Biomedical Engineering Reference
In-Depth Information
function
f(x)
, then any
M
<M
is also a lower bound of this function
and, therefore, any function bounded from below has an infinite set of lower
bounds. This set has a maximum, that is, there exists a lower bound
M
∗
such that
M<M
∗
for all other lower bounds. The number
M
∗
is called the
greatest lower bound (or infimum) of the function
f(x)
, which is denoted
as
M
∗
=
inf
x
∈
D
f(x)
.
Recall now the definition of the minimum of a function
f(x)
. A func-
tion
f(x)
is said to have a minimum if there exists
x
0
∈
D
such that
f(x
0
)
D
. The point
x
0
is called a point of minimum
of the function
f(x)
and the value
f(x
0
)
is called the minimum of this
function over the domain
D
, which is denoted as
f(x
0
)
≤
f(x)
for any
x
∈
min
x
∈
D
f(x)
.
The minimum of any function, if it exists, coincides with the greatest lower
bound of this function. A minimum, however, may not exist for a function
bounded from below. As an example, consider the function
f(x)
=
=
|
|
1
/
x
,
which is defined for any
x
=
0 and is positive in the domain of defini-
tion. The greatest lower bound for this function is equal to zero, because
for any positive
M
, the inequality
f(x)<M
holds for all
x
such that
|
x
|
>
1
/M
. However, this function does not have a minimum, since for
any
x
=
0, there exists
x
=
|
x
|
0, satisfying the inequality
>
|
x
|
, such that
f(x
)<f(x)
.
This problem requires that the maximum of the absolute value of the dis-
placement of body 2 relative to the base be minimized under the constraint
imposed on the maximum magnitude of the force allowed to be transmitted
to body 1. This is a typical problem encountered in the basic design of shock
isolation systems for engineering structures. For example, when designing a
safety system for automobile occupants, the designer must know the phys-
ical limitations for the reduction of the relative displacement of the seats,
provided that the load transmitted to the occupants in a crash with impact
velocities from a given range does not exceed the magnitude that humans
can tolerate without serious or fatal injuries.
Problem 4.1 is a limiting performance problem. The concept of
limiting performance analysis for shock isolation introduced in Section
3.2 for single-degree-of-freedom systems is readily generalized to
multi-degree-of-freedom systems. The limiting performance analysis does
not impose any constraints on the design configuration of the isolation
system or on the content of information to be utilized in the feedback loop
of the controller. In the limiting performance analysis, the action of the
isolator is characterized by a generic control force depending only on time
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