SLAPPING MOTIONS. S. Balakrishnan and H. Rahnejat. Wolfson School of Mechanical & Manufacturing Engineering. Loughborough University. Loughborough ...
VIETRI SUL MARE, SALERNO, ITALY, 18-20 SEPTEMBER
TRANSIENT ELASTOHYDRODYNAMIC LUBRICATION OF PISTON SKIRT TO CYLINDER LINER UNDER COMBINED RECIPROCATING AND SLAPPING MOTIONS S. Balakrishnan and H. Rahnejat Wolfson School of Mechanical & Manufacturing Engineering Loughborough University Loughborough, UK Keywords: piston skirt lubrication, conformal contact, elastohydrodynamics Nomenclature a, b contact half-widths along circumferential and skirt length respectively (m)
c F
clearance (m) applied load (N)
h film thickness (m) rxp , ryp radii of piston in x and y direction (m) rxc , ryc radii of bore in x and y direction (m) Pd
piston diametre (m)
Pl
piston skirt length (m)
s u
profile speed of entraining motion (m/s)
x y
entraining direction rypϕ
α β δ η
piezo-viscosity index (Pa -1 ) piston tilt angle (degrees) deflection (m) viscosity (Pa s)
ρ ϕ
density (kg/m3 ) circumferential direction (rad) TDC top dead centre BDC bottom dead centre Non-dimensional parameters:
η=
hRy uη0 x F y p η ρ , x= , W= , h= 2 , u= , y= , ρ= , P= 2 a E ′Ry a E ′Ry a Pmax η0 ρ0
ABSTRACT The paper presents detailed quasi-static analysis of elastohydrodynamic conjunction between piston skirt and cylinder bore for high performance internal combustion engines. The high combustion pressures and the discontinuous nature of the axial profile of the piston skirt causes pressure spikes in the vicinity of contact
extremities, which are reminiscent of finite line contacts in non-conforming configurations. The secondary motion of the piston causes a moment about the gudgeon pin, resulting in a misaligned contact. 1- INTRODUCTION Early study on lubrication of piston skirts was carried out by Knoll and Peeken [1] to determine the hydrodynamic load on cylinder liner due to piston secondary motion and concluded that the generated load was very much dependent on the profile of the piston, gudgeon pin eccentricity and piston skirt length-to-diameter ratio. Oh et al [2] investigated this problem by including the deformation of piston skirt. His conclusion further reinforced the importance of skirt profile in improving lubrication and reduction of friction between the piston skirt and cylinder liner. In this paper, the authors investigate lubrication between piston skirt and cylinder liner for a high performance engine, where the loads are considerably higher and the lubricant film formed in the conjunction is much thinner than that generated in passenger vehicles. Apart from the higher loads experienced by the piston, this adverse condition is attributed to the profile and the piston skirt length to diameter ratio. A P P typical piston in passenger vehicle has l ≈ 0.9 − 1.0 , whereas for the investigated piston, the l ≈ 0.60 − 0.75 . Pd Pd This ratio translates to the contact patch dimension. 2- THEORETICAL FORMULATION: 2.1- Geometrical considerations and elastic film shape Tilting motion of the piston is caused by the eccentric position of the gudgeon pin with respect to the centre-line of the piston itself. The conformal contact of the piston skirt to cylinder liner is affected by this tilt angle, causing a misaligned contact. The approach of the contiguous bodies is obtained by geometric considerations. At the TDC, the rigid body motion of the skirt towards the liner is determined by its tilt. The elastic film shape is given as:
h = c + s tan β + δ
(1)
Once the value of the tilt angle and the corresponding contact force is calculated by dynamic analysis of piston motion for articulation of the connecting rod at given values of crankshaft position and the corresponding gas force, then lubricated contact conditions can be studied at any instant of time. This approach leads to a quasistatic analysis of combined sliding and normal approach of the contiguous bodies in contact. This method is an approximation of the actual transient conditions. A rectangular-type contact results, which can be visualised by unwrapping the deformed shell. The equivalent model may be that of a a cylinder with a very large radius of curvature against a flat plane. In most vehicles the piston skirt is either barrelled or has chamfered relief radii towards its extremities (i.e. crowned or dubbed-off). The equivalent radii in x and y directions are given as:
1 1 1 = + Rx rxp rxc 1 1 1 = − Ry ryp ryc 1 = 0 , thus: Rx = rxp . rxc ryp (ryp + c) In the circumferential direction: ryc = ryp + c, ∴ Ry = . c Where for the cylinder bore: rxc = ∞, ∴
(2)
Typically, the crown radius, rxp = 500 → 3000 m and due to small clearance in the range: , c = 5 → 30 µ m , the equivalent radius Ry = 200 → 600 m . It can be noted that the two radii are similar for an aligned contact leading to a sizeable rectangular contact. The undeformed profile of the contiguous bodies in the circumferential direction is: s =
y2 and in the axial 2 Ry
x2 direction for a crowned or barrelled shape piston is given as: s = . Thus: 2 Rx
si , j
(x =
i, j
− m)
2 Rx
2
(y +
i, j
− n)
2
(3)
2 Ry
Note that the circumferential direction is denoted by y = Ryϕ . A typical profile is shown in figure 1.
Figure 1: Piston Axial Profile 2.2- Calculation of contact deformation Contact deflection must be obtained at any location x,y in the contact domain to be used in equation (1). This is obtained by the solution of general contact elasticity integral as:
2p δ= π E'
dx1dy1 1 ∫− a% ∫−b% 2 2 2 ( y − y1 ) ( x − x1 ) a%
b%
(4)
where the deflection at any contact location x,y is due to all the columnar elements of pressure p, of area dx1dy1 at locations x1,y1. Integrating equation (4) yields:
δz =
2
π
PD*
where the influence coefficient matrix D* is obtained by Johnson [3] as:
(5)
(
)
(
)
1
2 2 2 ( y + a% ) + ( y + a% ) + x + b% D* = x + b% ln 1 2 2 2 ( y − a% ) + ( y − a% ) + x + b%
(
)
(
)
(
)
(
)
(
)
2
2
(
)
2
2
(
)
2
2
2 ( y − a% ) + ( y − a% ) + x − b% + x − b% ln 2 ( y + a% ) + ( y + a% ) + x − b%
(
1
2
2 x + b% + ( y + a% ) + x + b% + ( y + a% ) ln 2 x − b% + ( y + a% ) + x − b%
)
(
)
(
)
(
)
(
)
2 1
1
1
1
2 2 2 x − b% + ( y − a% ) + x − b% + ( y − a% ) ln 1 2 2 2 % % x + b + ( y − a% ) + x + b 1 1 where a% = and b% = ∆y ∆x
(6)
Equations (3) and (4) are substituted for profile and deflection in equation (1) to yield the elastic film shape at any nodal computation position: i,j. 2.3- Reynolds’ equation The generated pressure distribution is obtained by solution of Reynolds’ hydrodynamic equation. For the contact described here the two dimensional Reynolds’ equation for sliding motion taking place in the x-direction in nondimensional form is given as: 3 3 ry 3 E ′u ∂ a 2 ∂ ρ h ∂P ∂ ρ h ∂P + = 12 2 ρh 2 a bph ∂ x b ∂ x η ∂ x ∂ y η ∂ y
( )
(7)
The Couette flow term on the right-hand side of the equation is due to relative sliding motion of the piston with respect to the cylinder liner. The speed of entraining motion is u =
1 x& .The second term on the right hand side 2
of equation (7) gives the rate of change of elastic film shape. This is the squeeze film velocity.
2.4- Boundary conditions There are two boundary conditions employed to limit the solution domain. Firstly pressure elements at the edges of the computational boundaries are set to zero. Secondly, to avoid the generation of negative pressures in the fluid film, at the cavitation boundary, the Reynolds’ condition: P =
∂P ∂P = = 0 is employed. This condition ∂x ∂y
prevails by setting any value of P to zero every time a negative pressure value is encountered during the iteration procedure.
2.5- Lubricant Rheology Lubricant viscosity-pressure dependence is given by Barus [4] as:
η = η0 eα p
(8)
Lubricant density variation with pressure is given by Dowson and Higginson [5] as:
ρ = 1+
0.6 p 1 + 1.7 p
(9)
3- METHOD OF SOLUTION The Newton-Raphson method is applied for the solution of the Reynolds’ equation in the following numerical form: my −1 mx −1
∑ ∑ J ki ,,lj ∆Pk ,l = − F i , j
(10)
l =2 k =2
where, Fi , j is the Reynolds’ equation and it is referred to as the residual term. The Jacobian matrix is a tensorial quantity, given in terms of the residual derivatives as:
J ki ,,lj =
∂Fi. j
(11)
∂Pk ,l
The Jacobian matrix is a tri-diagonal matrix. Using the Gauss-Seidel iteration method, the system state equation can be written as:
∆Pkn,l = (− Fi , j − J ki ,−j1,l ∆Pkn−1,l − J ki ,+j1,l ∆Pkn+−11,l − J ki ,,lj−1 ∆Pkn,l −1 − J ki ,,lj+1 ∆Pkn,l−+11 ) / J ki ,,lj
(12)
where n is the iteration counter in the above recursive equation. For the reason of good numerical stability an under-relaxation factor is employed to update the pressure according to:
Pi ,nj = Pi.nj−1 + Ω∆Pi ,nj
(13)
where Ω is the under-relaxation factor, typically 0.1, chosen under the reported conditions in this paper. 3.1- Convergence criteria There are two convergence criteria. Firstly, pressure convergence is sought according to the following:
∑∑ ( P n − P n −1 )2 i j i, j i, j ≤ 10−4 N 0.5
(14)
Where: N = nx n y If the above condition is not satisfied the iteration index, n is updated and the whole procedure mentioned above is repeated (see equation (12)).
The convergence criterion on load balance is given as:
∫ ∫ P( x , y )dxdy − W
≤ 10−4
(15)
If the above condition is unsatisfied the lubricant film thickness is adjusted and the above procedure is repeated. This is carried out using the following relationship:
h = c + ζ W − ∫∫ Pdxdy
(16)
Where ζ is a damping parameter, chosen in the range: 0.01 → 0.08 . 4- RESULTS AND DISCUSSION The methodology highlighted above is used for quasi-static analysis of piston skirt-to-cylinder liner contact at various positions in a stroke of piston. The simulated conditions relate to a typical engine with short piston skirt. The combustion pressure reaches a maximum of 48 MPa, which occurs approximately 8º (in terms of crankangle) past TDC. The piston’s modulus of elasticity is 211 GPa with a Poisson’s ratio of 0.3. The piston nominal principal radius is 44.5 mm with the skirt length of 28 mm. The nominal clearance is of the order of 20 µm. The piston skirt axial profile is asymmetrical, with the top and bottom end sections having radii that deviate from the principal nominal radius by 80 µm on the top and 46 µm at the bottom. The bottom section is merely a chamfer. This chamfer creates the wedge effect for lubricant entrainment as the piston traverses toward the BDC. The actual profile is shown in figure 1. This represents an aligned conforming profile of the skirt against the cylinder bore. The speed of piston in this engine reaches a maximum of 23.6 m/s. The entraining motion ceases at TDC and BDC momentarily and the lubricant is retained in these instances by entrapment, squeeze film motion and rapid replenishment due to short lived stop time. The contact domain is quite large in many instances and in order to obtain convergence and realistic pressure distributions it is necessary to use as refined a computational mesh as possible. The results reported here are based on a mesh density of 180 X 80 nodes, with the former being in the axial direction of piston motion. The computation time depends on a number of factors. These include mesh density, applied contact load, the degree of misalignment and the speed of entraining motion. With the exception of entraining motion, an increase in any of the other parameters results in longer computation times. The longest computation time occurs in the vicinity of maximum combustion and is of the order of 6 hours using Intel Fortran compiler for Linux on a Pentium IV 1.8 GHz machine.
Velocity (m/s)
20
10
0 50
100
150
200
-10
-20
Crank Angle (Degrees)
Figure 2: Piston Velocity
250
300
360
Figure 2 shows the speed of translational motion of the piston. The downward sliding velocity is designated to be positive. The position investigated corresponds to the crank-angle of 8º, where the maximum combustion pressure of 48 MPa is applied, resulting in an extreme case, with misalignment to a contact force of 2.5 KN.
Figure 3: 3-Dimensional Pressure Distribution Figure 3 shows the three dimensional pressure distribution. The contact area is extended circumferentially over approximately a π -film and in the axial direction over the skirt length of 28 mm.
Figure 4: Oil Film at Maximum Pressure The pressure spikes observed here are similar to the edge discontinuities often encountered in finite line counterformal contacts of rollers to races in rolling element bearing or cam-to-follower, at the extremities of the
flat region of the skirt before the relief radii at the top or bottom sections are encountered. Note that most of the deformation takes place in this region as the piston skirt is deformed under pressure to conform to the shape of the bore.
Figure 5: 3-Dimensional Pressure Distribution for tilt of 0.085º There are changes of profile at the extremities of this flat region, where the relief radii commence. The minimum film thickness of 1.9 µm, as shown in figure 4, occurs when pressure is relieved at the outlet region. This is due to the fact that there is not enough pressure to cause deformation.
Figure 6: Oil Film at Maximum Pressure for tilt of 0.085º The above example assumes an aligned conformal contact between the piston skirt and the cylinder bore. In fact, for the position used in this analysis, the piston is subjected to secondary tilting motion and assumes a tilt angle of 0.085º. This renders a misaligned conformal contact, which alters the elastic gap shape and, therefore, the pressure distribution. Figure 5 shows the three dimensional pressure distribution.
Due to the tilt, being in the direction of bottom of the skirt, the pressure is now distributed over a larger area and more evenly distributed at the two extremities of the contact, when compared to the previous case. Comparing figures 4 and 6 (the axial plot for film through the maximum pressure regions), the value of minimum film has reduced to 1.52µm but its location remains at the outlet region. Higher pressures generated in the central region of the contact deform the curved surface of the piston (along the skirt) to almost a flat surface. This corresponds to bending of the hollow skirt to conform better to the cylinder bore profile. 5 – CONCLUSION From the results obtained, it was observed, by bringing the aligned contiguous bodies close together, pressures generated by the lubricant is higher when there is more abrupt change in profile, thus leading to a higher pressure at the inlet region than that at the outlet region. The tilt of the piston about the gudgeon, had inadvertently, caused a smoother profile transition, hence yielded pressure of lower magnitude than that of a non tilted profile. A repercussion of this finding is that with a reduced clearance, the degree of contiguity is enhanced, leading to a reduction in the secondary pressure peaks, in addition to reduced impact momenta, which is responsible for slapping noise. This concurs with the conclusion made by Knoll and Peeken [1], Oh et al [2] and Xie et al [6]. 6 – REFERENCES [1] Knoll G.D. and Peeken H.J., “Hydrodynamic lubrication of piston skirts”, Trans. ASME J. Lubrication Technology, Vol. 104, 1982, pp. 504-509 [2] Oh, K.P. et al, “Elastohydrodynamic lubrication of piston skirts”, ASME Journal of Tribology, Vol. 109, April 1987, pp. 356-362 [3] Johnson, K.L., Contact Mechanics, Cambridge University Press, Cambrige, 1985 [4] Barus C, “Isothermal, Isopiestics and Isometrics relative to Viscosity”, Amer. J. Science., 1893, pp. 87
Vol. 45,
[5] Dowson, D. and Higginson, G.R., “A numerical solution to the elastohydrodynamic problem”, Journal of Mechanical Eng. Science, Part 1, Vol. 6, 1959 [6] Xie, Y.B. et al, “A Comprehensive study of the friction and dynamic motion of the piston assembly”, Proc. Of Instn. Mech. Engrs. Part J, Vol 212, 1998, pp.221-226
