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4.5 CONTACT STRESS AND WEAR: PROGRAMMING
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Let us consider the general case of two cylinderlike surfaces in contact.They are represented by a cam and a follower. The radius of curvature of the follower 1 is equal to the radius of the roller Rr for the roller follower, and it goes to infinity for a flat
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CAM MECHANISMS 4.26
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follower. The radius of the cam s curvature 2 can be found from the following equation [compare to Eq. (4.23)]: 2 = R0 + s + s (4.36)
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Equating d 2/d = s + s to zero, we can find the position of the cam, where a minimum of 2 occurs, and find its value from Eq. (4.36). Assuming perfect alignment of the contacting bodies, we have conditions described by Hertz and can check maximum compressive stress c from his well-known equation c = 0.558 where P(1/ 1 + 1/ 2) L[(1 2 )/E1 + (1 2 )/E2] 1 2 (4.37)
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P = normal load between cam and follower L = actual thickness of contacting follower and cam 1, 2 = Poisson s ratios for follower and cam, respectively E1, E2 = moduli of elasticity of follower and cam, respectively
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Some selected data concerning properties of materials for cams and followers are given in [4.7]. Equation (4.37) may be used for finding the minimum permissible value min of the cam s profile radius of curvature. We recall that min was necessary for calculating the values of R0 and Rr from Eqs. (4.23) and (4.26), respectively. Rearranging Eq. (4.37) gives, for a roller follower,
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2 min 3.2 S c
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1 2 1 2 1 2 + E1 E2
1 Rr
(4.38)
The same equation holds true for a flat-faced follower, where 1/Rr = 0. Using Eq. (4.38), we can easily check to see if the commonly recommended and used value min = 0.25 in (6 mm) is justified in the particular design. Elements of a cam system, as well as of other machine parts, are subject to wear. The proper choice of metal combinations may increase the life of kinematic pairs of the cam system and decrease their wear. Some experience is necessary in choosing materials to fulfill the requirements of satisfactory cam action with low wear over a long period. Designers, as a rule, prefer to make the follower of softer or first-worn-out material, since manufacturing of the follower is less expensive than manufacturing of the cam profile. There are, however, cases where the cam is cheaper and thus is made of softer material.
4.5.1 Programming of Cam Systems The steps shown in Fig. 4.20 are as follows: 1. Make a preliminary sketch of your cam system, and estimate the dynamic factor 2 d = me c /ke, according to Eq. (4.8). 2. If your system is a positive drive, go to step 3; otherwise go to step 4. 3. Choose a proper symmetric diagram s ( ), and write equations for s ( ), s ( ), and s( ). Write a computer program for and s( ) with increments of equal to 1 . 4. Choose a proper unsymmetric diagram s ( ), and proceed as in step 3. 5. Print table of and s( ).
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CAM MECHANISMS 4.27
CAM MECHANISMS
FIGURE 4.20 Programming of cam systems.
6. If you decided to use a roller follower, go to step 7a; otherwise go to step 7b. 7. a. Establish the radius of the roller Rr and the value of maximum pressure angle max ( max 30 for the reciprocating follower and max 40 for the oscillating one). Find the value of for which s /tan max s Rr is maximum, by equating s /tan max s to zero. Calculate the value of the radius R0 of the prime circle from Eq. (4.26). Go to step 8a. b. Find the value of for which min s s is maximum, by setting (s + s ) to zero. Calculate the value of the radius R0 of the prime circle from Eq. (4.23). Calculate the face width F from Eq. (4.24). Go to step 8b. 8. a. Check undercutting of the cam profile from Eq. (4.27). b. Find the spring rate of the retaining spring. Go to step 10. 9. If the assumed value Rr is satisfactory, go to step 10; otherwise, assume a new value of Rr and return to step 7a.
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