Civil Engineering Reference
In-Depth Information
FIGURE 5.26
Tool discretization using arcs and lines.
FIGURE 5.27
Spline discretization of the tool shapes.
x
s
,
x
s
The shape co-ordinates functions
(
)
are defined according to:
2
d
x
s
k
,
n
u
k
n
()
C
x
1
(
)
n
n
1
:
k
1
n
,
(5.49)
2
d
x
s
k
,
n
u
k
n
()
C
x
2
(
)
k
1
where
1
u
k
k
1
--------------------
()
(5.50)
(
k
1
)
!
k
,
n
k
,
n
2d
1 is the degree of the spline function which is generally taken as three,
C
x
1
and
C
x
2
are the spline
coefficients which are defined in order that:
x
s
n
()
x
s
x
s
n
()
x
n
and
p
(5.51)
d x
is also continuous
p
d
p
1, 2
x
n
,
x
n
are the characteristic points which define the spline.
The optimization parameters are the displacements of some of these characteristic points along selected
A shortcoming of cubic splines is that the displacement of a single point at the one hand of the curve
modifies the curve everywhere, so cubic Bspline functions can also be used, as the displacement of a
(
)
n
