Biomedical Engineering Reference
In-Depth Information
Instantaneous Shock Pulse
The concept of an instantaneous shock
pulse was introduced and discussed in Chapter 2 (Section 2.1.3). This pulse
produces a change in the velocity of the base with an infinitesimal duration.
This model applies when the duration of the shock pulse is much less than
the other characteristic times of the system, for example, when the pulse
duration is less than the natural period of oscillations of the object that is
attached to the base by a spring or spring-and-dashpot isolator. For a base
moving along a straight line, the instantaneous pulse is represented by the
Dirac delta function
v(t)
=
V δ(t),
(3.38)
where
V
is the change in the velocity of the base due to the pulse.
As was shown in Chapter 2 (Example 2.5), an instantaneous pulse can
be regarded as a limiting case of a rectangular pulse. The rectangular pulse
defined by Eq. (3.24) approaches an instantaneous pulse as
τ
→
0and
aτ
=
V
. Therefore, the solutions of Problems 3.1 and 3.2 for the instantaneous
pulse represented by Eq. (3.38) can be obtained as the limits of Eqs. (3.31),
(3.32), (3.36), and (3.37) as
τ
→
0. Then, for Problem 3.1,
V
2
2
U
,
J
1
(u
0
)
=
(3.39)
⎧
⎨
V
U
,
−
U
if 0
≤
t
≤
u
0
(t)
=
(3.40)
⎩
if
t >
V
0
U
,
and for Problem 3.2,
V
2
2
D
,
J
2
(u
0
D
)
=
(3.41)
⎧
⎨
⎩
V
2
2
D
2
D
V
−
if 0
≤
t
≤
,
u
0
D
(t)
=
(3.42)
if
t >
2
D
V
0
.
3.2.3
Limiting Performance Curve
The function
g(U)
of Eq. (3.10) that was introduced in Section 3.1.4 in
connection with the duality of optimal shock isolation problems plays an
important role in limiting performance analysis. Since the criterion
J
2
attains
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