Page 441 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 429

CAM SYSTEM DYNAMICS—RESPONSE 429

1 +
V* l () = v* h () ◊ e - ilh dh
- Ú 0
b+
Vd,,bl) = v d,,q) ◊ e - ilq d . (13.60)
(
q
(
b
- Ú 0
In a similar manner, Y*(l), A*(l), Y(d,b,l), and A(d,b,l) are defined as the Fourier
transforms of y*(h), a*(h), y(d,b,q), and a(d,b,q) respectively, although for Y*(l) and
Y(d,b,q), the upper limit of the transform integral must extend to +•. The analog of Eq.
(13.59) in the transform plane is as follows:
(
(
b
Yd,, bl) = d Y* bl)
(
(
Vd,, bl) = dV* bl)
Ê d ˆ
(
(
Ad,, bl) = Á ˜ A* bl) . (13.61)
Ë b ¯
Note that Y*(l) = Y(1,1,l).
The dynamic error Z is defined as the difference between the actual and the static fol-
lower responses. Let k = w n/w, where w n is the natural frequency of a one degree-of-
freedom elastic model of the cam-follower system and w is the rotational speed of the
cam. Neglecting damping, the governing equation of Z is assumed to be
2
Z ¢¢ + k Z = ( a d,,bq ). (13.62)
The absence of a damping term in Eq. 13.62, required for the method developed here, is
usually conservative. The amplitude of the residual vibration induced by the rise curve is
then
1
Rk () = Ad ( ,,b k). (13.63a)
k
Because v is defined to be zero for q < 0 and q > b, Eq. (13.63a) becomes
Rk () = ( b k). (13.63b)
V d,,
The residual response spectrum is thus the Fourier spectrum of the velocity curve v(d,b,q).
The area of an admissible velocity curve between q = 0 and b is equal to the follower rise d.
Convolution, h(q), of two functions f(q) and g(q) is defined as

+•
g q t)
f g
h q () =* = -• Ú f t () ◊ ( - dt. (13.64a)
In the transform plane,
G l
H l () = () ◊ (). (13.64b)
F l
It can be easily verified that the area under the curve h is equal to the product of the areas
under the curves f and g, i.e.,
( area) = ( area) ◊( area) . (13.65)
h f g
Also if f = 0 when q < 0 and q > q f , and g = 0 when q < 0 and q > q g , then h = 0 when q
< 0 and q > q h, where
q = q + q . (13.66)
h f g