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•.
vibration are interrelated, then if one of these quantities is kept constant,
the magnitude of the remaining two can be determined. For example, if we
consider the general expression for the acceleration of a periodic vibration
which is given by:
a = A
0
sin (wt)
then the velocity will be given by:
v
=
fa
dt
and the displacement by:
- Ao cos (wt)
w
d =
J J
a dt · dt =
f
v dt
-A
=~sin
(wt)
w
4.1.
4.2
.
4.3.
It can be seen therefore, that by
integrating.
the acceleration signal once
with respect to time we divide it by
w
to obtain the velocity·, whilst by
integrating twice with respect to time we divide by w
2
to obtain the dis-
placement. This is shown in Fig.4.3 in which the velocity and displacement
spectra are derived from integrating an acceleration spectrum of constant
magnitude.
Both the velocity and the displacement spectra decrease in pro-
portion with the frequency and have a slope of 20 dB/decade and
40 dB/decade respectively (6 dB/Octave and 12 dB/Octave respectively).
ii:l
+5
~
~
-
5
c:
0
-10
a.
~
-
15
'; -20
>
·~
-
25
Ill
ai
-
:ll
c:::
-
35
-
40
-
45
-
50
-
55
-
60
-
65
-
70
1--
)I
I
"""
I
v
"
lfi/
r'\.
vv
"
1
/
-
Acceleration = a
"r--.
..........
i\.
t.....~locity =~
'
'
"""!'-..
\
'r--
r--.
'r--.
1
\.Displacement
=
a
2
'I'-..
'\.1
I
I
I
I
I
I
I
IW
'r--
1\.
I
I
~".....
'\
I
I
-
75
1
10
10
2
10
3
2
10
4
Hz 2
Frequency (Hz)
nu~·
Fig.4.3. Frequency dependence of velocity and displacement upon acceler-
a~n
·
17
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