Page 134 - Cam Design Handbook

THB5 8/15/03 1:52 PM Page 122

122 CAM DESIGN HANDBOOK

n n È n ˘ 2
x Â
x Â
1 ()
1 ()
1 ()
x
x
x
R () = WN () WN () - WN () WN () ÍÂ WN () ˙ , (5.15)
x
jk ,
j k ,
j
j k ,
j
j
jk ,
jk ,
j
j k ,
j
=
=
=
j 1 j 1 Î j 1 ˚
n
2 ()
2 ()
x Â
R () = W N () W N () -
x
x
jk , j jk , j jk ,
j 1
=
n
n
È 1 () x  1 () x  2 ( ) ˘
x
x
Í 2 WN () WN ()+ WN () WN () ˙
j k ,
j k ,
j k ,
j k ,
j
j
j
j
Î j 1 j 1 ˚
=
=
È n ˘ 2 È n () 1 ˘ 2
x
˙
Í Í Â WN () + 2WN j k , () x ÍÂ WN j k , () x ˙
j
j
j
j k ,
Î j 1 ˚ Î =1 ˚
j
=
3
È n ˘
ÍÂ WN j k , () x ˙
j
Î =1 ˚
j
(5.16)
The derivatives of the B-splines required above can be evaluated using the recurrence rela-
th
tionships described earlier. The m derivative of rational B-splines at a given point x satisfy
the following relationship:
n
 R () = 0 . (5.17)
()
m
x
jk ,
j=1
This equation affords a convenient accuracy check when constructing the derivatives of
rational B-splines.
Implementation of the Procedure. The systematic procedure for implementing the
rational B-spline approach is very similar to the scheme for applying B-splines described
earlier. The process is as follows, assuming the motion constraints have already been
established:
1. Select the appropriate order of the rational B-splines to be used.
2. Choose the weight sequence.
3. Establish a knot sequence.
4. Determine the values of each rational B-spline or rational B-spline derivative at all
points where motion constraints are imposed.
5. Collect the values of rational B-splines and/or rational B-spline derivatives and form
a linear system of equations using the motion constraints in Eqs. (5.11) and (5.12).
6. Solve the solution of the linear system equations formed in the step above for the
coefficients A j.
7. Evaluate rational B-splines and/or their derivatives as needed to determine displace-
ment and/or its derivatives between motion constraints using Eqs. (5.11) and (5.12).
Example Applications. The examples that follow illustrate the application of rational B-
splines to the synthesis of the rise portion of a DRD motion program. The cases presented
illustrate the effects that adjustments to the rational B-spline parameters have on the syn-
thesized motion programs.
Note that in the cases below, normalized values for both displacement and time are
used to provide a convenient basis for comparison of results. Accordingly in the examples
¯
S c denotes a normalized value for cam displacement and S the follower output motion dis-