Biomedical Engineering Reference
In-Depth Information
satisfies the inequality
J
2
(F )
≤
P.
(5.74)
Problem 5.4 is an auxiliary problem for Problem 5.3, which is an ana-
logue of Problem 4.9 for Problem 4.8.
This problem coincides with Problem 5.1, except that
J
1
, rather than
J
1
,
is used to denote the criterion to be minimized in Problem 5.4. Therefore,
in accordance with Eq. (5.32),
1
1
exp
C
T
.
m
1
V
2
2
P
Pτ
m
1
V
P
K
K
J
min
=
−
−
−
−
(5.75)
1
The numerical value of this quantity for the parameters adopted for the
model of the occupant and the shock pulse is given by Eq. (5.33), that is,
J
min
=
19
.
4cm
.
(5.76)
1
The optimal control
F
0
(t)
is given by Eqs. (5.29) - (5.31).
The optimal time history of the occupant's lower torso relative to the base
(airframe) on the interval 0
≤
t
≤
T
,where
T
=
m
1
V/P
, is expressed as
[Eqs. (5.25) and (5.27)]
1
exp
C
t
,
P
K
K
y(t)
=
x
0
(t)
−
−
−
0
≤
t
≤
T,
(5.77)
where
⎧
⎨
t
Ut
2
2
V
2
π
sin
πt
τ
−
−
for 0
≤
t
≤
τ,
τ
x
0
(t)
˜
=
(5.78)
⎩
Ut
2
2
Vτ
2
−
−
≤
Vt
for
τ<t
T.
Minimum Value of Criterion J
1
in Problem 5.4
Substitute
S
of
Eq. (5.58) for
D
3
into Eq. (4.170) and calculate, with reference to Eq.
(5.76),
J
min
J
min
=
−
S
=
4
.
4cm
.
(5.79)
1
1
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