 0.27 0.35 0.40

0.0 0.03619 0.08044 0.09526 1.5

TABLE 3 Comparison of results (configuration Specimen

Ed

JPODIS’

1 1

0.0 0.0

0 1

2 2

0.25 0.25

3 3

IV ; iTsz = 10.0; v = 0.3;&

a,

= 2.666; ;,, = 0.1)

130,

Fo

c

s

9.97250 9.97250

0.750687 0 750687

0.49972 0.49972

0.03958 0.03958

0.37376 0.37376

0 1

11.70018 11.77316

0.707496 0.705671

0.48274 0.48079

0.03258 0.03274

0.40516 0.40408

0.50 0.50

0 1

13.32604 13.44890

0.666849 0.663778

0.46320 0.45982

0.02806 0.02826

0.42226 0.419714

4 4

0.75 0.75

0 1

14.81230 14.96860

0.629693 0.625780

0.443240 0.438860

0.02490 0.02508

0.429344 0.425380

5 5

1.00 1.00

0 1

16.07608 16.32384

0.596196 0.591904

0.423940 0.419000

0.02256 0.02272

0.429780 0.424608

aJPODIS = 0, pocket zone excluded; JPODIS = 1, pocket zone included.

zone is excluded (Fig. 3(e)), only one layer of elements has been used in the 2 direction. For these two types of discretization the results have been compared in Figs. 5(a) and 5(b) and Table 3. The difference in the two sets of results is insignificant but the computer time required in the first case is approximately twice that required in the second case for one set of data on a Digital Equipment Corporation DEC 2050 computer. Therefore, wherever the results presented in the paper are not specified otherwise, they have been computed for the second case (the pocket zone excluded) to reduce the computational cost. Pressure distributions have been presented in Figs. 5 and 6 for various values of deformation coefficient cd, Poisson’s ratio v and restrictor design parameter cSz. With increase in cd, the pocket pressure j& decreases but the

Pressure

Distribution

along

AA

Fig. 5. (a) Pocket pressure (configuration II; & ) and with the pocket zone zone included (ratio on pocket pressures (configuration II; LR distribution and deformation (configuration III; along AA (11) and along CC (22) for cd = 0 (- - - -).

= 1.0; fh = 0.1; Y = 0.3) with the pocket excluded (- - -& (b) effect of Poisson’s = 1.0; Th = 0.1; Cp = lS.O);ic) pressure ER -- 4.0; ?h = 0.1; v = 0.3;Ca_= 20.00) ), Cd = 0.6 (- - -) and Cd = 1.00 (-

pressure on the land does not follow the same pattern (Figs. S(c) and 6). Surface compression of the bearing pad is a maximum on the pocket boundary and a minimum on the external boundary of the thrust pad (Fig. 5(c)) for all bearing configurations.

51

1.0

0.75

0.50

0.25

0

0.25

0.50

0.75

1.0

Radius

Radius

Fig. 6. Effect of deformation on pressure distribution along a diametral plane for a bydrostatic bearing circular pocket (configuration I; cd = 30; rz/rl = 2.0; th/r2 = 0.1; v = 0.3; = 1.0). i;, = l.o;ZR

04

I 0

I

I

0.25 Dsformotlon

1

0.75

0.50 coefficient

I

I.00

Fd

Fig. 7. Load capacity (configuration ) and with the pocket included (-

II; ZR = 1.0; fh = 0.1; zone excluded (- - -).

v =

0.3) with the pocket

zone

The resultant effect of the variations in pressures on the land and in the pocket is reflected in the results of the load capacity F, (Fig. 7). As Poisson’s ratio v increases, the pocket pressure increases (Fig. 5(b)) and, to obtain the maximum load capacity of a capillary-compensated hydrostatic thrust

52

I 0

i 0.25

t 0.50 Oeformotion

coefficient

Fig. 8. Bearing flow (configuration

t 0.75

1 100

&j

II; &

= 1.0; 8, = 0.1; csz = 15.0).

bearing for a selected geometry, a proper combination of the three parameters &, & and v should be selected. The results of bearing flow (a, = 0,/a) are presented in Figs. 8 - IQ, The bearingflow increasesas both CSzand cd increase. The results of the fluid film stiffness coefficient s presented in Figs. 11-13 indicate that, to obtain maximum stiffness s at a particular load capacity FO, a proper selection of cSsz,& and v is needed but, to achieve an overall maximum fiuid film stiffnesscoefficient 3, a proper design of bearing in which all the four parameterscSsz,&, Yand Fe are considered is essential. The damping coefficient c decreases with increase in cd and E)iS2 but increases with increase in v (Figs. 14 - 16). A high value of damping coefficient c is desirable to damp out the vibration quickly. Therefore, for a particularload to be supported by a compensated hydrostatic thrustbearing cpera@g under dynamic conditions, a proper selection of bearinggeometry, CSz, Cd and v is essential, in order that a high stiffness, adequate damping and reasonableflow requirementare achieved,

6. Conclusions On the basis of the results reported in this paper the fo~ow~g conclusions can be drawn.

1

0.0

01

3.0-

5.0 -

7.0-

9.0.

I

0.1

Load

Fig. 10. Bearing flow (configuration

0.3

4

0.L

= 2.666;

ih = 0.1;

V = 0.3).

= 4.0; Eh = 0.1; V = 0.3).

To

IV; &

III; &

capacity

0.2

Fig. 9. Bearing flow (configuration

4 IA

1;

11.0.

13.0.

15.0r

16.0 -

2 .ol 0.1

4.0 -

6.0.

6.0 -

lO.O-

% L 12.0-

1;

16.0-

16.0 -

zoo-

20.0-

zc.o-

1 0.2

C;22=

05

20.0 , rd

=

I 0.6

1.00

(4

0.L

LOAD CAPACITY

I

0.3

\

0.2

To

0.5

Fig. 11. (a) Stiffness coefficients (configuration II; & (configuration II; L, = 1.0; ih = 0.1; co = 15.0).

0.11 0.1

t

0.5

(b)

C1.1 L 0

= 1.0; -h t = 0.1; v = 0.3); (b) effect

0.6

c

,2 -

of Poisson’s

Deformolion

Fd

ratio on stiffness

coefficient

I

0.75

I

0.50

I

0.25

coefficients

1

I 00

55

56

Cd

(configuration

coefficient

I

0.75

(configuration

Deformation

I

0.50

I

0.25

Fig. 15. Damping coefficients

-

Fig. 14. Damping coefficients

0.06

0

III; LR = 4.0; rh = 0.1; v = 0.3).

II; ER = 1.0; fh = 0.1; CQ = 15.0).

I

I.00

0

I

0.1 Load

capacity

0.2 ?‘,

0.3

8

0.0

1.06-

.07 -

.06-

.05 -

.04 _

.03 -

.02 -

I

0.2

0.3 Load

0.1 capocuty

0.5

0.6

.?.a

Fig. 16. Damping coefficients (configuration

IV; &

= 2.666;

& = 0.1; v = 0.3).

(1) The pocket pressure& decreaseswith increase in c* but increases with increasein cS2and v. (2) The bearing flow decreases and the damping coefficient increases with increasein Poisson’s ratio. (3) In general,for a particularbearinggeometry the maximum valuesof load capacity and the fluid film stiffness would not be attainedat one particular combination of &,, cd and V, and so a judicious selection of three parametersis essential. (4) For a particularload a proper selection of bearinggeometry, cSs2,cd and ZJis needed to achieve an optimum design.

References 1 H. C. Rippel, Cast Bronze Hydrostatic Institute, Cleveland, OH, 1963.

Bearing Design Manual, Cast Bronze Bearing

59 2 W. B. Rowe and J. P. O’Donoghue, A review of hydrostatic bearing design, Proc. Conf. Externally Pressurized Bearings, Institution of Mechanical Engineers, London, 1971, pp. 167 - 187. 3 D. Dowson, Elastohydrostatic lubrication of circular plate thrust bearings, J. Lubr. Technol., 89 (1967) 237 - 244. 4 V. Castelli, G. K. Rightmire and D. D. Fuller, On the analytical and experimental investigations of a hydrostatic axisymmetric compliant surface thrust bearing, J. Lubr. Technol., 89 (1967) 510 - 520. 5 C. Singh, T. S. Naiwal and P. Sinha, Elastohydrostatic lubrication of circular plate thrust bearing with power law lubricants, J. Lubr. Technol., 104 (1982) 243 - 247. 6 N. Acherkan, V. Push, N. Ignatyov and V. Kudinov, Machine Tool Design, Vol. 3, Mir, Moscow, 1969, p. 217. 7 R. Sinhasan, S. C. Jain and S. C. Sharma, Elastohydrostatic lubrication of capillary compensated circular pad thrust bearings, Wear, 91 (1983) 131 - 147. 8 A. Cameron, The Principles of Lubrication, Longmans, London, 1966. 9 0. C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London, 1977. 10 S. A. Morsi, Passively and actively controlled externally pressurized oil film bearings, J. Lubr. Technol., 94 (1972) 56 - 63. 11 R. C. Ghai, D. V. Singh and R. Sinhasan, Load capacity and stiffness considerations for hydrostatic journal bearings, J. Lubr. Technol., 98 (1976) 629 - 634. Engineering, Mc12 J. J. O’Connor and J. Boyd, Standard Handbook of Lubrication Graw-Hill, New York, 1968, pp. 5.32, 9.4. 13 F. M. Stansfield, Hydrostatic Bearings, Machinery Publishing, London, 1970, p. 82.

Appendix A A.1. Closed form expressions for a capillary-compensated rigid thnrst pad bearing The flow of incompressible lubricant in the clearance space of a circular thrust pad bearing is given by [Al, A21

By integrating eqn. (Al) and equating the flow through the capillary restrictor and the bearing, the expression for the pocket pressure p, is derived. A.l.l. Pocketpressurepc The pocket pressure DCfor a dynamically loaded (a/i/& # 0) capillary compensated circular thrust bearing is given by -

aii/aT jj, = csl- -K,, K,hT3

WV

+ csI

where 2 ifl, = 12s ‘1 i r2 11

(r2Pd2 2 lofO2lrJ

-

1

_

1

(A3) I

60

(A4) (A5) A. 1.2. Resultant fluid film reaction P The resultant fluid fihn reaction F is given by

ai;

-

F = K&-

Kf2 ar

WI

where nt 1 - (r&-2)2I

&, =

2

Kf2=

(AT)

logdr2Pl)

$[[I-

(;i’i’+f#

-

-

11 -

(rdr2)212

lw&2lrl)

I (A@

For the equilibrium case (%/a~ = 0) the load-carrying capacity F0 is given by (A9

FQ = K&x A.1.3. Fluid film stiffness s The fluid film stiffness coefficient 8 is obtained as

3Khr2

1 - (rIlr2)2

= ’ (1 + Kh,3)2 2 log,(r2/rl)

(AW

where K=

_ C,,

2lr

(All)

log&2lri)

A.1.4. Damping coefficient c The fluid film damping coefficient c is defined as +-z

=-Kfl

ai

a@,

L +Kf2 ah

(A=)

where (-413)

61

References for Appendix A Al A. Cameron, The Principles of Lubrication, Longmans, London, 1966. A2 M. K. Ghosh and B. C. Mazumdar, Dynamic stiffness and damping characteristics of compensated hydrostatic thrust bearings, J. Lubr. Technol., 104 (1982) 491 - 496.

Appendix B: Nomenclature a C E

radius of the capillary fluid film damping coefficient modulus of elasticity for the bearing material load capacity FO film force due to the squeeze velocity FT h film thickness squeeze velocity ah/at rigid bearing film thickness h, film thickness in the equilibrium position (ah/at = 0) h0 L capillary length Ll , L2, 2 thrust pad parameters (Fig. 2) number of elements in the elastic deformation field % number of elements in the lubricant flow field 4 number of pockets nP pressure (gauge) P pocket PESSUE (ah/at f 0) PC supply pressure PS Q,/2, bearing flow sum of the nodal flows for the nodes on the external boundary : restrictor flow Qi recess radius (circular pocket, Fig. 2) r1 outer radius of the pad r2 S fluid fihn stiffness coefficient t time thickness of the thrust pad th u, u, w deformations in the X, Y and 2 directions X, Y, 2 orthogonal cartesian coordinate system (Fig. 3) Non-dimensional parameters -e A area of the eth element pocket area (j = 1, . . . , np) j&i c Ch,3/r24p for configuration I (Fig. 2(a)); Ch,3/@13L2 tions II, III and IV (Figs. 2(b) - 2(d)) Cd

(f',/E)(th/h,)

CSl

WWa4/hr3L)

C S2

(~dWdL2)

for confii-

62

F,/rz2p, for configuration I (Fig. 2(a)); F,/L1 L2pS for configurations II, III and IV (Figs. 2(b) - 2(d)) h/h, a&/a7 LiIL2 (Fig. 2) PIP, PCIPS

QnP (Q,~lhhNLIL~)

&(12p/P,h,3) Sh,/r,*p, for configuration I (Fig. 2(a)); Sh,/p,L1L2 tions II, III and IV (Figs. 2(b) - 2(d)) th/b

for configura-

(= th/r2)

Greek symbols XILI YIL, ; viscosity cc Poisson’s ratio V 7 WL 12/k2p,) Subscripts and superscripts bearing b pockets C eth element pad thickness “h rigid bearing r restrictor R supply pressure S the transpose of a matrix T the corresponding non-dimensional parameters static equilibrium position or steady state position 0 squeeze terms IMatrices El @I {RI {a,} ($1

global global global global global

fluidity matrix stiffness matrix for the three-dimensional elasticity problem nodal force matrix squeeze term matrix nodal deformation matrix