Biomedical Engineering Reference
In-Depth Information
L
x
∆
L
v
FIGURE 7.22
Beam “lengthening.”
x
1
x
2
x
3
x
v
1
v
2
d
2
d
3
v
3
v
FIGURE 7.23
Segmented beam.
To rectify this discrepancy, the straight-line displacement of each portion of the beam
is summed together until the length of the curved beam equals that of the original beam,
as shown in figure 7.23. Once that length is reached, the rest of the beam is not included
in the deflection plot. As a result of this beam “reshortening,” the deflection of the
hypothetical beam shown in figure 7.23 will eventually look as shown in figure 7.24.
For the static case, there is a distinct final deflection solution for each moment
value. The plot in figure 7.25 shows the distinct solutions for various moment values.
7.4.3
D
YNAMIC
C
ASE
The previous section describes the initial and final positions of the beam, but does
not describe the dynamics it undergoes to reach the final position. This section will
describe the effort to build the dynamic model.
For a step input of 2 V on a 1-
0.25 in.-strip (0.2-mm thick), the moment will
be constant; such a constant moment will produce a distinct final deflection. The
step response of the tip to such a deflection command is shown in Shahinpoor (2003).
From that step response, it would appear that the beam could be modeled as a simple
second-order transfer function.
The step response of a second-order transfer function of the following form will
approximate the step response if
×
ω
= 0.364 Hz and
ζ
= 0.3:
G
(
s
) =
ω
2
/
s
2
+ 2
ζω
s +
ω
(7.18)
2
The step response of this model is shown in figure 7.26.
