Page 194 - Cam Design Handbook

THB7 8/15/03 1:58 PM Page 182

182 CAM DESIGN HANDBOOK

d dl p
p ¢() ∫q = ¢()e (7.12a)
q
l
dq dl t
p ¢¢() = ¢¢() + ¢ () eq l q e l 2 q k (7.12b)
t n
where Eq. (7.10) has been used.
An expression for k can now be derived in terms of the geometric variables of the cam
profile from Eqs. (7.12a 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 dot-
multiplying those two sides, the term in k will vanish because vectors e n and e t are mutu-
ally orthogonal. An alternative consists in first rotating vector e t through an angle of 90°
counterclockwise, which can be done by multiplying both sides of Eq. (7.12b) by matrix
E, defined as
È 0 - ˘ 1
E = . (7.13)
Í Î1 0 ˙ ˚
Thus,
2
l
l
q
Ep¢¢() = ¢¢() q Ee + ¢ () q k Ee n (7.14)
t
where
Ee = e and Ee = - . (7.15)
e
t n n t
Hence,
l
l
Eq¢¢() =- ¢ () q k e + ¢¢() q e (7.16)
2
q
t n
Second, on dot-multiplying the corresponding sides of Eqs. (7.12a) and (7.16), we
obtain
q
p ¢() Epq T ¢¢() =- ¢ () ql 3 k (7.17)
and, if we recall Eq. (7.3),
p
q
q
p ¢() Epq T ¢¢() =- ¢() q 3 sgn [l ¢()]k (7.18)
where sgn(·) is the signum function of (·), which is defined as +1 if its argument is posi-
tive and -1 if its argument is negative. If the argument vanishes, then we can define sgn(·)
arbitrarily as zero. Therefore,
T
p ¢() q Ep ¢¢() q
k =-sgn [ l¢()] 3 (7.19a)
q
p ¢() q
T
or, if we realize that E is skew symmetric, i.e., E =-E, then we can also write
T
p ¢¢() q Ep ¢() q
q
k = sgn [ l¢()] 3 . (7.19b)
p ¢() q
Either of Eqs. (7.19a and b) is the expression sought for the curvature of the cam profile.
A visualization of Eq. (7.19b) is displayed in Fig. 7.5. In that figure, notice that k > 0 at
P 1, which indicates a convexity at this point. Likewise, k < 0 at P 2, thereby indicating a
concavity at P 2.