Biomedical Engineering Reference
In-Depth Information
where
J
1
(u)
=
max
)
|
x
1
−
x
v
|
,
t
∈
[0
,
∞
J
2
(u)
=
max
t
∈
[0
,
∞
)
|
x
2
−
x
1
|
,
J
3
(u)
=
t
∈
[0
,
∞
)
|
x
2
|
,
max
(6.35)
=
t
∈
[0
,
∞
)
|˙
−˙
|
J
4
(u)
max
x
2
x
1
,
J
5
(u)
=
t
∈
[0
,
∞
)
|
max
(
x
2
˙
−˙
x
1
)(x
2
−
x
1
)
|
,
and
D
i
(i
=
2
,
3
,
4
,
5
)
are the maximum allowable values (threshold val-
ues) of the corresponding system performance criteria.
The performance criterion
J
1
represents the maximum excursion of the
occupant in the vehicle,
J
2
the maximum chest compression,
J
3
the max-
imum chest acceleration, since
m
2
comprises almost the total mass of the
thorax,
J
4
the maximum rate of chest compression, and
J
5
the maximum
chest viscous response.
The parameters of the thoracic injury model are (Lobdell et al., 1973)
m
1
=
0
.
3kg
,
m
2
=
18 kg
,
k
11
=
10
,
522 N/m,
k
12
=
71901N/m,
δ
0
=0
.
03 m,
(6.36)
=
·
=
·
c
11
403
.
3N
s/m,
c
12
2192
.
1N
s/m,
=
=
·
k
2
13
,
153N/m,
c
2
175
.
4N
s/m.
The upper bounds for the constrained injury criteria are defined as
0
.
229 m
2
/s.
(6.37)
The problem will be solved for an impact velocity of 48 km/h, which rep-
resents the U.S. government's (NHTSA) frontal impact safety standard test
(FMVSS 208), and pulse durations of 0.08, 0.1, and 0.12 s.
D
2
=
0
.
046 m,
D
3
=
80
g,
D
4
=
6m
/
s
,
=
5
6.3.3
Solution Plan: An Auxiliary Problem
In Problem 6.1, the control
u(t)
to be found is not subjected to constraints,
which can complicate the numerical solution of this problem. This can
be addressed by transforming the equations of motion [Eq. (6.26)] and
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