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FIGURE 72
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Notation for pressure-angle determination
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723 Cam Curvature The shape of a planar curve G depends on the rate of change of the direction of its tangent with respect to the arc length, a measure that is called the curvature of G, and designated hereafter as k This variable plays an important role in the design of cam mechanisms, for it is directly related to the occurrence of cusps and the already mentioned effect known as undercutting The reciprocal of the curvature k is the radius of curvature r, ie, r= 1 k (79)
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The radius of curvature at a point of the cam pro le is the radius of a circle tangent at that point to the cam pro le, on the concave side, as shown in Fig 73 The curvature of that circle is the same as that of the cam pro le For our purposes, the radius of curvature is positive if the center K of the circle is located between the center of rotation O and the point of tangency Q; otherwise, the radius of curvature is negative In the discussion that follows, formulas for calculating the curvature at any point of a cam pro le are derived Let us consider a planar curve, as shown in Fig 74 At any point P on the curve, whose position vector is denoted by p, a unique orthonormal pair of vectors is de ned, namely, the tangent and the normal vectors indicated in that gure as et and en Let l measure the arc length as in Fig 74 Unit vectors et and en, their derivatives with respect to l, and the curvature k are related by the Frenet-Serret formulas, Brand (1965), namely,
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GEOMETRY OF PLANAR CAM PROFILES
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de t = k en dl de n = -k e t dl
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(710) (711)
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Now, let the position vector p be a function of a parameter q that, in general, is different from l Differentiation of p with respect to q twice yields
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G Q K g T q O
FIGURE 73 Visualization of the radius of curvature
P en l
FIGURE 74 Planar curve and the orthonormal vectors et and en
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p ( q )
dl dp = l ( q )e t dq dl
(712a) (712b)
p ( q ) = l ( q )e t + l 2 ( q )k e n
where Eq (710) has been used An expression for k can now be derived in terms of the geometric variables of the cam pro le from Eqs (712a and b) It is apparent that the expression sought is a scalar, which calls for a scalar operation between the vectors of those equations However, the dot product of the right-hand side of those equations will not be very helpful Indeed, on dotmultiplying those two sides, the term in k will vanish because vectors en and et are mutually orthogonal An alternative consists in rst rotating vector et through an angle of 90 counterclockwise, which can be done by multiplying both sides of Eq (712b) by matrix E, de ned as 0 -1 E= 1 0 Thus, Ep ( q ) = l ( q )Ee t + l 2 ( q )kEe n where Ee t = e n Hence, Eq ( q ) = - l 2 ( q )k e t + l ( q )e n (716) and Ee n = - e t (715) (714) (713)
Second, on dot-multiplying the corresponding sides of Eqs (712a) and (716), we obtain p ( q ) Ep ( q ) = - l 3 ( q )k and, if we recall Eq (73), p ( q ) Ep ( q ) = - p ( q ) sgn[l ( q )]k
T 3 T
(717)
(718)
where sgn( ) is the signum function of ( ), which is de ned as +1 if its argument is positive and -1 if its argument is negative If the argument vanishes, then we can de ne sgn( ) arbitrarily as zero Therefore, k = - sgn[l ( q )] p ( q ) Ep ( q ) 3 p ( q )
(719a)
or, if we realize that E is skew symmetric, ie, ET = -E, then we can also write k = sgn[l ( q )] p ( q ) Ep ( q ) 3 p ( q )
(719b)
Either of Eqs (719a and b) is the expression sought for the curvature of the cam pro le A visualization of Eq (719b) is displayed in Fig 75 In that gure, notice that k > 0 at P1, which indicates a convexity at this point Likewise, k < 0 at P2, thereby indicating a concavity at P2
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