Biomedical Engineering Reference
In-Depth Information
unconstrained and, as is indicated by calculations, can assume rather large
values on some time intervals. These facts make it easier to get a numerical
solution for Problem 6.2 as compared with Problem 6.1.
Having solved Problem 6.2, one can readily calculate the optimal control
force
u
0
(t)
for Problem 6.1. To do that, express
x
1
(t)
=
x
2
(t)
−
ξ(t)
using
Eq. (6.41) and substitute this expression and
W
=
m
2
w
0
(t)
into Eq. (6.39).
This yields
−
ξ(t))
u
0
(t)
=
m
1
(
x
2
(t)
¨
+
m
2
w
0
(t),
(6.47)
where the functions
x
2
(t)
and
ξ(t)
have been obtained by solving Prob-
lem 6.2.
To solve Problem 6.2 a nonlinear programming technique was used,
for which the continuous control function
w(t)
was approximated by a
piecewise constant function defined on a discretized time interval. These
discrete values of the control function were utilized as the design variables.
For further details about the reduction of the optimal control prob-
lem associated with the limiting performance analysis of shock isolation
systems to linear or nonlinear programming, see Balandin, Bolotnik, and
Pilkey (2001). For single-degree-of-freedom systems such a reduction was
described in Section 3.3.
6.3.4
Numerical Results
Figure 6.2 shows the results of the solution of Problem 6.1 depending
on the duration of the sinusoidal deceleration pulse for a given impact
velocity (
v
0
48 km
/
h). This figure indicates time histories of the optimal
control force [
u
0
(t)
], the chest compression [
x
2
(t)
=
−
x
1
(t)
], the chest accel-
eration [
x
2
(t)
], the rate of the chest compression [
¨
x
2
(t)
˙
−˙
x
1
(t)
], the chest
˙
−˙
−
viscous response ([
x
2
(t)
x
1
(t)
][
x
2
(t)
x
1
(t)
]), and the occupant excur-
−
=
sion [
x
1
(t)
0
.
08
,
0
.
1
,
0
.
12 s. The corresponding half-sine
pulses are shown in Fig. 6.3. The curves of Fig. 6.2 indicate that from all
injury criteria subjected to constraints only the chest compression reaches
its limiting value, while the other criteria remain below their upper bounds.
The behavior of the control force and all criteria, apart from the occu-
pant excursion, during the occupant deceleration interval has only weak
dependence on the pulse duration for 0
.
08 s
x
v
(t)
]for
T
p
0
.
12 s. The deceler-
ation interval is the time from the impact to the instant when the thorax
excursion (the quantity
x
1
≤
T
p
≤
x
v
) reaches a maximum. The occupant excur-
sion increases as the deceleration pulse decreases. This increase is about
130% as
T
p
decreases from 0.12 to 0
.
08 s. The motion of the occupant's
thorax occurs at almost a constant control force during a substantial segment
−
Search WWH ::
Custom Search