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t m = tan -1
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E 0500 = tan -1 = 171 deg M 1625
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The eccentric is a type of cam It is a circular member having its center of rotation offset from its geometric center The follower may be at-faced or roller type A translating atfaced follower will be discussed The eccentric represents the simplest cam mechanism possible Its zero pressure angle eliminates the problem of the follower jamming In Fig 148a we see an eccentric rotating about point A with its geometric center at a distance E In Fig 148b we see the equivalent mechanism which is the Scotch yoke mechanism in which the follower movement is a simple harmonic function Let y = follower displacement, in ye = equivalent mechanism follower displacement, in q = cam angle of rotation for displacement y, rad e = crank angle of rotation (equivalent mechanism) for displacement ye = rad E = distance from cam center to circular-arc center of curvature, in h = 2E = maximum displacement of follower, in w = cam and equivalent mechanism angular velocity, rad/sec In this example, y = ye and q = e From Fig 148b we see that the follower displacement
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(a) Cam and follower
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(b) Equivalent mechanism
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ye = E(1 - cos e ) in Differentiating, we nd the velocity and acceleration y = Ew sin e ips y = Ew 2 cos e in sec 2
(1410) (1411)
We see that the eccentric circle size has no effect on the follower action; only the eccentricity, E does Furthermore, offsetting the line of follower motion from the cam center of rotation does not change the follower movement
For the radial cams developed elsewhere in this book the desired displacement characteristics are initially established and then the shape of the cam is mathematically determined The analysis of the shape also includes the study of the geometric pressure angle and curvature of the cam Then the cam-follower dynamics are investigated if necessary In this section, we will establish the cam contours from known geometric shapes (sometimes blended with other shapes) with limited control of the cam-follower system dynamics These shapes are rarely utilized in design In producing a radial cam we can apply any curve or combination of curves such as straight lines, circular arcs, Archimedes spirals, involute, logarithmic spirals, ellipses, parabolas, and hyperbolas As cam-follower mechanisms the curves can be utilized as partial or complete rotating bodies in contact with the follower 1471 Special Contour In Fig 149 we see some combinations of curves that have been used in cam mechanisms A circular arc (dwell) and a circular arc nose have sometimes been combined with the
Radial cam composed of contour combinations
Nose rn
Dwell flank (a) Triple-arc cam
FIGURE 1410 Circular arc cams
(b) Tangent cam
other aforementioned curves This blending of curves is not acceptable for high-speed action, since the dynamic characteristics are poor due to the discontinuities in either the velocity or the acceleration of the follower Note that the least complex conics applied to cams such as ellipses, parabolas, and hyperbolas have a continuous evolute, ie, continuous locus of the center of curvature, to give the acceleration curve continuity This ensures more acceptable high-speed characteristics However, discontinuities in evolute and acceleration curves exist when blended with circular noses and anks In addition, the Archimedes spiral, logarithmic spiral, and involute start with an impractical, abrupt slope in which a bump occurs with a discontinuity in the velocity of the follower Blending curves have been employed to correct this theoretical in nite acceleration In the past, triple curve cams having a circular arc nose, involute anks, and a harmonic or parabolic blend into the base circle were popular in the automotive eld Note that the logarithmic spiral has inherent qualities that make it desirable for all sorts of bodies in contact Applied to the cam form of Fig 149 it provides the smallest radial cam for a given pressure angle Moreover, the maximum pressure angle is constant during the action 1472 Circular Arc Cams In the past, the cams were of a combination of circular arcs with or without tangent straight lines Even now, some designers utilize these cams regardless of their poor dynamic properties In Fig 1410a, we see the blending of a circular arc cam having three different sized circles with r the radius of curvature for the circles In Fig 1410b the circles are blended with straight lines (tangent cam) All cams may provide motion to roller followers or convex curved-faced followers
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