The document discusses different methods of analyzing vibration signals, including frequency analysis and overall level measurements. It explains that frequency analysis provides more detailed information about signal sources than time signals alone. Both linear and logarithmic scales can be used to present vibration data, with logarithmic scales better able to show changes across a wide frequency range. Different types of filters, such as those with constant bandwidth or constant percentage bandwidth, can be used in frequency analysis.

1. 9LEUDWLRQ
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l Why Frequency Analysis
l Spectrum or Overall Level
l Filters
l Linear vs. Log Scaling
l Amplitude Scales
l Vibration Parameters
l The Detector/Averager
l Signal vs. System analysis
BA 7676-12, 1
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The lecture explains the different ways signals can be treated using
detectors and filters/analysers. It discusses the presentation of data using
different axis and the way to combine analysis type with scale type. The
fundamental rule of the BT product and selection of filter/analysis type is
also covered together with the choice of parameter. Finally signal vs.
system analysis is briefly discussed.
Copyright© 1998
Brüel & Kjær Sound and Vibration Measurement A/S
All Rights Reserved
/(&785(127(
English BA 7676-12

2. 7KH0HDVXUHPHQWKDLQ
Transducer Preamplifier Detector/
Filter(s) Output
7KH0HDVXUHPHQWKDLQ
5HPHPEHU The system is never stronger than the weakest link in the chain.
Having covered the transducers and preamplifiers in the previous lecture, we
have come to the last part of the chain, the ways of analysing the output
signals of the preamplifier.
After analysis, which today can have many different forms, the result will be
presented as an output to screen, paper or storage medium.
As this is what the user normally will see, it is imperative to choose a suitable
output format as discussed later in this lecture.
Page 2
BA 7676-12, 2
Averager

3. :K0DNHD)UHTXHQF$QDOVLV
Page 3
D
BA 7676-12, 3
E Vibration
A
B C
Amplitude
Time
Frequency
A
B CD E
Amplitude
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The frequency spectrum gives in many cases a detailed information about
the signal sources which cannot be obtained from the time signal. The
example shows measurement and frequency analysis of the vibration signal
measured on a gearbox. The frequency spectrum gives information on the
vibration level caused by rotating parts and tooth meshing. It hereby
becomes a valuable aid in locating sources of increased or undesirable
vibration from these and other sources.

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The process of Frequency Analysis is as follows: By sending a signal
through a filter and at the same time sweeping the filter over the frequency
range of interest (or having a bank of filters) it is possible to get a measure of
the signal level at different frequencies. The result is called a Frequency
Spectrum.
Page 4
BA 7676-12, 4
Acc.
Level
Frequency

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The simplest way to express the condition of any system is to assign only
one number to it. This is often done using the RMS detector output, and this
gives a number expressing the vibration energy level. However it does not
give much possibility to make any kind of diagnosis. For that we need more
parameters.
7KHUROHRIIUHTXHQFDQDOVLV
The frequency spectrum gives in many cases a detailed information about
the signal sources which cannot be obtained from the time signal. This
permits many kind of diagnoses to be made. The frequency content can be
found in many different ways, using scanning filter, filter banks or as it is the
case mostly today a digital treatment of a record using Fourier
Transformation.
Page 5
BA 7676-12, 5
Overall
Level
Frequency
Spectrum
$YHUDJHU
)LOWHUV

6. )UHTXHQF6SHFWUXPRU2YHUDOO/HYHO
Fan
Vibration
Gearbox
Frequency Spectrum Overall Level
)UHTXHQF6SHFWUXPRU2YHUDOO/HYHO
To decide whether monitoring or testing of the overall level is sufficient or a
complete frequency spectrum is required, the test engineer must know his
machine and something about the most likely faults to occur or which part of
the object is of interest.
The illustration shows two different situations in monitoring, but it might as
well be testing:
Monitoring of a fan: The most likely fault to occur is unbalance, which will
give an increase in the vibration level at the speed of rotation. This will
normally also be the highest level in the spectrum. To see if unbalance is
developing, it is therefore sufficient to measure the overall level at regular
intervals. The overall level will reflect the increase just as well as the
spectrum.
Monitoring of a gearbox: Damaged or worn gears will show up as an
increase in the vibration level at the tooth meshing frequencies (shaft RPM
number of teeth) and their harmonics. The levels at these frequencies are
normally much lower than the highest level in the frequency spectrum, so it
is necessary to use a full spectrum comparison to reveal a developing fault.
A general rule is overall measurements are permissible for simple, non
critical machines, while more complex, more critical machinery requires
spectral analysis.
Page 6
BA 7676-12, 6
Date
Frequency
Vibration
Frequency Date

7. Page 7
3UHVHQWLQJWKH'DWD
BA 7676-12, 7
l Linear vs. Log Scaling
l Amplitude in dB?
l Linear and Logarithmic Frequency Scales
– Decades
– Octaves
3UHVHQWLQJWKH'DWD
It is always important to put some effort into choosing the best way to present
data.
The simplest is to choose linear scales with ranges dictated by the range of
data, but often this does not permit important data to be clearly seen.
Therefore logarithmic scales are often used, sometimes for level using dBs
or just with appropriate numbers at the tick marks, sometimes for the
frequency maybe with decades or octaves as indications.

8. /LQHDUYV/RJDULWKPLF6FDOHV
1/10 1/2
Page 8
BA 7676-12, 8
0 1/2 1
Empty Full
0 0.25 0.5 0.75 1
0 1/5 1
Full
0.05 0.1 0.2 0.5 1
6FDOHV
Let us forget vibration for a few minutes.
The situation shown here is well known to most people. Often it is difficult to
judge the amount of petrol left in the car when the petrol gauge has a linear
scale. If the petrol gauge had a logarithmic scale, the lower end of the scale
would be “stretched“ so that the amount of petrol left in the tank could be
more easily seen. Note that the logarithmic scale has no zero.

9. /LQHDUYV/RJDULWKPLF6FDOHV
0.01 1 Decade
1
Page 9
BA 7676-12, 9
0 10 20 30 40 50 60 70 80 90 100
0 0.1 0.5 1
Linear
1 Decade
1 2 5 10 25 50
1
2
2
5
5
10
10
0.01 0.1 1 10 100
1 Decade 1 Decade 1 Decade 1 Decade
20
20
50
50
100
100
Logarithmic
6FDOHV
Which scale to use depends on the unit to be scaled. Distance and time
scales are typical examples of linear scales, but for scaling of units where the
ratio between two values is of more interest than the absolute value it is an
advantage to use logarithmic scales e.g. the different coins and notes in our
monetary systems have values which, when plotted on a logarithmic scale,
show approximately equal “distance” between adjacent values.

10. /LQHDUYV/RJDULWKPLF)UHTXHQF6FDOHV
120 Hz 50 Hz
Page 10
Vibration
Level
BA 7676-12, 10
200 400 600 800 1K 1,2K 1,4K 1,6K 1,8K 2K Hz
20 50 100 200 500 1K 2K 5K 10K 20K
Linear
Frequency
Logarithmic
Frequency
0
Vibration
Level
/LQHDUYV/RJDULWKPLF)UHTXHQF6FDOHV
Both linear and logarithmic frequency scales are used in connection with
vibration measurement. The linear frequency scale has the advantage that it
is easy to identify harmonically related components in the signal. The
logarithmic scale, however, has the advantage that a much wider frequency
range can be covered in a reasonable space and each decade is given the
same emphasis. The signal shown here is the vibration signal from a
gearbox plotted with the two different scales. The harmonically related
components in the signal are easily identified on the linear scale and the
logarithmic scale gives many details in the low end while it covers a 10 times
wider frequency range at the same time.

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B
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Frequency
Frequency Frequency
f1 f0 f2
f1 f0 f2
Bandwidth = f2 – f1
Centre Frequency = f0
f1 f0 f2
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Having chosen a frequency scale, the next step is to choose the form of the
many individual filters which are used to make the analysis. This illustration
shows the properties of an ideal filter and real filters.
An ideal band pass filter will only allow signals with frequencies within the
pass band (bandwidth B = f2 - f1 ) to pass. Ideal filters do not exist, however.
In practical filters, signals with frequencies outside the passband will also go
through, although in an attenuated form. The further away from the pass
band, the more attenuated the signals will be. The bandwidth of practical
filters can be specified in two different ways:
1. The 3dB (or half power) Bandwidth
2. The Effective Noise Bandwidth
The 3 dB bandwidth and the effective noise bandwidth are almost identical
for most practical filters with good selectivity.

12. Constant Bandwidth Constant Percentage Bandwidth
Linear
Frequency
Page 12
)LOWHU7SHV
BA 7676-12, 12
B = x Hz
B = 31,6 Hz
B = 10 Hz
B = 3,16 Hz
or Relative Bandwidth
B = 1 octave
B = 1/3 octave
B = 3%
B = y% =
y ´ f0
100
0 20 40 60 80 50 70 100 150 200
Logarithmic
Frequency
7SHVRI%DQGSDVV)LOWHUV
Two types of band pass filters are used for frequency analysis.
1. Constant Bandwidth filters where the bandwidth is constant and
independent of the centre frequency of the filter.
2. Constant Percentage Bandwidth (CPB) filters, where the bandwidth is
specified as a certain percentage of the centre frequency i.e. the
bandwidth is increasing for an increase in centre frequency. CPB filters
are some times called Relative Bandwidth filters.

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1 2 5 10 20 50 100 200 500 1k 2k 5k 10k
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When using constant bandwidth filters it is recommended to use linear
frequency scales, when the result is to be displayed. See what happens if a
logarithmic frequency scale is used!

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125 8k
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Frequency, Hz
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When using constant percentage bandwidth filters it is recommended to use
logarithmic frequency scales, when the result is to be displayed. See what
happens if a linear frequency scale is used!

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120 Hz 50 Hz
Page 15
Vibration
Level
BA 7676-12, 15
200 400 600 800 1K 1,2K 1,4K 1,6K 1,8K 2K Hz
20 50 100 200 500 1K 2K 5K 10K 20K
Linear
Frequency
Logarithmic
Frequency
0
Vibration
Level
/LQHDUYV/RJDULWKPLF)UHTXHQF6FDOHV
If the right combination of filter type and frequency scale is chosen, the
frequency spectra look alike.
To know if the analysis is done with constant bandwith filters or with CPB
filters just have a look at the scaling of the frequency axis.
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Depends on the actual application.
5XOHRIWKXPE: Analysis with constant bandwidth filters (and linear scales) is
mainly used in connection with vibration measurements, because signals
from mechanical structures (especially machines) often contain harmonic
series and sideband structures. These are most easily identified on a linear
frequency scale.
Analysis with CPB filters (and logarithmic scales) is almost always used in
connection with acoustic measurements, because it gives a fairly close
approximation to how the human ear responds. In connection with vibration
measurements, the CPB bandwidth filter is used for measurement of
structural responses, and for survey of the condition of machines (3 decades
can easily be covered by CPB bandwidth filters).

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Frequency
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The narrower the bandwidth used, the more detailed is the information
obtained.
The filter bandwidth need to be selected in such a way that the important
frequency components can be distinguished from one another. However it
also has to be large enough to get the analysis done in a reasonable time.
The more detailed (narrow band) analysis however requires a longer
analysis time.
Page 16
BA 7676-12, 16
Vibration
Level
Frequency
Vibration
Level
Frequency
Filter
width
Frequency
Spectrum
Frequency

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B = bandwidth
T = time
%7 ³ 1
Page 17
BA 7676-12, 17
(often called the Uncertainty Principle)
7KH%73URGXFW
The statement BT ³ 1 sometimes called the Uncertainty Principle, tells us
that if we choose a very small bandwidth B, then we need a corresponding
measurement time T which is very large, and there is no way around this
basic principle. We can also explain it by saying that if we want to know
whether there is a signal at 1 Hz (i.e. with a 1 second period) then we need
at least to wait for one period of the signal before we can say much.

18. /LQHDUYV/RJDULWKPLF$PSOLWXGH6FDOHV
1000 1000
Page 18
BA 7676-12, 18
´ 3.16
Frequency
´ 3.16
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l Constant factor changes are equally displayed for all levels
l Optimal way of displaying a large dynamic range
/LQHDUDQG/RJDULWKPLF$PSOLWXGH6FDOHV
It is often the case that interesting frequency components in a vibration
spectrum have a much lower amplitude than the dominant components.
These low levels will hardly be registered if a linear amplitude scale is used.
It is therefore common practice to use logarithmic amplitude scales. The
logarithmic amplitude scale may be marked off in mechanical units, such as
ms-2, but often the decibel (dB) scale is used.
*HWWLQJ0RUH2XWRI'DWD
An obvious consequence of the logarithmic amplitude scale is the chance to
get the most out of existing data, since more can be seen at one time,
without needing to change the display. This display also has the added
advantage of compressing the effect of random fluctuations, both in machine
vibration signal, and in noise.

19. 1 G% 2 10
Page 19
7KHG%6FDOH
Acceleration
BA 7676-12, 19
dB
re. 10-6 m/s2
Acceleration
m/s2
´
G%
´
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æ
D
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æ
ö
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ö
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The dB scale reduces the considerable numerical span of the normal log
scale to a compact linear numbering system. The dB scale is such that a
given percentage interval in, say, acceleration level is represented by a given
number of dB’s. This is a great advantage when dealing with vibration
signals, since we are often very interested in a percentage change in the
vibration level rather than in the actual levels. Zero dB on the dB scale can
be chosen for any vibration level e.g. 1ms-2. The level 10-6 ms-2 has,
however, been internationally chosen as the reference level for acceleration.
(Be careful however, some US and CDN military applications use 10-6 g as a
reference!)
7KHG%)RUPXOD
Conversion from actual level to dB or vice versa can easily be performed
using this formula. The calculation can easily be carried out with a pocket
calculator.

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System
Response
(Mobility)
+ =
8 dB
8 dB
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Critical vibration components usually occur at low amplitudes compared to the
rotational frequency vibration. These components are not revealed on a linear
amplitude scale because low amplitudes are compressed at the bottom of the
scale. But a logarithmic scale shows prominent vibration components equally
well at any amplitude. Moreover, percent change in amplitude may be read
directly as dB change. Therefore noise and vibration frequency analyses are
usually plotted on a logarithmic amplitude scale.
What determines the magnitude of vibration?
What is creating the vibration?
It must be understood that when we measure vibration, it is often a
compromise. We would much rather measure the forces generating the
vibration directly. This is practically impossible. So we measure the result of
the force, which is the vibration. The vibration spectrum, and even the overall
level, is indirectly linked to the force spectrum, or overall level, via the mobility
function. The diagram shows an interesting mobility phenomenon. The force
spectrum contains a peak at the frequency shown. However, because the
mobility has an “anti-resonance” at that same frequency, the vibration
spectrum contains no significant peak at that frequency. This shows that it is
not only the largest peaks in a spectrum that we should be interested in. But
note that an 8 dB increase in force will still show as an 8 dB increase in
vibration.
Page 20
BA 7676-12, 20
891875
Input + = Vibration
Forces
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l Imbalance
l Shock
l Friction
l Acoustic
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l Mass
l Stiffness
l Damping
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l Acceleration
l Velocity
l Displacement
Frequency
Frequency Frequency

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ms-2 dB
Page 21
BA 7676-12, 21
1 000 000
1000
1
0.001
0.000 001
240
180
120
60
0
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With piezoelectric accelerometers we are able to detect a vibration amplitude
range of almost 100 000 Million to 1 (1011:1); with a dB scale this range is
reduced to a manageable 220 dB. The dynamic range of a single
accelerometer will, however, typically be 108.

22. 9LEUDWLRQ3DUDPHWHUV
Relative Amplitude
100 000
10 000
1000
10
100
1000
10 000
:KLFK3DUDPHWHUWRKRRVH
If the type of measurement being carried out does not call for a particular
parameter to be measured e.g. due to some standard, the general rule is that
the parameter giving the flattest response over the frequency range of
interest should be chosen. This will give the biggest dynamic range of the
whole measurement set up. If the frequency response is not known start by
choosing velocity.
An advantage of the accelerometer is that its electrical output can be
integrated to give velocity and displacement signals.
This is important since it is best to perform the analysis on the signal which
has the flattest spectrum. If a spectrum is not reasonably flat, the contribution
of components lying well below the mean level, will be less noticeable. In the
case of overall measurements, smaller components might pass completely
undetected.
Page 22
BA 7676-12, 22
0.1 1 10 100 1 k 10 kHz
Frequency
10
1
$FFHOHUDWLRQ
100
9HORFLW
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100 000

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Page 23
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BA 7676-12, 23
Acc.
$
Choose
Vel.
Disp.
Displacement
Choose
Velocity
Acc.
Vel.
Disp.
0HDVXUHPHQW
Acc.
Vel.
Disp.
Choose
Acceleration
8VHWKH)ODWWHVW6SHFWUXP
In most cases this will mean that velocity is used as the detection parameter
on machine measurements. On some occasions acceleration may also be
suitable, although most machines will exhibit large vibration accelerations
only at high frequencies. It is rare to find displacement spectra which are flat
over a wide frequency range, since most machines will only exhibit large
vibration displacements at low frequencies.
In the absence of frequency analysis instrumentation to initially check the
spectra, it is safest to make velocity measurements (but still using the
accelerometer, of course, since even the integrated accelerometer signal
gives a better dynamic and frequency range than the velocity transducer
signal).

24. Page 24
BA 7676-12, 24
Vibration
Level
7KH'HWHFWRU$YHUDJHU
Peak-Peak
Peak-Peak
Time
RMS
Peak Peak-
Peak
Time
Hold
Peak
RMS
Vibration
'HWHFWRU$YHUDJHU
The final link in the measurement chain before the display is the
Detector/Averager which converts the vibration signal into a level which can
be shown on the display.
The example here shows the output level (RMS, Peak, Peak-Peak or max.
Hold) for a burst of a sinusoidal signal of constant amplitude applied to the
input.
Notice how the output level decays when the input level drops giving rise to
fluctuations in the signal. The amount of fluctuation and decay is determined
by the averaging time chosen.

25. Page 25
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BA 7676-12, 25
Time
Averaging Time = 10 s
Averaging Time = 1 s
Time
Vibration
Level
(Peak)
Vibration
$YHUDJLQJ7LPH
With a short averaging time the detector will follow the level of a varying
signal very closely, in some cases making it difficult to read a result off the
display. If a longer time constant is used however some information might be
lost. This is especially true if the signal contains some impulses.

26. 6LJQDOYV6VWHP$QDOVLV
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In most of the preceding slides it has been assumed that a vibration existed,
generated in some way by forces present in the system itself. When this
vibration signal is analyzer we call it Signal Analysis.
During development of new structures, and in some cases to analyse in
detail existing structures, it is a requirement to try to make a model of the
structure, in such a way that if input forces are given the output vibration can
be calculated.
The illustration shows such an application measuring the mobility by
introducing forces at different positions and measuring the input force
together with the output vibration. These types of measurements are used to
make a modal model of the structure, which can then be used to predict the
behaviour of the structure under given circumstances. The model can also
be used to predict the effect of changes in the structure, especially if it is
combined with Finite Element Modelling (FEM). This type of analysis is
called System Analysis, but it is beyond the scope of this lecture to cover this
in more detail.
Page 26
BA 7676-12, 26
Vibration
signal
Excitation
(Input)
Vibration
Response
(Output)
Signal Analysis System Analysis

27. This lecture should provide you with sufficient
information to:
l Choose the right vibration parameters to measure
l Present the measured data in a suitable way
l Understand the basic filter and analysis parameters
and limitations
l Understand the difference between signal and
system analysis
Page 27
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BA 7676-12, 27

28. /LWHUDWXUHIRU)XUWKHU5HDGLQJ
Page 28
BA 7676-12, 28
l Shock and Vibration Handbook (Harris and Crede, McGraw-Hill 1976)
l Frequency Analysis (Brüel Kjær Handbook BT 0007-11)
l Structural Testing Part 1 and 2
(Brüel Kjær Booklets BR 0458-12 and BR 0507-11)
l Brüel Kjær Technical Review
– No.2 - 1996 (BV 0049-11)
– No.2 - 1995 (BV 0047-11)
– No.2 - 1994 (BV 0045-11)
– No.1 - 1994 (BV 0044-11)
– No.1 - 1988 (BV 0033-11)
– No.4 - 1987 (BV 0032-11)