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A
V+
Figure 3
V+
A
A =
r
Dx
¢
3
9
2 - 0 0 0 -
4
4
3 1
3
-
0 0 0
4 2
4
3 3 3 3
0 0 p+ k p+ k - p 0
8 4 8 4
3 3 3 3
0 0 p+ k p+
k 0 0
4 8 2 8
3
3 3
0 0 - p 0 p+
k 0
4
2 8
9 3
-
0 0 0 3
4 4
Dx
Dy
Dz
Dqx
Dq
y
Dq
z
f
f
3
L
Et w A f
=
3
t
t
t
x
y
z
x
y
z
3
L r
¢
Et w Af 3
(1)
(2)
B
V- V-
Figure 4
3.2 Compliance matrix
V 0
The shape of the flexure can be used to
determine its compliance matrix. The flexure
can be considered to be constructed of four
identical L’s (See figure 2). The compliance
matrix of these L’s is determined, assuming small
deflections and simple stress distributions within
the cross-section of the beam, as in [7]. The
compliance matrix for one L is given below.
B
V
where D r x is the displacement, L is the length of
one side of the square sheet, E, t, and w are the
modulus of elasticity, thickness, and width of the
material, p=t 2 /h 2 is the aspect ratio squared,
k
E
G
= is the ratio of the modulus of elasticity
and
r
the modulus of rigidity of the material, and
f is the applied force. These compliance
matrices are translated and rotated so that they
are positioned as in figure 2. Then the
compliance matrix of the whole flexure
assembly, C, is found by combining the
compliances of the individual L’s
C
C
=
= A
-1
-
i
Í
i
1
We find that
(3)
1
20
0 0 0 0 0
0
1
20
0 0 0 0
0 0
k + p
p
4( k + 4p)
0 0 0
0 0 0
3( k + p)( k + 4p)
p
2
12k + 80kp + 40p
0 0
2
0 0 0 0
3( k + p)( k + 4p)
p
2
12k + 80kp + 40 p
0
2
0 0 0 0 0
3
112
(4)
Dx
r
L r
Et w Cf . (5)
3
= 3
A number of insights can be gained from this
compliance matrix. First, we notice that it is