Steady state analysis of noncircular worn journal bearings in nonlaminar lubrication regimes

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Tribology International Vol. 29? No. 6, pp. 493-498, 1996 Cotwiaht 0 1996 Elscvier Science Ltd . , CT u

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state analysis wurn journal

in nonlaminar lubrication regimes

Anjani Kumar and S. S. Mishra

The steady state behaviour of non-circular worn journal bearings is analysed for various wear depth parameters (&,), following Constantinescu’s turbulent lubrication theory. Computed r?sults are compared with published results. It is observed that geometric change caused by wear has a significant effect on the steady state characteristics of bearings. Copyright 0 1996 Elsevier Science Ltd

Keywords: worn 6earings, non-circdar bearings, htrbdent regime, sready sure performance, nodaminar regime

Introduction

There has been a great deal of interest in predicting the behaviour of bearings operating in the nonlaminar lubrication regime. Such a flow occurs in bearings because of the low viscosity process fluids and the high surface speed of journals.

The hydrodynamic plain circular journal bearings supporting high speed turbomachinery become noncircular after being used for long periods of time (over 10 years) due to wear on the bearing surface. Local changes in the bearing geometry due to wear strongly affect the hydrodynamic lubrication of bearings. There- fore, for reliable operation of rotor-bearing systems operating at high speeds, it becomes important to determine the effects of the geometric change (due to wear) on the performance of bearings.

Many researcherslm5 have examined the modes of bearing failure due to wear both qualitatively and quantitatively. Dufrane et ~1.~ were the first to investigate the worn journal bearings used in steam turbine generators and established a model of wear geometry for use in the analysis of such bearings. They subsequently analysed the effects of geometric

Mechanical Engineering Department, Regional Institute of Tech- nology, Jamshedpur-831 014, India. Received 29 March 1994; revised 20 September 1994; accepted 18 May 1995

change due to wear on bearing lubrication at low operating speeds. Vaidyanathan and Keith7 have analysed the performance characteristics of non-circular journal bearings for four different bearing profiles, namely circular, worn circular, two lobe and elliptical, considering the effects of turbulence and cavitation. Hasimoto et ai.* have examined theoretically and experimentally the effects of geometric change due to wear on the steady state characteristics including turbulence, in terms of film pressure, attitude angle and Sommerfeld number. Hashimoto ef uZ.~ have further determined the effects of wear on the dynamic coefficients and whirl onset speed of a rigid rotor theoretically in both laminar and turbulent regimes.

The aim of the present work is to present detailed numerical results on the effect of geometric change due to wear on the performance of noncircular worn journal bearings in the nonlaminar lubrication regime. The equations for hydrodynamic worn journal bearings have been solved numerically using a finite difference method with a successive over-relaxation scheme.

Analysis

Governing equations

The wear pattern and geometry of a worn journal bearing are shown schematically in Fig 1. Dufrane et ~1.~ observed a uniform wear pattern in width along

Tribology International Volume 29 Number 6 1996 493

Non-circular worn journal bearings: A. Kumar and S. S. Mishra

Nomenclature

L LID N POZ

radial clearance bearing diameter wear depth maximum wear depth eccentricity friction coefficient normalized friction force shear force on the journal surface,

dimensionless shgr force on the journal surface (Fs = 2CFslpIJRL)

turbulence coefficients local film thickness, dimsnsionless

local film thickness (hO = hJC) dimensionless film thickness at

beginning of the cavitated zone bearing length aspect ratio rotating shaft speed local film pressure above ambient,

dimensionless fluid pressure in the

film region (iO =s)

end flow from the bearing, dimensionless end flow

QJ2

(Q=g&) R journal radius Re, Re* mean Reynolds number (Re = pVC/

t.~), local Reynolds number (Re* = pVh/p)

V W,E

shaft surface velocity (V = RIO) load on bearing, dimensionless load

Sommerfeld number

(S =

mz 2wc2 pAJR2L

circumferential, radial and axial coordinates

diensionless axial coordinate

Greek symbols extent angle of the worn region

03 = of - es) wear depth parameter (&, = do/C) eccentricity ratio (eO = eJC) absolute viscosity of fluid dimensionless circumferential

coordinate (0 = x/R) angular extent of uncavitated film circumferential location of the

starting point and final point of the worn region

fluid density shear stress Couette surface shear stress,

nondimensional Couette shearing Th

stress ic = L c 1 PV

attitude angle angular velocity of journal

Fig 1 Schematic diagram of worn journal bearing

X0 = i + eocOse (11 and for the worn region in nondimensional form as:

7io = so + co c0se - c0se (e + +o) (2) where

The starting and final points of the worn region OS and Of, can be estimated by solving the following equation:

c0ge + c$~) = a0 - i (3) For the steady state condition, the generalized differen- tial equation for the film region, in the theory of hydrodynamic lubrication, can be written in nondimensional form as:

where G0 and GZ are turbulent coefficients. Constantinescul” suggested the following expressions for Go and GZ.

l/G0 = 12 + 0.0260 (Re*)“.8265

l/G= = 12 + 0.0198 (Re*)“.741 (3

the length of worn journal bearings, used in steam turbine generators, and almost exactly symmetrically located at the bottom (as shown in Fig 1).

The film thickness for the non-worn region is given in nondimensional form as:

494 Tribology international Volume 29 Number 6 1996


For the laminar regime, l/G6 = l/G= = 12.

Constantinescu and Galetusell have shown that the shearing stress acting on the surface can be represented by:

where, - ,rc = 1 + 0.0023 (Re*)*.=5 m

For laminar flow the value of yc is unity.

The boundary conditions in the film region are:

iO(O, ? 1) ‘= 0 (ambient)

$$(0,0) = 0 (symmetric)

j&3,i) = p<, (0+27&Y)

F0 = 0 if i0 < 0 (in the cavitation zone f3 2 tQ

Equation (4) is solved numerically using the finite difference method, satisfying the boundary conditions given by Eq. (8). A successive over-relaxation scheme is adopted to accelerate the convergence-of the iteration method and the pressure distribution p0 is obtained. The steady state characteristics are calculated from the pressure distribution.

Results and discussion The results obtained from the present analysis have been compared with the published results, obtained by the semi-analytical finite element method of Hashim- oto et ~1.~. The results are in excellent agreement with the published work (Fig 2).

The effect of geometric change due to wear on the steady state characteristics of journal bearings such as Sommerfeld number, frictional power consumption and flow rate have been computed and the results are presented in Figs 3 through 10.

The steady state performance of bearings depends on parameters such as Re, l 0, L/D. In the present analysis a parametric study has been carried out by varying all the parameters mentioned above.

Load-carrying capacity

Figure 3 shows the variation of Sommerfeld number with eccentricity ratio for various values of wear depth parameter for Re = 20000 and 30000 with LID ratio 0.5. The increase in wear depth parameter decreases the load capacity (i.e. increase in Sommerfeld number) in turbulent bearings. The effect of turbulence is to increase the load capacity and the effect is more pronounced at higher values of the wear depth parameter. The eccentricity ratio, under the constant Sommerfeld number, increases as the wear depth parameter increases.

The variation of Sommerfeld number with eccentricity ratio for difference values of L/D ratio is shown in Fig 4, for both unworn (&, = 0) and worn journal bearings. The load parameter is reduced in worn

L/D = 1.0 Re = 5000

- - - - Hashimoto et d. results (l9S6, Rd. 8) - Present analysis

l- .

11 111 I I I l l l l l l I I I IllIll I 0.005 0.01 0.1 1.0 2.0

Sommerfeld number (So)

Fig 2 Sommerfeld number versus eccentricity ratio for different values of wear depth parameter, &,

10.0 E LID = 0.5

- Re = 20000 4\ - - - - Re = 30000

0.01 1 I I I I y 0.0 0.2 (I.4 0.6 0.8 0.9

Eccentricity ratio (CO)

Fig 3 Sommerfeld number versus eccentricity ratio for different values of wear depth parameter, &,

bearings and the tendency becomes more pronounced as the LID ratio decreases.


- - - - so = 0.0 - so = 0.5

Re = I.5000

0.01

k \’

0.004 I I I .

0.0 0.2 0.4 0.6 0.8 0.9

Eccentricity ratio (q,)

Fig 4 Sommerfeld number versus eccentricity ratio for different values of L/D ratio

LID = 0.5 ---- Re=lOOO - Re = 20000


- - - - so = 0.0 - so = 0.5

Re = 15000

1.0 -.- 0.0 0.2 0.4 0.6 0.8 0.9


Fig 6 Coejjkient of friction versus eccentric@ ratio for different values of L/D ratio

L/D = 0.S - Re = 20000

P - - - - Re = 50000

0.1 1 I I I I I 0.0 0.2 0.4 0.6 0.8 0.9


ll 0.6 0.8 0.9

Eccentricity ratio (&J

Fig 7 Flow rate versus eccentricity ratio for different values of wear depth parameter, &,

Fig 5 Coejj%ient of friction versus eccentricity ratio for different values of weur depth parameter, &, the constant friction coefficient increases as the wear

depth parameter increases (Fig 5).

Friction coefficient

The variation of friction coefficient with eccentricity for different values of wear depth parameter and for different L/D ratios is shown in Fig 5 and Fig 6, respectively.

The effect of wear in increasing the friction coefficient of turbulent bearings is more pronounced at lower L/D ratios (Fig 6).

End flow

Friction coefficient is increased with increasing wear as well as with turbulence. The eccentricity ratio under

The variation of end flow with l O for different values of &, and for different L/D ratios is demonstrated in Fig 7 and Fig 8, respectively. End flow increases with increases in wear as well as with increases in Re. The

496 Tribology International Volume 29 Number 6 1996


100.0 =

2 t4 10.0 2

q

?

II IO

u 5

z iz 1.0 -

---- ho = 0.0 - so = 0.5

Re = ISOC lm = ‘O “2.0 --

1.5

fi

-*-- **--

.,

/ , ’

N ,’ ,

/

/ 1 ,

, ,

,

**-- 47.0 **---- J- ,’ *- **-- , 4 , .’ **-- **-- ---

+$ ,. J4 4 -0.5 , , , **-- **-- // .. f**----

.’ /. , , , , ,

0.1 1 I I I I I 0.0 0.2 0.4 0.6 0.8 0.9

Eccentricity ratio (Q)

Fig 8 Flow rate versus eccentricity ratio for different values of L/D ratio

I on r LID = 0.5 Re = 40000

I. so = 0.0 \

2. so = 0.2 2

3. 8. = 0.4 \ 4. 8” = 0.5

2

I I I I I 0 0.2 0.4 0.6 0.8 I.0

Eccentricity ratio (E,,)

Fig 9 Attitude angle versus eccentricity ratio for different values of wear depth parameter, &,

effect of turbulence is more pronounced at higher values of wear (Fig 7). The effect of wear on the end flow rate is increased with increases in L/D ratio of the bearings (Fig 8).

Attitude angle

The variation of attitude angle with eccentricity ratio for different values of wear depth parameter is shown in Fig 9; and for different L/D ratios, it is depicted in Fig 10.

4. LID = 2.0

I I I I I n cl.2 0.4 0.6 0.8 I .o

Eccentricity ratio (co)

Fig 10 Attitude angle versus eccentric@ ratio for different values of L/D ratio

Attitude angle decreases with increases in wear and the effect is slightly greater at larger eccentricity ratios (Fig 9). Figure 10 shows that change in attitude angle due to wear is uniform for all values of L/D ratio.

Conclusions The following conclusions may be drawn from the above analysis and discussion:

The effect of wear is to decrease the load- carrying capacity, and to increase the frictional drag and flow rate of journal bearings operating under turbulent flow conditions. The effect of wear on load-carrying capacity and friction coefficient is greater at lower L/D ratios. Turbulence increases load-carrying capacity, friction coefficient and end flow and the effect of turbulence is more pronounced at higher values of wear depth parameter. The eccentricity ratio increases with wear depth parameter under constant load parameter and also under constant friction parameter.

References 1.

2.

3.

4.

5.

bckworth W.E. and Forrester P.B. Wear of lubricated joumaJ bearings. Conf. on Lubrication and Wear, Proc. Inst. Mech. Engs. London, October 1957, 714-719 Forrester P.B. Bearing and Journal Wear. Symp on Wear in the Gasoline Engine. Thornton Research Centre, October 1960, 75-91 Katzenmeier G. The influence of materials and surface quality on wear behaviour and load capacity of journal bearings. Tribology Convention, Proc. Inst. Mech. Engs. Paper No. Cl lOOl72, 1972, 80-84 Mekhtar M.O., Howartb R.B. and Davies P.B. Wear characteristics of plain hydrodynamic journal bearings during repeated starting and stopping. ASLE Trans. 1978, 20, 3, 191-194 Szerf A.Z. Ttibology - Friction, Lubrication, and Wear, McGraw-Hills New York, 1980, .507-536



6. Dufraue K.F., Kannel J.W. and McCbkey T.H. Wear of steam turbine journal bearings at low operating speeds. J. L&r. Technol., Trans. ASME, 1983, 105, 3, 313-317

7. Vaidyanathan K. and Keith Jr. T.G. Performance characteristics of cavitated noncircular journal bearings in the turbulent flow regime. Trib. Trans. 1991, 34, 1, 35-44

8. Hasimoto H., Wada S. and Nojima K. Performance characteristics of worn journal bearings in both laminar and turbulent regimes. Part I: steady-state characteristics. ASLE Trans. 1986, 29, 4, 565-571

9. Hashiioto H., Wada S. and Nojima K. Performance characteristics of worn journal bearings in both laminar and turbulent regimes, Part II: Dynamic characteristics. ASLE Trans. 1986, 29, 4, 572-577

10. Constantinescu V.N. Theory of turbulent lubrication. Paper No. AEC-TR-6959, US Atomic Energy Commission, Division of Technical information, 1968

11. Constantinescu V.N. and Galetuse S. On the determination of friction forces in turbulent lubrication. ASLE Trans. 1%5, 8, 367

498 Tribology International Volume 29 Number 6 1996

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Steady state analysis of noncircular worn journal bearings in nonlaminar lubrication regimes