In this lecture, we will look at the value of Fmax. Fmax is the maximum frequency that we are interested
to capture for the purposes of analysis.
One basic principle is that if we are looking at a high operating speed, the Fmax will be higher.
And if we are looking at a lower operating speed, the Fmax will be lower.
If you are looking at a machine which does not have any blades
teeth
call it fingers
the Fmax value of 10 times the operating speed should be sufficient.
If we have
a machine which is fingered elements such as gears fans, then we use 3 times the operating speed
into number of fingers as a rule of thumb.
So if we have
a machine operating at 50 cycles,
Has 6 fingers, so this comes to 300 and
3 times that is 900. We need to have Fmax of 900 hertz.
If the operating speed is high, if Max required is high, this is the basic rule.
The sampling, which converts the analog vibration signal into digitized signal
is called a time record.
Sampling rate is the number of samples that are made in each second.
And there is a Nyquist criteria, which is that if we
sample a waveform by 2.56 times the frequency of the waveform, then we can capture the waveform in
its original shape.
So this means we want to
collect a Fmax of
1000 hertz, then the sampling rate should be
2560 Hertz.
The sampling time is 't sub ', and the sampling frequency would be 'f sub s' . 'f sub s' should be greater
than or equal to 2.56 times the Fmax as said earlier. In the signal processing, we use filters
the function of filtering is to
remove the undesired portions from the input data and to retain the desired frequencies. We have
4 types of filters.
1 is the low pass filter, which allow lower frequencies to pass through.
Band pass filters, allow frequencies within a certain band to pass through.
And we have band stop filters. These stop frequencies within a given band. High pass filters allow
high frequencies to pass through.
Depending on the requirement, we can use any of these filters or combination of filters to
capture the desired frequency range and to discard the undesired frequency range from the incoming
signal.
Filters have two issues.
One is the settling times
if we do not give enough settling time, then we will have noise in the output and the second is the
filter characteristics.
The filters do not cleanly cut the frequency where we desire, but it has a slanting characteristics
and therefore some other means, have to be applied to remove the undesired portion of the
signals which
creep into the desired part of this signal collection.
There are some rules.
Rule no 1, is the sampling rate should be 2.56 times Fmax, so this is one of the basic rules.
and
the number of samples which we take
based on the sampling rate in the Time Waveform, appear as bandwidth or resolution in the spectrum.
And the bandwidth for resolution is set by the lines chosen, and these are as lines of resolution or
LOR.
So this is something that we come across more frequently in the
signal processing.
So if you have
a high sampling rate
we will have smaller bandwidth or high resolution.
And if we resolve the signal, we will be able to read
events which are taking place at
close intervals.
We need to have high resolution in order to capture all the events in the input signal
The bandwidth is equal to Fmax/ LOR.
So the lines of resolution are high, the bandwidth will be narrow and are resolution would be high.
Rule # 2 is the number of samples is proportional to the resolution.
If the resolution is high, the number of samples should be high.
And the number of samples should be 2.56 times the resolution.
We look at an example.
We have a signal of 200 Hz, we want to have an Fmax of 200 Hz.
If the lines of resolution are 200 than the width of the bin, we call it between two
samples is 1 hertz or the lines of resolution is 1 Hertz and the time required to compute the spectrum
is 1 second.
If we have 400 lines of resolution, then the bandwidth is half
and the resolution would be double, and this would take 2 second. Anything which is happening
at a frequency greater than 0.5 hertz would be captured here.
And if we have lines of resolution of 800, then the resolution is even finer
It is 0.25, and the time required to compute the total spectrum would be 4 seconds.
So we look at another example here, the lines of resolution are 400, 800
We have a fixed Fmax of 1000 hertz
As we increase the lines of resolution, the resolution also increases the sampling rate remains the
same.
The number of samples we take are more in the sampling time increases.
So normally the lines of resolution are 400, 800, 1600, 3200, and
6400.
So they double and if Fmax is fixed at 1000 Hz.
in this example, the resolution keeps improving as we increase the LOR, the sampling rate remains the same
The number of samples increase to increase the resolution and the time it takes to sample the signal
also increases.
This has an implication that if you want to have a signal which is highly resolved , then it will take
more time to capture the signal.
And this would make sense only if we need to have the kind of resolution
if we do not need a very high resolution, then there's no need to increase the lines of resolution.
There's the terminology, which is dynamic range.
We look at the x-axis and we look at the frequencies and we look at the lines of resolution in the
frequency.
But we are also interested in the amplitude, so resolution of amplitude of vibration is given very
dynamic range. In the FFT spectrum
we are mostly interested in the high peaks which are over and above or close to the acceptance level
or to the maximum permissible vibration limits.
However, we are also interested to look at some of the
low amplitude harmonics, which appear as warning signs much before the problem is aggravated.
One of them is the bearing tones.
We looked at it in the lecture on bearings, and these harmonics of bearing tone provide advanced
useful information. So we need to have a high dynamic range in order to
resolve the amplitude of vibrations
The dynamic range of the instrument depends on the number of bits in the analog to digital converting
digitizer.
The dynamic range is given by 20 in to log of the smallest divide by the biggest signal.
So this is the dynamic range definition. And
if we have a dynamic range of 96 dB, this can detect signals up to 0.015 mV
which is good in this corresponds to 16 bit
analog to digital converter.
Windowing is another signal processing activity, which is performed by the
vibration analyzer.
It is selected by the operator.
There is a leakage in the signals when we are converting these signals from Time Waveform to FFT spectrum
Windows is a process of stopping this leakage
We have several types of windows.
The most common one is that 'Hanning' window it is applicable to all signals which are periodic or repetitive.
We have a signal over here.
And we measured the signal from the start at the end, once I cycle complete and we will get in FFT
spectrum when 1 spike over here.
But normally
we will not be able to do this for each and every waveform that is present in the incoming signal.
So if we have a truncation of the signal, which is not at zero at the end, but of some amplitude,
this will translate into a spectrum which is very wide at the base and to control this phenomena
we have to convert ends of the input signal to zero for each and every frequency in this application of.
The mathematical process of weighing function is known as windowing, and if we have a signal which is
passed to the 'Hanning" window, we'll have output like this
and if we do not use a window, we will have a
signal like this, there is a lot of noise over here, which is not real and which is just crept in
because we have not used the windows.
Averaging is very important to
reduce the impact of noise in the signals.
If you take several inputs and average them, and whatever is a non-representative function or noise
that would be reduced by the average. Assuming that we have different frequencies, some frequencies
which are coming from noise are present in this spectrum, some are in this spectrum and some are in
this spectrum.
And when we add them and divide by the number of samples, then those
unique frequencies would be reduced in amplitude.
The time synchronous averaging is one of the averaging process, which is used most commonly and
As a result of the averaging, the spikes which are present in all the spectrums will remain and the
noise signal will be diminished.
It will not be completely removed so that average spectrum represents
something which is close to the true vibration behavior.
And in the signals, we also get beats
so if we have
2 frequencies which are very close to each other of the same amplitude, or sometimes not of the same
amplitude when they are in synchronization, then they produce a signal, which is a high amplitude,
and when these frequencies are,
at a phase where they are opposite each other, they cancel each other and they produce a signal which
is of a very small amplitude.
So this appears as a beating phenomena and we can see it in the Time Waveform very clearly and beating
tells us
that there are frequencies which are operating close to each other.
And if we have a good resolution
then we could resolve these frequencies in the FFT spectrum, if the resolution is lower, the bandwidth
is high, then these two signals will merge together and in the FFT spectrum, they will appear as
one signal and the beating will not be explained.
Fmax and LOR , we have already covered . LOR of 400
are normally used for low Fmax, and a higher Fmax is required for high Fmax to resolve the details
in the signal and output where the Fmax was small
400 LOR OK, but this is a large Fmax and therefore the LOR required were larger
to resolve this signal.
The
measuring speed
depends on the Fmax and frequency range. If the Fmax is higher which means the frequency range
is higher, the measuring speed would increase and if the LOR is high, the quantity of data will
increase and the measuring speed will go down.
We have seen that in the previous slides.
In the signal processing, sometimes to clearly see the input from the lower amplitude but important
frequencies we use logarithmic scale to
reduce the impact of the
high amplitudes and to increase the impact of the very low amplitudes.
So in this way, we can
see things which are not
visible in the normally linear scale.
So this is also part of the signal processing.
This is the end of this lecture, thank you.