Civil Engineering Reference
In-Depth Information
2
frequency given by
m
is an important property of the system, so we mod-
ify the characteristic equation to
=
k
/
c
m
s
s
2
2
+
+
=
0
(10.128)
Solving Equation 10.128 by the quadratic formula yields two roots, as expected,
given by
c
m
c
m
2
2
1
2
s
1
=
−
−
4
2
(10.129a)
c
m
c
m
2
2
1
2
s
2
=
+
−
4
2
(10.129b)
The most important characteristic of the roots is the value of
(
c
/
m
)
2
2
,
and
−
4
there are three cases of importance:
If
(
c
/
m
)
2
2
1.
0,
the roots are real, distinct, and negative; and the
displacement response is the sum of decaying exponentials.
−
4
>
If
(
c
/
m
)
2
2
2.
0,
we have a case of repeated roots; for this situation,
the displacement is also shown to be a decaying exponential. It is
convenient to define this as a critical case and let the value of the damping
coefficient
c
correspond to the so-called
critical damping coefficient
.
Hence,
c
c
=
4
−
4
=
2
m
2
or
c
c
=
2
m
.
If
(
c
/
m
)
2
2
3.
0,
the roots of the characteristic equation are
imaginary; this case can be shown [2] to represent decaying sinusoidal
oscillations.
−
4
<
Regardless of the amount of damping present, the free-vibration response, as
shown by the preceding analysis, is an exponentially decaying function in time.
This gives more credence to our previous discussion of harmonic response, in
which we ignored the free vibrations. In general, a system response is defined
primarily by the applied forcing functions, as the natural (free, principal) vibra-
tions die out with damping. The response of a damped spring-mass system cor-
responding to each of the three cases of damping is depicted in Figure 10.14.
We now define the
damping ratio
as
=
c
/
2
m
and note that, if
>
1
, we
have what is known as
overdamped
motion; if
=
1,
the motion is said to be
critically damped;
and if
<
1,
the motion is
underdamped
. As most structural
systems are underdamped, we focus on the case of
<
1
.
For this situation, it is
readily shown [2] that the response of a damped harmonic oscillator is described
by
e
−
t
(
A
sin
x
(
t
)
=
d
t
+
B
cos
d
t
)
(10.130)
