Biomedical Engineering Reference
In-Depth Information
Problem 4.4 is an optimal shock isolation problem for a single-degree-
of-freedom system that was discussed in detail in Chapter 3. The solu-
tion of Problem 4.4 for the single-degree-of-freedom (rigid) model can
be used as a basis for constructing an exact or approximate solution for
Problem 4.2. In the next section, a procedure for constructing the exact
solution will be presented for the case where the optimal behavior of the
single-degree-of-freedom model is provided by a constant-force control. In
Section 4.1.4, it will be shown that if the stiffness
K
of the spring con-
necting bodies 1 and 2 is sufficiently large, the solution of Problem 4.4
gives a good approximation to the solution of Problem 4.2 in terms of the
performance index. Accordingly, the solution of Problem 4.3 gives a good
approximation to the solution of Problem 4.1.
4.1.3 Construction of Optimal Control for Two-Body Model
Based on Optimal Control for the Rigid Model
Notation and Preliminary Calculations
x
0
(t)
be the
optimal control and the corresponding optimal time history of the coordinate
x
for the rigid-body model. Solve Eq. (4.14) subject to the initial condition
ξ(
0
)
=
Let
u
0
(t)
and
0, implied by Eq. (4.15), to obtain
exp
τ)
u(τ ) dτ.
t
1
c
k
c
(t
=−
−
−
ξ(t)
(4.32)
0
ξ
0
(t)
and introduce the
u
0
(t)
as
Denote the function
ξ(t)
for the control
variable
−
ξ
0
(t).
y
0
(t)
˜
=˜
x
0
(t)
(4.33)
Equation (4.28) governing the dynamics of the rigid object coincides
with Eq. (4.13) governing the motion of body 1 in the two-body model.
The variables
x
,
y
,and
ξ
in the two-body model are related by
y
=
x
−
ξ.
(4.34)
This relationship coincides with that of Eq. (4.33) for the variables
y
0
,
and
ξ
0
. The variable
y
in Eq. (4.34) measures the coordinate of body 2 of
the two-body model relative to the base. Therefore, the function
x
0
,
˜
˜
y
0
can be
interpreted as the time history of the coordinate
y
of the two-body model,
provided that the motion of body 1 coincides with the optimal motion of
the object in the rigid model.
˜
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