Vol. 44 Supp.
SCIENCE IN CHINA (Series A)
August 2001
Numerical simulation of piston ring in the mixed lubrication WANG Wenzhong, HU Yuanzhong, WANG Hui & LIU Yuchuan State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China Received July 5, 2001
Abstract Piston and piston ring lubrication is a factor that strongly affects the performance of the reciprocating internal combustion engine. When the oil film thickness becomes smaller than a certain value, depending upon the roughness of the surfaces in contact, mixed lubrication has to be considered. This paper analyzes the lubrication of piston ring and predicts pressure distribution, lubricant film thickness and surface deformation. The work is based on a unified numerical approach assuming that the pressure distribution obeys Reynolds equation in hydrodynamic lubrication regions while in asperities contact regions, the contact pressure can be obtained through the so-called reduced Reynolds equation. The computation experience shows the unified equation system is stable and can deal with severe operating conditions. Keywords: mixed lubrication, elastohydrodynamic lubrication, piston ring, surface roughness.
Piston-ring lubrication plays an important role in engine fuel and oil consumption. The studies on the friction and lubrication behavior of piston-ring pack can improve the performance, efficiency and prolong durability of engines. Substantial work has been done over the years to understand the ring pack system involved in the interactions of various phenomena such as ring twist, gas flows through gaps and lands/grooves, hydrodynamic and mixed lubrication etc. However, various aspects of ring pack behavior, e.g., the inter-relationships between lubrication and ring friction and wear, are not as well understood. It has been widely recognized that hydrodynamic lubrication is prevailing between the piston rings and cylinder bore throughout much of the stroke in well reciprocating engines. The theory for a simple case was given by Lloyd[1], and others tried to extend this theory to take account of the various complications that might be expected to occur in the lubrication of a practical engine. The piston ring surface profile is an important factor. Castleman[2] made some pioneering calculations by considering ring surface as a convex, symmetrical profile, and predicted oil film thickness of 10 µm which is surprisingly in good agreement with more calculations. Other forms of the ring face profile were adopted by some investigations, such as Furuhama[3] et al. Dowson et al.[4] presented a significant contribution to the numerical analysis of piston ring lubrication. The cyclic variation of film thickness was predicted by a full numerical solution and the analysis was extended to a complete ring pack, in which the role of inter-ring pressures and lubricant starvation is considered. Although the lubrication of piston ring is hydrodynamic over most of a stroke, mixed lubrication appears in the locations near TDC and BDC. The study on the lubrication of piston ring has to be extended to the mixed lubrication regime. By combining the Average Reynolds equation
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developed by Patir and Cheng[5] with Greenwood and Tripp’s[6] asperity contact model, Rohde[7] applied a mixture friction model to piston ring lubrication in order to examine the dependence of film thickness and friction power loss on surface topography. The mixed lubrication of piston ring was also considered by Jeng[8] and other researchers. In these early analyses, a conventional way was to predict average pressure and average film thickness by means of statistical models. However, the local information on pressure fluctuations and asperity deformation would be lost during the process. On the other hand, there have been significant progresses in deterministic solutions of rough surface EHL. Especially, Hu and Zhu[9] recently presented a unified approach in which the Reynolds equation was consistently used in both hydrodynamic and contact regions. This deterministic model allows one to simulate the physical process of mixed lubrication and to collect information on local pressure and stress fluctuations. The objective of the present paper is to develop a more realistic model based on the unified numerical approach, for predicting lubrication behavior of piston ring, including the effects of various factors such as surface roughness, variation of gas pressure in inter-ring space et al. This computer program package is capable of simulating the entire transition from the full film EHL, mixed, down to boundary lubrication. 1 Theory and control equation 1.1 Geometry and coordinate system. The ring surface in contact is assumed to be a round profile (fig. 1). Considering the computation efficiency, it is not necessary to solve the Reynolds equation over the entire contact surface. For this reason, only a narrow piece is taken from the piston ring/cylinder assembly. The computation domain thus takes a rectangular form (the shadow area in fig. 2). Comparing with the radius of ring, the width of computation domain in circumferential direction is so small that circumferential radius of ring can be assumed to be infinite in considered segment. When the load is large, the deformations of ring and bore should be taken into account. Eventually, the expression of film thickness between ring face and cylinder wall is
h( x, y, t ) = h0 (t ) + hs ( x, y ) + δ 1( x, y, t ) + δ 2( x, y, t ) + V ( x, y, t ) ,
Fig. 1. Geometry and coordinate system.
Fig. 2. Pressure domain and computation domain.
(1)
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where hs ( x, y ) a function for the profile of piston ring.
δ1(x, y,t) , δ 2(x, y,t) roughness amplitude of piston ring and cylinder wall respectively. V ( x, y, t ) — surface deformation calculated by
V ( x, y , t ) =
2 πE ′
∫∫ Ω
P(ξ , ζ ) (x − ξ )2 + ( y − ζ )2
dξdζ .
(2)
It should be noted that the pressures outside of computation domain contributes to the surface deformations on the locations inside the domain. Under the assumption that the pressure distribution outside of the domain is a symmetrical image of that in the domain, an extended pressure domain, whose sides are as two times long as that of the computation domain, can be obtained as showed in fig. 2. 1.2 Reynolds equation The 2-D Reynolds equation has been used in the present study, in which surface roughness effects are included. The dimensionless form of the Reynolds equation is: ∂ ∂X
∂P ∂ ∂P ∂ ( ρ H ) ∂ ( ρH ) + = ε y + ε x ∂X ∂T ∂X ∂Y ∂Y
(3)
with boundary condition: P = P1(t) X = X0, P = P2(t)
X= Xe and
∂P =0. ∂x
Here: X0 is the inlet distance. Xe is the exit distance that must satisfy Reynolds boundary condition. P1(t) is inlet gas pressure; P2(t) is outlet gas pressure. P1(t) and P2(t) will change in an entire work circle of engine. Where
a 3 Ph aPh ρh 3 , εy = ε x = 12b 2 uη 12uη 0 η 0
ρ h 3 η
εy = K
Dimensionless variable is defined below: H=
p ρ η y h x ut , X = , Y = , P= . ρ= , η = , T= . a a b Ph ρ0 η0 a
1.3 Load equation
W (t ) =
∫∫ P( x, y, t )dxdy .
(4)
W (t ) load acting on ring face of one segment, which consists of several components as follows: (i) Hydrodynamic force on the bearing area of ring face. (ii) Asperity contact force on bearing area.
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(iii) Gas force acting on non-bearing area of ring face. (iv) Elastic force of the ring. 2
Numerical solution
The prediction of lubrication and contact characteristics is based on an approach developed by Hu[9] for full numerical solution of mixed lubrication. The new approach is to use the same Reynolds equation consistently in both hydrodynamic and contact regions. The idea is based on the belief that the solution of Reynolds equation under the constraint of h=0 will give the same result as that from the contact mechanics. As the film thickness becomes zero in the contact regions, the pressure flow vanishes and the Reynolds equation is reduced to the following form: U=
∂h ∂h + =0 ∂x ∂t
At h = 0 .
(5)
As eq. (5) is actually a special case of eq. (3), a unified equation system and numerical scheme can be applied to both hydrodynamic and contact regions. A full numerical solution over the entire computation domain is thus obtained by solving one equation system. It is not necessary to know the information about the contact borders and boundary condition, and the approach can reach stable and convergent solution under very severe operating conditions. 3 Results and discussions
All the equations in previous sections are solved simultaneously by the unified numerical approach, the discretization of the Reynolds equation and iteration scheme is similar to those described by Hu[9], but some modifications have been made to fit the ring pack. A computation domain of –2.5∆X1.5 and –1.5∆Y1.5 is employed. The smooth surface is first investigated, and as the first step of studying roughness effects, isotropic sinusoidal surfaces are considered. In order to limit the amount of work involved in computation, we artificially designed a “small” engine with only one ring for checking the performance of developed program. The geometric parameters of the engine are listed in table 1. Table 1 Structural and material parameters of a small-sized engine D = 3D0 mm R =10 mm L = 35 mm N = 1000 rpm E = 0.5 mm
Nominal diameter for the cylinder bore Crank radius Connecting rod length Rotational frequency of the engine Ring end gap
E = 200 Gpa µ = 0.3 α = 1.82E+4(m/N) η0 = 0.096 PaCs
Young’s modulus Poisson ratio Pressure-viscosity exponent Dynamic viscosity
3.1 Smooth surface Fig. 3(a) depicts the pressure distribution and film thickness at a location near TDC where the load is large and velocity close to zero. At this position where wear and failure is easy to occur,
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F i lm t h ic k n e s s P re s s u re
S l id i n g d ir e c t io n
H/a
P/Ph
F ilm th ic k n e s s P re s s u re
H/a
S lid in g d ire c tio n
P/Ph
ring and bore are in contact to each other, giving a pressure distribution similar to Hertz distribution. Fig. 3(b) gives the same items but ring and bore are in full hydrodynamic lubrication.
(a) (b) Fig. 3. the film thickness and pressure distribution of smooth surface. (a) at TDC (Load = 10.5 N/mm, V = 0.007 mm/s), (b) load =10.5 N/mm, V = 500 mm/s.
3.2 Isotropic sinusoidal surface The pressure distribution and oil film thickness between a smooth surface and a sinusoidal surface are given in fig. 4, in which the ring position corresponds to the TDC and one can
Fig. 4. 3D views of film shape and pressure distribution for piston ring and cylinder wall in rough surface contact. Load = 10.5
M inim um Film thickness
N/mm, V = 23.5 mm/s, σ = 0.085 µm.
0.0 25 0.0 20 0.0 15 0.0 10 0.0 05 0.0 00
0
2 00
40 0
6 00
80 0
C ra n k a n gle
Fig. 5. The minimum film thickness over an entire work circle.
see asperity contacts. Fig. 5 displays the variations of the minimum film thickness, hmin over an entire work circle of the engine. It is shown that film thickness approaches zero in the vicinity of either top or bottom dead center due to the very small sliding velocity at the two locations. Fig. 6 gives the load ratio, defined as the load supported by asperity over the load supported by lubricant, as well as the contact area ratio. It has been confirmed again that there is direct contact between ring and bore near TDC and BDC.
0.5
Rq=0.50 Rq=0.25
0.4 0.3 0.2 0.1 0.0
0
30 60 90 120 150 180
Crank angle
contact load ratio
NUMERICAL SIMULATION OF PISTON RING IN THE MIXED LUBRICATION
contact area ratio
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0.5 Rq=0.50 Rq=0.25
0.4 0.3 0.2 0.1 0.0
0
30 60 90 120 150 180
Crank angle
(a) (b) Fig. 6. (a) The ratio of real contact area and nominal contact area; (b) the ratio of load supported by real contact area and the applied load.
4 Conclusions
A numerical analysis of piston ring lubrication has been performed based on a unified approach. The calculation cases presented in this study have demonstrated that this new unified approach is capable of predicting the contact and lubrication characteristics of piston ring under various operating conditions. It appears to be a useful engineering tool for industrial applications. The results also show that the piston ring is hydrodynamic lubrication for the most part of stroke, and mixed lubrication takes place only in the positions near TDC and BDC where the direct contact area grows with the increasing roughness amplitude. This means that careful lubrication design is necessary to prevent ring and bore from failure such as sever wear or scuffing near TDC and BDC. References 1. 2. 3.
Lloyd, T. The hydrodynamic lubrication of piston rings, Proc. Instn mech. Engrs 1968-69 183(Pt 3B), 28-34. Castleman, R. A., 1936, A Hydrodynamic Theory of Piston Ring Lubrication, Physics, Vol. 7, Sept., pp. 364-367. Furuhama, S., 1959, A Dynamic theory of Piston Ring Lubrication, 1st Report-Calculation, Bull. JSME, Vol. 2, p. 423.
4.
Dowson, D., Economou, P. N. Ruddy, B. L., Strachan, P. J., and Baker, A. J. S., 1979, Piston Ring Lubrication—Part .
5. 6. 7. 8. 9.
Theoretical Analysis of a Single Ring and a Complete Ring Pack, Energy Conservation through Fluid Film Lubrication Technology: Frontiers in Research and Design. Winter Annual Meeting of ASME, pp 23-52 Patir, N., and Cheng, H. S., 1978, An Average Flow Model for Determining the Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication, ASME Journal of Lubrication Technology, Vol. 100, No. 1, p. 12. Greenwood, J. A., and Tripp, J. H., 1971, The Contact of Two Nominally Flat Surfaces, Pro. Inst. Mech. Engrs., Vol. 185, p. 625. Rohde, S. M., 1980, A Mixed Friction Model for Dynamically Loaded Contact with Application to Piston Ring Lubrication, Surface Roughness Effects in Hydrodynamic and Mixed Lubrication. ASME Publication, pp. 141-153. Jeng, Y. R., 1992, Friction and Lubrication Analysis of a Piston-Ring Pack, SAE Paper 920492. Hu, Y.Z., and Zhu, D., 2000, A Full Numerical Solution to the Mixed Lubrication in Point Contacts, ASME Journal of Tribology, 122, pp. 1-9.
