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If the entire assembly is vendor-supplied as a needle-bearing cam follower, then the average load is dictated by the cam-follower force F23. The follower makes m turns per cam revolution. The follower s first turn has an average load to the a power of (F23)a = 1 1 2 /m
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where d = cam angular displacement. The subsequent averages are
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a (F23)2 =
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The global average to the a power is
a Fglobal =
1 1 (F23)a = i mi = 1 2
2 a F23 d 0
Consequently, the roller average radial load is identical to the cam s, but the follower makes m times as many turns. The follower contact surface between the cam and follower has an endurance strength cycles-to-failure relation of the form of S 1/bN = K, and so the average hertzian stress can be written H = 1 2
2 1/b H d 0 b
Cp = w
1 2
[F23(Kc + Kr)] 1/2b d
(29.45)
where = cam rotation angle, b = slope of the rectified SN locus, w = width of the roller or cam (whichever is less), Cp = a joint materials constant Cp = 1 1 1 2 2 + E1 E2
(29.46)
and the parameters Kc and Kr = the curvatures of the cam and roller surfaces, respectively. One surface location on the cam sees the most intense hertzian stress every revolution. That spot has a hertzian stress of H = Cp [F23(Kc + Kr)]1/2 max w (29.47)
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STRENGTH UNDER DYNAMIC CONDITIONS 29.35
STRENGTH UNDER DYNAMIC CONDITIONS
Relative strengths can be assessed by noting that the cam requires a strength of (Sfe)mN at the critical location in order to survive N cycles. The roller sees the average stress everywhere on its periphery mN times, and its strength requirement is (Sfe)mN. The strength ratio is [(Sfe)N]cam = [(Sfe)mN]roller [F23(Kc + Kr)]1/2 max 1 2 (
2 b
[F23(Kc + Kr)] 1/2b d
0 max
= If cam and roller are steel,
)avg
(Sfe)mN = mb(Sfe)N enabling us to place endurance strengths on the same life basis, namely, N, which is convenient when consulting tables of Buckingham load-strength data giving K or its equivalent. Thus, mb [(Sfe)N]cam = ( [(Sfe)N]roller
)avg
For steel, a 108 cycle expression, that is, (Sfe) = 0.4HB 10 kpsi, can be used. Using bhn for roller Brinell hardness and BHN for cam Brinell hardness, we can write BHN = mb (
)avg
(bhn 25) + 25
(29.48)
This form is convenient because the roller follower is often a vendor-supplied item )avg, this and the cam is manufactured elsewhere. Since max is larger than ( alone tends to require that the cam be harder, but since the roller endures more turns, the roller should be harder, since m > 1 and b < 0. Matching (tuning) the respective hardnesses so that the cam and roller will wear out together is often a design goal. Design factor can be introduced by reducing strength rather than increasing load (not equivalent when stress is not directly proportional to load). Since the loads are often more accurately known than strengths in these applications, design factors are applied to strength. The relative hardnesses are unaffected by design factor, but the necessary widths are w= Cp Sfe/n
2 max cam
Cp Sfe/n
roller
)avg m 2b
(29.49)
Either equation may be used. The width decision controls the median cycles to failure.
REFERENCES
29.1 R. C. Juvinall, Stress Strain and Strength, McGraw-Hill, New York, 1967, p. 218. 29.2 E. M. Prot, Fatigue Testing under Progressive Loading: A New Technique for Testing Materials, E. J. Ward (trans.), Wright Air Development Center Tech. Rep., TR52-148, September 1952.
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STRENGTH UNDER DYNAMIC CONDITIONS 29.36
LOAD CAPABILITY CONSIDERATIONS
29.3 J. A. Collins, Failure of Materials in Mechanical Design, Wiley-Interscience, New York, 1981, chap. 10. 29.4 W. J. Dixon and F. J. Massey, Jr., Introduction to Statistical Analysis, 3d ed., McGraw-Hill, New York, 1969, p. 380. 29.5 J. T. Ransom, Statistical Aspects of Fatigue, Special Technical Publication No. 121, American Society for Testing Materials, Philadelphia, Pa., 1952, p. 61. 29.6 J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, 5th ed., McGraw-Hill, New York, 1989. 29.7 C. R. Mischke, A Probabilistic Model of Size Effect in Fatigue Strength of Rounds in Bending and Torsion, Transactions of ASME, Journal of Machine Design, vol. 102, no. 1, January 1980, pp. 32 37. 29.8 C. R. Mischke, A Rationale for Mechanical Design to a Reliability Specification, Proceedings of the Design Engineering Technical Conference of ASME, American Society of Mechanical Engineers, New York, 1974, pp. 221 248. 29.9 L. Sors, Fatigue Design of Machine Components, Part 1, Pergamon Press, Oxford, 1971, pp. 9 13. 29.10 H. J. Grover, S. A. Gordon, and L. R. Jackson, Fatigue of Metals and Structures, Bureau of Naval Weapons Document NAVWEPS 00-25-534, Washington, D.C., 1960, pp. 282 314. 29.11 C. Lipson and R. C. Juvinall, Handbook of Stress and Strength, Macmillan, New York, 1963. 29.12 R. C. Juvinall, Fundamentals of Machine Component Design, John Wiley & Sons, New York, 1983. 29.13 A. D. Deutschman, W. J. Michels, and C. E. Wilson, Machine Design, MacMillan, New York, 1975. 29.14 E. B. Haugen, Probabilistic Mechanical Design, John Wiley & Sons, New York, 1980. 29.15 C. R. Mischke, Mathematical Model Building, 2d rev. ed., Iowa State University Press, Ames, 1980. 29.16 G. Sines and J. L. Waisman (eds.), Metal Fatigue, McGraw-Hill, New York, 1959. 29.17 H. O. Fuchs and R. I. Stephens, Metal Fatigue in Engineering, John Wiley & Sons, New York, 1980.
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