Tài liệu Friction and Lubrication in Mechanical Design P2 doc

Nonetheless the idealized case of elastic bodies with smooth sur- faces is considered in this chapter as the theoretical reference for the contact between rough surfaces.. CONTACT Case

20 Chapter I

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Dyson, A., “Frictional Traction and Lubricant Rheology in Elastohydrodynamic Lubrication,” Phil Trans Roy Soc., Lond., 1970, Vol Crook, A W., “The Lubrication of Rollers,” Phil Trans Roy Soc., Lond.,

1961, Vol A254, p 237

Crook, A W., “The Lubrication of Rollers,” Phil Trans Roy Soc., Lond.,

1963, Vol A255, p 281

Cheng, H.S., “A Refined Solution to the Thermal Elastohydrodynamic Lubrication of Rolling and Sliding Cylinders,” ASLE Trans., 1965, Vol 8,

pp 397410

Johnson, K L., and Cameron, R., “Shear Behavior of Elastohydrodynamic Oil Films at High Rolling Contact Pressures,” Proc Inst Mech Engrs, 1967-68,

Vol 182, Pt 1, No 14

Dowson, D., and Whitaker, A V., “A Numerical Procedure for the Solution of the Elastohydrodynamic Problem of Rolling and Sliding Contacts Lubricated

by a Newtonian Fluid,” Proc Inst Mech Engrs, 196546, Vol 180, Pt 3B,

Plint, M A., “Traction in Elastohydrodynamic Contacts,” Proc Inst Mech Engrs, 1967-68, Vol 182, Pt 1, No 14, p 300

O’Donoghue, J P., and Cameron, A., “Friction and Temperature in Rolling

Sliding Contacts,” ASLE Trans., 1966, Vol 9, pp 186-194

Benedict, G H., and Kelley, B W., “Instaneous Coefficients of Gear Tooth

Friction,” ASLE Trans., 1961, Vol 4, pp 59-70

Misharin, J A., “Influence of the Friction Conditions on the Magnitude of the

Friction Coefficient in the Case of Rolling with Sliding,” International Conference on Gearing, Proceedings, Sept 1958

Hirst, W , and Moore, A J., “Non-Newtonian Behavior in Elasto-hydrody-

namic Lubrication,” Proc Roy Soc., 1974, Vol A337, pp 101-121

Johnson, K L., and Tevaarwerk, J L., “Shear Behavior of Elastohydrodynamic Oil Films,” Proc Roy Soc., 1977, Vol A356, pp Conry, T F., Johnson, K L., and Owen, S., “Viscosity in the Thermal Regime

of Elastohydrodynamic Traction,” 6th Lubrication Symposium, Lyon, Sept.,

1979

Trachman, E G., and Cheng, H S., “Thermal and Non-Newtonian Effects on Traction in Elastohydrodynamic Contacts,” Elastohydrodynamic Lubrication,

1972 Symposium, p 142

Trachman, E G., and Cheng, H S., “Traction in Elastohydrodynamic Line

Contacts for Two Synthesized Hydrocarbon Fluids,” ASLE Trans., 19?, Vol

Winer, W O., “Regimes of Traction in Concentrated Contact Lubrication,” Trans ASME, 1982, Vol 104, p 382

Sasaki, T., Okamura, K., and Isogal, R., “Fundamental Research on Gear Lubrication,” Bull JSME, 1961, Vol 4(14), p 382

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17(4), pp 271-279

Introduction 21

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Sasaki, T Okamura, K Konishi, T., and Nishizawa, Y “Fundamental Research on Gear Lubrication,” Bull JSME, 1962, Vol 5( 19), p 561 Drozdov, Y N., and Gavrikov, Y A., “Friction and Scoring Under the Conditions of Simultaneous Rolling and Sliding of Bodies,” Wear, 1968, Vol 11, p 291

Kelley, B W., and Lemaski, A.J., “Lubrication of Involute Gearing,” Proc Inst Mech Engrs, 1967-68, Vol 182, Pt 3A, p 173

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to the Lubrication of Concentrated Contacts, Special Publication No NASA- SP-237, National Aeronautics and Space Administration, Washington, D.C.,

1970, p 34

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on EHD Film Thickness,” ASME J Lubr Technol., Apr 1983, Vol 105,

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J Appl Mech., March 1967, p 153

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1977, pp 628-637

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Szeri, A Z., Tribology: Friction, Lubrication and Wear, Hemisphere, New York, NY, 1980

2

The Contact Between Smooth Surfaces

It is well known that no surface, natural or manufactured, is perfectly smooth Nonetheless the idealized case of elastic bodies with smooth sur- faces is considered in this chapter as the theoretical reference for the contact between rough surfaces The latter will be discussed in Chapter 4 and used

as the basis for evaluating the frictional resistance

The equations governing the pressure distribution due to normal loads are given without detailed derivations Readers interested in detailed deriva- tions can find them in some of the books and publications given in the

references at the end of the chapter [l-391

CONTACT

Case 1 : Concentrated Normal Load on the Boundary of a Semi-Infinite Solid

The fundamental problem in the field of surface mechanics is that of a concentrated, normal force P acting on the boundary of a semi-infinite body as shown in Fig 2.1 The solution of the problem was given by Boussinesq [I] as:

22

The Contact Between Smooth Surfaces 23

2

Figure 2.1 Concentrated load on a semi-infinite elastic solid

a, = horizontal stress at any point

I

= P((1 2n - 2v)[$ - z v2 (E 2n ~ 3 ( , 2 ~ 2 ) - 5 / 2 ) - ” ~ ] - 3r2Z(r2 + 2 2 ) - 5 / 2

az = vertical stress at any point

- 3 p 2 3 ( , 2 ~ 2 ) - 5 / 2

2n

rrz = shear stress at any point

- - _ 3 p , z 2 ( , 2 + z 2 ) - 5 / 2

2n

where

v = Poisson’s ratio

The resultant principal stress passes through the origin and has a magnitude:

3 P

3P

= 4 = 2 4 9 + 2 2 ) - 2nd2

The displacements produced in the semi-infinite solid can be calculated from:

24 Chapter 2

U = horizontal displacement

and

w = vertical displacement

where

E = elastic modulus

At the surface where 2 = 0, the equations for the displacements become:

P(l - 2 )

(tt’)Z,0 =

(1 - 2u)(l + u)P

which increase without limit as Y approaches zero Finite values, however, can be obtained by replacing the concentrated load by a statically equivalent distributed load over a small hemispherical surface at the origin

Case 2: Uniform Pressure over a Circular Area on the Surface of a Semi-Infinite Solid

The solution for this case can be obtained from the solution for the con-

centrated load by superposition When a uniform pressure q is distributed over a circular area of radius a (as shown in Fig 2.2) the stresses and

deflections are found to be:

(w)~=(, = deflection at the boundary of the loaded circle

- - 4( 1 - u2)qa JrE ( w ) ~ = ” = deflection at the center of the loaded circle

- 2( 1 - u2)qa

-

E

(a2 + z3 z*)3/2 1

- 4 - I +

- (

The Contact Between Smooth Surfaces 25

(I?

Figure 2.2 Uniform pressure on a circular area

= horizontal stress at any point on the Z-axis

1

( t ) r = O = 5 (or - oz)r=o

= maximum shear stress at any point on the Z-axis

From the above equation it can be shown that the maximum combined shear stress occurs at a point given by:

and its value is:

26 Chapter 2

Case 3: Uniform Pressure over a Rectangular Area on the Surface of a Semi-Infinite Solid

In this case (Fig 2.3) the average deflection under the uniform pressure q is

calculated from:

W,,e = k(1 - u 2 ) ; JA

where

A = area of rectangle

k = factor dependent on the ratio b/a as shown in Table 2.1

It should be noted that the maximum deflection occurs at the center of the rectangle and the minimum deflection occurs at the corners For the case of

a square area (6 = a) the maximum and minimum deflections are given by:

Figure 2.3 Uniform pressure over a rectangular area

The Contact Between Smooth Surfaces

Table 2.1 Values of Factor k

1

1.5

2

3

5

10

100

0.95 0.94 0.92 0.88 0.82 0.71 0.37

27

Case 4: A Rigid Circular Cylinder Pressed Against a Semi-Infinite Solid

In this case, which is shown in Fig 2.4, the displacement of the rigid cylinder

is calculated from:

P(1 - ”*)

w = -

2aE

P

Figure 2.4 Rigid cylinder over a semi-infinite elastic solid

28 Chapter 2

where

p = total load on the cylinder

a = radius of the cylinder

The pressure distribution under the cylinder is given by

which indicates that the maximum pressure occurs at the boundary (Y = a)

where localized yielding is expected The minimum pressure occurs at the

center of the contact area ( r = 0) and has half the value of the average pressure

Case 5: Two Spherical Bodies in Contact

In this case (Fig 2.5) the area of contact is circular with radius a given by

a = 0.88 /: R ,

Figure 2.5 Spherical bodies in contact

The Contact Betwven Smooth Surfaces

assuming a Poisson’s ratio U = 0.3 and the pressure distribution over this

29

area is:

The radial tensile stress and maximum combined shear stress at the bound- ary of the contact area can be calculated as:

where

P = total load

_ - +-

E, El E2

E , , E2 = modulus of elasticity for the two materials

Case 6: Two Cylindrical Bodies in Contact

The area of contact in this case (Fig 2.6) is a rectangle with width b and

length equal to the length of the cylinders The design relationships in this case are:

q = pressure on the area of contact

where

P’ = load per unit length of the cylinders

R, RI R2

_ - - - +-