Civil Engineering Reference
In-Depth Information
(a)
(b)
Figure 7.33
Cut a surface into pieces: (a) two pieces; (b) three pieces.
indicative but can be exact. Let's say that we would like to divide a surface into exactly 17
pieces of roughly equal area. The surface is first divided into two pieces, which are then
adjusted on the common boundary such that the ratio of the area between the two pieces is
about 8:9. Similarly the procedure is repeated on the two surface pieces to divide the large
one into nine pieces and the smaller one into eight pieces, as shown in Figure 7.34.
In case there is no clue as where is the optimal cut, the discretised surface will be bisected
by a plane passing through its centre of gravity with an orientation normal to one of its
principal axes of inertia. Cutting along the centre of gravity can give us roughly two equal
pieces without much calculation, which can be adjusted along the cut boundary for further
balancing of any desirable geometrical quantities. If the centre of gravity is not inside the
object, it will still work well by selecting the best cut along one of the principal directions.
1
Δ
∑
∑
Centre of gravity
c
=
(, ,)
xyz
=
Δ
(, ,)
x
yz
where
Δ
=
Δ
c
c
c
i
i
i
i
i
i
=
1
,
N
i
=
1
,
N
f
f
2
2
yz
+
−
xy
−
zx
i
i
ii
ii
∑
2
2
Moment of inertia
I
=
Δ
−
xy
z
+
x
−
y z
i
ii
i
i
ii
i
=
1,
N
f
2
2
−
zx
−
yz
xy
+
ii
ii
i
i
where Δ
i
and (x
i
, y
i
, z
i
) are, respectively, the area and the centroid of triangular facet Si,
i
, and
(x
= −
are the co-ordinates of the centroid relative to the centre
of gravity. The principal axes of inertia are given by the eigenvectors of
I
.
Based on pure geometrical consideration on the shape of the resulting cut pieces, the cut-
ting plane normal to the minor principal axis is preferable to the cutting plane normal to the
major principal axis, as shown in Figure 7.35. However, when there is a change of topology,
the cut that leads to a simple topology (for instance, from multi-connected to simply con-
nected) is preferred, as shown in Figure 7.36. The cut along the longitudinal axis, normal to
,,)(
yz
x,y,z
)(
x,y ,z
)
i
i
i
i
i
i
c
c
c
Weight = 9
Weight = 5
Weight = 3
Weight = 2
Weight = 4
Equally divided into 4 pieces
Weight = 8
Equally divided into 8 pieces
Figure 7.34
How a surface is divided into n = 17 pieces.
