 So, far you know we have studied about modulation and signal beating, but the question to us is you know how do I find out this modulated signal which has been carried by a carrier signal. We talked about amplitude modulated signal, how a carrier frequency carries a modulated frequency. The information from the machinery to us is actually the modulated signal. So, when you looked at the amplitude modulated signal in time domain, we had a signal of this nature wherein this is the high frequency carrier signal. This may not be of interest to us, but it is this envelope. If you look at this envelope which is a low frequency signal as you can see from the time period is actually the signal which contains information which will be useful for machinery diagnostics. The question before us is you know if we go to a real machinery we would be having such signals on top of it there will be lot of noise. So, question is once I do an FFT of such a signal which has the high frequency signal which has the noise etcetera in the frequency spectrum the significance of this green signal in our case will be lost. It will be kind of buried in a high frequency noise. There will be lot of high frequency noise and then how do I identify this or well intuitively you can write a say you know why do not we filter it. We will filter it filter the high frequency signals. So, that the green signals in the low frequency green signals are available that is fine. It is people can do it that way and people also do that way that putting a high pass filter where they when sorry rather a low pass filter wherein they will remove certain high frequency components and look into the low frequency signal. But sometimes it happens that even in this low frequencies there is lot of noise or other phenomena happening which kind of obscures this green signal. So, we have to find out ways to find out this low frequency envelope. So, this happens because of signal modulation. So, at the at the generation point of the signal there is a modulation effect happening because of the frequency of one being related to other or one being caused by the other like the case of the rottering fans which I was telling you about. So, signal has got modulated it has come to us as a modulated signal. So, our objective on how is to demodulate the signal. So, this demodulation has to be done by one is I can put a set of low pass filters and then there are another another technique which you will be covering mostly today is this Hilbert transform. But the problem with low pass filter as you will realize is we do not know the frequency content of the signal. So, what will be my cutoff frequency of the low pass filter is a very difficult thing to establish, but you know diagnosticians or analysts who do with certain signals they play around with the low pass frequency the cutoff frequency of the low pass filter and see whether any meaningful information is being obtained in the low pass low side of the frequency spectrum that is one way wherein just by analog filtering we can do some sort of a demodulation. But now there are well established digital signal processing techniques using Hilbert transform which will demodulated demodulate the signal to find out its envelope. So, envelope which we call as envelope analysis is nothing, but it is also signal demodulation because I need to find out the low frequency signal which is coming as an envelope I can demodulate the demodulate the measured signal and then I can get the information about the signal. Once this signal has been demodulated or envelope analyzed I can then do my traditional FFT to find out its frequency content. So, demodulation itself is in time domain or the low pass filters you know we set it in all the signals from this filters are again in time domain after the Hilbert transform also the signals are in time domain. So, and after we have got this time domain signal and that was that green envelope signal which I talked about we can do further signal processing like an FFT analysis. Now I will before I go to the process of doing demodulation or envelope analysis let me give you one example as to how this demodulation this modulation occurs in a real machinery and then how we can demodulate it to find out the defect frequencies of that machinery. Let us take the case of a simple ball bearing. So, ball bearing actually has you know many elements this is known as the outer rays which is usually stationary and this is the inner rays and these are the rolling elements and these are the balls or rollers. Now there are possible scenarios this is rotating and this is usually fixed the outer rays is fixed in our case and this is rotating. Now there could be few possible cases that I have a defect located here in the outer rays. In this case if you will imagine if you look at the time signal and each time this balls basically this balls will spin and rotate around the inner rays and outer rays and they have this rotating frequencies that can be calculated from the dimensions which will come to later on. There are different because the shaft is rotating at n rpm all this will be relative to n rpm you know some will be little higher some will be little lower and so on. No matter what each time this rolling element or the ball passes over the outer rays defect it is as if you know you are going over a pit each time and if you if you measure the vibration created because of a ball hitting a pit each time I will get an impact and then I will get another impact another impact. As many times the balls hit this defect I will be getting a continuous pulse of impacts ok. Now this impact the frequency of the impact is actually inverse of the time period and that relates to the speed at which the balls are going over the defect ok. But the phenomena is actually inside this impact if you if you imagine if I one of one such impacts may be you will get a ringing signal like this. The frequency which is there or which is characteristic of this defect is actually inside this. This is one way of getting an uniform signal out of this bearing because of defect in the outer rays. On the other hand suppose the defect was there in the inner rays let me put it here ok and because the shaft carry a load and load is always acting downward some load double. So each time the defect comes under the load axis there is a great amount of force on this defect here and the outer inner rays and then once the inner rays moves away this force will decrease and so on. So the intensity of the vibration will be modulated as the as a function of the rotational speed and if I was to draw it here I will have one defect in time domain. This high peaks means the I R defect is right on the axis of the load on the shaft acting downwards and of course this is the vibration signal. So you will see that there is again a effect of modulation occurring here. So this is how physically a modulation also occurs ok. Now if I am talking about a gear box with intermediate shafts you can imagine there are about 6 to 7 bearings and you see because if I have put just one transducer on the machine and this transducer suppose it is to measure. Now I am going to explain to you the severity and the magnitude of the vibration levels and what kind of contamination a transducer may be having. Suppose this is my shaft this is my gear box ok. There are bearings here bearings in the holding the intermediate shaft and there could be gears this is my input shaft output shaft. If I just put one transducer somewhere here and these are all the gears bearing B1, B2, B3, B4, B5, B6 a simple gear box has in a B1 to B6 bearings G1 to G4 gears and I have one transducer T1. So the signal measured by this transducer T1 could have everything. The problem before me is to identify from signal processing which bearing is defective and this is what in reality we face the problem. So the effects of suppose there are few bearings which are having defect there are couple of gears which are having defect. So everything is a composite signal is what we get a sum effect of all problems are there in one signal. So the problem before us is how do I find out which bearing is defective. So we have to know the characteristic frequencies of each of this gears and bearings and specifically if I come to one bearing one bearing itself the inner rest there could be a defect outer rest there could be a defect in the rolling elements there could be a defect. So all these defects will have characteristic frequencies and these frequencies we will tell later how they can be calculated once you know the rotational speed and the dimensions and the number of rolling elements. So if the frequencies are known to us beforehand I am sure that in this signal these frequencies are buried they are there but then we have to have a right kind of eye or magnifying glass if you may say so to look into them and find out what these frequencies are. So the envelope analysis is one such technique or signal demodulation is one such technique by which I can find out these characteristic frequencies. So nowadays in the FFT analyzers which are available there is an option of doing FFT after envelope analysis we do the fast Fourier transform not of the total signal but only of the signal which is carrying the information. I am not interested to know about the carrier frequency okay I am not interested to know the rotational speed I am interested to know whether my defects which are showing as modulated signals which are showing as that envelope can be detected in the signal. So envelope analysis is that way a very very powerful technique for signal processing. Now as I was telling you filtering is one method of finding out the envelope in a signal but the another method is this technique known as the Hilbert transform. So before I go into Hilbert transform I just wanted to briefly tell you about this signals which are one is an amplitude modulated case if I have this amplitude as a function of cosine or a sine series this is the frequency of modulation and then this is the carrier frequency you will see that the amplitude the modulated signal can be represented as this equation and you will see the sum and difference signals also show up and that is what we did by what is known as the side bands occur because of a modulated signal okay and apart from the carrier frequency but the just to recall if the signals were beating you would not see an individual frequency you will only see the sum and difference frequencies but once they are modulating you will see an individual carrier frequency and the side bands around the carrier frequency as fc plus fm and fc minus fm and then once you have the frequency modulated you will see lot of symmetrical side bands okay and given by this number k here. So fc plus k k could be 1, 2, 3 so there are lot of symmetrical side bands will occur and this particularly occurs in gear boxes we will see practical examples later on the question is we know how to detect side bands because of signal modulation okay but question is if I am just interested to know fm it becomes a problem for me because fm could be a very very low frequency signal how do I find out that fm from signal analysis okay so that is the problem we have in front of us. So there is one mathematical technique which we do for example if the 2 signals are beating I will get this kind of waveform and amplitude modulated signal would also look like this as you would have seen so the this signals here are actually a high frequency carrier signals because this number of data points are 1000 here and that the time period here could be dependent on the sampling frequency whatever sampling frequency is taken here so number of data points times delta t so delta t will be total time period will be just recollect total time period nothing but n times delta t which is nothing but n times 1 by fs n by fs so total time period of a block if n is equal to 1000 fs is whatever you will get that many seconds here. Now the question is I am interested to find out the this envelope here how do I do that there is a technique which is very widely used Hilbert transform in this what happens is this is a mathematical expression of the Hilbert transform okay we generate an analytical signal z t here which is the original signal x t plus the imaginary part is the Hilbert transform of this signal x t okay. So this analytical signal actually if you do the fft of the envelope signal we will get what is known I mean sorry the analytical signal itself is going to give you the envelope of the signal and this can be done in the software MATLAB to use the function Hilbert on an amplitude modulated signal we will get the envelope. Now if you look here this is the envelope which was of the beating signal which has been obtained by just doing Hilbert transform of that beating signal and this you all can do in software like MATLAB etcetera very easily. So no matter what because this is the low frequency signal which has been carried by my high frequency carriers if you look at the high frequency carrier here okay this high frequency carrier has is carrying this modulated signal and I am interested to know this envelope and I will not go into the mathematical details of this transformation but it suffice to say by just by doing Hilbert transform of the signal you will get an analytical signal wherein you can get the instantaneous frequency of the signal you can get the phase angle of the signal I will not go into the details of this right now and then we will very easily get the envelope and this envelope can be you can do the frequency analysis of this envelope signal so that you can find out the properties of the signal frequency contents of the signal. So envelope analysis of the bearing signals can be done and then we can get the low frequency bearing defect signals because if you will recollect this bearings, bearings are elastic members say let me take the case of an outer race just an outer race say bearing is used in a machinery wherein the maximum rpm is 3000 rpm and that boils down to 3060 that is 50 hertz so any operation in this bearing is going to produce a 50 hertz excitation frequency right or few multiples of it at the most maybe if it is 10 times the fundamental rate this could be 500 hertz so the external forcing function external forcing frequency on the bearing is about 500 hertz imagine because this is an elastic material it has its mass it has its density it has its stiffness etcetera. So this rings also have their natural frequency we can model them itself some equivalent stiffness and mass obviously a bearing designer is not going to design a bearing whose natural frequencies are in this range otherwise you know every time the bearing is in operation it will get excited at its resonance so no designer would do that usually as a thumb rule the natural frequencies of the structural elements of the bearings are much much higher and above 20 kilo hertz ok so that during the operation of a bearing these do not get excited the bearing does not have large motions because of its resonating conditions but you can think of it another way suppose now this outer race there is a defect as a pit as if you are going on a road and suddenly there is a pit so if this balls are travelling spinning and travelling on this outer race what is going to happen that if I come across this pit on the outer race I am going to have an impulse this will be an impulse excitation as if something has hit the outer race as if you have banged on it like you go on a bump or a pothole on the road you feel that impulse so this bearing has got excited by an impulse read now I will draw your attention to your earlier theories on Fourier analysis any time domain signal 80 suppose this is a very small signal of dimension tau tau becomes very very small its response in the frequency domain will be very very large and you perhaps would have done this as a homework I believe and so on ok or if this is this tau tends to 0 I will have the frequency response of a similar to a broadband excitation that means here because of an impulse where time period is very small the frequency response is very very large time and frequency are inversely related so now if I go back to the example of my rolling element bearing is getting excited by a frequency because of a small defect which looks something like this or something like this ok almost close to an excitation where all frequencies are present ok so if this mechanical system which has a stiffness which has a natural frequency omega n k by m is getting excited by a forcing function wherein all the frequencies are getting excited obviously its natural frequency will get excited if I was to pose the problem another way suppose I can model the bearing by this set of equation by a forcing vector in the right hand side so this force is actually this in the frequency in the this in time domain but in the frequency domain this would look like this that means what the system in this case the bearing has got excited at all the frequencies and it has got excited at its resonant frequencies so I am going to have a resonant condition for it is known as the high frequency technique h f r t so in this case what happens if I go to the bearing signal on top of the amplitude modulated signal I will have if I was just to draw them visually on top of it I will have also large motion so I have a high frequency resonant technique a resonance phenomenon occurring on top of it I have the low frequency amplitude modulation and my transducer is not a mathematician transducer cannot understand all this transducer will faithfully represent my mechanical vibrations as an electrical signal and it will only give me this so it is we through our technique we have to understand whether resonant resonant has a resonance has occurred or whether a modulation is occurring because in such large motions very very low frequency small amplitude amplitude modulation will be obscured on top of it there will be so much of signal noise okay so we have to again find out by what method I can be sure that defect in the bearing has occur so one way is to monitor the high frequency vibrations anything greater than 20 kilo Hertz so because of an impact or because if you say because of a shock this is constituted as a shock because of a shock large motions occur and they occur at high frequencies so traditionally or conventionally there are many products one such is the shock pulse meter or SPM this essentially means to measure the bearings response at frequencies beyond 20 kilo Hertz to check whether the bearings resonance has been excited or not well who is exciting the bearings resonance it is not the normal operating function of a bearing but it is if there is a defect like a pit like defect in the inner s outer s because of this pit I am getting an impulse and this impulses frequency response is a wide band response which has excited the right hand side of the equation so I am getting large motions at the resonant frequencies of this bearing and as a bearing designer they would they always make sure that the bearings natural frequency are not in the operating frequencies of the bearing and they are at a frequencies much much higher than the operating zone and usually of the order of 20 kilo Hertz so in fact a vibration measurement at high frequencies is a sure indicator that a defect has occurred so in fact a shock pulse meter is nothing but a high frequency vibration meter so then we can say for sure that a defect has occurred but we have a little more diagnostic problem in the sense here well this defect is because of what is it because of an inner s is it because of a ball is it because of an outer s so from the geometry if I can calculate the defect frequencies okay this could be because of the inner s outer s ball or the fundamental trend frequency and I will tell you later on how this frequencies can be calculated once these frequencies are calculated we can then use the envelope analysis or signal demodulation to identify in the frequency spectrum whether these frequencies show up okay and then I will when we talk about the fault diagnostics and bearings I will give you certain actual measured values and measured spectrums of the inner s frequency outer s frequency after demodulation because if you will see I will just I will give you two frequencies spectrum one without envelope analysis one with envelope analysis one without envelope envelope analysis okay where this could be the defect frequencies but if I do it with the envelope analysis so this could be the IR defect this is 2 times x 3 times x and so on and these are IR this could be OR and this is what without envelope this is with envelope these are the vibration signals in fact the signals which have drawn here they are even much much I would say dirtier or noisier than this spectrum shown here so it becomes sometimes very very difficult to know whether it is an IR signal or a noise signal but once we do an FFT sorry envelope and then do an FFT all this modulated frequencies will very nicely show up but this is not to be confused with the shock pulse meter because shock pulse meter is a very very high frequency variation measurements which will only tell you about the level but with this envelope analysis I can find out the individual defect frequencies now I will give you another another example here as to you must have all realized this is this is from a plant very close to Kalapur where in this is a electrical motor it's a 3-4 electrical motor and then there is a blower basically they take in air and from the atmosphere and there are then there are series of blowers in this blower house there are about 5 such blowers and each one operating at close to 28-2900 rpm and then you will see all of this there are lot of valves here because this 5 blowers are in series and eventually they will take in the air and feed to the excuse me excuse me feed to the blast furnace okay now in this single room because there are 5 blowers what happens if one of them is having running at a frequency or a speed little less than even you know one rpm or a few rpms the frequency of these machines will be different and because they are close by you will happen they happen the phenomenon of beating is going to occur and if you enter such a place you will hear this kind of a loud you know decreasing increasing sound if you go to any plant wherein many machines are operating supposed to operate at the same speeds because of some reason or the other if some machine is running a little lower speed than the other you will hear this kind of this kind of waning voxing noise which happens even in the ship engine rooms also I believe because there will be machines which will be running not exactly at the same frequencies and this happens in the blowers also of course you know now the reason I wanted to show you this is this sometimes should not be confused as modulation okay because if you look at the response of a beating signal and a modulated signal they all look very similar in the time domain but if you look at the frequency contents of these rooms of these signals they are quite different in the amplitude modulated signals I will see side bands fm fc plus minus fm and also if I am interested to know fm I can do the signal demodulation or do the envelope analysis but if I want to do find out the two frequencies which are responsible for beating I can simply do a fast Fourier transform wherein I have to be sure that the delta f is very fine excuse me very fine okay to distinguish beating I need to have a frequency resolution where delta f has to be very very fine suppose we have one signal at so this is 10 hertz and this is 10 10.5 hertz if my delta f is 2 hertz I am never able to distinguish between these two signals isn't it but this signals would have created beating so in my frequency domain I can very easily mess this if delta f is kept at 2 hertz if delta f was equal to or less than 0.5 hertz I will be able to distinguish them so that is the problem while doing frequency analysis to identify beating otherwise I will just mess this and beating is because of independent independent frequencies because in this example this two blowers or two motors may not be having the same speed okay and they are independent one is not related to the other okay but then this will give rise to a beating phenomena as opposed to the case of the ball bearing wherein the defects and the frequencies which are related because if I have the inner rays rotating at one speed the outer rays the balls the their frequencies are all related to the inner rays frequencies okay and thus they modulate modulate because of the load modulate because of the speed and so on so amplitude modulation can be very nicely because you know if I if I have an FFT of a signal wherein there is a modulated frequency of a very low frequency FM this will be as I was telling you very easily obscured in other noises so envelope analysis is a excellent method to find out FM okay but there may not be any other frequencies next to FM if I have a finer resolutions I can check that but if I see other two frequencies 10.5 and 10 who are not related to each other then I can say the signals are independent and they are responsible for beating of the signal okay now in a in a plant like this and I should just mention it to you that in fact we at IIT Kharagpur had designed this enclosure for them just to kill this noise which was generated because of the beating because again if you see if I have one machine generating 100 decibel of noise another machine of course another 100 and 100 100 decibel of noise the sum effect will be 106 decibels but because of beating at some frequencies there will be a large amplitude and that amplitude will be very very loud so to reduce that we can do put an enclosure enclosure wherein you can put in lot of a lining material we will come to this later on but I will just show you another spectrum of this noise which was measured by the by which was measured and then we had analyzed this at that frequency now this what is here is a one-third octave spectrum we have talked about the different types of filter wherein I can have a fixed resolution filter wherein if a filter has an upper band and a lower band okay the bandwidth of the filter is upper band minus lower band in the fixed resolution case like we do in the narrow band when you do the narrow band analysis what happens is delta F we specify delta F how do you specify it is 1 by T 1 by n delta T it is equal to FS by n and that is fixed when we do the narrow band FFT analysis but if FFT analysis was not done but I am doing a frequency domain analysis by filtering I can set filters of different bandwidth I can set filters of constant bandwidth another world another filter is percentage bandwidth in percentage bandwidth there is a relationship between this upper frequency and lower frequency if the upper frequency is equal to twice of the lower frequency this is what is known as an octave band and its upper frequency is 2 to the power 1 by 3 FL this is a one-third octave band the reason we have octave band and one-third octave band there is a history behind it because you know our human ears can distinguish octaves of frequency very easily I may not my ear may not be able to distinguish 10 hertz and 10.5 hertz I may require an araband analyzer to distinguish between 10 and 10.5 hertz or 10 and 10.25 hertz but perhaps our human ear some of you can perhaps distinguish between 100 hertz or 200 hertz or 400 hertz or 1000 hertz because they are almost multiples so traditionally when FFT analysis was not being used people were using fixed band filters analog filters to do filtering and nowadays also since this has been predominantly used in the industry people stick to what is known as one-third octave band and if you look at this spectrum here this closely corresponds to about 500 600 some odd frequency which happens in this band you will see all the levels are pretty low but it sticks up around 600 this is 400 this is 1 kilo there is a one-third octave spectrum so there is a percentage bandwidth here around this is nothing but the electrical motors blade pass frequency because behind the electrical motor you know such large motors traditionally each of these motors was close to about 900 kilowatt of the power and you know this large motors and then the shaft which is rotating and this motors and running continuously and then they are all blowing in hot air you know this air finally goes to the furnace at a very very high flow rate so this motors and all become heated up so in fact in the rear of the motor there is a blower and this blower has a fan and this fan there are number of I think about 23 vanes so this air gets sucked in into the motor and this motor is getting cooled and then and this vane creates you know because it is a fan rotating at very very high speed this is what is responsible for the motor noise and this frequency has shown up here and if there are 5 such motors in that plant and there is a small variation in their speed this is giving rise to beating okay and obviously I cannot bring down the beating phenomena because for operational reasons some of the fans rotate at slightly lesser speed in the other one so all I can do is to reduce the effect of the large noise which is heard because of the beating is to put an enclosure on the motor of course there are other issues that this enclosure should not introduce you know should not increase the temperature inside the motor because the bearings which are there they are certain class they should not get overheated more than 75 degree Celsius because if you enclose it the cooling air capacity to the motor reduces and the bearings become hot so bearings may cease so you know it's there all it's like a domino effect so any plant if you see there are enough protections for the motor for the bearing for the vibration for the noise particularly for the bearing there we need to also have a round the clock temperature indicated to see whether the temperature is increasing or decreasing or staying still if the temperature increases we need to take precautions as to how to reduce the bearing temperature maybe we should have you know chillers on the enclosures when we have you know hot environment wherein the machines affect machines will get heated up because of an enclosure we need to have you know chilled cool air going in okay like an air conditioner small air conditioner put in fact such enclosures are there enclosures with having chillers then so the temperature does not increase okay so the purpose of this lecture was to demonstrate to you the two phenomena which occurs in machinery one is beating beating has occurred in this case of the blower and other is a signal modulation these are two phenomenons which occur and then we have to find out methods how to distinguish them to distinguish beating I have to have the frequency resolution reduced to distinguish amplitude modulation I have to do what is known as the envelope analysis envelope analysis could be done by low-pass filtering or it can be done by Hilbert transform okay and once we have understood the signals we can now diagnose the fault in the machine okay thank you