Stiffness optimization of a variable restrictor-compensated hydrostatic thrust bearing system

Wear, 44 (1977) 223 - 230 0 Elsevier Sequoia S.A., Lausanne

223 - Printed

in the Netherlands

STIFFNESS OPTIMIZATION OF A VARIABLE RESTRICTORCOMPENSATED HYDROSTATIC THRUST BEARING SYSTEM

C. K. SINGH and D. V. SINGH Mechanical and Industrial Engineering 247667 (India) (Received

Department,

May 26, 1976; in final form September

University of Roorkee,

Roorkee

23, 1976)

Summary

In hydrostatic bearing systems, proper matching of the parameters of the compensating restrictor to those of the bearing is important. Optimum values of the variable parameters of the bearing system for maximum stiffness have been obtained by the steepest gradient method of optimization for circular and rectangular pads. The analysis provides an optimum configuration of the bearing restrictor system.

Introduction

Hydrostatic bearing systems, if properly designed, could be as stiff as rolling element bearings with remarkably good performance characteristics for application in machine tools and other mechanical systems. Although restrictors of the capillary and orifice types are the simplest class and they are commonly used in hydrostatic bearings, many other types with variable fluid flow resistance have been designed [l - 41 to give superior performance characteristics. An important consideration in the design of good restrictorcompensated hydrostatic bearings is the proper matching of the restrictor with the bearing to give the optimum value to some performance parameter. Using the steepest gradient approach of optimization, the thrust bearing geometry and the parameters of a plunger-type variable resistance restrictor [4] have been determined to give maximum stiffness. The schematic arrangement of the bearing restrictor system to be optimized is shown in Fig. 1. Two configurations of the bearing pad have been studied: a circular pad with a concentric circular supply pocket for which an exact closed-form solution is available and a rectangular pad for which an approximate solution has been used.

224 Ps - Supply

pressure

~constontl

h

Spring

Bearing

Restrictor

Fig. 1. Schematic

diagram

showing

restrictor

and bearing.

Variable restrictor The restrictor consists of a cylinder with a plunger which has a sill. The plunger is retained by a spring with an initial compression F,. Oil is supplied to the cylinder through a pipe of radius rl . It is assumed that the pressure drop in the clearance space between the plunger sill and the cylinder head is large compared with that in the connecting pipes. The radius of the supply pipe is assumed to be specified and all the other restrictor parameters are determined to give the best matching to provide maximum stiffness. There is an upper limit to the initial compression of the plunger spring which depends on the supply pressure. This upper limit is identified by the condition that the sill, when it is in contact with the cylinder head, will block the flow against the supply pressure. This leads to the inequality constraint F,/AzP,

< 1

(1)

For the pressure ratio 0 the following

constraint

is obvious:

p < 1.0

(2)

The flow through given by

Q, =

the restrictor,

when the film thickness

7rh3(1 - 0) Wn

The equilibrium

over its sill is h, is

(3)

(r2/r.1) of the plunger gives

F, + Kh = nrfP,

Using the continuity

+

s r1

P2nrdr + (rs - rg)P,

of flow condition,

eqn. (4) yields

(4)

225

F, +Kh

=(l

-p)A,a,P,+PA,P,

From eqn. (5) the film thickness h = {--F,

+ (1 - fl)AIa,

(5)

h may be obtained

as

+ /3A,}P,/K

(f-3)

Circular pad The load capacity

of a circular pad with a concentric

pocket

is

W = /3a,A,P,

(7)

and the flow rate is nH3flP,

QB=

(9)

6yln (RI&,)

Equating the flows through the restrictor and the bearing and substituting for h from eqn. (6), the bearing load support can be expressed as {--F,

In (R/R,)

W = a,A,

+ (1 -fl)a,A1 H3

In (r2/r1) (’ -p)ps

The pad stiffness is obtained

+ PAZ}3

by differentiating

eqn. (9) with respect to

H:

K/34/3(1 - /3)2’3 ’ (l-/3)(rz

-rf)n

21n (r.2/rl 1

+7v$3--FFO-3/3(1-(j)

n(r$ 21n

(10)

r:)

(r2 PI

1I

where

Rectangular

pad

A reasonable approximation for the load-carrying capacity of a rectangular pad with a central pocket can be obtained by assuming that the pressure distribution of the sill is linear from the pocket to the outer boundary (Fig. 2): W = /3P,LB -&

and the flow is

(2LB + 21b + Lb + ZB)

(11)

226

Fig. 2. Pressure

distribution

(rectangular

pad).

(12)

PPs

Again equating the flow through the restrictor and the flow through the bearing and substituting for h from eqn. (6), the load support for a rectangular pad can be expressed as 1 L-l x

B-b

In (r2/rI)(l

f--F, + (1 -P)u,AI

-‘)”

’

+ P&l3

(13)

H3

3

The pad stiffness S of a rectangular bearing can likewise be obtained differentiating eqn. (12) with respect to H: S=

by

1 6(2LB + 21b + IB + Lb) ,34/3(1

_

p)2/3

2_ +

nr?j/3-F,

-

3p(l-0)

I

nr3 -

r2 2 In

2

rl @2/h

1

(14)

Non-dimensional

stiffness

As the stiffness of the pad increases monotonically with an increase in spring stiffness K, it is logical to define a non-dimensional bearing stiffness (I = S/K for seeking the optimum value of the bearing pad stiffness.

227

Optimization

of the bearing stiffness

Tbe stiffness expressions are non-linear and contain a large number of variables. Therefore principles of non-linear optimization have to be used for optimizing the stiffness of the bearing. There is no single technique which may be applied universally to all non-linear optimization problems. In this paper, the random search technique has been adopted, in which the search starts from some arbitrary point in an ~-Dimensions search space. In this attempt, the steepest ascent method [5, 61 provides a convenient organized iterative search procedure which is based on small perturbation theory stipulating the following: (1) for all values of the variables representing a space state, the stiffness is uniquely defined and finite; (2) for all values of the variables, every partial derivative of o with respect to each of the variables is uniquely defined, finite and continuous; (3) there is a finite maximum value of u. It is assumed that the optimum design problem prescribes that for the circular pad the size of the pad R and the size of the supply pipe rl are available as design data. The optimization has been done with reference to the parameters Ro, r2, r3, p and Fc. For the rectangular pad it is assumed that the bearing area is similar to that of the circular pad and that the value of r1 is also available. Optimization has therefore been done with reference to the parameters B, L, b, 1, rzr r3, fl and Fc. The iterative procedure for optimization involves the following steps. (1) An arbitrary set of parameters based on the possible sizes of the bearing and the restrictor is selected. (2) Partial derivatives (given in the Appendix) at trial points are successively computed with respect to each of the state variables Xj. If all the partials (au/&~), at the hth trial are zero or very small, iteration is terminated. Otherwise a direction dk at the hth point, where the step size tk increases the objective function (stiffness) along the steepest slope, is determined. (3) A new trial point identified by the state variables xfcik+’is calculated by the following equation: x:+i = xf + dk& Iteration is returned to step (2) and again a new trial point is computed. procedure is repeated until the condition in step (2) is satisfied.

(15) The

Results Table 1 gives the starting values and the optimum values of the parameters used for optimizing the stiffness of the circular pad. The starting values were intuitively selected and nine iterations were required for optimization Table 2 gives similar data for the rectangular pad; the starting values of the variables B, L, b and I were arbitrarily taken, keeping the area BL of the

228 TABLE

1

Circular

pad (R = 10 mm,

Initial Final

TABLE

=

3 mm)

RO

r2

‘3

(mm)

(mm)

(mm)

1 1.647

3.3 3.95

3.6 3.97

(7

0.5 0.42

8 8.051

0.2814 0.3835

2

Rectangular ____-

pad (rl = 3 mm) --

(mm)

L (mm)

Fmm)

imm)

;krn);krn)

15 15.0024

20 20.0017

5 5.0019

10 10.0012

3.3 3.32

B

Initial Final

rl

3.8 3.814

P

PO

CJ

0.5 0.481

8 8.0022

0.9154 0.9164

rectangular - pad similar to the area of the circular pad. The starting values of r2, p and F, were taken to be the same as the starting values of the circular pad and the starting value of r3 was arbitrarily taken to be closer to the optimum value of r3 for the circular pad. Four iterations were required for optimization. If the supply pressure and the lubricant viscosity are specified, the flow Q, is proportional to h3 (1 - /3)/ln (r2 - rl). The value of the film thickness h for the optimum configuration of the circular pad and of the rectangular pad can be found from eqn. (6) and the ratio of flow rate for these configurations can be calculated using data from Tables 1 and 2. The ratio of flow rates for optimum configurations of circular and of rectangular pads is 0.617. The load support for the optimum configurations can be computed from eqn. (7). The ratio of the load support for the optimum configuration of the circular and of the rectangular pad is 0.467.

Conclusions The optimization results reveal that to obtain the maximum bearing stiffness a flat faced plunger (Fig. 3) in the variable restrictor, rather than the type of plunger described by Churin [4] , should be used for a circular pad bearing. The stiffness of the bearings is directly proportional to the spring constant of the restrictor spring. It is therefore recommended that the stiffness of the spring in the restrictor should be as large as is feasible. It must be noted that, since there is an optimum value for the initial compression of the restrictor spring, it will be impracticable to use too high a value of the spring stiffness.

229

--

To bearing

Fig. 3. Optimum restrictor configuration

for circular pad.

The flow rate and load capacity for the optimum design of a circular pad are smaller than those for the optimum design of a rectangular pad. The optimum parameters do not violate the inequality constraint of eqn. (1).

Nomenclature 1 -R;/R’ OB

2 In (WRu)

(circular pad),

2LB + 21b + Lb + IB 6LB

(rectangular pad);

the bearing size coefficient

1 - (W2)2 2 In

AB

Al A2 b

B Fo h H K 1 L P PP Ps QB

9, ‘1

r2 r3

R

, the restrictor size coefficient

(r2lr1)

nR2 (circular pad), LB (rectangular pad); the effective area of the bearing nr$, the restrictor sill area nr$ the restrictor plunger area the width of the pocket of the rectangular pad the width of the rectangular pad the initial compression in the spring of the restrictor the film thickness in the restrictor the film thickness in the bearing the stiffness of the restrictor spring the length of the pocket of the rectangular pad the length of the rectangular pad the pressure the pocket pressure the supply line pressure the flow through the bearing the flow through the restrictor the fluid inlet pipe radius the plunger sill radius the plunger major radius the bearing radius (circular pad)

230 Ro S F u

the pocket radius (circular pad) the stiffness of the bearing the bearing load P,/P,, the pressure ratio S/K,the dimensionless stiffness of the bearing

References M. E. Mohsin, The use of controlled restrictors for compensating hydrostatic bearings, 3rd Int Machine Tool Des. Res. Conf., Birmingham, 1962, Pergamon Press, Oxford, p, 429. G. S. K. Wong, Interface restrictor hydrostatic bearings. In Advances in Machine Tool Design and Research, Pergamon Press, Oxford, 1965, p. 272. J. E. Mayer and M. C. Shaw, Characteristics of an externally pressurized bearing having variable external flow restriction, J. Lubr. Eng., 85 (1963) 291. I. N. Churin, Variable restrictor hydrostatic system, Mach. Tool. (USSR), 12 (1964) 2. D. J. Wilde, Optimum Seeking Methods, Prentice-Hall, Englewood Cliffs, N.J., 1964. C. Hadley, Non-linear and Dynamic Programming, Addison-Wesley, Reading, Mass., 1964.

Appendix The following terms, which are functions of the restrictor geometry and the pressure ratio p, are defined to lend brevity to the zlsebraic expressions for the partial derivatives: gl = (6 - $)/2rl

g2 = m-1 Ii-2

6f3 = In

g, =

0-2/h

)

d(l

-P)(l

--PP - WP3

+ W)w2/g3

Circular pad

au

-_._

_=2a

PLO _R2-II;

aRo

2

+3Roln

_._1

(R/Ro)(

Ret tangular pad

a0jaL au/al

= (2B + b)lg,

a0jaB

= (2L f 1)/g,

= (2b + BM,

a 0/a b = (21+ L)lg,

where g, = 2(LB + lb) + Lb + 1B. Circular and rectangular

a0

-=

ar2

1 -3r2g3

pads

n(l + 33 -

3P2)(r2g3 -g2)

g&3

a0

-=

ar3

4or3(1 -8) g4