Biomedical Engineering Reference
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K
Stiffness coefficient of the vertebral
column
m
1
Mass of the upper torso
m
2
Mass of the lower torso
V
Initial velocity of the seat pan, lower
torso, and upper torso
τ
Duration of the shock acceleration
pulse
Performance index
J
1
Maximum magnitude of the
displacement of the lower torso relative
to the seat pan
Performance criterion
subjected to a constraint
J
2
Maximum magnitude of the vertebral
column's compressive force
Constraint
P
Maximum allowable value for the
criterion
J
2
Optimal solution
F
0
Optimal control
F
J
1
(
F
0
)
Optimal value of the performance
index
J
1
For the system of Eqs. (5.12) - (5.14) with
z
specified by Eqs. (5.7) and
(5.10), find the optimal control force
F
0
(t)
that minimizes the criterion
J
1
, provided that the criterion
J
2
does not exceed a prescribed value
P
,
that is,
J
1
(F
0
)
=
min
F
{
J
1
(F )
|
J
2
(F )
≤
P
}
.
(5.17)
The statement of this problem coincides with that of Problem 4.1.
5.2.2
Solution
To illustrate the application of the technique introduced in Section 4.1 to a
limiting performance analysis of shock isolation systems for reducing the
risk of spinal injuries, Problem 5.1 will be solved with the parameters of
the shock pulse specified as
V
=
9
.
4m
/
s
,
τ
=
90 ms
.
(5.18)
These values approximately characterize the shock pulse applied to the
occupant's seat in a helicopter crash. The maximum compressive load
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