A full lubrication model for rough surface piston rings

Zhou Quan-bao, Zhu Tie-zhu and Wang Rong-sheng*

A model to simulate the effects of one dimensional roughness of the piston ring surface on lubrication and friction is developed based on stochastic theory. The applications of the model to an actual diesel engine indicate that about 8 .3%-9.4% increase in friction power loss can be expected when the rough surface (d = 0.6 #m) rather than the smooth surface is considered. The peak friction forces in compression and power stroke were predicted to occur very near the fired top dead centre (ftdc). This agrees with measured data. Some roughness contacts near the ftdc were observed.

Keywords: piston ring, surface roughness, lubrication, friction

Introduction

Owing to the intrinsic constraints of the machining procedure, the formed piston ring and cylinder liner surfaces are more or less rough. This kind of roughness, moreover, is random for an arbitrarily selected compon- ent. Only stochastic theory can be used to study their overall behaviour.

Use of the stochastic method is common in the study of bearing lubrication characteristics ~ -5. Its advantages are that it can reflect the effects of the surface roughness on oil film formation, oil film pressure and surface friction.

The following assumptions are required in the establish- ment of the rough surface lubrication model.

The piston ring face shape after running-in is shown in Fig 1. Both the ring and cylinder liner surfaces are rough. The actual distance between the two surfaces is

H(x, t) = h(x, t) + h~(x, 4) (1)

where h(x, t) is the nominal be expressed as

h r q- x2/(2R1) h(x, t) = h, + x2/(2R2)

oil film thickness, which can

- b l ~<x~<0 (2)

O < x < ~ b 2

hs(x, 4) is the random part of the oil film thickness. Its probability density function is given by 2

f(hs) = {305/32(c2 - h2)3/c 7 - c < hs < c else (3)

where c = 3d. d is the standard deviation of the random variable h s.

Each ring is full oil lubricated. The flow-in oil film thickness for each ring is large enough, so that no starved lubrication phenomena will occur.

The wet length of the piston ring extends from x = - b 1 to x = Xo (Fig 1, when the piston moves up). The oil film

* Department of Naval Architecture, Dalian University of Technology, Dalian, Republic of China

in this region is called the working film. The boundary conditions are

x = Xo dp/dx = O. (4)

x = Xo ~ = O. (5)

The ring surface and cylinder liner inner surface are one dimensionally rough, in the x direction only (Fig 1).

With these assumptions, the Reynolds equation can be expressed as

E(dp/dx) -- dP/dx

= 6 pU{E(1/H 2) + 2E(1/H3)} (6)

where

E(x)=flc xf(hs) dhs (7)

is an expectation value of x. 2 is an integrated constant. Integrating Eq (6), the oil film pressure distribution can be obtained

f x {p l _bl<<x<~O P = dP = (8) -b, P2 0 <( X ~ X 0

where

P1 = F(x, R l, ,~) - F ( - bx, R, , 2) (9)

P2 = F(x, R2, )~) -- F( -- bl, R,, 2) (10)

~ h L U Pe Pz

press lume Oil ring

Fig 1 Piston rings' shape, oil film and pressure distribution. U = piston speed

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Zhou et al - - lubrication mode/for rough surface piston rings

The details of the function F are described in the appendix. The integrated constant 2 and oil film length in a diverging channel, Xo, can be determined using boundary conditions (Eqs (4) and (5)). The oil film resultant load per unit length is then

fo f:o = ,0 dx = P1 dx + P2 dx (1 l) - -h i - h i

The load acting on the piston ring in the y direction is

w~ = (b~ + b~)eo + 0.5(b~ + 2b2 - Xo)(ez - P~) (12)

It consists of ring elastic pressure P~ and a part of the gas pressure difference (P~- P0. Another part of (P~- P 0 is offset by Ph which decreases linearly in the x direction as shown by the dotted line in Fig 1. From the force balance request We = Hip, the minimum oil film thickness, hr, under each ring at any crank angle can be solved step by step using the try and error method. This method, however, will be void when U = 0. The following Reynolds equation should be used in these cases.

d[~/dx = 12 #(dh/dt)(x - xo)E(1/n 3) (13)

where (dh/dt) is the rings' radial movement speed. The shear stress acting on the friction surface is then

= O. 5d~/dxE(1/HE)/E( 1/H 3) + # U {4E(1/H)

- 3(E(1/HE))2/E(1/H3)} (14)

and the friction force is

f Zo

Fr = ~Do ~ dx (15) - b l

Resu l ts a n d d iscuss ions

Obviously, the model set up above is general. The input data include cylinder bore, Do; oil viscosity, #; engine speed, n; rod length, Rt; half stroke, Ro; gas pressure, Pz, PI and P2; ring geometric parameters (eg bl, b2, RI, R2 and t); ring elastic pressure Pc; and the standard deviation d of the random variable, hs. Table 1 gives an example of the 6135ZG engine.

The gas pressure in the cylinder (P~) and in the inter-ring space (P~, P2) can be obtained from the measured data or from the cycle simulation program. A plot of them as a function of crank angle is shown in Fig 2. The oil viscosity depends on the oil temperature. For a Chinese made # 14 diesel engine lubrication oil (similar to SAE 40), oil viscosities of 0.021 N s m -z, 0.017 N s m -2 and 0.013 N s m -2 correspond to oil temperatures of 80°C, 85°C and 95 °C respectively.

Fig 3 shows the instantaneous minimum oil film thickness versus crank angle diagram at different oil temperatures.

Table 1 The input data of the 6135ZG engine

D O = 1 35 mm n = 1500 r / m i n Rj = 280 mm R o = 70 mm R1 = 2.0 m R2 = 2.5 m

Pe = 2 0 0 0 0 0 N m -2 (gas r ing) 5 0 0 0 0 0 N m -2 (o i l r ing)

b x t = 3 x 5 mm (gas r ing) 2 x 5.2 mm (oi l r ing)

bl = b /2 d = 0.6 #m

12

10 L\\\',l Pz

] / ~ T°pgas l E ~ 8 ring

/ ~Pz 2nd gas ~ 6 ring / / \ / P2 4 / ~ 3rd gas ~\\\-,]

ring I

0 m " ~ " " " " ~ m " ' l " " I I I t [ -40 ° ftdc 40 ° 80 °

Crank angle

Fig 2 Pz, P1 and P2 as a function o f crank angle.

P1

120 °

6

E =L 4

2 0 ~I0° --- ' I ~ " ~I , , ,

7r (ftdc) 21r 3~ 4rr Crank angle

Fig 3 Minimum oil film thickness o f the top gas ring. Oil temperature: • 80°C, • 85°C and x 95°C.

The minimum oil film thickness in the cycle occurs near ftdc, Many calculations indicate that the oil film thickness iteration near the ftdc (about 10°CA (crank angle) in all) will not converge if the limit of h, 4: c is given. But if h~ is allowed to take a value less than c, the expectations such as E(1/H), E(1/H 2) and E(I/H 3) will be divergent. The reason is that the real roughness contacts exist in this engine near ftdc while the present model is invalid in this case. Mixed lubrication models which can deter- mine the load sharing between film and asperities are needed. Therefore, the dotted line near ftdc in Fig 3 is not obtained by Eq (6) or by Eq (13). It is drawn by assuming that the oil film in the short time period near ftdc varies linearly.

The relationship between friction force and crank angle at different oil temperatures is shown in Fig 4, where Fig 4(a) shows the results of the rough surface model (simply called rsm) and Fig 4(b) shows the results of the smooth surface model (ssm) under the same conditions. In both cases, the lower the oil temperature (or the greater the oil viscosity), the thicker the oil film, and the greater the friction force. The differences between rsm and ssm are obvious: they have different friction force-crank angle diagram shapes and different peak friction force values. In ssm, the peak friction force in each stroke is approxi- mateJy all in the middle of the stroke, with the maximum value of the top gas ring about 18-25 N (which corres- ponds to an oil temperature of 80-95 °C) occurring in the power stroke. In rsm, however, the peak friction force of the single piston ring is reached very near the ftdc, both in the compression stroke and power stroke. For the top gas ring, it peaks to 34-42 N, 16-17 N higher than that

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of the ssm. That the maximum friction force occurs at or near ftdc is reasonable and coincides with the measured friction curve by Furahama and Sasaki 6.

Table 2 lists the peak friction force values from rsm and ssm at three different oil temperatures. The higher the oil temperature, the greater the relative error.

In speaking of the friction power loss, use of the ring assembly total friction power loss might be more suitable. Fig 5 shows the piston ring assembly's friction power loss diagram obtained with the ssm and rsm at different oil temperatures. Their difference varies from 8.30/0 - 9.4 %. For the engine 6135ZG, the measured total friction power loss for each cylinder is about 4.43 kW. Since 20-56 % of it is contributed by the piston ring assembly, according to a lot of reports 7'8, these computat ions do agree with the lower end of this range.

Z

LL

30

20

10

0

10

20

30

40

/ ~ Crank angle

30

20

10

0 10

20

30

/ ~ Crank angle

V b

Fig 4 Friction force diagram of the rough surface model and smooth surface model (top gas ring). (a) Rough surface and ( b ) smooth surface. Oil temperature: • 80°C, • 85°C and x 95°C.

Zhou et al - - lubrication mode/for rough surface piston rings

0.8

0.6

0.4

0.2 i 80 85

Rough surface

Smooth surface

I I

90 95

0il temperature, °C

Fig 5 Friction power loss o f the ring assembly at different oil temperatures.

C o n c l u s i o n s

• A full oil lubricated rough surface piston ring/cylinder liner friction model has been set up. It is a base from which to study oil film formation and friction force under rough surface conditions.

• Better prediction accuracy and a more reasonable friction force curve can be expected if the piston ring/ cylinder liner surface roughness effects are included in the model. Friction power loss of the rsm is 8.3 % - 9 . 4 % greater than that of ssm.

• Real roughness contacts near fired top dead centre are observed in the 6135ZG engine. In this case, the mixed lubrication model will be more suitable.

• The model presented here is equally applicable to a reciprocating piston engine or a compressor.

• A mixed lubrication model may be added to the present model without great difficulty.

R e f e r e n c e s

1. Tzeng S. T. and Saibel E. Surface roughness effect on slider beating lubrication. ASLE Trans., 1967, 10, 334-338.

2. Christensen H. Stochastic models for hydrodynamic lubrication of rough surfaces. Proc. I. Mech. E., Pt 1, 184 (55), 1969-1970

3. Chriatensen H. and Tonder K. The hydrodynamic lubrication of rough bearing surfaces of finite width. Trans. ASME, July 1971, 324-330

4. Christensen H. A theory of mixed lubrication. Proc. I. Mech. E., 186 (41), 1972, 421-430

5. Lebeck A. O. A study of mixed lubrication in contacting mechanical face seals in Surface roughness effects in lubrication. Dowson D., Taylor C. M., Godet M., Berthe D. (eds), 1978

Table 2 M a x i m u m fr ict ion forces f rom rsm and ssm

Oil temperature, °C Maximum friction force, N

rsm ssm Error (abs.) Error (rel.)

80 42.0 25.0 17.0 85 38.5 22.0 1 6.5 95 34.0 1 8.0 1 6.0

40.5% 42.9 % 47.1%

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Zhou et al - - lubrication m o d e / f o r rough surface piston rings

6. Furuhama S. and Sasaki S. New device for the measurement of piston frictional forces in small engines. S A E paper 831284

7. Rohde S. M. A mixed friction model for dynamically loaded contacts with application to piston ring lubrication. Proceedings o f the 7th Leeds-Lyon Symposium on Tribology, 1980

8. Zhou Q. B., Wei X. Y. and Wang R. S. A starved lubrication model for the piston rings. Paper submitted to 1988 A S M E / A S L E Joint Tribology Conference

A p p e n d i x

F r o m Eq (7) c a n be o b t a i n e d

E ( 1 / H ) = 3 5 / 3 2 / c 7 { ( c 2 - hZ)31n((h + c) / (h - c))

+ 2 / 1 5 c h ( 1 5 h 4 - 40(ch) 2 + 33c4)} (A1)

E ( 1 / H 2) = 3 5 / 3 2 / c 7 { 6 ( c 2 - h2)2h ln((h + c) / (h - c))

- 0.8c(15h 4 - 25c2h 2 + 8c4)} (A2)

E ( 1 / H 3) = 3 5 / 3 2 / c 7 { 3 ( 5 h 2 - c2)(c 2 - h 2)

x ln((h + c) / (h - c)) + 2ch(15h 2 - 13c2)} (A3)

F r o m Eqs (2), (A2), (A3) a n d (6), a n d t e m p o r a r i l y r ep l ac ing R~ a n d R 2 wi th R, is o b t a i n e d

d P = 6 l t U 3 5 / 3 2 / c 7 D i Q t - 0.8c Di + 6 x 2 i - 1 "= i = 1

= F(x, R, 2) (A4)

where

D 1 --- hr 5 +h, c4 _ 2c2h, 3

9 2 = (c 4 + 5h~ - 6 c 2 h 2 ) / 2 R

D 3 = (5h 3 - 3 c 2 h , ) / 2 / R 2

D4 = (5h~ - c 2 ) / 4 / R 3

D~ = 5 h , / 1 6 / R 4

D6 = 1 / 3 2 / R 5

D 7 = 15h 4 - 2 5 c 2 h 2 + 8c 4

D a = (30h 3 _ 2 5 h r c 2 ) / 3 / R

D 9 = (90hr 2 - 2 5 c 2 ) / 2 0 / R 2

D l o = 1 5 h J 1 4 / R 3

D l l = 1 5 / 1 4 4 / R 4

Dx2 = 3 ( 6 c c h 2 - 5h 4 - c 4)

D 13 = 3(6cchr - l O h 3 ) / R

Dx4 = 3(3cc - 15h2)/2/R 2

D 1 s = - 1 5 h , / 2 / R 3

D16 = - 1 5 / 1 6 / R 4

D17 = 30char - 26c3hr

D 1 s = (45chr 2 _ 13c3)/3/R

D19 = 9 c h ~ / 2 / R 2

D2o--- 1 5 c / 2 8 / R 3

Qi = x 2 i - 1/( 2 i - 1) ln((h + c) / (h - c)) - ((2R)~) 2i+ 1

/ ( 2 i - 1 ) /R(T2 , p - T2, ) i = 1, 2 , . . . , 6 (A5)

T,p = (x / (2g)½) " - 1/(n - 1) - (hr + c)T~_ 2p

T2v = x / ( 2 g ) ½ - (h~ + c) ½ a r c t g ( x / ( 2 R ( h , + c)) ~)

7", = (x / (2R)½) " - 1/(n - 1) - (h~ - c)T~_ 2

7"2 = x / ( 2 R ) ~ - (hr - c ) ~ a r c t g ( x / ( 2 R ( h , - c)) t )

Therefore

P I = dPlx~<0 = F(x, R t, 2 ) - F ( - b 1, R1, 2) --hi

P2 = d P = d P + d P I ~ > o - b l Z > 0 - b l

= P1 [ z=o + F ( x , R2, ~,) - F(0, R2, 2)

= F ( x , R 2, 2 ) - F ( - b x , Rx, 2)

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