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FIGURE 4.12 Constant-breadth circular-arc cam.
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CAM MECHANISMS 4.17
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CAM MECHANISMS
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where b = cos 0.25 0/cos 0.75 0. Cam motions for full rise (0 0) are described in Table 4.4. Such cams are symmetric; therefore, 0(rise) = 0(return), and the two dwells d1 and d2 are the same and equal to 180 0. Table 4.4 can also be used for calculation of full-return motion. Dimensions of the cam (R1) and maximum values of the acceleration increase with a decrease in 0. Acceleration diagrams for different values of 0 are shown in Fig. 4.13.
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TABLE 4.4 Basic Equations for a Constant-Breadth Circular-Arc Cam, Using A = R1
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FIGURE 4.13 Acceleration diagrams.
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4.3 LAYOUT AND DESIGN; MANUFACTURING CONSIDERATIONS
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The cam profile is an inner envelope of the working surface of the follower. After the displacement diagram is determined, the cam layout can be found by using the usual graphical approach or by computer graphics with a rather simple computer program. In the design of a plate cam with a reciprocating flat-face follower, the geometric parameters necessary for its layout are the prime-circle radius R0, the minimum
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CAM MECHANISMS 4.18
MACHINE ELEMENTS IN MOTION
width of the follower face F, and the offset e of the follower face. The value R0 can be found from R0 > ( min s s)max (4.23)
where min is a minimum value of the radius of the cam-profile curvature. Its value for such practical reasons as contact stresses might be assumed equal to 0.2 to 0.25 in [5 to 6 millimeters (mm)]. Since s is always positive, we should examine that part of the follower acceleration diagram for the rise motion where acceleration is negative. The face width F can be calculated from F > smax smin (4.24)
To avoid undercutting cams with a roller follower, the radius Rr of the roller must always be smaller than | |, where is the radius of curvature. The pressure angle (Fig. 4.14) is an angle between a common normal to both the roller and the cam profile and the direction of the follower motion. This angle can be calculated from tan = s s + R0 + Rr (4.25)
FIGURE 4.14 Cam mechanism with reciprocating roller follower.
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CAM MECHANISMS 4.19
CAM MECHANISMS
It is a common rule of thumb to assume for the preliminary calculation that max is not greater than 30 for the reciprocating follower motion (or 45 for the oscillating one). Acceptable values of max that can be used without causing difficulties depend, however, on the particular cam mechanism design and should be found for any actual mechanism from the dynamic analysis. After establishing the value of max and Rr in accordance with the preliminary layout of the mechanism, we can find the value of the prime-circle radius R0 from the equation R0 s s Rr tan max (4.26)
Now check whether the assumed value of Rr is small enough to avoid undercutting of the cam profile. It can be done ([4.7]) by using Eq. (4.27): |s | s 1 1 +2 3 sin max |s | tan max |s | min
Rr
sin3 max
(4.27)
The primary choice of the follower motion should always be guided by a good understanding of the planned manufacturing technique. Tracer cutting and incremental cutting are two very common methods of cam manufacture. Incremental cutting consists of manufacturing the profile by intermittent cuts based on a table with accurate values of angular cam displacement (cam blank) and linear displacement s( ) of the follower (cutter). This method is used for making master cams or cams in small numbers. In the tracer control cutting method, the cam surface is milled, shaped, or ground, with the cutter or grinder guided continuously by either a master cam or a computer system. This is the best method for producing large numbers of accurate cam profiles. In the process of cam and follower manufacturing, several surface imperfections may occur, such as errors, waviness, and roughness. These surface irregularities may induce shock, noise, wear, and vibrations of the cam and follower systems. Imperfections of actual profile cannot exceed an accepted level. Therefore, highly accurate inspection equipment is commonly used in production inspection. Actual displacements of the follower are measured as a function of the cam rotation; then the resulting data can be compared with tabulated theoretical values. By application of the method of finite differences (Sec. 4.3.1), these data can be transformed to actual acceleration curves and compared with theoretical ones. There is, however, a drawback in such a method in that it is based on static measurements. An example of results obtained from a widely used production inspection method is shown in Fig. 4.15 ([4.5]). Line 1 was obtained from some accurate data from a table of values and the corresponding s( ) values. Next, two boundary curves were obtained from the basis curve by adding and subtracting 10 percent. This was an arbitrary decision, it being assumed that any acceleration curve contained between such boundaries would be satisfactory. These are shown as upper and lower bounds in Fig. 4.15. The main drawback of the method is that only maximum values of actual acceleration diagrams have been taken into account. It is important to realize that waviness of the real acceleration curve may cause more vibration troubles than will single local surpassing of boundary curves. A much better method is that of measuring the real acceleration of the follower in an actual cam mechanism at the operating speed of the cam by means of high-quality
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