Civil Engineering Reference
In-Depth Information
function analysis, requires two-channel instruments. When performing so-called modal
analysis, where the purpose is to map the global vibration pattern of a mechanical
system, a multi-channel instrument will be appropriate. As pointed out above these types
of instruments are, owing to modern digital analysis methods, commonly available.
However, the demand for knowledge to enable them to be used properly is by no means
diminishing.
The main type of practical analysis in the fields of acoustics and vibration is
frequency domain analysis. The emphasis will therefore mainly be placed on this type of
analysis. The fundamental bases for this method are Fourier series and Fourier
transforms. This chapter aims at giving an overview of the mathematical basis and
furthermore on the modifications necessary for treating data in a digital form, i.e.
performing digital signal analysis. Simulations and use of data in digital form are used to
produce the illustrations below.
1.4
FOURIER ANALYSIS (SPECTRAL ANALYSIS)
Frequency domain analysis of acoustic and vibration signals is normally denoted by
spectral analysis as we want to extract information, in more or less detail, of the
frequency or spectral content. The analysis technique, based on Fourier series and
Fourier transforms, will be demonstrated using a number of different types of signal:
periodic, transient as well as stochastic.
1.4.1
Periodic signals. Fourier series
We now assume that we have a function
x
(
t
) that varies periodically with time period
T
,
i.e. we may write
x
(
t
) =
x
(
t
+
kT
) where
k
is a whole number. According to Fourier's
theorem such a function may be represented by the series
∞
kt
2
π
kt
2
π
⎛
⎟
.
∑
(1.1)
xt
()
=+
a
a
cos
+
b
sin
⎝
0
k
k
T
T
⎠
k
=
1
The so-called fundamental frequency
f
1
(in Hz) is given by the inverse of the period time
T
. Equation
(1.1)
therefore tells us that the function
x
(
t
) may be expressed by the
fundamental frequency and its harmonic components, hence
k
ω
π
1
f
=== ,
f
k
(1.2)
k
1
T
2
where ω
1
is the fundamental frequency expressed by its angular frequency in radians per
second. Using the latter leads to the equation in its simplest form but we shall, as far as
possible use the most common measurement variable, the frequency in Hertz (Hz).
Equation
(1.1)
may then be written
∞
∑
(
)
,
(1.3)
x t
()
=+
a
a
cos2
π
f t
+
b
sin2
π
f t
0
k
k
k
k
k
=
1
where the coefficients
a
k
and
b
k
are given by the integrals
