Page 438 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 426

426 CAM DESIGN HANDBOOK

T
2 ˙˙
J = W F + W F 2 ˙˙˙
Ú 0 1 fe 2 fe
(13.54)
2
Y() -
+ Y()[ t Y ()] + V () t Y () t dt
t
t
2
i d 2
where weight functions V 1(t) and V 2(t) are normal distributions given by
k - n
,
v t () = e 2 i =12 (13.55)
i
S
d
with
Ê t - ˆ 1 2
n = Á ˜
Ë S ¯
d
k ª10 4
S ª10 -5
d
and the function Y d (t) is a polynomial given by
1
Y t () = 6 t - 5 t 6 0 £ t £ ¸
5
d
Y t () = 6 S - 5 S 6 1 < t £ T Ô
5
d
Ô
where T >1 ˝ (13.56)
Ô
-
T t
S = Ô
T -1 ˛
The boundary conditions of the “interior point” at time t = 1.0 are now indirectly spec-
ified through the introduction of weight functions V 1 (t) and V 2 (t) in the cost functional.
To ensure convergence of the variation-of-extremals algorithm as well as reducing numer-
ical errors in the integration, it is necessary to start the optimization with a relatively large
-2
standard deviation of about 10 . With succeeding iteration, constant S d is gradually
reduced to a point at which the weighting functions V i(t), i = 1,2 approximate a delta func-
tion. These weighting functions force the boundary conditions at this interior point to be:
Y 1 () = .
1 0
˙
0 0
and Y 1 () = .. (13.57)
These boundary conditions are not expected to give a cam displacement, Y c(1), of unity
at time t = 1 since the system equation (Eq. [13.30]) is of second order. If this is a concern,
2
an additional term V 3(t)Y ¨ (t) may be introduced in the cost functional to specify the inte-
rior boundary condition Y ¨ (1) = 0.0 indirectly. Weight function V 3 (t) is defined identically
to V 1 (t) and V 2 (t).
With the definition of T (see symbols), the previously used normalizations remain
unchanged. This minimizes the amount of modification needed in problem formulation
and computation.
The formulation of the optimization problem for a tuned D-R-R-D cam can now be
given as follows.
Minimize
T
2 ˙˙
J = W F + WF 2
Ú 0 1 fe fe
(13.58)
2 ˙
2
t
+ V ()[ t Y ()] + V () t Y () t dt
t
Y() -
1 d 2