in this class, we will talk about something what is known as sub strum analysis. Well by now you know that in the signals there could be side bands because of modulation ok. We have just seen that side bands could be because of frequency modulations, side band because of amplitude modulation. In the case of amplitude modulation I have the side bands you know fc plus minus fm ok. And in the case of frequency modulation I have you know fc plus k times fm where there will be multiples of the modulation frequencies. Now imagine if there are not one source of modulation but there are several source of modulations in a signal particularly a multi stage gearbox. So there will be a family of side bands ok. So in a signal if I have a lot of side bands apart from the fundamentals you can imagine the problem I will have in the frequency spectrum as to relate to which frequency or which side bands came from which machine ok that is the problem I have in front of me ok. So this sub strum is a technique by which we can cluster a group of side bands coming from one source as a single parameter. Particularly sub strum for its you know utility actually what happened in a sub strum is such a powerful signal technique analysis technique in which it can distinguish between side bands very easily. A lot of this earthquake signals or seismic signals they have lot of side bands. Human voice or speech signals they have lot of side bands ok to understand between different pronunciation different words which are coming out of a human being particularly for speech processing sub strum is very very useful. So sub strum started its origin is from earthquake signals and from human voice signals and then when later on we as a machinery diagnostic engineers found out that our gear box signals also have lot of side bands. So why not we use sub strum to understand more about this gear box signals for the fact that they can distinguish and identify different groups of side bands coming from a gear box. So what is this sub strum? If you look into this equation here it says that sub strum of a signal is the inverse Fourier transform you know F minus to the power minus 1 is the inverse Fourier transform of the Fourier transform of a signal and its logarithm of that Fourier transform ok. So because you are doing an inverse transform here the sub strum is again back to the time domain ok. So let me graphically represent this to you. Suppose I have a signal x t I do its Fourier transform I will get a signal x as a function of F which can be written as x real as a function of F plus x imaginary as a function of F ok. Now if this frequency signal has for the sake of argument has lot of side bands and these are the side bands around a particular frequency and then on top of it there will be noise this becomes very difficult to understand as you will see in an example I will show you ok. So now if I do a inverse Fourier transform of this I will get back x sub strum in another time domain and this is what is known as the sub strum. Well how this is done because to do the inverse Fourier transform because I am having a real signal real quantity here and I am having an imaginary quantity here. So to do the inverse Fourier transform you take the logarithm of the amplitude of this as the real part and the imaginary part will be the phase. But this phase will always be from minus pi to pi. So this will be a discontinuous function. So to smoothen it this has to be made a continuous function. This continuous function for the phase is done by a technique which is known as phase unwrapping and there are many algorithms available for doing this phase unwrapping. So once I have this real and imaginary of the frequency converted one the real part being the logarithm of the amplitude and the imaginary part being the continuous phase. Once I have such an edited spectrum then I do what is known as the inverse Fourier transform to go back to the frequency to the time domain. Now in the time domain what happens because in if you look at substrum the way it comes it is just the spectrum C replaced by S. When we had a frequency as the x axis in the spectrum here it will be it is known as the Q frenzy. So I will get different Q frenzy here. Now this Q frenzy I will get certain peaks here. Now this is in time domain. So say suppose this is T 1 time and then say this is T 2 time F 1 will corresponds to 1 by T 1, F 2 will corresponds to 1 by T 2. So if there is a high amplitude at a particular Q frenzy of T 1 that means in this spectrum I had high amount of side bands at frequency F 1 because otherwise it would be very difficult if I go back here. It would be very difficult I will show you an example it will be very very difficult to find out which frequencies are predominant which side bands are predominant from such an FFT analysis. Because FFT analysis you know for our class we have seen a nice pure tone and nice mathematical functions we get neat Fourier transforms. But in reality once you go to the machines we will have so many unknown frequencies it becomes very difficult for us to find out which frequencies are present which are the defect frequencies and so on. So if I do this substrum I can do find out whether F 1 is strong enough or F 2 strong enough or F 3 strong enough or so on depending on as many Q frenzes you have. Now question is you would have seen FFT can be done by the n log n algorithm and it can be implemented in real time ok. Unlike FFT substrum cannot be done in real time because of block of data has to be taken they have to processed to find out their log amplitudes they have to be processed to find the unwrapped phase as a continuous signal. So once we have an FFT then once you go for a substrum analysis there are software who will do this calculation for you but then it is not real time it will take some time for computation. Once such computation is done and then once you do the substrum you will see the harmonics or the amplitudes or the harmonics as they are called at different Q frenzes ok and then you can find out whether a particular group of side bands is present or not. Another effect of substrum is it deconvolutes or removes the effects of the path because if the two signals are convoluted you know you will or the effects are there suppose a signal x multiplies with signal y ok but if I take the logarithm because I am taking the logarithm it will be log x plus log y. So the effect of the path is actually added to a signal which can be nicely removed once you do substrum. So that is why substrum had an important application in seismic analysis of signal because if an earthquake occurs the signal which is recorded by us is signal x times signal y, y could be because of the path from the epicenter of the earthquake to the location where your transducers were put. This effects could be added and removed and their effects can be eliminated once you do substrum analysis because of this logarithm property of logarithm ok. So that is another advantage of substrum analysis but you know here we will see same also happens suppose I have a large gear box. Lot of signal contamination occurs because of the path whether it is coming from location bearing through bearing A or through bearing A B C all those effects will be lost or easily removed once you do substrum. So to summarize substrum is a powerful method to find out side bands in modulated signals. Now we will go to one example wherein we will see how substrum analysis helps us diagnose a fault you know cement plant. If I was to show you here actually what happens in a cement plant this is a large rotary kiln ok. This is about if I was to explain you the cement plant here. This is a kiln wherein all the mixtures for making the cement you know the mortar mix the limestone the etcetera they are put in this drum and in this drum there are very very hard stones or rolling kind of rocks which will be it is like a tumbler it is like a tumbler and this drum this diameter could be about you know 2 to 3 meters ok this diameter and this length could be about you know 7 to 8 meters and this there is a small inclination between this feeding end and the delivery end and this could be about 2 to 3 degrees slope. So you put in the raw materials and this the beauty of this is or the unique part of this drum rotates at about 2 to 3 rpm it rotates very very slowly and this media along with this grinding media along with this feed is mixed and finally end of here you will have the dry cement powder coming out ok. So, cement mixing cement manufacturing is very very easy in that sense if you have the right proportions of the ingredients you just grind them and you will get cement powder it could be a mixture of fly ash, limestone, gypsum etcetera and then you just grind them. So this is nothing but a grinding mechanism and same is same also happens to your you know your washing machine detergent etcetera they are also manufactured in the same process. We will have some ingredients they could be temperature controlled and then you have a grinding media, but from a mechanical engineers point of view you see that this is running at 2 to 3 rpm and I have this big motor here this could be about you know 900 horsepower this motor is running at about you know 1200 rpm or 1400 rpm or 2800 rpm that depends on the number of poles in the motor. So obviously from 1400 rpm if I was to go down to 2 to 3 rpm you can imagine the amount of speed reduction required in this gear box. So a very critical thing it is such a cement plant is this gear box. So this gear box is a heavy gear box with such a large speed reduction. Obviously I cannot get such large speed reduction in just one gear. So there are intermediate set of gears and they are helical gear they can take lot of load and so on and then we have this gear box which is having the speed reduced and then we have sometimes this speed is also not to 2 rpm. We have another which is known as the bull gear here all around the drum there is the ring gear and then there is the pinion gear and then finally this brings about the speed reduction and then so that the there is a torsion shaft here and this is the weakest member here. So if there is any failure the torsion shaft will shear and fail okay just a safety measure in a sense you know we do not want any of the bearings to fail or the gears to fail but just let this torsion shaft shear okay and then they can take in lot of sometimes this is held by plumber blocks here and the problem here is this bearings in this gear box suppose I put a transistor here they will be capturing phenomena of all the gears capturing phenomena of some of the gears here and bearings. So now question is which bearing has a fault and that is what if you do the analysis you will see that the same problem occurs at which side band corresponds to where and how do I find that. So this is just sorry this is just to give you a small example of the dimensions of this plant this drum has an ODE of 2500 millimeter ID of 2350 millimeter and it is 12 meter long and it weighs about 70 tons you can imagine the magnitude of the cement plant mill or the drum the girth gear has 150 teeth the girth pinion has 21 teeth the girth gear or the ring gear okay and then there is a speed reduction from 21 teeth to 150 teeth the torsion shaft has an ID of 395 sorry ODE of 395 ODE there are couple of gear couplings and in this gear boxes there is a pinion gear and all these different modules are there and the number of teeth are also mentioned here if you can read that 17 teeth in the gear stage 1 pinion 61 teeth in the stage 1 gear 62 sorry 22 in the stage 2 pinion 34 teeth in the stage 2 gear and then you can work it out. The motor is a three phase in a induction motor with a 600 kilowatt and it runs at 985 rpm and bearings for location 36 are given and then the just for your benefit the grinding and mixture material in drum is about 39 tons is the mixture material so we get about 4000 kg of cement in it because this 39 tons is the weight of the grinding media and then the cement raw material flow is about 12 tons per hour and the drum filling ratio is about 25 to 40 percent drum is of course not entirely filled it is like a hollow drum which is churning with this grinding media and the raw material so we are getting about 12 tons an hour of cement from this plant okay and this is what it is there in a cement plant so as you will realize be it a cement plant be it detergent plant this gear box is very very critical to the plant this gear box fails or if this motor fails the plant is shut down okay and this is some of the views of this this is the torsion shaft and this plant is very close to us you know we are done some measurements there after a failure and then we have to do a diagnosis also this is the pinion and this is the drum which is rotating okay this is the other side and the ring gear or the girth gear is inside this enclosure okay and the pinion is here and this is the torsion shaft and this is the gear box okay this is the motor here so motor is driving the input stage intermediate stage and then this side goes to the torsion shaft and because of the I would say defects because of obvious reason there are certain defects so we put axillary meter at the bearing locations axillary meter near the foundations and we had taken the measurements of this viruses simultaneously and obviously this measurements cannot I mean this analysis as I was telling you substrum cannot be done real-time on the fly so we have to record the signals in a data recorder wherein you can record almost 8 channels of vibration simultaneously to a bandwidth of 20 kilowatts you can record them in the state and then you know because in such a dusty environment we cannot be cement dust if you go to that plant lot of cement dust everywhere we cannot be putting computers and sophisticated equipments and then doing an analysis setting to area of you know cement dust so all you do is you just record them in that media bring them to the lab and then use your softwares of MATLAB you can be doing FFT and then do this substrum analysis as per the algorithm I just mentioned and then of course you know another study which we did just to see the dynamics of the bearings here this is another view of the cement mill is you know this is the stage 1 pinion the rotor shaft and the compound gear train torsion shaft girth gear and this is the drum so you can imagine the dimensions of the problem okay now these are the vibrations of the cement plant at different different locations and all I have shown you here is the spectrum these are just the FFT spectrum and these are real FFT spectrum obtained from the eight different locations on the cement plant while the operations were going on and if you look at any one spectrum it becomes very very difficult for anybody to find out which kind of side bands are there and if I so this is very very difficult for us from a signal analysis part of you to find out the group of side bands so this is just to tell you the severity of the problem we have in hand so now this is the vibration spectrum the top one is the vibration spectrum this is in kilohertz near the bearing of the multistage gearbox and then you will see a group of side bands occurring okay and then I would not know so if I if I come to the second plot here they are all in time domain and if you look here the corresponding frequency will be shown and you will know while this corresponds to a particular rotational shaft of the gear and then we know its frequency content and from this frequency we will know well this means this has a high amplitude means this is the strongest side band and that is what substrum helps us in I will give you another example here just from this gearbox so suppose I have opinion so this is having 20 teeth and so this is having 50 teeth suppose this is running at 1200 rpm and this corresponds to 1200 by 60 that is 20 hertz so this will be running at a speed of 20 into 20 by 50 that is equal to 400 by 50 that is equal to 8 hertz so the gear meshing frequency is nothing but n times t1 this is 20 times 20 times 8 times 50 that is 400 hertz now suppose you know the sake of measurements I put this set in a gear box this is the gear box casing and I put one transducer to do the measurements okay it means so happen if I look into the time history of the signal which I have measured I will get something here but at most the analysis I do in time domain is I can find out their RMS mean kurtosis etc but this will only give me a relative measure of the strength of the signal and that is all I cannot pinpoint whether my fault has occurred in opinion or a gear but if I do a FFT analysis what is going to happen is I may see on 8 hertz signal 20 hertz signal and few other signals but what is going to happen is and this is very very important this could be this could be 8 hertz this could be corresponding to the rotational speed of the gear this could be 20 hertz corresponding to the rotational speed of the pinion this is the gear meshing frequency of 400 hertz but what I will also see is the side bands around the gear meshing frequencies and I will draw another family here this spacing in one case will be 8 hertz in another case it will be 20 hertz why has this occurred because modulation is there because of modulation side bands have occurred so I have got a group of side bands okay now this is very nicely drawn here but if you go back to my actual measurement and this is what it is looks like it will be very difficult for me to find out I may find out the gear meshing frequencies but here other than the gear meshing frequencies there will be frequencies coming from the bearings frequency coming from many other machineries so it is very difficult to for me to know whether the side band is because of opinion or because of a gear so the power of the substrum is to identify which side band is more strongly predominant for example in this gearbox I will know if a side band occurs at 8 hertz that means I have signal coming from my gear if my side band occurs at the 20 hertz I know this will be from my pinion okay so what is what is going to happen if there is a defect it may so happen that the amplitude of the side bands around the gearbox is going to increase the frequencies are not going to change frequency affects because of the phenomena so this amplitude of the side bands will increase and this is very nicely explained here but in real-world signal it will be very difficult first to identify the side bands forget about whether to know whether they have increased or decreased but if I do a substrum see the cupherency and know well you know it is this frequency which corresponds to the pinion which is increasing I can say for sure that the side bands of the pinion has increased so pinion is at a fault and that is the power of substrum analysis so sometimes if you have done a good measurement a good kind of frequency analysis if you have a good delta f you can do a good resolution and then you may be able to distinguish them but substrum otherwise is a very very powerful technique to determine modulations in signals okay but this will be again complicated by the fact that while doing all these operations suppose the speed of your machine changes then what is going to happen we will have frequency smearing okay because once it was one hard say 10 hertz next time it was 10.1 hertz next time it is 9.1 hertz so instead of getting and then if you do multiple averages instead of getting a sharp peak you will land up with a signal like this now you will be confused now what what has gone wrong with my machine is it this frequency is this frequency is this frequency so this is very important while doing all this analysis we have to ensure that the speed is not changing or otherwise we do what is known as the synchronous averaging which we do what is known as the order analysis order tracking by order I mean one rotation or one rotational speed no matter what so it is the time taken to rotate by one revolution and its inverse okay so order analysis if the speed is fluctuating the signal will be taken for the complete revolution so we will not miss that and that is something we will discuss in the later class but in subterm analysis the advantage is it it will detect the side bands and also remove the effect of the path see if I was to go back again to the vibration spectrum of the girth gear support this is how it is the real-world signal is full of so many frequencies because this is the frequency domain it becomes very difficult to identify well whether it is this frequency this frequency and so on okay but once you do the subterm you will see the important group of frequencies getting clustered into this differences okay so again in MATLAB you know the software which we are using in our class you could look into the commands like seps term c-seps okay if you do not take the logarithm of the amplitude now you may also land up with you know different forms of the substance whether it is a complex seps term or a magnitude seps term so there are many ways by which people represent the data out of a seps term okay and sometimes they call it as a power spectrum if you are dealing with a power signal and auto power spectrum then you will have a power seps term some because you know if there is a voltage square term or an amplitude square term I can call it as an auto power spectrum if I do a seps term of an auto power spectrum it will become what is known as a power seps term okay there are different ways of representing this data but we can generate signals in MATLAB I can briefly tell you how to generate a signal in MATLAB for example we can first is you know in MATLAB we can write your own M file say for t is equal to if I take del t as a 0.01 hertz that means f s is equal to 1000 hertz I can generate a signal x is equal to 10.0 sin 2.0 star pi star f say may be 10.0 star del t star t so this will generate a signal x okay I could modulate this signal by x m as 10.0 sin of 2.0 star pi star times sin of 2.0 star pi star so if you generate such a signal I have a carrier frequency of 50 hertz a modulated frequency of 10 hertz a sampling frequency of 1000 hertz a modulated and the sampling frequency so once you do the fft of such a signal x m how do you do that fft of such a signal fm suppose you write x f is equal to 2.0 by suppose you have taken 1000 data points star absolute value of fft of x m you will get the amplitude in the frequency domain and then in such a spectrum you will see the frequency of fc plus fm you will see fc minus fm you will see fc if you do this substrum of this signal you need not take the fft so you have to look into the algorithm c seps of x f x c seps trum of x f so fx this is in the frequency domain and this is in the time domain so you will get all the signals which has which corresponds to this will give you in the q frequency so you will get the q frequencies corresponding to the frequencies of fc minus fm and fc plus fm in the substrum and this you all can do in program like MATLAB my first to come back to my example here again look at the viruses spectrum and substrum at gear box gear box base if you are just to compare the gear box base because of a very very poor foundation which is so many frequencies are coming up in the spectrum and even if I asked anyone to find out the frequencies contained in the signal you will be all scratching your heads getting lost what are the frequencies here but the same spectrum converted to substrum you will see the group of side bands coming up a q frequencies clustered side bands clustered into their families and you will see whichever q frequency has the highest amplitude you know that has a strong presence in that signal and that is how substrum helps us identify signals helps us finding out the important frequencies in a spectrum the effect of the path etcetera is lost because you know if you look here at the gear box base lot of effects of the path etcetera was there the gear box base was loose in fact we had discovered many looseness in the gear box foundation itself and all those effects are nullified once you do the substrum those effects can be easily removed and similarly when we did on the motor and gear box only we could see the spectrum of different natures altogether and from this example actually what happened we could find out the characteristic frequencies of the signal which were not apparent in the spectrum but once we do these did the substrum we could find out the characteristics q frequency in the substrum and we could pretty well conclude that the bearing 2 was loose on the foundation bearing 1 and the millware missile line and pinion on the gear box was damaged in fact later on we removed the gear box and subsequently it was reported that one of the pinions in the gear box had a damaged tooth.  Spectrum from a real world machinery signal has lot of frequencies which we may not be able to account for because of all I know is the defect frequency but there are so many extraneous signals which could be from nearby machines which could be from machinery parameters which I am not even aware of but if I know I am looking for a particular family of side bands I can very easily do substrum analysis and then see them as the q frequencies because one thing you should recall recollect q frequency is 1 by or inversely I will not say 1 by but is related to this 1 by the frequency so q frequency is in time domain usually in milliseconds and this is in hertz. So, there has to be an inverse relationship. So, we have a time to spectrum to substrum. So, this is in the frequency domain and this is in the time domain of course this is to this is to this is the relationship and this is time means q frequency of course. So, the effect of all the path properties are lost once we do substrum. So, substrum applications are also speech seismic and then of course in machinery agnostics this is a very important not a real time operation this is on a stationary signal I may not be able to do substrum on a non stationary signal for example, one transient it will be very difficult but if this transient is captured we can do some sort of an analysis of it. But this has given way to nowadays you know what is known as simultaneous or joint frequency analysis and there are few modern methods of signal processing like the wavelet analysis, short time, Fourier transform which has been nowadays you know used in machinery for fall diagnostics particularly for the non stationary signals. But in this very preliminary or the first course on machinery fall diagnostics and signal processing I will not be focusing on these aspects of signal processing, but just to it will suffice that there are methods available today wherein we can do analysis of non stationary signals by the wavelet analysis, short time Fourier transform wherein the signals frequency is changing from sample to sample and then we I need to know in three dimensions you know other function of time, other function of frequency these techniques are available wherein we can see for example, I will give you examples of non stationary signal something breaking cracking impulse door slamming etcetera these are phenomena something hitting rubbing this can be very easily understood by wavelet analysis, wavelet analysis also there are two types discrete wavelet transform and the continuous wavelet transform and then short time Fourier transform because you know we can and then there are multi resolution Fourier transform also. But we will not stress on this in the this course being the very first course in machinery fall diagnostics and signal processing, but this techniques are available nowadays in the literature and people have been practicing them for using them in condition monitoring and we will later on see some applications when we talk about applications how some of this techniques are used particularly for fall diagnostics and bearings and gears and rotating machines where the shafts have cracks etcetera how this techniques can be used to find out their faults.