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# Fundamentals of elastohidrodyanamic lubrication

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## Fundamentals of elastohidrodyanamic lubricationDocument Transcript

• VIII Component Performance and Design DataCopyright © 1997 CRC Press, LLC.
• 58 Fundamentals of Lubrication Elastohydrodynamic Michael M. Khonsari and D. Y. Hua CONTENTS Nomenclature.................................................................................................................................611 Geometry of Contact....................................................................................................................613 Dry Contact....................................................................................................................................614 Elastohydrodynamic Line Contact..............................................................................................616 Elastohydrodynamic Elliptical Contact.......................................................................................621 Starvation........................................................................................................................................625 Thermal Correction.......................................................................................................................625 Partial-Film EHL............................................................................................................................627 Traction............................................................................................................................................627 Examples.........................................................................................................................................630 References.......................................................................................................................................636 NOMENCLATURECopyright © 1997 CRC Press, LLC.
• GEOMETRY OF CONTACT A general Hertzian contact between two bodies is shown in Figure 1.1 Two principal planes are used to characterize the geometry at the point of contact. Rxl, Ryl, and Rx2, Ry2 are principal radii for body 1 and body 2, respectively. In general, the principal planes of body 1 and body 2 may not coincide. However, for most engineering machine elements, the principal radii Rxl and Rx2, as well as Ry1, and Ry2 lie in the same plane. In this chapter, the following equivalent radii and equivalent modulus of elasticity are introduced. FIGURE 1 Geometry of elliptical contact.1 The equivalent radius in x direction is and the equivalent radius in y direction isCopyright © 1997 CRC Press, LLC.
• where “+” and “-” represent convex and concave of the surface 2, respectively. Then, the curvature sum in x and y direction is defined as The equivalent elastic modulus is The above equations are valid for the general case of an elliptical contact as formed between two ellip- soids with aligned principal axes, two crowned cylinders, or two cylinders that cross at right angle. The elliptical contact can be reduced to two special cases: Circular contact — when Rx1 = Ryl = R1 and Rx2 = Ry2 = R2, i.e., contact between two spheres. In this case, R = 1/(1/R1 + 1/R2). Line contact — both Ryl and Ry2 are infinity. Then, Ry → ∞ and the curvature sum R = Rx. (cf. Figure 2). FIGURE 2 Line contact: (a) nonconformal; (b) conformal; (c) equivalent elastic cylinder and rigid surface. DRY CONTACT LINE CONTACT Two cylinders pressed against one another under a normal load will produce a plane rectangular con- tact area. If the cylinders are unequal, the contact area is not truly rectangular. Nevertheless, the plane contact is a reasonable assumption. Under a normal load, w, the “contact patch” will have width of 2b. In the absence of lubricant, the normal load is parabolically distributed over this area. The half- width of contact and the maximum Hertzian contact pressure are functions of the load per unit length, the equivalent curvature radius, R, and the equivalent elastic modulus, E. TheCopyright © 1997 CRC Press, LLC.
• Hertzian predictions of mean pressure, the magnitude and location of the maximum shear stress, as well as the normal approach of the centers are listed in Table 1. CIRCULAR CONTACT The contact between two spheres forms a circular region whose diameter is 2a. The radius of the contact and the maximum pressure in terms of the load, radii of the spheres, and elastici- ty modulus are given in Table 1 along with mean pressure, maximum shear stress, maximum tensile stress, and the normal approach of the center. ELLIPTICAL CONTACT The geometry of an elliptical contact is shown in Figure 1. The elliptic parameter k is defined as the ratio of the ellipse semimajor axis a to that of semiminor axis b. In general, the ellipti- cal parameter requires solving the first and the second elliptical integrals. The approximation of the elliptical parameter and the integrals can be used to simplify the expression which is related to the radius ratio.2 The definition and the approximation equations are listed in Table 2. These approximations are valid for the range of 1 ≤ Ry/Rx≤ 100, or 1 ≤ κ ≤ 18.Copyright © 1997 CRC Press, LLC.
• The radii of the elliptical contact, a and b, as well as the maximum Hertzian pressure, pH, are functions of several parameters such as load, equivalent radius of the bodies, and the elasticity mod- ulus, as well as the elliptic parameter and the elliptic integral. The appropriate equations are listed in Table 1. The contact deformation at the center of the contact is also provided in Table 1. ELASTOHYDRODYNAMIC LINE CONTACT FILM SHAPE AND PRESSURE DISTRIBUTION A typical film shape and pressure distribution of elastohydrodynamic lubrication (EHL) is shown in Figure 3. Generally, EHL pressure distribution closely resembles the dry Hertzian contact with the major exception of a pressure build-up in the inlet region and a pressure spike in the exit region. Existence of the sharp pressure spike accompanied by a film constriction at the exit region are important characteristics of the elastohydrodynamic lubrication regime. Several trends in EHL may be noted. First, increasing speed or decreasing load tend to increase the magnitude of pressure spike and move its location towards the inlet region. Under very heavy loading, the pressure spike tends to decrease and eventually vanish, i.e., the pressure profile approaches that of the dry Hertzian. In EHL applications, both the maximum Hertzian contact pressure and the pressure spike are important parameters. Although the pressure spike is very nar- row, its occurrence is very important since it may produce high subsurface stresses that directly affect the rolling element bearing fatigue life. The minimum film thickness at the film constriction compared to surface roughness dictates whether the lubrication film is thick enough to protect the surfaces. The central film thickness (essentially the parallel central region) is also a useful parameter in engineering design. The film thickness is reduced by starvation of the lubricant and by inlet heating as discussed in sections on “Starvation” and “Thermal Correction.” The appropriate EHL equations can be conveniently grouped in terms of the following dimen- sionless parameters:Copyright © 1997 CRC Press, LLC.
• FIGURE 3 © 1997shapePress, pressure distribution of line contact. Copyright Film CRC and LLC.
• where the viscosity–pressure coefficient is defined as In nonconformal contacts, it is important to include the variation of viscosity with pressure. There are two general relationship. The Barus viscosity–pressure relation is The typical values of viscosity-pressure coefficient a for several lubricants are listed in Table 3.3 The other relation due to Roelands4 is given below: The typical value for z is 0.6, S0 is 1.1 and a is 5.1 × 10-9 The EHL formulae reported in this chapter are based on Barus’ equation unless otherwise spec- ified. REGIMES OF FLUID FILM LUBRICATION Many expressions for evaluating EHL film thickness are available in the literature. These are obtained using curve fitting techniques to the numerical solutions of the governing equations that involve the Reynolds equation coupled with surface deformation. These expressions, however, only apply to a particular range of operation conditions and cannot be extrapolated into different regimes. It is, therefore, necessary to define the regimes for appropriate usage of the film thickness expressions. Referring to Figure 4, the following regimes may be defined:5 • Rigid-isoviscous, load is not high enough to produce either an appreciable viscosity change or elas- tic deformation of contact surfaces • Rigid-viscous, significant viscosity increase occurs due to high pressure but the elastic deformation of contact surfaces is negligible • Elastic-isoviscous, elastic deformation of contact surfaces is quite large compared to the film thick- ness but the viscosity change due to pressure is negligible • Elastic-viscous,6 viscosity changes due to pressure and elastic deformation of contact surfaces play important roles. This is the regime of “full” EHL FILM THICKNESS FORMULAE The following dimensionless groups conveniently categorize the appropriate regime:Copyright © 1997 CRC Press, LLC.
• Film thickness formulae for the above-mentioned regimes are listed in Table 4. PRESSURE SPIKE FORMULAE Pressure spike amplitude and its locations are also determined by curve fitting the results of numeri- cal simulations. Data which were used in curve fitting covered a wide range of operating parameters with dimensionless load W varying from 0.2045 × 10-4, dimensionless speed U varying from 0.1 × 10-11 tp 5.0 × 10-11, and values of dimensionless materials parameter G of 2504, 5007, and 7511. One must check to make certain that these restrictions are satisfied for a given application. The pressure spike magnitude and its location are determined from the following expressions,7Copyright © 1997 CRC Press, LLC.
• FIGURE 4 Lubrication regimes of line contact.4 (From Roelands, D.J.A., Correlational Aspects of the Velocity-Temperature-Pressure Relationship of Lubrication Oils, Druk, V.R.B., Groningen, Netherlands, 1966.) Pressure spike location is The center of pressure (the location of the center of pressure indicates the position at which the resulting force acts) is given by: Another form of minimum film thickness expression is also available,7 In dimensional form where w is the load-per-width, minimum film thickness isCopyright © 1997 CRC Press, LLC.
• The central film thickness is ELASTOHYDRODYNAMIC ELLIPTICAL CONTACT The characteristic film shape and pressure distribution of an elliptical EHL is similar to that of the line contact. Some typical pressure and film thickness profiles predicted by the EHL theo- ry are shown in Figure 5.8 The maximum Hertzian contact pressure, pressure spike, and mini- mum film thickness, as well as central film thickness are of interest. FIGURE 5 Typical contour plot of film thickness (left) and pressure profile (right) for a cir- cular contact.8 In order to show the different regimes of lubrication problems, the dimensionless parameters defined in Equations 5 to 9 are used. The four regimes of rigid-isoviscous, rigid-viscous, elastic-iso- viscous and elastic-viscous are illustrated in Figure 6.9 FILM THICKNESS FORMULAE To determine the appropriate regime, the following dimensionless parameter groups are defined as: Film thickness formulae in these different regimes are summarized in Table 4 and Table 5. The minimum film thickness for more general consideration of the velocity vector is:14Copyright © 1997 CRC Press, LLC.
• FIGURE 6 Lubrication regimes of elliptical contact.8 (a) k = 111; (b) k = 1; (c) k = 3; (d) k = 6. whereCopyright © 1997 CRC Press, LLC.