Biomedical Engineering Reference
In-Depth Information
where
ξ
is a number from the interval [
−
ε, ε
]. For the limit as
ε
→
0, this
becomes
∞
δ(t)f(t) dt
=
f(
0
).
(2.23)
−∞
This relation can be used as a formal definition of the delta function. The
integral on the left-hand side should be regarded as conventional notation
because the delta function is not a function in the proper sense. A function
must be finite at any point of its domain, which is not the case for the
delta function, which is “equal to infinity” at
t
=
0. From the mathematical
point of view, the delta function is a linear functional over the space of
continuous functions. The value of this functional for any function
f(t)
is
defined as
f(
0
)
.
The delta function is a convenient mathematical instrument to character-
ize transient processes occurring during a small time interval, such as shock
disturbances.
An instantaneous shock can be obtained as the limiting case of the rectan-
gular pulse of Example 2.1 or the half-sine pulse of Example 2.2 as
T
→
0,
while
A
=
S/T
for Example 2.1 or
A
=
πS/(
2
T)
for Example 2.2.
The mechanical effect of the instantaneous shock is an instantaneous
(stepwise) change in the linear and/or angular momentum of the mechanical
system subjected to the shock pulse at the time instant
t
=
t
0
.
In practice, the shock disturbance can be regarded as instantaneous if
the duration of the pulse is much less than the characteristic time of the
system, for example, the period of the natural vibrations or the time interval
on which the motion is of interest. To validate this statement, consider
two examples for one-degree-of-freedom systems subjected to a rectangular
shock pulse.
Example 2.5 Instantaneous Shock as the Limiting Case of the
Rectangular Pulse for a Free Particle
Let a free particle of mass
m
moving along a straight line with the
velocity
v
0
undergo a rectangular dynamic shock pulse
⎧
⎨
S
τ
if 0
≤
t
≤
τ,
F(t)
=
(1)
⎩
0if
t > τ,
Search WWH ::
Custom Search