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ELEMENTS OF CAM PROFILE GEOMETRY
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4 (b) Cam layout
FIGURE 61 Cam with oscillating follower
CAM DESIGN HANDBOOK
63 CAM PRESSURE ANGLE
631 Introduction The cam pressure angle indicates the steepness and forces of the cam surface The pressure angle is the angle (at any point) between the normal to the pitch pro le and the direction of follower motion; see Fig 62 The size of the cam directly affects the pressure angle, the curvature of the cam pro le, and the proportions of the supporting cam shaft and hub In all machinery design minimum size is desired to reduce weight and the inertia effects of all moving parts So it is too with cam-follower systems As the cam is made smaller the pressure angle increases and the radius of curvature decreases The pressure angle is limited to minimize forces and de ections in the machine and for CNC increment manufacturing Let us discuss the transferring of the cam displacement curve to the radial plate cam, Fig 63 If the pitch curve of the displacement diagram is plotted on a radial cam, we see that it is distorted toward the cam center As an example, in Fig 63a, let us take a straightline pitch curve OB in the displacement diagram This pitch curve has a constant pressure angle a for a total rise h in b cam degrees Let point T be the midpoint of rise OB, and, in Fig 63b, let A be the cam center By trial and error, choose a radius AT so that the pressure angle of the pitch curve at point T is equal to a The pressure angle is changed when laid out on the radial cam It is larger than a below point T and smaller than a above point T It can be reduced by using a larger cam
of rec m tion ot ion
um l e im ax ang , m re a m ssu pre
Direction of motion Trace point
Norm al
Follower
Pitch point
g Tan ent
Pitch profile ro Cam profile
Prime profile
rb rp
Base circle Pitch circle
FIGURE 62
Cam nomenclature
ELEMENTS OF CAM PROFILE GEOMETRY
Less than a
Normal Total rise, h
Pitch curve
a B Midpoint, T
Mo re th a an
Pitch curve Cam angle b (a) Displacement diagram
Midpoint, T
A Cam center
(b) Radial cam layout
FIGURE 63 Radial cam layout distortion
632 Pressure Angle Forces Translating Roller Follower Let us analyze the side thrust due to an excessive pressure angle on a translating radial follower It will be shown that the allowable pressure angle is limited by the length of the follower overhang, its guide bearing length, the coef cient of friction of the follower, and the follower stem rigidity and backlash First assume that the effect of the roller bearing and its rolling on the cam are negligible Let A = follower overhang, in B = follower bearing length, in F = force normal to cam pro le, lb Lo = external load on follower (includes working load, weight, spring force, inertia and friction, lb) N1 and N2 = forces normal to follower stem, lb W = radius of follower stem, in a = pressure angle, deg am = maximum pressure angle, deg m = coef cient of friction Figure 64 shows the direction of cam rotation, the normal forces on the follower, and the frictional forces opposing the motion of the follower For static equilibrium, the sum of forces along the vertical axis is SFy = 0 = - L0 + F cosa - mN1 - mN2 (61)
Let point p and q be the intersection of N1 and N2 on the line of follower motion From statics the sum of the moments is
CAM DESIGN HANDBOOK
FIGURE 64 Radial cam translating rollerfollower force distribution
SM p = 0 = - FA sina + NB - mN1W + mN2 W SMq = 0 = - F( A + B) sin a + N2 B - mN1W + mN2W
(62) (63)
Simplifying Eq (62) and Eq (63) and assuming mN1W and mN2W equal zero since they are negligible yields N1 = N1 = A F sina B (64) (65)
A+ B F sin a B
Substituting Eq (64) in Eq (61) and eliminating N1 0 = - L0 + F cosa - mF Substituting gives the external load 2A + B L0 = F cosa - m sin a B A sin a - mN2 B (66)
ELEMENTS OF CAM PROFILE GEOMETRY
Solving for the force normal to the cam F= L0 2A + B cosa - m B sin a (67)
The normal force F is a maximum (equals in nity) which means that the follower will jam in its guide when the denominator of Eq (67) equals zero Therefore, cosa m - m 2A + B sin am = 0 B
The maximum pressure angle without locking the follower in its guide is am = tan -1 B m ( 2 A + B) (68)
Let us substitute some trial values to compare the magnitude of the results If we let A = B and assume the values for the coef cient of friction of bronze on steel to be m ( kinetic) = 010 and m (static) = 015 Substituting in Eq (68), we nd the maximum pressure angle for each condition: am = tan -1 am = tan -1 B = 73 degrees for m = 010 ( 010)(2 B + B) B = 66 degrees for m = 015 ( 015)(2 B + B)
Note that these values and the derivation of Eq (68) are based on the ideal assumption that the follower is perfectly rigid Thus, the coef cient of friction may actually reach a value of 025 or more depending on relative elasticity and backlash of the follower A exible stem may dig into the lower corner of the bearing Therefore, the suggested guide in practice is to keep the coef cient of friction m, the follower overhang A, and the backlash as small as possible with the bearing length B as large as possible, in the range of B = 2A Generally the safe limiting pressure angle in practice is 30 degrees However, for light loads with accurate low-friction bearings, the author was successful using a pressure angle as high as 48 degrees Note that commercially available ball bushings for the linear moving stem have provided low friction and little backlash; see Chap 10 We may observe that the follower jamming is of concern only when the follower moves in the direction opposite that of the external load L As shown in Figure 64, jamming occurs during the rise period only During the fall period, the size of the maximum pressure angle is generally not limited in proper cam design However, the author has seen machine installations in which the follower drove the cam during the fall action It occurred with a chain-driven cam and a spring-loaded follower The spring force, acting on an excessive pressure angle of fall, produced detrimental shock and uctuating action in absorbing the backlash of the system Also, if the load Lo varies according to the inertia acceleration in both the rise and fall, the pressure angle should be considered in both 9 discusses load uctuations and re ected torque curve which should be an initial point of analysis in all pressure angle investigations
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