Civil Engineering Reference
In-Depth Information
viscous damping small enough to make it oscillate. We assume that
x
(
t
) is the
displacement of the mass when we initially move it from its stable position and then
release it. We may write
−
at
xt
( )
=
Ae
cos(2
π
ft
)
for
t
>
0
0
and
xt
( )
=
0 otherwise.
How quickly the motion “dies out” is determined by the constant
a
and
Figure 1.7
shows
a time section where
a
is 1 s
-1
and 50 s
-1
, respectively. The amplitude
A
is set equal to 1.0
be
1
2
⎡
⎤
2
2
2
⎢
⎥
a
+
4
π
f
Xf
()
=⋅
⎢
A
.
⎥
2
(
)
⎡
2
2
2
2
⎤
2
2
2
⎢
a
+
4
π
f
−
f
+
16
π
a
f
⎥
⎣
0
⎦
⎣
⎦
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
0.00
0.05
0.10
0.15
0.20
0.25
Time
t
(s)
Figure 1.7
Time function of a damped oscillation (see the expression for
x(t)
above). The constant
a
is equal to
1 (thin line) and 50 (thick line), respectively.
This expression is shown in
Figure 1.8
for the two values of
a
. It should be noted
that the ordinate scale is logarithmic in contrast to that used in
Figure 1.6
. As expected
we do get a very narrow spectrum around
f
0
when the system has low damping. Setting
a
= 1 s
-1
reduces the amplitude to 1/10 of the starting value after a time 2.3 seconds, i.e.
after some 60 periods. When
a
is 50 s
-1
the amplitude is down to 1/10 just after one
period and the spectrum is then much broader.
1
The frequency
f
0
will not be independent of the damping of a real system. However, with the chosen variation
in the damping coefficient
a
the variation in
f
0
will be approximately 5%.
