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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING NORTH-HOLLAND PUBLISHING COMPANY

NUMERICAL

31(1982) 179-187

METHODS FOR CALCULATING PRESSURE DI!3TRIBUTION IN GAS BEARINGS Zi-Cai LI* and E. DAI

Shanghai Institute of Computer Technology, Shanghai, China Received 24 August 1981

In gas bearings, the pressure distribution is governed by a non-linear Reynolds equation. In order to solve this equation two numerical methods, the conservative difference scheme and the finite element method, are provided in this paper. They are superior to the finite difference. method of Colemman [2]. Use of the finite element method is advocated because of its flexibility in solving the Reynolds equation.

1. Introduction

In order to obtain a numerical solution of the pressure distribution in gas bearings, Colemman [2] provided a finite difference method. However, it is not very feasible in complex problems. For example, it is very difficult to construct a difference equation on the nodes of the interior boundary, especially on the corner boundary point. Therefore the present paper will provide two methods-the conservative difference scheme and the finite element method. The evidence from a numerical example is that the results obtained by both methods are more accurate than those obtained by the finite difference method of Colemman [2].

2. The nonlinear Reynolds equation and its boundary conditions In this paper we take the gas bearing of a spiral groove thrust plate as an example. In the stable case the pressure distribution of the gas bearing will satisfy a dimensionless Reynolds equation:

where r and 8 are polar coordinates, p is the pressure, A is a compressibility clearance having different values in different regions (Fig. 1):

number, H is the

*Present address: Department of Computer Science, University of Toronto, Toronto, Ontario MSS lA7, Canada.

00457825/82/OOOO-OOOO/!IO2.75 @ North-Holland

Z.-C. Li, E. Dai, Numerical

180

methods for gas bearings

Fig. 1. Gas bearing with a spiral groove thrust plate.

H = HI Hz

in the groove region, in the ridge and seal regions,

where HI and Hz are positive constants. Obviously (1) is a nonlinear equation, its solution domain is a circular ring (Ri I r I R,), where Ri and R, are the radii of the interior and outer circles (Fig. 1). Because of the symmetry it is enough to solve in the groove and its neighbourhood ridge and seal regions (Fig. 1). Let y=lnr,

u = p2.

(2)

Eq. (1) becomes

Under the transformations (2) a symmetric solution region in Fig. 1 transforms into S of Fig. 2. The width along the direction 0 of that figure is 21r/z (z is the total number of grooves).

Fig. 2. The solution region S = SI US2 with the coordinate

system Y-O-0.

Z.-C. Li, E. Dai, Numerical

methods for gas bearings

181

Since on physical grounds, the pressure is always positive, d/u in (3) exists. In this paper we assume that u has a positive lower bound, &, namely,

(4)

U~So>O.

Eq. (3) is a quasi-linear conditions are as follows. u=l,

elliptic equation.

Obviously,

(3) is simpler than (1). Its boundary

ony=y,=lnR,,

au/ay = 0 )

(5)

on y = yi = In Ri

(6)

and u(y, 8 + 277-/z)= u(y, 0)

for all 8.

(7)

Eq. (3) is also equivalent to fl ezy du,

d2U, + *=

ay2

ae2

H:di

de ’

(8)

1=1,2,

where u1 and u2 represent the solution, u, in regions S1 and S2, respectively. On the interior boundary T’,,,there are two conditions of continuity for flux,

= 2A ezyd/u(H, - Hz) cos cr ,

on DB

(9

and

H3au,=H3au,

l

JY

* dy

onm ’

(10)

3

where cr is the spiral groove angle and where the solution u is continuous on the interior boundary, i.e. u = u1 = u2. Colemann [2] suggests that the finite difference method be used for solving (8~(10). However, the interior boundary BD is not parallel to the coordinate axes 0 and Y, so the discrete difference equations in [2] for the condition (9) are too crude. Moreover, it is difficult to see which of (9) or (10) should be used for the difference equation at the corner, D, of the interior boundary F’,. So, we seek some better numerical methods.

3. The conservative

difference

scheme

Eqs. (9) and (10) can be combined as a single equation H: 2

- H; 2

= 2A e”d/u(H,

- Hz) cos(n, 0))

(11)

182

Z.-C. Li, E. Dai, Numerical methods for gas bearings

where II denotes the direction of the normal to the interior boundary. that the boundary conditions (5j(7) and the equation:

f ri

H3 g

We can easily prove

dl = -2A 36 e2yHv/U dy rt

(12)

are equivalent to (3), @k(7) and (ll), where 17_is the boundary of any subregion Si C S, n is the outward normal direction of Tim Eq. (12) represents conservation of gas mass and can be used for constructing the difference equation. Consequently, it is not necessary specially to consider the troublesome interior boundary conditions (11). The resulting conservative difference scheme, though easily obtained, is more reasonable than the finite difference method of Colemann [2]. The solution region S is divided into many right triangles and acute triangles (Fig. 3). Suppose that ml and m2 are the numbers of equidistant nodes along the direction 0, ytl and n2 are the numbers of equidistant nodes along the direction Y. Denote m = ml + pnz, n = IZ~+ n2. In order that the triangle division is into right triangles and acute triangles, the ratio of mesh spacings Ayj/AOi must satisfy the following inequalities: i sin 2c~5 Ayi~A~i5 ctg at,

(13)

where Ayi and AOi are the mesh spacings along the directions Y and 0, respectively. From the partitioning in Fig. 3 the integers ml, m2 and ~1~must satisfy i sin 2a zz k.

(14)

~sin2~~k*(l+~)~~ctg~

(15)

and

where /I is the width ratio of groove to ridge. The constant k. is given by ko = (z/2n) In(RJR,) , and R, is the radius of the middle circle in Fig. 1.

Fig. 3. The triangulation.

Z.-C. Li, E. Dai, Numerical methods for gas bearings

183

When conditions (14) and (15) hold, the partitioning in Fig. 3 creates right triangles or acute triangles, ensuring that the origin Ai of the escricircle of a triangle is in or on the triangle itself. The region enclosed by the straight lines AiAi of origins is taken as a mesh region. Let us take an interior point 0 and its mesh region (Fig. 4) as an example for discretization of (12). The left-hand side (12) can be approximated as

(16) where the coefficients aj = [AjCjH& + C,A,-,H$_,]/OJ

.

(17)

A, = A6, AjC;. also denotes the length of the straight line AjCp When the coordinates of the vertices of the triangle A,, are known, we can easily obtain the coordinates of the origin A of the es&bed circle (Fig. 5): XA =

-(ydi + y$j + ymdm)/4A

J’A =

-(Xidi + Xjdj + X,dm)/4A

7

wheredi=x~-x~+yf-y~,dj=x2,--x:+y$triangle is

l A=$1 1

Xi

y?, d, = X: -

XT + xf - XT,

and the area of the

yi

xj yj Xfn ym *

(18)

Thus the coefficients aj of (17) are obtained from the cooginates of origins Ai. The right-hand side of (12) containing the nonlinear v/u can be approximated

as

fQ Q

In

j

*A

T

Fig. 4. The interior

point

0 and its mesh region

Si.

Fig. 5. A triangle

P

element,

e.

Z.-C. Li. E. Dai. Numerical methods for gas bearings

184

I‘ e2yH~/U dy = i= pj(~/ui +

-2A

f.

~\/uo) ,

(19)

j=l

where the coefficients on the right are given by pj = -A

ezy’, [HA,(YA,

HA,-,(Y, -

- YG) +

b-1)1.

Then the difference equation of seven nodes is obtained from (12) (16) and (19):

5

cYj(U,

-

U(j)= f:

pj(d<+

A&).

(20)

j=l

/=o

Obviously, the conservative difference scheme (20) has contained the interior boundary conditions (11). In addition, there are the boundary equations from (5)-(7). We obtain m X yt difference equations in the m x it unknowns ui, and denote them in the simple operational form F(V)=

K,

(21)

where V is an unknown vector with the components ui, F is a nonlinear known vector. Eq. (21) can be easily solved by the Newton method [4].

operator,

K is a

4. The finite element method It is worth noting that (14) and (15) must hold for the stable numerical solution of the conservative difference scheme (20). However, the finite element method in this paragraph need not satisfy these limitations and is better because of its flexibility. Suppose that the exact solution U* of (3), (5)-(7) and (11) exists. Then, we construct two equations

(22) and H: 2

- Hz $f

= 2A e2yd/u*(Hl

- Hz) cos(n, 0)

(23)

on DB and CB. Obviously, the solution of (22), (5)-(7) and (23) is still u*. If we temporarily regard u* as a constant solution, (22) and (23) are ‘formally’ linear. Define a functional I(U)

Then,

among

=

I 1 [iH3[

($7

all admissible

+ (ST]-

functions

2~ ezyHd/

satisfying

$}

dy df3.

(5) and (7), the solution

(24) U* makes the

Z.-C. Li, E. Dai, Numerical methods for gas bearings

185

minimum of the functional I(U), i.e., the first order variation of I(U) is equal to zero: aI

= 0.

(25)

Below, we obtain the difference equation by the finite element method linear function in a triangle element (Fig. 5):

[l, 3,5]. Take the

U = NiUi + Niui + N~u, 3

(26)

where Ni = (ai + bix + ciy)/2A, ai = Xjy, - X,yj, bi = yj - ym, Ci = X, - ~j. The coefficients Uj, bj, cj, a,, b, and c, can be similarly expressed. First, we apply the linear finite element method regardless of the presence of q/u*. Then, it follows that if e denotes the triangular element in Fig. 5,

= & H3[(biui + bjuj + b,u,)bi + (CiUi+ CjUj+ cmUm)Ci]

-$bi/I e H

e2yd?

dy de.

(27)

Second, since u* = u, the integral containing vu*

-$bi//e H e2yd/u*

dy de = -t

-

=

bi

can be approximated

as

H e2Yd/u dy de

-

-+AbiH[e2YTduUT+ e2ypdup + e2yQd/uo]

z -&4biH[(e 2yr+ ezyp)V/ui + (e 2YP+ e2Yo)d/ui + (ezyO+ e2YT)d/um] ,

V-3)

where the points P, Q and T are shown in Fig. 5. Then, we obtain

$P’(u)=$ [(bf + I

-

Finally, the nonlinear (29):

Cf)Ui

+

(bibi + CiCj)Uj

+

(bib, + CiCm)Um]

$4biH[(eZY’ + e2yp)VG+ (e2yp+ e2yo)d/ui + (ezyO+ e2YT)duu,] .

difference equation

at the interior point 0 is obtained

(29)

from (25) and

(30)

Z.-C. Li, E. Dai, Numerical

186

methods for gas bearings

where &j and Bj are the coefficients independent of U. In addition, there are the boundary conditions (5) and (7). Then, the numerical solution for u can be easily obtained by the Newton method [4]. After the numerical solution for u has been obtained, the integral of the pressure distribution is evaluated from (d/u-

1) e2y dy df3.

(31)

We can prove the convergence of the finite element method and provide the error estimate of its numerical solution according to the analytical method of the nonlinear finite element method [l].

5. A numerical example

Consider a gas bearing of a spiral groove trust plate with the parameters z = 15, (Y= 72”, p = 1.3, HI = 5.5, Hz = 1, Ri = 0.4, R, = 0.7, R,= 1. Take ml = 4, m2 = 3, nl = 20, n2 = 15. Then (14) and (15) hold. All-the triangles divided are right triangles or acute triangles. We calculate the pressure p = d/u by the conservative difference scheme and the finite element method, then obtain the value of w from (31). Table 1 presents the calculated results for various compressibilities, A. It is seen from Table 1 that the difference between two methods is very small, the largest relative error being only 0.2%. This suggests how reasonable and consistent the two methods are. The quantity w of (31) above, evaluated by either method is closer to the experiment value than is that obtained by the finite difference method of Colemann [2]. Indeed, the finite element method is the best of these three methods.

Acknowledgment

This work was supported by the Shanghai Institute of Computer Technology. Also it is with pleasure that we express our acknowledgment to Professor L. Endrenyi and Dr. G.L. Feng for their valuable suggestions. Table 1 The integrated value of the calculated pressure distribution

A

The finite element method

The conservative difference scheme

5 20 130 500

0.130685 0.524077 3.393624 12.491314

0.130686 0.524142 3.394879 12.513590

Z.-C. Li, E. Dai, Numerical methods for gas bearings

187

References [l] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, [2] R. Colemann, The numerical solution of linear elliptic equations, Trans. ASME, [3] J.T. Oden and J.N. Reddy, An Introduction to the Mathematical Theory New York, 1976). [4] J.M. Ortega and W.C. Rhinbodt, Iterative Solution of Nonlinear Equations in Press, New York, 1970). [5] G. Strang and G.J. Fix, An Analysis of the Finite Element Method (Prentice-Hall,

Amsterdam, 1978). Ser. F 90 (1968) 773-776. of Finite Elements (Wiley, Several

Variables

Englewood

(Academic

Cliffs, NJ, 1973).