28

Paper 5

THE HYDRODYNAMIC LUBRICATION OF PISTON RINGS By T.Lloyd* Although the piston ring is a critical component in a reciprocating engine and although its successful functioning depends upon adequate lubrication, very little work has been done on the analysis of piston ring operation as a phenomenon of hydrodynamic lubrication. This paper outlines briefly the theoretical and experimental work that has been done, and produces a new analysis particularly suited to solution by computer. The analysis assumes : (a) that the shape of the piston ring face is parabolic, but not necessarily symmetrical; (b) that the lubricant viscosity is constant under the ring, but may vary in any fashion over the piston stroke; (c) that oil film disruption may be represented by calculating oil pressure assuming no disruption and then discarding negative pressures; ( d ) that the pressure upstream and downstream of the ring and the load on the ring may vary cyclically over the stroke; (e) that there is circumferential symmetry; and (f)that the motion of the ring parallel to the cylinder face is that of the piston, while the motion perpendicular to the cylinder face satisfies the force balance.

Using the simple example of the symmetrical ring face, isoviscous lubricant, constant load, zero upstream and downstream pressures, and sinusoidal piston motion, it is shown that there exists a parabolic shape which maximizes the minimum oil film thickness. Comprehensive results are presented for this case. The use of the more general analysis is illustrated for a diesel engine design for which the ring face shape has been measured.

INTRODUCTION

A

is performed by the piston ring of the internal combustion engine. I t must prevent the cylinder gases from escaping past the piston while contributing little to the friction losses of the engine and, at the same time, the oil that keeps down the friction must not be allowed to rise above the piston or it will be burnt. To do this the ring must slide at high speed over the cylinder liner surface, conforming closely to the liner shape, which may deviate significantly from circular. Much of its time is spent at high temperatures, so that lubrication is difficult. Because of the critical nature of its job, the piston ring has been the subject of many theoretical and experimental investigations from many different standpoints :

(3) stress analysis of the ring to give ring shapes having constant wall pressure or twisting under pressure; (4) vibration characteristics of piston rings ; (5) engine oil consumption with different ring packs; (6) gas blow-by with different ring packs; (7) friction losses attributable to the piston rings; (8) heat conduction from the piston to the liner through the piston rings.

DIFFICULT TASK

(1) liner-ring material properties to minimize wear; (2) porous materials to carry oil within the ring itself; The M S . of this paper was received at the Institution on 4th November 1968 and accepted for publication on 6th Decemba: IY68. 34 * Research Fellow, Department of Mechanical Engineering, University of hrottingham, Nottingham. Proc Instn Mech Engrs 1968-69

The basis for the analysis presented in this paper is that hydrodynamic lubrication plays an important role in determining the behaviour of the piston ring. This approach is now new; it was suggested as long ago as 1936 by Castleman (I)+. Nineteen years later Eilon and Saunders wrote a paper on piston ring lubrication ( 2 ) . These two papers, and indeed almost all other papers published in the field, suffer from the limitation that the effect of changes of ring velocity are neglected. Exceptions

t

References are given in Appendix -7.1. Vo1183 Pt 3P

THE HYDRODYNAMIC LUBRICATION OF PISTON RINGS

to this are the three papers by Furuhama (3)-(5), in which the piston ring was treated as a bearing subject to cyclic fluctuations of load and velocity. He chose a bearing geometry consisting of two circular arcs separated by a flat section, parallel to the liner. T o make the equations easier to solve, the effect of the trailing circular arc (the region where cavitation might be expected) on the pressure distribution was neglected. Reference (5) supports the theory, by correlation with experiment of the oil film thickness at mid-stroke. The object of this paper is to describe a theoretical method for predicting the oil film thickness between the piston ring and the liner wall for (1) a ring surface geometry defined by an offset parabola, and (2) a cyclic variation of load, velocity, viscosity, and upstream and downstream pressure. Using the simple example of the symmetrical ring face, isoviscous lubricant, constant load, zero upstream and downstream pressures, and sinusoidal piston motion, it is shown that there exists a parabolic shape which maximizes the minimum oil film thickness. The use of the more general analysis is illustrated for a diesel engine design for which the ring face shapes are known.

Notation b ‘Wedge’ coefficient, ~/2/ (2h’~r’). b’ ‘Squeeze’ coefficient, (dh’,ldf)jh’,. Ci A constant of integration. H Non-dimensional film thickness, hlh,. Film thickness at the position x. h h0 Minimum film thckness. h‘0 (= ho/R). (= dh’,/d?). K l A constant of integration. L Connecting rod length. I Piston ring length. 4 Offset of the parabola. P Non-dimensional oil film pressure, ph’o;2170~r‘. P” A constant of integration. 1’: Non-dimensional upstream pressure (pressure/2~w). p 2 Non-dimensional downstream pressure (pressure/2~w). Non-dimensional external load on the piston pa ring per unit length (load/4@). P Oil film pressure. R Crank radius. Radius of curvature, a t x = 0, of the parabola r used to define the shape of the ring face. Non-dimensional radius of curvature, r/R. r‘ S Speed modifying - - factor R sin i cos t 1L.t’l-[(R/L) cos 6J2 t Time. i Non-dimensional time, tw . Velocity of the piston (= Rws). Li h10

{

Proc Instn Mech Engrs 1968-69

I.

29

Velocity of the ring in its groove normal to the direction of piston motion. Non-dimensional distance along the ring 1 \/(2hkJ * An integration constant. Distance along the ring, from the point of minimum film thickness. Non-dimensional oil film force (force/47wR). Oil viscosity. Substitution variable (X = tan 4). Engine speed in radiunit time.

V

X

[i

XO X

A

; w

Subscripts 1,2,3,4

The condition at the point marked (l), (2), (3), and (4) in Fig. 5.1. THEORY

The theory is based upon all of the assumptions implicit in using the Reynolds equation to find the pressure distribution in the oil film between the piston ring and the liner, together with the following particular assumptions :

(1) The oil viscosity does not vary along the ring face, although it may vary with temperature at different positions along the liner. (2) The ring does not tilt in its groove. ( 3 ) There is sufficient oil available for the hydrodynamic equations to apply. ( 4 ) Oil film disruption may be accounted for by calculating the pressure distribution for a continuous oil film and then setting all negative pressures to zero. (5) An off-centred parabola is sufficiently general to represent the bearing shapes encountered in practice. (6) Lubrication is hydrodynamic over the complete stroke. (7) There is circumferential symmetry, so that the problem may be treated as if it were one-dimensional. The piston ring geometry is shown in Fig. 5.1. r is the radius of curvature, at x = 0, of the parabola used to define the shape of the ring face. The Reynolds equation for this situation is : dh = -6U-+12~ . (5.1) dx This may be integrated once with respect to x and put into non-dimensional form,

dP- --6 Jb(H-H”)

- 2b’( X-Xo)+ KI} (5.2) H3‘ dX Integration of equation (5.2) is made easier by using the substitution, X = tan 4. Integrating, P = -& ([8(sin 24+2#1)-sec~ 4” x(sin4~+8sin2~+124)+Po]b -[Cl -2Xo(sin4 4 s 8 sin 2+ 12+)- 16 C O S ~4]b’ +K,(sin 4++8 sin 24+124)+K2} (5.3)

+

. .

Yo1183 Pt 3P

T. LLOYD

30

\

Liner

\

Fig. 5.1. Piston ring geometry The constants of integration, sec2 4*, Po, X,, and C,, may be found from the requirement that the wedge and the squeeze components of the pressure must both be zero at position (1) and position ( 2 ) . Kl and K, arise from the presence of upstream and downstream pressures, and may be eliminated by the conditions that P = Plh’o/r‘ and P = Pzh’o/r‘ at positions (1) and ( 2 ) respectively. Before the pressure distribution given by equation (5.3) can be integrated to give the oil film force, any region of negative pressure must be identified and excluded from the integration. Using the position of minimum pressure, the position of maximum pressure, and one end of the ring as starting points, the two internal points at which P = 0 are found by the secant rule. The oil film force may be calculated by integrating the formula from position (1) to (3) and from (4) to (2) as indicated in Fig. 5.1 :

A = f(63)-f(+1>+f(+Z>

-f(44)

.

(5.4)

where -J2*’3!2 I ~

= 162/(2h’,)

([16$ tan 4-secZ

4*

x (12$ tan 4 - 2 cos 24) +Po tan 4 ] b -[Cltan +2X0(124 tan 4 - 2 cos 24) -84 -4 sin 241 b’ + K , (124 tan # - 2 cos 2Q.)+Kz tan +} . (5.5) Proc Instn Mech Engrs 1968-69

d = A,b+O,b‘+O,K1+d,Kz . . (5.6) In evaluating the constants of integration, Kl and K,, it is found that: Let

Kl = ( a l ~ l + u 2 P 2 ) ~ ’ o / r ’ .} K , = (a3P1+ua,PL)h’,/r‘

.

(5*7)

where a,, a,, u3, and u4 are constants. The parameters which affect the oil film force, and which may vary with time, are s, KO,hf0, Plyand P2. I n order to generate the locus of h’, over the stroke, the partial derivatives of the oil film force with respect to these five parameters must be known. All but one may be obtained from equation (5.6) : % A_ - A1 as r’d(2h’0/7.’) i?A A, -- hfO . (5.8) 2A A3a,h’0 d4a3hr0 _ ?PI r’ +- r

1

ah’o

I

If inertia effects are neglected, then a t all times the internal oil film force must equal the external load, Pa. VoZ 183 Pt 3P

31

T H E HYDRODYNAMIC LUBRICATION OF PISTON RINGS

Because this parameter includes viscosity, then viscosity as well as external load may vary cyclically and be taken into the same non-dimensional term without increasing the complexity of the analysis. The only assumption that must be made is that the viscosity of the lubricant between the ring and the liner is constant. Satisfaction of the force balance at time i + 8 i is expressed by the equation

(Pa)ftdt =Ll,+84,

. . . .

(5.9)

Sd, is built up from the derivatives given in equations (5.8), while the remaining derivative (ad/?h’,) is found by knowing the total change (O,-d,-,,) over the previous step and subtracting from it changes due to the other known partial derivatives. When equation (5.9) is built up in this way, it may be put in the form dlh’o+d2&’o+d3= 0 . . (5.10) and solved by: -d3+(kO)f (d,/S2-0.5d2) (jG)L7h t = d,/ 6, +03d2 (h‘o)i+ot = (hfo)i+O.5[(h’o),+,,+ ( A ’ O ) J ~ ~ This analysis enables a prediction of to be made a t time i+ Si from conditions at i, and so a locus of h10 may be marched out. At the end of each time step, the internal and the external forces should be equal and this is used to test the adequacy of the time step and to change it when necessary. The solution is marched from mid-stroke, when the squeeze effect can be ignored, and solution is continued until the locus is cyclic.

T H E P A R A B O L I C RING S H A P E GIVING T H E G R E A T E S T MINIMUM OIL FILM THICKNESS

Fig. 5.2 shows piston ring loci for the simple case of a symmetrical ring face, isoviscous lubricant, constant load, zero upstream and downstream pressures, and sinusoidal piston motion for three curvatures of ring face. The general shapes of the loci may be explained in the following way. A highly curved ring face will develop a large pressure by wedge action, while a flat ring face would generate no pressure at all by wedge action. The reverse is true for the development of pressure by squeeze action, since a highly curved surface allows the oil under it to escape easily as the ring falls towards the liner surface, while a flat surface does not. In Fig. 5.2 it can be seen that the highly curved ring has a very large minimum oil film at mid-stroke, when the action is predominantly wedge, but that it falls very rapidly near the ends of the stroke when the action changes to squeeze. The flattest ring never achieves a very large minimum of oil film even at midstroke, but the fall at the end of the stroke is not so severe. Two conclusions can be drawn from this figure. First, even with a very large change in curvature (a factor of 20) the range of minimum oil films is not very great (a factor of 1.6), and second, there exists a parabolic ring face shape which maximizes the minimum oil film thickness. If the mechanism that creates a suitable ring face shape is that a poor shape will come into contact with the liner and with any small particles in the lubricant, then the ring may be expected to wear until its shape is sufficiently close to the optimum as to slow down the rate of wear to some low value. Thus the optimum ring shape is of particular interest. Optimum values of radius of curvature and the corresponding values of minimum oil film thickness are given in Fig. 5.3. This figure has been reduced to two lines by the following similarity analysis,

Mid-

0L 15

I

P

20

25

30

0

35

a/ I

Fig. 5.2. Typical piston ring loci for steady load: sinusoidal motion Proc Instn Mech Engrs 1968-69

Fig. 5.3. The parabolic ring shape giving the greatest minimum oil Jilm thickness Vol I83 P t 3P

32

T. LLOYD No. 3 Ring

No. 2 Ring

No. I Ring

/-

I

- - Talysurf proflie - _ _ _ Approximating p a r a b o l a Fig. 5.4. Diesel engine piston ring projiles Equation (5.5) shows that two cases will give the same values of d ~ ’ ( h ’ ~ / rprovided ’~) that b and also 4 at x = &Z/2 are the same. Now

b=

S

. .

v’(2h’0Y’) This requires that I Z ‘ ~be Y ’identical.

(5.11)

]

[-2R

. (5.12) ~l(2h’~~’) As I Z ’ ~must Y ’ be constant, equation (5.12) implies that 1;R must also be constant. For a given Z/R: 4s=1,2= tan-1

d

J>ccd?~’,”cc A Y

. .

around each ring were taken, and the measurements were repeated with the rings reversed in the rig. In all, 240 profiles were obtained. The conclusion was that the forms shown in Fig. 5.4 represented the three ring shapes. This figure also shows the offset parabolas used to represent these shapes. T h e combustion chamber pressure distribution and geometrical details were supplied by the manufacturers, so that the ‘blow-by’ analysis described by Englisch (7) could be used to give the pressure distribution above and below each ring over the stroke. This analysis assumes that blow-by is unsteady adiabatic flow through orifices (the ring gap) into containers (the volume trapped between

(5.13)

The latter two functions are plotted against l!R In Fig. 5.3, and so enable r’ and li0 to be evaluated for any combination of parameters. From the graph of dh’02 against RIZ, it may be concluded that the minimum film decreases with increasing load and with decreasing engine speed, crankthrow, ring length, and oil viscosity. ANALYSIS A P P L I E D TO A DIESEL ENGINE

As a demonstration, the full analysis has been applied to an existing engine-a Ruston and Hornsby Limited AT type. This is a four-stroke diesel engine, running at 500 rev/min, with a bore and stroke of 12.5 and 14.5 in respectively. Each piston has three compression rings. Three of these rings were taken from an engine after service and their face shapes were carefully measured. This is described in detail by Ferrett (6). The profiles were measured with a Talysurf machine, six stations Proc Instn Mech Engrs 1968-69

Pressure above t h e 2nd r i n g Pressure above

0 ) bdc

I

I

I

I

tdc

b d c.

tdc

bdc

C R A N K POSITION

Fig, 5.5. The pressure distribution between the piston rings over the cycle Vol 183 Pt 3P

THE HYDRODYNAMIC LUBRICATION OF PISTON RINGS

0

10 DISTANCE D O W N THE LlNEFi--rn

20

adjacent rings). Inertia effects of the rings are neglected, the axial positions of the rings wifhin the groove being determined solely by the pressure differentials across them. The pressure distributions obtained are shown in Fig. 5.5. The temperature distribution down the liner wall was supplied by the manufacturers. This has been converted into a viscosity distribution by assuming that the lubricant has the properties of SAE 30, and this is plotted in Fig. 5.6. The importance of the variation can be gauged from the fact that it changes by a factor of 3 in the region of interest. The pressure exerted by the ring was calculated from the diarnetral closing force. The resultant ring loci are shown in Fig. 5.7. It can be seen that the two rotations of the four-stroke cycle, with the associated pressures, produce quite different loops in the ring loci. The offset of the parabolas used for the ring shapes results in loci very unlike the symmetrical figure of eight loci in Fig. 5.2. Loci such as these indicate considerable scraping of the oil from the top of the liner to the bottom. From the appearance of the shape of Ring 1, it seems very probable that it ran tilted. Indeed, the computer programme would not work at all when the offset was No. 2 ring

t.d c

Mid stroke

included, and with no offset the minimum oil film thickness was only 0.000 006 in. T h e minimum oil film thicknesses for Ring 2 and Ring 3 of 0.000 10 and 0.000 09 in respectively should be compared with the Talysurf records for the ring shapes in Fig. 5.4. The minimum film of Ring 2 is large compared with the surface irregularities, but that of Ring 3 is of the same order as the surface roughness. CONCLUSIONS

Fig. 5.6. Cylinder wall oil viscosity

No.1 r i n g ( n o o f f s e t )

33

Consideration of the piston ring as a hydrodynamic bearing increases our understanding of its mode of operation, and the computer makes it possible to deal easily with many of the complexities involved in such an analysis. A problem which must be solved before the technique can be used as a design tool is the factor which determines the ring shape of a ‘run in’ piston ring, so that it can be predicted at the design stage. This paper suggests one approach to the problem, by demonstrating that a shape of ring exists which maximizes the minimum oil film thickness. It also shows how the analysis may be used when the ring geometry is known. ACKNOWLEDGEMENTS

The author would like to thank Ruston and Hornsby Limited for supplying the engine details and the piston rings used in this investigation; Mr E. Ferrett for the careful measurements of the piston ring profiles; and Professor A. G. Smith for placing the facilitics of the Department of Mechanical Engineering at his disposal. APPENDIX 5.1 REFERENCES

(I)

CASTLEMAK, R. A. ‘A hydrodynamic theory of piston lubrication’, IJhpics 1936 7, 364. No. 3 rirg

tdc

_- 1

Mid-

stroke

0.001

M I N I M U M OIL in

FILN-

b.d.c

b.d.c

Fig. 5.7. The minimum oiljlm loci of the diesel engine piston rings Proc Insrn Afech Engrs 1968-69

Vol183 Pt 3P

34

T. LLOYD

EILON,S . and SAUKXIERS, 0. A. ‘A study of piston ring lubrication’, Proc. Znstn mech. Engrs 1957 171,427. (3) FURUHAMA, S. ‘A dynamic theory of piston-ring lubrication (First report: calculation)’, Bul1.J.S.M.E. I959 2,423. (4) FURUHAMA, S. ‘A dynamic theory of piston-ring lubrication (Second report: experiment)’, Bul2.J.S.M.E. 1960 3,291. ( 5 ) FURUHAMA, S . ‘A dynamic theory of piston-ring lubrication

(2)

Proc Instn Mech Engrs 1968-69

(Third report: measurement of oil film thickness)’, Bul1.J.S.M.E. 1961 4, 744. (6) FERRETT, E. F. C. ‘The lubrication of piston rings’, B.Sc. thesis, University of Nortingham, 1967. (7) ENGLISCH, C . ‘Abdichtungsverhaltnissevon Kolbenringen in Verbrennungskraftmaschinen’, Auto.-tech. 2. 1938 41, 579.

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