Biomedical Engineering Reference
In-Depth Information
where the square and curly brackets denote the integer and fractional parts
of the quantity within these brackets, respectively. Since [
t/π
] increases
without limit as
t
increases, the integral of Eq. (2.11) tends to infinity as
t
→∞
, and hence the condition of Eq. (2.9) does not hold.
2.1.3
Instantaneous Shock
An important class of dynamic shock disturbances can be modeled as a
short-duration action of a force of considerable magnitude, constant in
direction, and having a finite impulse defined by the integral
t
0
+
τ
S
=
F
(t) dt,
(2.12)
t
0
where
F
(t)
is the vector function expressing the time history of the force
while
t
0
and
τ
are the initial instant and duration of the shock pulse, respec-
tively. It is assumed that
F
(t)
=
0for
t
∈
(t
0
,t
0
+
τ)
and
F
(t)
≡
0for
t/
τ).
For instance, the shock disturbances of Examples 2.1 and
2.2 belong to the class under consideration. For these disturbances,
t
0
∈
(t
0
,t
0
+
=
0
and
τ
T
.
In physics, the integral
S
of Eq. (2.12) has a special name. It is called
the
impulse
of the force
F
over the time interval [
t
0
,t
0
=
τ
]. Knowing the
impulse of a force acting on a point mass for some time, one can calculate
an incremental change in velocity of the mass produced by this force during
this time. The increment is independent of the time history of the force
F
within the interval [
t
0
,t
0
+
+
τ
] and is given by
S
m
,
v
=
(2.13)
where
m
is the magnitude of the mass. This relation is a consequence of
Newton's second law,
m
˙
v
=
F
,
(2.14)
where
v
is the velocity of the point mass. To obtain Eq. (2.13) from Eq.
(2.14), integrate the latter equation with respect to time from
t
0
to
t
0
+
τ
and divide the resulting expression by
m
.
Newton's second law and, hence, the relation of Eq. (2.13) can be applied
to an arbitrary body or a system of bodies. In this case,
F
should be
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