Static and dynamic performance characteristics of an orifice compensated hydrostatic journal bearing with non-Newtonian lubricants

Tribology

ELSEVlER SCIENCE?

ln~ermkmal Vol. 29, No. 6, pp. 515-526, 1996 Copyright @ 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0301-679X/96/$lSMl +O.OO

0301-679x(95)00115-8

Static and dynamic performance chara’cteristics of an orifice compensated hydrostatic journal b with non-Newtonian lubricants FL Sinhasan

and P. L. Sah

This paper presents a theoretical study of the performance characteristics of hydrostatic rigid orifice compensated multirecess journal bearings using non-Newtonian lubricants. The generalized Reynolds equation governing the flow of lubricant having variable viscosity has been solved using the finite element method and iterative procedure. The static and dynamic performance characteristics are presented for non-Newtonian lubricants of which constitutive equation has been represented by the cubic shear stress law. The non-linearity factor (K) in the cubic shear stress law significantly influences the bearing performance characteristics, particularly the dynamic characteristics. Copyright 0 1996 Elsevier Science Ltd Keywords: method

hydrostatic

bearing,

non-Newtonian

lubricant,

Introduction Hydrostatic bearings are being used in a large number of applications. Their popularity stems mainly from the fact that they provide large fluid film thickness, high fluid film stiffness, good vibration damping characteristics, smooth relative motion even at very low speeds, high accuracy of location and precision and good dynamic performance. Hydrostatic bearings are widely used. The literature available1 in the area is abundant. The available literature is generally for journal bearings with a Newtonian lubricant. However, to meet the specific requirement of industries the lubricants are loaded with additives. The polymer-thickened oil behaves as pseudoplastic or dilatant fluids. The viscosity of such lubricants loaded Department of Mechanical Roorkee, India (247667). Received 16 March 1995; 1995.

and Industrial revised

6 July

Engineering, 1995;

accepted

University

of

25 September

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finite e/ement

with additives is not constant and usually has some nonlinear relation between shear stress and shear strain rates. This can be represented by different non-Newtonian models. Most of the non-Newtonian lubricants follow the cubic shear stress law2. The present method of solution for obtaining bearing characteristics is general and can accommodate any constitutive equation for non-Newtonian fluids and is simpler than the method proposed by Najji et ~1.~ Both methods of solution are iterative in nature. The present work is aimed at studying the effects of non-Newtonian lubricants on bearing performance characteristics. The finite element method and a suitable iteration scheme have been used. The Reynolds equation has been modified for lubricants with variable viscosity. The non-Newtonian effect is introduced by modifying the v&cosity- term using the cubic shear stress law (? + W = y) in each iteration and converged solutions of pressure have been obtained. The static performance characteristics in terms of maximum International

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and P. L. Sah

Notation radial clearance damping coefficient (i, j = 1,2) diameter of orifice journal diameter journal eccentricity fluid film thickness minimum fluid film thickness non-linearity factor bearing axial length pocket axial length critical journal mass number of 2-D element in the discretized flow field rotational speed resultant fluid film reaction (dh/,% # 0) resultant fluid film reaction (dh/dt = 0) x and z components of fluid film reaction (8h/& = 0) pressure pocket pressure

N P

P PC

Pcl,

h/c h,,.,k

tl( pRf/c2ps) WJpsRf XjlC ZjlC

eic XlRj circumferential coordinate YlRj axial coordinate

PC2

PC37 PC4

P max PS ;.R S;

In

pocket pressure in pockets 1, 2, 3, 4, respectively maximum pressure supply pressure total bearing flow flow through restrictor radius of journal fluid film stiffness coefficient (i, j = 1, 2) time (~j~j) journal surface speed external load Cartesian coordinate system, Fig 1 journal centre coordinates reference viscosity rotational speed of journal threshold speed damped frequency of whirl density of lubricant coefficient of discharge for orifice restrictor shear strain rate shear stress attitude angle

Subscripts and superscripts b bearing C pocket journal j static equilibrium ii restrictor S supply pressure corresponding non-dimensional parameter Matrices

PI

Dimensionless parameters

w Ha I&d { R~j} ,{ R=j}

pressure, minimum film thickness, bearing flow and attitude angle and the dynamic characteristics in terms of fluid film stiffness, damping coefficients, threshold speed and frequency-of whirl for a wide range of nondimensional load ( WO), restrictor design parameter 516

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fluidity matrix nodal pressure vector nodal flow vector column vector owing to hydrodynamic terms global right hand side vectors owing to journal centre velocities

( cs2), speed par ameter (0) and non-linearity factor (5) are presented. The effects of non-linearity factor (K) on the bearing performance characteristics are studied and it will help the designer to obtain the optimum bearing design. 6 1996

Orifice compensated

hydrostatic

Analysis The Reynolds equation governing the flow of an incompressible lubricant with variable viscosity in the clearance space of a finite journal bearing (Fig 1) is obtained in non-dimensional form as4: &(PF*

g)

+ $(PF2$)

4(~(1-~)~)+~

(1)

where p,,, FI and & are the viscosity functions: &=j;$

&=j-;$f

journal

bearing:

R. Sinhasan

and P. L. Sah

The film thickness in a rigid bearing with journal and bearing axes parallel is given by: Ii0 = 1 - Xj

COW

-

~j

sino

01

The lubricant flow field has been discretized using four noded quadrilateral isoparametric elements. Using the orthogonality condition of Galerkin’s method and Reynolds equation (1) and following the usual assembly procedure, the following system equation is derived for the discretized lubricant flow field:

mn.mvLxI = @Lx1+ wLLx~ +

~j{~~j}

nxl

+

(31

zj{R*j}nxl

where n is the total number of nodes in the discretized lubricant~ow~eld~

&=/-;;(+d2 Constant supply prc2ss”rc

Fig 1 Multirecess hydrostatic journal

bearing Triboiogy

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Compute Newton

R. Sinhasan

and P. L. Sah

nodal pressure using Raphson iterative method

pzyzzq I

Compute

Compute

9 IECH

velocity

strain

gradient

rate

;w = 1 -.-y-yzq

KNNT

= KNNT

+ 1

Fig 2 Solution scheme. ZMlJ = apparent viscosiQ; PERR = percentage error on journal centre position; TLEJ = percentage tolerance on journal centre position; INEWTN = index for Newtonian and non-Newtonian solution; IECH = index for journal centre position-convergence; ICNTN = index for convergence of non-Newtonian solution

Restrictor

flow

The equation in non-dimensional form for the flow (QR) of lubricant through the orifice restrictor is expressed as: QR = csz (1 --pC)r’* where es2 is the restrictor 518

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Non-Newtonian model The cubic shear stress law, which gives the non-linear relationship between shear stress and shear strain rate for the non-Newtonian lubricant, is written in nondimensional form as? ++&.&ji (9 The viscosity of a non-Newtonian lubricant is described by the apparent viscosity (pa). The apparent viscosity 6 1996

Orifice compensated

is a function as: /ia = f/T=

hydrostatic

journal

bearing:

R. Sinhasan

and P. L. Sah

of shear strain rate (9) and is defined &(5)

(f-5)

For an incompressible fluid the shear strain rate (5) is defined as a function of the second strain invariant5 which eliminates the direction dependency. y =

i

1 l/2

(au/a%)2

+

(adazy

In the non-dimensional form, the shear strain rate ($) at a point in the fluid-film is expressed as:

Boundary

conditions

Boundary conditions field are as follows: (i) (ii) (iii) (iv)

relevant

to the lubricant

flow

At the external boundary nodal pressures are zero. The nodal pressures for nodes on the pocket boundary are equal. The nodal flows are zero at internal nodes except for those nodes which are situated on pocket and external boundaries. The flow of lubricant through the restrictor is equal to the bearing input flow.

Solution

procedure

The complete solution domain (fluid-film zone) has been discretized using four-noded isoparametric elements. The matrices involved in Eq. (3) have been obtained after assembling the contributions of each element. These matrices for each element have been obtained using the Gauss quadrature numerical integration rule6. This requires numerical integration of each function FO, PI, F2 at all Gauss points in every element. The total number of elements used for discretizing the solution domain was 20.

Fig 3 Load capacity

F2 have been made across the fluid-film thickness at each Gauss point in every element using Simpson’s rule. (4) In the current iteration, the solution of Eq. (3) gives a solution of nodal pressure and nodal flows for the apparent viscosity (i&J distribution in the fluid film, obtained in the previous iteration. (5) The nodal pressures thus obtained in the mth iteration are compared with the corresponding values obtained in (m - 1)th iteration. (6) The iterative procedure is terminated when the difference in nodal pressure at each node in the successive iteration becomes less than the predefined tolerance:

L/D = I .o, F. = 0.5, 0 = 0.0, ii = 0 -

Pre.se.n~ work

---

[ll]

The iterative procedure (Fig 2) used in the present study to obtain the converged solution is described briefly below. (1) The solution for the Newtonian lubricant is obtained as the initial trial solution to be used for the non-Newtonian case. (2) For the solution of the non-Newtonian lubricant case, first the pressure derivatives @/ao and ap/apare computed, at each Gauss point in every element. (3) The shear strain rate ($) is calculated using Eq. (8) and the corresponding equivalent shear stress ?- is obtained from Eq. (5) using the Newton-Raphson method. The apparent viscosity (i&) is then computed at all sampling points using Eq. (6). The numerical integrations of FO, Fl and Tribology

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3

2 e

4

5

s2

Fig 4 Sti@aess coefficient International

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Orifice compensated

IU

hydrostatic

journal

bearing:

R. Sinhasan

and P. L. Sah

As results for the performance characteristics of an orifice compensated hydrosiatic journal bearing having non-Newtonian lubricant (/c # 0) are not available (to the best of the authors’ knowledge), the computer program developed using the analysis and the solution algorithms presented earlier in this paper for nonNewtonian lubricants have been used to compute the bgaring characteristics for Newtonian lubricant (k = 0) and are compared with the published results9-11, Figs 3-5. They also compare well.

2 1.5 L/D = I .o, K.

I.0

-

l’resen~

---

[II]

Using the converged solution of nodal pressures for bearings having a non-Newtonian lubricant, the performance characteristics are computedll.

= 0.5. S-J = 0.0, ii = 0 work

The static and dynamic performance characteristics of orifice compensated rigid hydrostatic journal bearings have been presented for the following parameters: L/D = 1.0 0 = 0.0,l.O Fig 5 Damping

coeficient

Iv* = 0.1,l.O

Otherwise the iteration process continues starting from step (2). (7) After the converged solution (p) is obtained the static and dynamic performance characteristics of the bearing are computed. Results

and discussion

The concept of apparent viscosity (j&) has been used earlier7p8 to compute the performance characteristics of a hydrodynamic journal bearing having non-Newtonian lubricant and the computed results were compared with the available experimental results2. They compared well. Table 1 Maximum

K

Difference (%I

1.0

1.0

0.0

0.58

1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0

1.0 1.0 0.58 1.0 0.58 1.0 0.58 1.0

-1.225 10.27 16.27 -1.95 29.90 -22.06 -43.11 65.29 -39.50 -30.72 -32.89 -44.19 -22.48 51.07

W0

Es*

0

Emax

1.0

1.0

hi”

1.0

2.4

-2*, 322 -11 c -12 c

0.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.5 2.0 1.5 1.5 2.4 2.0 2.0 2.0

G2

The variation of maximum pressure is shown in Fig 6 for high load. The variation in maximum pressure is marginal due to the effects of the speed parameter (0) and non-linearity factor (E), both at low and high loads. The maximum percentage reduction in maximum pressure is of the order of 1.225.

percentage

Parameter

-c M ?!th wi

and for different values of restrictor design parameter ( Cx2) and non-linearity factor (K). The maximum percentage variations in the results are presented in Table 1.

0

(Performance

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Fig 6 Maximum 6 1996

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I

I

I

I

I

0.5

1.0

1.5

2.0

2.5

3.0

pressure

Orifice compensated

hydrostatic

journal

bearing:

R. Sinhasan

and P. L. Sah

The minimum fluid film thickness under constant load is shown in Fig 7. It is okserved that the effects of the non-linearity factor (K) on the. minimum fluid film thickness is not very significant. The Faximum variation of minimum fluid film thickness (/q,,) is of the order of 10.27%, Table 1. The variation of flow (@) with the different values of non-linearity factor (K) and speed parameter (0) is shown in Fig 8. The flow decreases as the speed parameter increases but the effect is very marginal. The maximum percentage increase in flow is of the order of 16.27, Table 1. The variation of attitude angle (4) with the restrictor design parameter (&) is marginal, both for Newtonian and non-Newtonian lubricants, Fig 9. The maximum percentage reduction of attitude angle is of the order of 1.95, Table 1. In general, the effects of the non-linearity factor (Z?) on the msimum pressure (&J, minimum film thickness (&,), bearing flow-(Q) and attitude angle (4) under constant load ( Wo) are only marginal. Therefore, only few resuJs have been presented for these parameters &,,aX, hm, Q, 4). At constant load (wO) and -for different values of restrictor design parameter (GJ, non-linearity factor (K) and speed parameter (a), the fluid film stiffness coefficients (!&I and szz) are presented in Figs lOa, lob and lla, llb. From the results presented, it is seen that for an orifice compensated four-pocket hydrostatic jourqal bearing system the effects of nonlinearity factor (K) on the fluid film stiffness coefficients

,, , 1)

I

I

I

I

0s

I0

Is

2 II

I 2s

T %2 Fig 8 Bearing flow

0

0.5

I .o 6 5 4

0.7

v? IU I.5 50° 400

2.0 300 200

2.5 w

100

no

0.6

4, 5. 6

I, 2- 3

l-GE

0.5

Fig 9 Attitude angle 0.4 0

I

I

I

I

I

0.5

1.0

1.s

2.0

2.5

(Sll, S& are significant. The maxipum percentage increase in the stiffness coefficient (&) due to nonNewtonian effects is of the order of 65.29. Therefore, for a journal bearing system with a specified g?omeJry, the optimum value of stiffness coefficients (&, &)

-

Fig 7 Minimum

film thickness Tribology

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IX Sinhasan

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L/D=

l.O,%o=O.l

4.0

4.0

EEb 2

0.0

5

1.0

3

0.0

6

I.0

O.SR 1.0

3.0

1*=

2.0

2.0

1.0 0

I

I

I

I

I

0.5

I .o

1.S

2.0

2.5

c.

1.c 0.5

1 .o

1.5 Es

$2

2

Fig 10 Stiffness coejjicient

4.0 4.0

3.0 3.0 2 IVJ

lv?

2.0 2.c

I

I

I

I

I

0.5

1.0

I.5

2.0

2.5

l.( 0

E

I .a 0

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1 .o

1.s E.

s2

Fig 11 Stifjness coeficient

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2.0

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Orifice compensated

hydrostatic

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2.0

IiJ

2

I.5

I .n

Fig 12 Damping

I

I

I

I

I

I 0

coefficient

L/D

L/D

I

I

= I .O, i6”

I

= 1.0, i6,> = 1.I.l

= I .O

I

I 0

I

I

I

I

I

n.5

I .o

13

2.0

2 .s

-

Fig 13 Damping

coeftkient Tribology

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L/D

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= I .o, Q = 1 .n

4

3.n

I 5 6 2 3

,m

z

2.0

I

2 I

3 I .n

0.5

I .o

1.s

2.n

2,s

0

I

I

I

I

I

I

0 s

I ,o

1,s

2.0

2.5

3.0

cs

c.

2

s2

Fig 14 Stiffness coefficient

L/D=l.O,il=l.O 4

A

-

4 i

f’,,we

N.,

4 =

7

5 I

6 II

6 5

L/l1 = I .o. Cl = I .o

-

I

2 3 o.nn1

-

; 3

0.0001

L 1)

3

I 0.5

I I.0

I

I

I

1.5

2.0

2.5

I 0.5

-

I

I

I

I

1.n

1.S

2.0

2.5

c.5 2

Fig 15 Damping

coeficient Fig 16 Critical journal

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mass

Orifice compensated

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can be obtained by a judicious selection of all relevant parameters (WO, & and K). The results of damping coefficients (Cii, C& are presented in Figs 12a, 12b, 13a and 13b. The values of damping coefficients decrease as the non-Newtonian effect (K) is increased. The maximum percentage reduction in the damping coefficient (Cii) is of the order of 39.5. From the viewpoint of vibration of the journal under dynamic loading, it is essential to have proper values of damping coefficients (Cl1 , CZZ) to minimize oscillations. So, the non-Newtonian characteristic of the lubricant is not beneficial from the damping viewpoint. When the speed parameter (0) is zero, the attitude angle, cross coupled stiffness and damping coefficients are zero for all values of restrictor design parameter (CS~), but for the speed parameter, 0 = 1.0, the crosscoupled -stiffness (&, &r) and damping coefficients ( CiZ = C*i) are non-zero and are presented in Figs 14a, 14b and 15. The cross-coupled stiffness coefficients (,?i*, &r) decrease as the value of the non-linearity factor (K) increases both for low and high loads up to some value of restrictor design parameter ($ 5 2.0). The cross-coupled damping coefficients (Cl2 = C*r) do not show a definite trend with respect to the non-linearity factor (K). The results of critical journal mass (tic) and threshold speed (G&J with restrictor design parameter ( CS~) are presented in Figs 16 and 17. It is observed that both critical journal mass and threshold speed decrease as the value of the non-linearity factor (j?) increases at both low and high loads. The critical journal mass and threshold speed are less for the non-Newtonian lubricants than for the corresponding bearing with

Fig 17 Threshold speed

journal

bearing:

IX Sinhasan

and P. L. Sah

Newtonian lubricants. The maximum percentage reduction of critical journal mass (MC) and threshold speed (&,) are of order of 44.19 and 22.48 respectively, Table 1. Figure 18 shows the variation of whirl frequency ratio (&) with the r estrictor design parameter for different values of K and W,,. It is found that the whirl frequency is reduced if load is increased but it is more for nonNewtonian lubricants (i? # 0) both at low and high loads. The maximum percentage increase of whirl frequency ratio ( GJ~)is of the order of 51.07% obtained at WO = 1.0 and K = 1.0, Table 1. In general, the maximum variation in bearing performance characteristics is observed at high loads and higher values of non-linearity factor (E). Conclusions On the basis of the above results obtained using the theoretical analysis and solution algorithm presented in the paper, the following conclusions are drawn. (1) The effect of the non-linearity factor (Z?) is margiEa for the bearing characteristics parameters (2) k&LZk&EeZr?d film stiffness coefficient ($i) increases with the non-linearity factor (K) for C.Q > 1.5 and ,(.$,,) increases with (K) when CsZ B 1.0. At lower values of CsZ the effects of K on sii and & do not show a definite trend. (3) The coupled stiffness coefficients (&, $i) show an almost decreasing trend with increases of f?. (4) The damping coefficients (Cii, Q-decrease as the value of the non-linearity factor (K) increases.

Fig 18 Frequency of whirl Tribology

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(5) For hybrid bearings threshold speed decreases and whirl frequency ratio increases with increases in non-linearity factor (K) . (6) The whirl -frequency ratio reduces wi!h increases in load ( WO) bo!h for Newtonian (K = 0) and non-Newtonian (K # 0) lubricants.

5. Darby R. Viscoelastic Fluids; An Introduction and Behavior, Marcel Dekker, 1976

References

8. Siiasan R. and Goyal K.C. Elastohydrodynamic studies of circular journal bearings with non-Newtonian lubricants. Tribal. lnt. 1990, 23, 6, 419-428

1. Slnhasan R. and Sah P.L. Hydrostatic bearing - A review. Proc. XI National Conference of Industrial Tribology (NCIT95), New Delhi, India, 22-25 January 1995 2. Wada S. and Hayashi H. Hydrodynamic lubrication of journal bearings by psedu-plastic lubricants (Part II, experimental studies). Bull. JSME, 1971, 14, 279-286 3. N@ji B., Bou-said B. and Berthe D. New formulation for lubrication with non-Newtonian fluids. Trans. ASME J. Tribal. 1989, 111, 29-34 4. Sinhasan R. and Chandrawat H.N. Analysis of a two-axialgroove journal bearing including thermoelasto hydrodynamic effects. Tribal. lnt. 1989, 22, 5, 347-353

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6. Zienkiewicz O.C. The Finite Element Methods, Tata McGraw Hill, New Delhi, 1979 7. Malik M., Das B. and Sinhasan R. The analysis of hydrodynamic journal bearings using non-Newtonian lubricants by viscosity averaging across the film. ASLE Tram, 1983, 26, 1, 125-131

9. Raimondi A.A. and Boyd J. An analysis of orifice and capillary compensated hydrostatic journal bearing. ASME Joint Conf. on Lub., Baltimore, October 1954 10. Smgh D.V., Sinhasan R. and Ghai R.C. Finite element analysis of orifice compensated hydrostatic journal bearing. Tribal. Znt. December 1976, 281-284 11. Sinhasan R., Sharma S.C. and Jain S.C. Performance characteristic of externally pressurized orifice compensated flexible journal bearing. Tribal. Trans. 1991, 34, 46S-471

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