Mixed Elastohydrodynamic Lubrication in Finite Roller ...

Dong Zhu Jiaxu Wang State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400044, P. R. C.

Ning Ren1 Q. Jane Wang Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208

Mixed Elastohydrodynamic Lubrication in Finite Roller Contacts Involving Realistic Geometry and Surface Roughness Concentrated (or counterformal) contacts are found in many mechanical components that transmit significant power. Traditionally, concentrated contacts can be roughly categorized to point and line contacts. In point contacts, the contact area is small in both principal directions, while in line contacts, it is small in one direction but assumed to be infinitely long in the other direction. However, these two types of geometry are results of simplification that does not precisely cover all the contact conditions in engineering practice. Actually most line contact components are purposely designed to have a crown in the contact length direction in order to accommodate possible non-uniform load distribution and misalignment. Moreover, the contact length is always finite, and at two ends of the contact there usually exist round corners or chamfers to reduce stress concentration. In the present work, the deterministic mixed EHL model developed previously has been modified to take into account the realistic geometry. Sample cases have been analyzed to investigate the effects of contact length, crowning, and end corners (or chamfers) on the EHL film thickness and the stress concentration, and also to demonstrate the entire transition from full-film and mixed EHL down to a practically dry contact under severe operating conditions with real machined roughness. It appears that this modified model can be used as an engineering tool for roller design optimization through in-depth mixed EHL performance evaluation. [DOI: 10.1115/1.4005952] Keywords: elastohydrodynamic lubrication (EHL), mixed EHL, concentrated contact, line contact, roller contact, stress concentration, roughness effect

Introduction Elastohydrodynamic lubrication (EHL) is known as a mode of fluid-film lubrication, in which the lubrication formation mechanism is enhanced by the surface elastic deformation and the viscosity increase due to high pressure. This mechanism widely exists in concentrated (or counterformal) contacts that can be found in many mechanical components, such as various gears, rolling element bearings, cam-follower systems, vane pumps, ball screws, continuously variable transmissions and some metalforming tools, and so on. These components often transmit significant power and motion, so the study of EHL performance is vital to components design, durability/reliability improvement, and power loss minimization. Traditionally concentrated contacts are approximately categorized into two types, point and line contacts. In point contacts, the area of surface interaction is small in both principal directions, while in line contacts, it is small in one direction but assumed to be infinitely long in the other direction. Point contact geometry is usually simplified into two radii of curvature so that the contact zone is an ellipse, whereas the line-contact zone is commonly treated as an infinitely long uniform narrow band. The foundation of the line contact EHL theory was laid by Petrusevich [1], Dowson and Higginson [2,3] and many others, and that of point contact EHL by Ranger et al. [4], Hamrock and Dowson [5–7], and others. Although significant contributions 1 Currently employed by Ashland, Inc., Lexington, Kentucky. Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received July 9, 2011; final manuscript received January 30, 2012; published online March 5, 2012. Assoc. Editor: Michael Khonsari.

Journal of Tribology

have been made by many researchers since the 1960s–1970s, the contact geometry has been treated in the same way, being simplified to either an infinitely long line contact or an elliptical point contact in most numerical analyses. However, these two types of contact geometry are results of simplification that do not precisely cover all the contact conditions in engineering applications. Actually most “line contact” components, including gear teeth, cams, and rolling bearing elements, are purposely designed to have a crown in the contact length direction in order to accommodate possible non-uniform load distribution, fabrication errors, and misalignment. Moreover, the contact length is always finite, and at two ends of the contact zone, there usually exist round corners or chamfers for reduction of edge stress concentration. These design practices can be found everywhere. In fact, infinitely long uniform line contact can hardly be seen in reality. It is well known that, in an early stage of science/technology development, complex practical problems were usually simplified with a number of assumptions in order to utilize available tools for getting approximate but meaningful solutions. As the theory and technology advance, research efforts are always made to approach complicated engineering reality, and the simplification assumptions are inchmeal removed one after another. Although experimental measurements were conducted as early as 1967–1974 by Gohar and Cameron [8], Wymer and Cameron [9], and Bahadoran and Gohar [10] to investigate the effects of straight roller and modified roller geometry on the EHL characteristics, only a limited number of numerical simulations have been presented. Note that in experimental investigations, line contact specimens have to be finite in length due to practical reasons, and infinitely long line contact can hardly

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be achieved. However, in numerical simulations, finite “line” contact with a realistic generatrix is certainly more difficult to model. Early solutions of finite roller contact EHL include those by Mostofi and Gohar [11], Kuroda and Arai [12], Xu et al. [13], and others using the full Reynolds equation with either straight or profiled rollers of finite length under light or moderate loads. It was found that the minimum film thickness is always located near the ends of the roller due to pressure concentrations, but the influence on the central film thickness is relatively limited. Park and Kim [14] revealed, via their numerical solutions, that the maximum pressure peak and the minimum film thickness are dependent highly on the detailed profile of the roller. Chen et al. [15] specifically studied the EHL characteristics in logarithmic profiled roller contact and the effect of crown value. More recently, Liu and Yang [16] and Sun and Chen [17] presented their thermal EHL analyses for finite roller contacts, and Kushwaha et al. [18] investigated the effect of possible misalignment on the finite roller EHL. All the solutions mentioned above were developed under the fullfilm EHL conditions without consideration of surface roughness and possible asperity contacts. The present study employs a deterministic rough surface mixed EHL model recently developed by Zhu and Hu [19] and others in order to develop a more generic approach that can be used to solve engineering problems under practical conditions with real machined roughness. The contact geometry is considered as a roller contact, taking into account possible crowning, finite contact length, and round corners or chamfers at two ends. The traditional line contact and elliptical contact can, therefore, be considered as special cases of the roller contact with more geometric constraints. Numerical examples of the roller contact EHL are analyzed in order to evaluate the effect of detailed realistic contact geometry stated above, and demonstrate the capability of the model, which appears to be a useful tool in engineering practice for design optimization.

Contact Geometry A sketch of an idealized steady-state EHL line contact is given in Fig. 1, in which it is assumed that the contact line is infinitely long in the y-direction, and the load is uniformly distributed, so that the problem can be simplified to that of two-dimensional (2D) plain strain if the surfaces are assumed to be smooth. The original geometric gap between the two surfaces without deformation can be approximated by the following expression based on its Taylor expansion series: f ðx; y; tÞ ¼

x2 2Rx

(1)

where Rx is the effective radius of curvature in the direction of rolling perpendicular to the contact line. An idealized EHL point contact is illustrated in Fig. 2, which is three-dimensional (3D) in nature. In the same way the original gap between the two surfaces can be given roughly by f ðx; y; tÞ ¼

x2 y2 þ 2Rx 2Ry

(2)

However, in reality the contact geometry of mechanical components is often more complicated. Typical “line contact” components, such as gear teeth and bearing rollers, are usually designed to have a crown along the contact length direction in order to accommodate possible misalignment, fabrication errors, and nonuniform load distribution. Also, the contact length is always finite with round corners or chamfers at its two ends to minimize edge stress concentration. Therefore, the roller contact geometry illustrated in Fig. 3 appears to be more generic, precise, and practical. The mathematic simplification based on the Taylor expansion series may not be necessary because the realistic geometry is already sufficiently simple and easy to model. Definitions of relevant geometric parameters can also be found in Fig. 3. In the present study a roller in contact with a plane is assumed. Contact between two rollers can be readily handled in a similar way. It is important to note that both idealized line contact and point contact shown in Figs. 1 and 2 are actually special cases of the roller contact illustrated in Fig. 3.

Basic Equations The roller contact EHL model is a modification of the deterministic mixed EHL model originally presented by Zhu and Hu [19], and Hu and Zhu [20], and continuously modified by Wang et al. [21], Liu et al. [22], Zhu [23], and others. Most recently, it has been further improved by Zhu et al. [24] to consider possible surface evolution due to wear, and by Ren et al. to take into account the effect of plastic deformation when necessary [25,26]. The model is capable of simulating the entire spectrum of lubrication regimes, from full-film and mixed lubrication to dry contact, with a unified formulation and numerical approach. The pressure within the entire solution domain is governed by the Reynolds equation expressed as @ qh3 @p @ qh3 @p u1 þ u2 @ ðqhÞ @ ðqhÞ þ ¼ þ (3) @x 12g @x @y 12g @y @x @t 2

Fig. 1 Idealized line contact EHL

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Transactions of the ASME


Fig. 2

Idealized point/elliptical contact EHL

W2(x, y, t) are the surface evolution due to wear for Surfaces 1 and 2, respectively (see Ref. [24] for a detailed description). The surface elastic deformation can be calculated by the integral as follows: ve ðx; y; tÞ ¼

2 pE0

ðð X

pðn; 1; tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn d1 ðx nÞ2 þ ðy 1Þ2

(5)

The plastic deformation often occurs in reality due to either heavy external loading or surface asperity contacts or a combination of both. Its effect on the EHL performance may become significant in many cases (see Refs. [25,26]). According to Jacq, et al. [27], Chen and Wang [28], and others, the plastic deformation can be computed by

Fig. 3

p ðx; y; tÞ ¼ 2l

Geometry of a typical roller

V

In this equation, the lubricant can be either Newtonian or nonNewtonian (when g is considered as the effective viscosity defined in [19,20]). The x-coordinate is chosen to coincide with the rolling direction. For a roller contact EHL problem with two moving rough contacting surfaces, the local lubricant film thickness/gap is time-dependent and can be computed by h ¼ h0 ðtÞ þ f ðx; y; tÞ þ d1 ðx; y; tÞ þ d2 ðx; y; tÞ þ Ve ðx; y; tÞ þ Vp ðx; y; tÞ þ W1 ðx; y; tÞ þ W2 ðx; y; tÞ

(4)

Note that in Eq. (4) h0(t) is due to the normal approach of the two bodies, f(x, y, t) is the original contact geometry before deformation, Ve(x, y, t) is the surface elastic deformation, Vp(x, y, t) is the possible plastic deformation, d1(x, y, t) and d2(x, y, t) are the threedimensional original roughness profiles, and W1(x, y, t) and Journal of Tribology

ððð

epij ðn; 1; uÞeij ðn x; 1 y; uÞdnd1du

(6)

where eij is the elastic strain component at (n, f, u) due to a unit normal force applied on the surface at (x, y), epij is the corresponding plastic strain component, l is the material shear modulus, and V is the plastic strain affected volume. Please refer to Refs. [27,28] for details. The lubricant viscosity is assumed to be dependent on pressure, and one of the commonly used viscosity equations is the exponential law given below: g ¼ g0 eaP

(7)

Other viscosity laws can also be used here to readily replace Eq. (7) when necessary. Detailed discussion on the effect of different viscosity models is beyond the scope of the present study. The density of lubricant is also a function of pressure, commonly calculated by JANUARY 2012, Vol. 134 / 011504-3


q ¼ qo 1 þ

0:6 109 p 1 þ 1:7 109 p

(8)

The applied load is balanced by the integral of the pressure over the entire solution domain, X, i.e., ðð pðx; y; tÞdxdy (9) wðtÞ ¼ X

When looking at the Reynolds Eq. (3), one can easily realize that the pressure solution is extremely sensitive to the detailed geometric gap h(x, y, t). Any small deviation of the gap may cause significant change in pressure distribution. The geometric gap h(x, y, t), however, is computed by Eq. (4), from which one can readily estimate the relative importance of each term on the right hand side. Typically, the surface roughness terms, d1 and d2, are often in the orders of 10100 nm, the elastic and plastic deformation, Ve(x, y, t) and Vp(x, y, t), and normal approach h0(t) are in the orders of 0.110 lm, the surface evolution due to wear, W1 and W2, often not exceeding a few microns. One can obviously see that in reality deviation of the original geometry from the simplified profiles described by Eq. (1) or Eq. (2) can possibly be one or a few orders of magnitude greater than the other terms in the film thickness Eq. (4). More precisely defined contact geometry is certainly needed in the EHL modeling. Note that the original geometry, f(x, y, t), consists of simple shapes and their combination, to be used and shown in later sections. Its detailed mathematic descriptions for various types of roller geometry are sufficiently simple in nature but possibly lengthy, so we will not present them all here. However, giving a typical example may still be a good idea. For a crowned roller with rounded end corners (see Fig. 3), one can have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (10) f ðx; y; tÞ ¼ Rx D2 x2 where D¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2y y2 ðRy Rx Þ;

if j yj

L lc 2

or D ¼ Rc þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 d2

d ¼ j yj y c L r 1 lc yc ¼ 2 Ry qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rc ¼ ðRy rÞ2 y2c ðRy Rx Þ if L L l c < j yj 2 2 Figure 4 demonstrates a numerical example according to Eqs. (4) and (10), in which the machined surface RMS roughness is 0.5 lm, which is typical in engineering practice. It can be seen that the deviation of true surface profile due to the crowning and the round corners, the solid line, from that described by Eq. (1), the dashed line, is much greater than the roughness. It is of the same order of magnitude as, or greater than, elastic and plastic deformation and possible wear. It is certainly unreasonable to ignore the effects of realistic geometric modifications, such as crowning and round corners, while considering those of less important terms, elastic and plastic deformation, surface roughness and possible wear, in numerical modeling. 011504-4 / Vol. 134, JANUARY 2012

Fig. 4 Deviation of a roller profile from straight cylinder. The solid line represents the true surface profile, while the dashed line shows the simplified geometry described by Eq. (1).

Preliminary Results In order to investigate the basic characteristics of roller contact EHL, a set of sample cases are numerically analyzed. In these cases the material properties are E0 ¼ 219.78 GPa, go ¼ 0.096 Pa s, and a ¼ 18.2 GPa1. The nominal roller radius is Rx ¼ 19.05 mm, the nominal roller length is L ¼ 5 mm, the width of possible round corners or chamfers at the roller ends is lc ¼ 0.75 mm. The applied load is fixed at W ¼ 3855.6 N. The maximum Hertzian pressure is Ph ¼ 1.190 GPa, and the half width of the Hertzian zone is a ¼ 0.4126 mm under the given conditions if assuming a straight cylindrical contact. The rolling velocity is U ¼ 0.625 m/s, and the slide-to-roll ratio is S ¼ 0.25. Corresponding dimensionless EHL parameters are G* ¼ 4000, W* ¼ 1.842 104 and U* ¼ 1.433 1011. The solution domain is determined as 3:1 X 1:3 and 1:05 Y 1:05. The computational grid covering the domain consists of 257*257 nodes equally spaced. This corresponds to a spatial mesh size of DX ¼ 0.01719 and DY ¼ 0.008203. The progressive mesh densification (PMD) method is used to speed up the solution process, and its description can be found in Ref. [23]. The descriptions and obtained results are summarized in Table 1 and Fig. 5. Note that in the preliminary study the plastic deformation and the surface wear are not considered in the analyses in order to avoid excessive complexity. Plastic deformation and wear can be computed when needed, and their numerical methods have been clearly described in Refs. [24–26]. Also, Case (a) is for the corresponding infinitely long straight cylinder contact at the same Hertzian pressure as a comparison reference, Case (b) is a straight roller contact with a finite nominal length of 5 mm, Case (c) is a straight roller with the same finite length but the two ends are rounded and the corner radius is r ¼ 12.7 mm, Case (d) is the same as Case (c) but a crown of Ry ¼ 254 mm is added, and Case (e) is the same as Case (d) but the round corners are changed to chamfers of h ¼ 15 degree. Note that the use of the model system presented in this paper is for a relative comparison, and the error when using Eq. (5) for rollers with sharp edges is not considered. The values of the maximum pressure peak and von Mises stress, as well as the film thickness at the contact center, are also given in Table 1 for comparison. This set of cases covers a good variety of roller contact types so that the results are representative. It should be indicated that in Fig. 5 the first row of the graphs is the film thickness and pressure contours for the five cases, the second row is the film thickness and pressure distributions along the y-axis perpendicular to the direction of motion, and the third row is the subsurface von Mises stress fields on the y-z plane. This figure reveals that the infinitely long straight roller, Type (a), yields the lowest pressure peak and von Mises stress values, but, unfortunately, this type of roller hardly exists in reality. Type (b) is the straight Transactions of the ASME


Table 1

Case no. a b c d e

Fig. 5

Summary of results for five different types of roller contact

Description Infinitely long straight roller Finite straight roller, L ¼ 5 mm Finite straight roller with round corners, L ¼ 5 mm, r ¼ 12.7 mm Finite roller with crowning and round corners, L ¼ 5 mm, Ry ¼ 254 mm, r ¼ 12.7 mm Finite roller with crowning and chamfers, L ¼ 5 mm, Ry ¼ 254 mm, h ¼ 15 degree

Max. pressure (GPa)

Max. von mises stress (GPa)

Film thickness at contact center (nm)

1.186 5.936 1.839

0.662 3.181 1.073

380 401 389

1.474

0.862

361

4.554

2.442

358

Five different types of roller contact (H – film thickness contour, P – pressure contour)

Fig. 6 Effect of roller length – straight rollers

roller with finite length, causing the highest pressure peaks and von Mises stresses, as well as local surface contact (minimum film thickness becomes zero) at the roller ends, due to the edge effect, so it is not preferred in practice. Type (c) adds round corners to the roller ends, significantly reducing the pressure peaks and stress concentraJournal of Tribology

tions. Type (d) further modifies the roller geometry with an adequate crowning so that the design appears to be optimal. Type (e) is the same as Type (d) but replacing the round corners with the chamfers. It is obvious that the chamfers do not work well due to the existence of a geometric higher-order discontinuity. It is also important to JANUARY 2012, Vol. 134 / 011504-5


Fig. 7

Effect of roller length – straight rollers with round corners

notice that the change in film thickness at the contact center hc is rather limited in this set of cases, within 66% if using that of infinitely long roller as a comparison baseline. This clearly indicates that the detailed roller geometry design may have a significant influence on the contact and lubrication characteristics, and Type (d) and Type (c) are desirable in engineering practice. It should also be noticed that the stress concentrations due to pressure peaks look similar to those from dry roller contact solutions, e.g., those presented in Ref. [29]. Actually, dry contact is nothing but a special case of lubricated contact under extreme conditions such as ultra-low viscosity/density of lubricant. The recent mixed lubrication studies, such as Refs. [19,20,23] and specifically Ref. [30], have already demonstrated that EHL solutions gradually approach those of dry contact when the entraining speed approaches zero and the hydrodynamic effect vanishes. Please see Ref. [30] for detailed explanations.

Fig. 8

Effect of Roller Length The effect of roller length is of great importance in engineering design. As a preliminary study, two sets of cases have been analyzed, and the results are summarized in Figs. 6 and 7. The applied load is set to 771.1 N/mm for all the cases so that the nominal Hertzian pressure remains constant, Ph ¼ 1.190 GPa. Other input parameters are kept the same as those given above. The results show that for a set of straight rollers with no round corners or chamfers, the effect of roller length appears to be significant, the longer the roller, the lower the maximum pressure peak and von Mises stress. However, when the round corners are added, this effect becomes considerably less significant. The roller length does not seem to have a great influence on the lubricant central film thickness for the cases studied.

Effect of Crown Radius. Load 1200 N/mm, Roller Length 10 mm, Ph 5 1.4844 GPa.

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Fig. 9 Effect of crown radius – two more sets of cases. (a) Load 1200 N/mm, roller length 5 mm, Ph 5 1.4844 GPa, and (b): load 600 N/mm, roller length 10 mm, Ph 5 1.0496 GPa.

Effect of Crown Radius Crown radius is a design parameter relatively more difficult to determine. It relies on many factors, such as possible fabrication errors and misalignment, design geometry, operating conditions, system dynamics, and concerns related to contact stresses, lubrication performance, and durability. The present study focuses on its influence on the contact and lubrication characteristics. In the numerical cases shown in Fig. 8, the roller length is set to be 10 mm, the load increased to 1200 N/mm, and the round corner radius

r ¼ 12.7 mm. All the other input parameters remain the same. As a result, the nominal Hertzian pressure becomes 1.484 GPa and the load parameter W* ¼ 2.8661 104. It is observed that when the crown radius is small, it may become an elliptical contact, and the maximum pressure located at the center of contact may be much higher than the nominal line contact Hertzian pressure due to the elliptical contact geometry. As the crown radius increases, the maximum pressure decreases and gradually the two ends of the roller get in contact. If the radius is further increased, the maximum pressure inchmeal moves to the edge locations, and the

Fig. 10 Effect of round corner radius, load 1200 N/mm, roller length 5 mm, crown radius 1270 mm, Ph 5 1.4844 GPa


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solution approaches that of a straight roller with no crowning. Figure 8 suggests that there is an optimal range of the crown radius, within which both the maximum pressure and von Mises stress remain low. The most desirable radius appears to be about Ry ¼ 762 mm for this set of cases. The result also reveals that the effect of crown radius on the film thickness is relatively limited if the radius is not very small and the two roller ends still remain in contact with the mating plane. It is important to note that, although the basic trends remain the same or similar, the optimized range of crown radius depends largely on the specific design geometry and the operating conditions. Figure 9 demonstrates two more sets of cases, one with a changed roller length and the other under a reduced load. When the roller length is 5 mm, the optimized range is shifted to the left and the most desirable radius is decreased to 254 mm or so. If the roller length stays the same, 10 mm, but the load is reduced to 600 N/mm, the optimized range moves upwards and the preferred radius becomes about 1143 mm. It can be concluded that the crown radius does have a significant influence on the contact and lubrication characteristics and a careful optimization procedure is necessary.

Effect of Round Corner Radius Similarly, the effect of round corner radius relies heavily on the specific design and the operating conditions. There does not seem

to be a simple generic solution to design optimization problems. However, the acceptable data range may be somewhat wider. This can be seen from Fig. 10, which summaries the results from a set of typical cases, whose input data are the same as those given above for the cases shown in Fig. 9(a) except the changing radius of round corners from zero to about 101.6 mm. It is clear that the highest pressure peaks and von Mises stresses are located around the two edges if the corner radius is small or large. When the radius falls in the optimized range in the middle, the maximum pressure may be found at the center. The most desirable radius appears to be around 3850 mm. Also, the corner radius does not seem to have significant impact to the lubricant central film thickness.

Transition of Roller Contact Interface Status The current roller contact EHL model is developed based on the mixed EHL solution method presented in Refs. [19–23], which is capable of simulating the entire transition from full-film and mixed EHL down to dry contact, using three-dimensional machined rough surfaces with relative motion. It is well known that in reality no surface is ideally smooth, and the root mean square (RMS) roughness is typically of the same order of magnitude as, or greater than, the average lubricant film thickness under concentrated contact. Therefore, rough surface asperity contacts usually exist, and most power-transmitting components operate in

Fig. 11 Continuous transition from full-film and Mixed EHL to dry contact with two shaved surfaces

Fig. 12

Contact load ratio and friction as functions of speed or film thickness ratio over the entire transition

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the mixed lubrication regime. Figure 11 illustrates a set of cases, for which the input parameters are the same as those shown in Fig. 10 except that the round corner radius is fixed to 6.35 mm and the speed is changed continuously in a wide range from U* ¼ 2.2929 1020 to 1.1465 109 covering eleven orders of magnitude. In this set of cases, two shaved surfaces with nearly isotropic roughness topography are rubbing against each other, and the composite RMS roughness is 0.50 micron. The lightyellow colored areas in the film thickness contours indicate the asperity contacts, and the film thickness ratio k is defined as the ratio of the average film thickness over the composite RMS roughness, which is a good measurement of lubrication effectiveness. Assuming friction coefficient of boundary lubrication is known, say 0.15 in these cases, one can readily predict the friction coefficient in mixed EHL using the method described in Ref. [31]. Figure 12 summarizes the major results from this set of cases, plotting the contact load ratio and friction coefficient variations against the speed and the k ratio, respectively. Note that the contact load ratio Wc is defined as the load supported by the asperity contacts divided by the total load. When Wc ¼ 0, it is the full-film lubrication with no asperity contact. If Wc becomes greater than 0.80.90, it can be considered as dry contact (or boundary lubrication) with no significant hydrodynamic action. The friction curves in Fig. 12 can be considered as the “Stribeck” curves, showing the continuous variation of friction from the full-film and mixed to boundary lubrication and demonstrating typical characteristics.

Conclusions The mixed lubrication model is extended to investigate roller contact EHL problems. Based on the results and discussion presented above, the following conclusions can be drawn: •

•

•

•

•

•

Various types of contact geometry widely exist in practice. In-depth EHL analyses based on actual geometry more accurate than parabolic approximations, Eqs. (1) and (2), are certainly needed. In reality, line contact in many machine elements can be more precisely described as roller contacts, having limited contact length, crowning in axial direction and round corners or chamfers at roller ends in order to accommodate possible non-uniform load distribution, fabrication errors and misalignments, and avoid severe edge stress concentrations. These geometric modifications may greatly improve load distribution, reduce pressure peaks, release stress concentrations, and as a result, increase anti-wear anti-fatigue life. However, their influence on EHL central film thickness appears to be relatively limited. Optimization of roller geometric design is a complicated task, involving many factors such as possible fabrication errors and misalignment, roller shape and size, operating conditions, system dynamics, and concerns related to materials and lubricant. It is often difficult to find a simple generic solution to design optimization problems, and full consideration of the realistic geometry and surface roughness is needed. The mixed EHL model presented appears to be capable of handling rollers with real machined roughness over a wide range of operating conditions. It can be employed as a useful tool for component geometric design optimization. The unified mixed lubrication models are now capable of simulating the entire transition of interfacial status from fullfilm and mixed lubrication down to dry contact with an integrated mathematic formulation and numerical approach. This has indeed bridged the two branches of engineering science, contact mechanics, and hydrodynamic lubrication theory, which have been traditionally separate since the 1880s mainly due to the lack of powerful analytical and numerical tools in the past. Based on the recent advancement, an evolving concept of “interfacial mechanics” is proposed, bridging, covering, and eventually unifying contact and lubrication mechanics/theories. See Ref. [30] for more details.


Acknowledgment The present study is partially supported by NSFC (National Science Foundation of China) Project 50735008.

Nomenclature a¼ E0 ¼ G* ¼ h¼ ha ¼ hc ¼ hm ¼ L¼ lc ¼ p¼ Ph ¼ r¼ Rx ¼ Ry ¼ S¼ t¼ U¼ U* ¼ u1, u2 ¼ Ve, Vp ¼ w¼ W¼ W* ¼ Wc ¼ W1, W2 ¼ x¼ y¼ a¼ d1, d2 ¼ g, go ¼ k¼ h¼ q, qo ¼ r¼

half width of Hertzian contact zone effective Young’s modulus aE0 , dimensionless material parameter local film thickness (or gap), H ¼ h/a, dimensionless film thickness average film thickness in the central area of Hertzian contact zone film thickness at the center of contact zone minimum film thickness nominal roller length width of round corner or chamfer pressure, P ¼ p/Ph, dimensionless pressure nominal maximum Hertzian contact pressure round corner radius nominal roller radius, or effective radius in x-direction across contact line crown radius, or effective radius in y-direction (u2 u1)/U, slide-to-roll ratio time (u1 þ u2)/2, rolling velocity gou/(E0 Rx), dimensionless speed parameter velocities of Surface 1 and Surface 2, respectively surface elastic and plastic deformation, respectively applied load per unit contact length applied load w/(E0 Rx), dimensionless load parameter contact load ratio, defined as the load supported by surface asperity contacts divided by the total load material removal due to wear on Surface 1 and Surface 2, respectively x-coordinate (rolling direction), X ¼ x/a, dimensionless x-coordinate y-coordinate pressure-viscosity exponent original roughness profiles of Surface 1 and Surface 2, respectively viscosity and viscosity under ambient condition ha/r, film thickness ratio chamfer angle density of lubricant and density under ambient condition composite RMS roughness of the two surfaces

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