Digital Signal Processing Reference
In-Depth Information
Fig. 2.6. Mechanical water
pumping system.
A
V1
F
in
=
kx
(
t
)
V2
h
(
t
)
F
out
=
ch
(
t
)
voltage
x
(
t
):
F
in
=
kx
(
t
)
,
(2.11)
where
k
is the linearity constant. Valve V2 is designed such that the outlet flow
rate
F
out
is given by
F
out
=
ch
(
t
)
,
(2.12)
where
c
denotes the outlet flow constant and
h
(
t
) is the height of the water
level. Denoting the total volume of the water inside the tank by
V
(
t
), the rate
of change in the volume of the stored water is d
V
/
d
t
, which must be equal to
the difference between the input flow rate, Eq. (2.11), and the outlet flow rate,
Eq. (2.12). The resulting equation is as follows:
d
V
d
t
=
F
in
−
F
out
=
kx
(
t
)
−
ch
(
t
)
.
(2.13)
Expressing
V
(
t
) as the product of the cross-sectional area
A
of the water tank
and the height
h
(
t
) of the water yields
A
d
h
d
t
+
ch
(
t
)
=
kx
(
t
)
,
(2.14)
which is a first-order, constant-coefficient differential equation describing the
relationship between the input current signal
x
(
t
) and height
h
(
t
) of water in
the mechanical pump. It may be noted that the input-output relationship in
the electrical circuit, discussed in Section 2.1.1, was also a constant-coefficient
differential equation. In fact, most CT linear systems are often modeled with
linear, constant-coefficient differential equations.
2.1.5 Mechanical spring damper system
The spring damping system shown in Fig. 2.7 is another classical example of a
linear mechanical system. An application of such a mechanical damping system
is in the shock absorber installed in an automobile. Figure 2.7 models a spring
damping system where mass
M
,
which is attached to a rigid body through a
mechanical spring with a spring constant of
k
, is pulled downward with force
x
(
t
). Assuming that the vertical displacement from the initial location of mass
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