Page 442 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 430

430 CAM DESIGN HANDBOOK

These properties of the convolution operator can be applied to cam curves as follows: let
v i (d,b,q) be an admissible velocity curve of area d corresponding to a follower motion of
rise d in cam rotation b. Let w(d,b,q) be another admissible velocity curve. Then a new
admissible velocity curve v i+1 (d,b,q) is constructed as
v d¢¢ )* (
v ( d,,bq ) = ( , ,b q w d¢¢ ¢¢ ) (13.67a)
,q
,b
i
i+1
where the symbol * represents convolution (see Eq. [13.64a]) d = d¢d≤ (see Eq. [13.65])
and b = b¢+ b≤ (see Eq. [13.66]). Like ordinary integration, convolution is a smoothing
operation. As a result, with each convolution additional derivatives of v vanish at the start
and end of the rise.
Fourier transform of Eq. (13.67a) leads to equation (13.67b)
◊ (
V d¢¢ )
V ( d,, k) = ( , , k W d¢¢ ¢¢, k). (13.67b)
b
b
b
,
i+1 i
Equation (13.67a) then gives the post convolution cam curve and Eq. (13.67b), in con-
junction with Eq. (13.63b), gives the postconvolution residual response. Some specific
applications of convolution to the development of cam motions are discussed in the
following sections.
13.6.4 Fixed Convolution
Convolution can be applied in several ways to improve cam profiles. The most effective
way that Gupta and Wiederrich (1983) identified they called fixed convolution. In this
method we set d¢= d, d≤= 1, b¢= b≤= b/2 in Eqs. (13.67a and b). In this case, the nature
of w is not changed during the convolutions. The velocity curve is obtained from
Eq. (13.67a) as
b
v ( d, ,q ) = v d ( ,b 2,q )* w(1,b 2,q ) (13.68)
i+1 i
and the Fourier transform from Eq. (13.67b) as
V ( d, , k) = ( 2, k W(1,b 2, k). (13.69)
)
V d,b
◊
b
i+1
i
The cumulative effect of m convolutions is obtained as follows:
(
b
k V d,b 2
k W ,b 4
W ,b 2
W ,b 2
Vd, , k) = (1 , ) ◊ ( 1 k , ) ◊◊◊ (1 m , ) ◊ ( m k , . ) (13.70)
m 0
It is seen from Eq. (13.63b) and (13.70) that the residual response, R m(d,b,k) =
|V m (d,b,k)|, after m convolutions is dominated by the modifier function w(d,b,q), and the
starting function v 0(d,b,q) has only a small influence.
From the definition of v(1, b, q), it follows that v(1, 0, q) is a delta function (area = 1,
function vanishes for q < 0- and q > 0+). Its transform |V(1,0,k,)| = 1. As m Æ•, a lim-
iting residual response is thus obtained from Eq. (13.70).
•
lim Rd ( , ,k) = d ◊ ’ W(1 ,b 2 r ,k) (13.71)
b
m Æ• m
r=1
As will be seen in the example below, the envelope of R i+1 always lies above that of R i
until the value of k exceeds some crossover value. Above this value of k, the envelope
of R i+1 lies below that of R i . In this case, a limit R • is reached which provides the
lowest envelope available by this method, although it is possible that other basic
curves may provide a lower envelope. For the high-speed systems in which vibration
is typically a major concern, however, k will be sufficiently low enough so that the lowest