Friction and Lubrication in Mechanical Design Episode 1 Part 5 pptx

Traction Distribution and Microslip in Frictional Contacts 81 of rigid body rotation are determined by this iterative procedure, as depicted in the flow chart Fig... Print the results F

Figure 3.19 (a) Tangential load sharing between the two contact zones at differ- ent applied loads (b) Distance between the line of action of the resultant frictional resistance and centroid at different applied loads

Traction Distribution and Microslip in Frictional Contacts 81

of rigid body rotation are determined by this iterative procedure, as depicted

in the flow chart (Fig 3.22)

82 Chapter 3

0.9 l

O ,

Traction Distribution and Microslip in Frictional Contacts

b

83

forces negligible? Print the results

Find the traction distribution

using the modified linear programming

Calculate the residual force

c

Assume the new center of rotation

1 using a linear interpolation scheme

-lbf, and the coefficient of friction is 0.1 A grid with 80 square elements is

used to discretize the circular contact area of 0.36628 x 10-' in radius

A comparison between Lubkin's theory (solid line) and the numerical results (symbol s) is plotted in Fig 3.24 and very good agreement can be

seen The angle of rigid rotation (0.10641 x 10-* rad) is also found to com- pare favorably with Lubkin's theory (0.1 11 19 x 10-* rad) with a deviation

of 4.30% The center of rotation is at the centroid

Traction Distribution and Microslip in Frictional Contacts 85

EXAMPLE 2: Elliptical Hertzian Contact The elliptical Hertzian con-

tact area with an aspect ratio of 2 is assumed to occur when a normal load of 21601bf is applied on two steel bodies The pressure distribution is assumed to be Hertzian in this case The coefficient of friction is taken

to be equal to 0.1 and a twisting moment of 3.8in.-lbf is applied on the

interface A grid with 80 rectangular elements of the side ratio of 2 is used

to discretize the elliptical contact area

The contours of the magnitude and the direction of the tractions are plotted in Figs 3.25 and 3.26 The border line between the no-slip region and the slip region is also shown as a broken line The centroid in this case is the center of rotation

Interpreted Magnitude of Stress

Figure 3.25 Contour plot for the magnitude of traction on the elliptical contact area

86 Chapter 3

Direction of Predirected Stress (N=80)

Figure 5.26 Contour plot for the direction of traction on the elliptical contact

EXAMPLE 3: Disconnected Contact Areas on Semi-Infinite Bodies Two disconnected square areas of the same size (0.6in x 0.6in.) on a semi- infinite steel body are assumed to be in contact with another semi-infinite steel body The centroids of the two squares are located 1 in apart (Fig 3.27) Uniform pressure is assumed on each contact region and the coeffi- cient of friction is 0.12 for both regions Two cases of normal loading are considered here

Case 1 P I = 10,OOOpsi and P2 = 10,OOOpsi;

Case 2 P I = 20,OOOpsi and P2 = 10,OOOpsi

Each region is discretized with 36 square elements to have 72 elements for the entire contact area

The contours of the magnitude and the direction of the tractions are plotted in Figs 3.27 and 3.28 for Case 1 with a twisting moment of 500 in.-lbf and in Figs 3.29 and 3.30 for Case 2 with a twisting moment of 700 in.-lbf It

A4 = 500in.-lbf and normal loading of

Case 1

I

I

f, = 0.12 P, = 10000 p8i

Figure 3.28 Contour plot of the

direction of traction on the contact area

magnitude of traction on the contact I

area of disconnected squares with I

M = 700in.-lbf and normal loading of

Figure 3.30 Contour plot for the

direction of traction on the contact area

of disconnected squares with

M = 700in.-lbf and normal loading of

Case 2

88

Traction Distribution and Microslip in Frictional Contacts 89

can be seen that the traction contours and the slip patterns for both regions

1 and 2 are identical and the center of rotation is the centroid for the symmetric normal loading As would be expected, the case of asymmetric normal loading shows different traction distributions in the two discon- nected contact areas and the center of rotation is consequently found to

be displaced from the centroid Also notice that region 2 reaches the state of

total slip for Case 2, with a twisting moment of 700in.-lbf, and circumfer- ential tractions are assumed for region 2

The center of rotation always occurred on the line connecting the centroids of two disconnected squares The x-distance between the center

of rotation and the centroid for Case 2 versus the applied twisting moment is plotted in Fig 3.31

The development of the slip region with the increasing twisting moment

is shown in Fig 3.32 for Case 2 It can be seen that region 2 reaches a state

of total slip at a twisting moment of 700 in.-lbf, and that gross slip occurs at

770 in.-lbf Some slip is also shown to occur in region 1 below 700 in.-lbf The compliance curve relating the angle of rigid rotation and the twist- ing moment is plotted in Fig 3.33a for Case 1 and in Fig 3.33b for Case 2

Progression of slip with increasing twisting for contact area of dis-

3.5 FRICTIONAL CONTACTS SUBJECTED T O A COMBINATION

OF TANGENTIAL FORCE A N D TWISTING MOMENT

3.5.1 Iterative Procedure

The analysis of the frictional contact problem under a combination of tan- gential force and twisting moment is a highly nonlinear problem The problem is piecewisely linearized using an iterative method and a modified linear programming technique is utilized at each iteration The procedure

followed in the iterative method is shown in Fig 3.34

Traction Distribution and Microslip in Frictional Contacts 91

circular contact area of 5.15 x 10-2in radius results from a normal load

of 30001bf and the coefficient of friction is taken as 0.1 The tangential

force of 1461bf and the twisting moment of 4.8in.-lbf are applied on the

contact surface A grid with 80 square elements is used to discretize the circular contact area

traction at each grid point as zero

4

[ i = l + l 1

Find the traction distribution due to

the tangential force T' using the

modified linear programming [3]

fll is the applied tangential force)

Find the traction distribution due to

the twisting moment M' using the

preprocessor and the modified linear

programming (M' is the applied lwisting moment)

Combine the traction distribution due to

T' and Mi with that of the previous iteration

1

1

1

Loop 100 for ail the grid points

IS the combined traction force Fbigger

than the limit value fP@t a grid point k?

No

[Adjust F to the limit valuer^, I

Calculate the residual force R: and

the residual moment RL due to the

exceeding traction forces (F: - c p ~ )

Calculate the residual force R:

the residual moment RL due to

exceeding traction forces (F: -

Figure 3.34 Flow chart for the iterative procedure

94 Chapter 3

The contours of the magnitude and the direction of the traction distribution using the iterative procedure are plotted in Figs 3.36 and 3.37 The border- line between the no-slip region and the slip region is also shown as a broken line The center of rotation is found to be located at the centroid

The rigid body movement and the angle of rigid rotation obtained by the iterative procedure (0.68224 x 10-4 in and 0.10670 x 10d2 rad) agree well with those obtained by using a nonlinear programming formulation

[241*

The elapse CPU time on a Harris 800 to obtain the above results using the iterative procedure is 14 min, whereas that using the nonlinear program- ming technique is 31 min when the solution obtained by the iterative procedure is used as an initial guess

EXAMPLE 2: Disconnected Contact Area on Semi-Infinite Bodies Consi-

der two disconnected square areas of the same size (0.6in x 0.6in.) on a

Figure 3.37

subjected to a combined load (using iterative procedure)

Contour plot for the direction of traction on the circular contact

Traction Distribution and Microslip in Frictional Contacts 95

semi-infinite steel body The centroids of the two squares are located 1 in apart (Fig 3.38) Uniform pressure is assumed on each contact region and the coefficient of friction is 0.12 for both regions Each region is dis- cretized with 36 square elements (72 elements for the entire contact area) Two cases of loading are considered here:

Case 1 P I = 20,000 psi, P2 = 10, !OOO psi, T = 500 Ibf, M = 300 in.-lbf Case 2 P I = lO,OOOpsi, P2 = 20,OOOpsi, T = 6001bf, M = 400in.-lbf

For Case 1, the contours of the magnitude and the direction of the traction distribution obtained by the iterative procedure are plotted in Figs 3.38 and 3.39 and found to compare favorably with those obtained by applying the nonlinear programming technique [24]

The corresponding results for Case 2 are shown in Figs 3.40 and 3.41 In this case, region 1 is found to be in a state of total slip

The rigid body motions from the iterative procedure (0.16436 x 10F4 in and 0.12323 x 10-4rad for Case 1 and 0.30509 x 10-4in and 0.32721 x

10-4rad for Case 2) compare favorably with those from the nonlinear

f, = 0.12 e, = loo00

f, - 0.12 P, = 20000 pi

Figure 3.38 Contour plot for the magnitude of traction on the contact area of disconnected squares for Case 1 (using iterative procedure)

f2=0.12 P*rlOO00pd midM

+M)OIbf

f, = 0.12 P, = 20000 pcri

Figure 3.39 Contour plot for the

direction of traction on the contact

area of disconnected squares for Case 1

(using iterative procedure)

Figure 3.40 Contour plot for the : s t r ~ s - 1 2 ~ p s i :

area of disconnected squares for Case 2

(using iterative procedure)

Traction Distribution and Microslip in Frictional Contacts 97

f, = 0.12 P, = 20000 psi 4oo in,M

-6OOIW

f, = 0.12 P, = 10000 psi

Figure 3.41

disconnected squares for Case 2 (using iterative procedure) Contour plot for the direction of traction on the contact area of

programming technique (0.16502 x 10-4 in and 0.12263 x 10-4 rad for Case 1 and 0.31257 x 10-4in and 0.30892 x 10-4rad for Case 2), with deviations of 2.39% and 0.49% for Case 1, and 2.31% and 5.92% for Case 2, respectively

The elapse CPU times on a Harris 800 to obtain the above results by the

iterative procedure are 2 min for Case 1, and 8 min for Case 2, whereas those

necessary to obtain the results from the nonlinear programming technique

are 18 min for Case 1, and 46 min for Case 2, respectively, when the solu- tions obtained by the iterative procedure are used as initial guesses

REFERENCES

2 Lundberg, G., “Elastische Beruhrung Zweier Halbraume,”Forsch Ingenieurw., 1939, Vol 10, pp 201-21 1

Greenwood, J A., and Tripp, J H., “The Elastic Contact of Rough Spheres,”

J Appl Mech., Trans ASME, March 1967, pp 153-159

Schwartz, J., and Harper, E Y., “On the Relative Approach of Two Dimensional Elastic Bodies in Contact,” Int J Solids Struct., Dec 1971, Tsai, K C., Dundurs, J., and Keer, L M., “Contact between an Elastic Layer with a Slightly Curved Bottom and a Substrate,” J Appl Mech., Trans ASME, Sept 1972, Ser E., Vol 39(3), pp 821-823

Kalker, J J., and Van Randen, Y., “Minimum Principle for Frictionless Elastic Contact with Application to Non-Hertzian Contact Problems,”J Eng Math., April 1972, Vol 6(2), pp 193-206

Conry, T F., and Seireg, A., “A Mathematical Programming Method for Design of Elastic Bodies in Contact,” J Appl Mech., Trans ASME, June Erdogan, F., and Ratwani, M., “Contact Problem for an Elastic Layer Supported by Two Elastic Quarter Planes,” J Appl Mech., Trans ASME, Sept 1974, Ser E, Vol 41(3), pp 673-678

Nuri, K A., “Normal Approach between Curved Surfaces in Contact,” Wear, Francavilla, A., and Zienkiewicz, 0 C., “Note on Numerical Computation of Elastic Contact Problems,” Int J Numer Meth Eng., 1975, Vol 9(4), pp Haug, E., Chand, R., and Pan, K., “Multibody Elastic Contact Analysis by Quadratic Programming,” J Optim Theory Appl., Feb 1977, Vol 21(2), pp Kravchuk, A S., “On the Hertz Problem for Linearly and Non-Linearly Elastic Bodies of Finite Dimensions,” Appl Math Mech., 1977, Vol 41(2),

Johnson, K L., “Surface Interaction Between Elastically Loaded Bodies Under Tangential Forces,” Proc Roy Soc (Lond.), 1955, A, Vol 230, pp Deresiewicz, H., “Oblique Contact of Non-Spherical Elastic Bodies,” J Appl Mech., Trans ASME, 1967, Vol 24, pp 623-624

Traction Distribution and Microslip in Frictional Contacfs 99

Timoshenko, S P., and Goodies, J N., Theory of Elasticity, 3rd ed., McGraw

Hill Book Company, New York, NY, 1970

Lubkin, J L., “The Torsion of Elastic Spheres in Contact,” J Appl Mech., Trans ASME, 1951, Vol 73, pp 183-187

Choi, D., “An Algorithmic Solution for Traction Distribution in Frictional Contacts,” Ph.D Thesis, The University of Wisconsin-Madison, 1986

The Contact Between Rough Surfaces

4.1 SURFACE ROUGHNESS

All surfaces, natural or manufactured, are not perfectly smooth The smoothest surface in natural bodies is that of the mica cleavage The mica cleavage has a roughness of approximately 0.08pin The roughness of manufactured surfaces vary from a few microinches to 1000 pin depending

on the cutting process and surface treatment Representative examples of some of these are given in Table 4.1

Roughness represents the deviation from a nominal surface and is a composite of waviness and asperities Both of these are shallow curved surfaces with the latter having wavelengths orders of magnitude smaller than the former Asperities can be also considered as wavy surfaces on a microscale with their height being in the order of 2-5% of the wavelength,

as illustrated in Fig 4.1

4.2 SURFACE ROUGHNESS GENERATION

Surface roughness plays an important role in machine design During the metal cutting operation, a machined surface is created as a result of the movement of the tool edge relative to the workpiece The quality of the surface is a factor of great importance in the evaluation of machine tool productivity The results from a large number of theoretical and experi- mental studies on surface roughness during turning are available in the

100

The Contact Between Rough Surfaces 10 I

500- 1000 500- 1000

literature [ 1-18] Although various factors affect the surface condition of a machined part, it is generally accepted that the cutting parameters such as speed, feed, rate, depth of cut, and tool nose radius have significant influence

on the surface geometry for a given machine tool and workpiece setup There is also general agreement that surface roughness improves with increasing machine tool stiffness, cutting speed, and tool nose radius, and

decreasing feed rate [l-31 It has also been reported [2, 91 that at speeds less

than a certain value, discontinuous or semidiscontinuous chips and built-up- edge formation may occur, which can give rise to poor surface finish At speeds above that specific value, the built-up-edge size decreases and the surface finish improves This specific speed limit depends on many factors such as workpiece, tool conditions, and the state of the machine tool Sata

[9] reported 22.86 m/min as the speed limit in his experiment Of the factors

influencing the surface roughness, the depth of cut was found to have the