 lecture is on order analysis, after having discussed the Fourier transform Fourier series Fourier analysis. Usually I would have seen that in Fourier analysis we dealt with stationary signals, where in the signal was not changing during the time the Fourier operation was being done. But in real life you will see instances where the machines speed do not remain constant and they change while we are doing either the data acquisition or doing the analysis. We have discussed about substance analysis, wherein we could find out the families of side bands. But today in the last series on signal processing that would be completing module 3, we will be discussing about order analysis. So, let us see what this order analysis is. To begin with I will just recall your discussion on ideas on signal. Signals could be periodic and very deterministic or sedium signals could be this could be an example of your stationary signal and another could be non-stationary in the sense the features of the signal over a given block of time are different like the mean standard deviation etcetera could be different over different blocks of time, which is the reason why we call them to be non-stationary. As opposed to stationary signals whose RMS amplitude whose mean whose standard deviation is the same no matter how long where you take the sample or take the signal. But let us take the case of a signal from a rotating machinery wherein the speed was changing and you will recollect may be the frequency was increasing again decreasing this kind of time period will be obtained. This is the high frequency band and this is the relatively low frequency and this is the amplitude. Now, how is such a signal generated is first we will see and then I will take you to the one experiment wherein I will show you how this has been collected. Usually if I have a rotating shaft and there is a usually a key way and the shaft is rotating. If I and of course you know this is held on bearings may be I will these are my bearings and of course there is a casing etcetera this could be any machine and if I put a transducer here this is my transducer. Now what would happen is this transducer if it was going to sense this gap every rotation what would happen is this time this is the voltage has generated by the sensor and this should be nothing but one revolution of that shaft. Now if the speed of the shaft was changing because of the dynamics of the machine what would happen this time period would either decrease or increase. But that transducer or which is sometimes known as a key phasor by some commercial companies gives the signal as to indicate whether the shaft has come just below or the key has come just below the transducer. So, any speed fluctuation is captured by just knowing that one rotation of the shaft has been completed. Now if while we are doing a data acquisition for example, if I had put an axiometer here or a vibration sensor here through the appropriate software a command was given that only when this transducer senses a spike then only you acquire the data. In that sense it always becomes that this is kind of synchronous to the location of the key wave below the transducer. So, if the speed is changing I did not worry each time I will be capturing my data just from one single point and this could be my vibration data and if this vibration data had noise this would look something like this and then if I repeatedly added such signals from the vibration sensor and did I mean I would get a neat signal like this and this is what we had perhaps discussed earlier this is known as synchronous time domain. So, such synchronous time domain averaging if we are collected by a key phasor triggering what we could do is we could remove the non-stationary non-stationarity out of a signal because how this order analysis helps us because I do not want to bring in a speed fluctuation effect is reduced or minimized frequency domain analysis well what does this mean what does this mean just to recollect I had told you about stationary signals and non-stationary signals. So, I want to remove the non-stationarity out of a signal and just find out this stationary component of the signal, but that does not mean that non-stationary signals have no use in condition monitoring that is not right. For example, if I take the case of a some impact I could do the frequency characteristics of such an impact by techniques of signal processing which are known as you know wavelet analysis, short time Fourier transform etc. Just to understand the frequency characteristics of such non-stationary signals. So, there are well established signal processing techniques like wavelet analysis, STFT or short time Fourier transform to find out the frequency characteristics of non-stationary signals or which are known as the time frequency characteristics we are not going to discuss about this, but we are rather going to discuss whether in a rotating machine if there is a small fluctuation of speed it leads to non-stationarity, but I am still interested to find out the stationary component of the signal. So, what kind of analysis we do like I just explained you we do the time synchronous averaging by a key phasor signal or by a trigger signal obtained from the key. Let us see if this speed was fluctuating while we are doing a Fourier frequency transformation what would happen. So, if I take such a signal and do the frequency domain analysis I will see just one peak perhaps because this is one signal, but imagine if for some reason there is a speed fluctuation in the machine. So, what is going to happen? So, in such a case what would have going to happen is you know we will start the low frequency may be there in the process of operation I will see in the FFT many frequencies one because of the low frequency one because of the high frequency. So, I have mischaracterized my signal because the signals time spacing or frequency had changed while I was doing the FFT operation. So, this is not right I would not like to have such an analysis where the frequency representation is a smeared bandwidth smeared frequency. So, rather we will now see how such a smearing can be avoided rather than looking at the frequencies had I looked at how many revolutions it had completed I could have got this. So, I will remove my time index from time to may be number of revolutions and that means actually orders how many orders are present. So, no matter whatever is the time taken as long as one complete revolution is not completed I will not be getting affected. So, you see in many machines we have moving in fixed parts in machines for example, this moving parts could be rotating or could be reciprocating rotating parts could be sharps gears reciprocating parts could be a piston etcetera. So, the frequency of this rotating parts all the reciprocating parts are always related to the rotational speed. If my rotational speed changes these frequencies would change, but there will be also certain fixed parts like the body panels and casings they will have certain resonance frequencies and these frequencies are not going to change no matter what happens to your speed. So, you can see that there are some frequencies related to rotational speed like say for example, gear mesh frequency or g m f this is nothing, but number of teeth times the rotational speed right. So, you see if the rotational speed changes the gear mesh frequency in the spectrum is going to change and if the machine there is a small fluctuation and speed this fluctuation and speed could be because of power supply variation could be because of the supply frequency variation could be because of the load on the machine. So, there will be momentarily a speed change. So, this is going to affect the gear mesh frequency because if you know if you are doing 100 averages of an FFT operation it takes certain time within that time taken to do 100 averages. If the speed changes changed you would have got a smeared frequency spectrum and that is what we want to avoid by order analysis rather than let us look into how many complete revolutions have occurred, but in there are also in machines certain fixed parts like the body panels casings and because they are structural components they have mass they have stiffness they will be also having a resonant frequency. So, you see in the same machine there are two classes of frequencies one is the one like example, the gear mesh frequency related on rotational speed and other resonant frequencies which is not related to rotational speed. So, this is very easy to detect. For example, in a frequency spectrum out of a machine suppose I had the liberty or the convenience of changing the rotational speed what would happen every frequency which we are not related to the rotational speed like a structural resonant frequency would be constant right they would not change, but anything which is related to the rotational speed will change this is one set maybe this is another set set one set two, but these are these are fixed these are fixed. So, very easily I can find out the resonant frequency or the sets of rotational speeds. So, usually when I have the opportunity of ramping up or run up or run up or coast up or run down coast down they mean the same thing. This means I am increasing the speed rotational speed speed from to a higher value and this is vice versa this is the reverse. That means if I was to plot the speed and say may be the virus in amplitude this would this would be a run up case. I will give an example where they use a lot of run ups. Let us take the case of a gas turbine. Gas turbines have to run at almost around 20,000 to 30,000 rpm. So, one has to start from start from rest from 0 hertz all the way up to 20,000 rpm. So, what happens in this 0 to 20,000 rpm you can understand that there will be many modes of natural frequency of this rotating shafts. So, imagine if I was to very slowly you know increasing the speed from 0, 100 hertz, 200 hertz, 300 hertz and then I am sure to pass through couple of resonances of this rotating speeds of the rotating shafts of the turbine. And if so happens if I stay for a longer time at one of these resonating speeds I may be I may be creating harm to the shaft in the sense the shaft could be subjected to high fatigue loads. In fact, when the switch on a gas turbine they usually ask you to know accelerate up to the rotational speed. Well, at the same time not stay at a particular speeds for till you reach the final speed for a longer time because as you know you will see in rotodynamics that it takes time for the amplitude to build up at resonance. So, you have to quickly pass through the resonance and if you have sat in aircrafts and the pilot was starting the engine you would have experienced that once it goes up to the highest RPM of the gas turbine you will go through regimes of different resonating frequencies and you will you can feel that kind of jerk or loud noise at particular frequencies that is when you are passing the resonances. So, that is what is known as a run up or a coast up and then similarly when you switch off a machine it will come down to rest and that will be coast down or run down. So, if I was to measure the acceleration of such a coast up or coast down and do a frequency domain analysis you will see that how the frequency distribution will get smeared. Rather than this if we measure just the order or the rotational of the machines you will see that the fixed speeds will be not affected by the rotational change in the rotational speed of the machine. So, why this vibration and sound mechanisms are generated because of the critical speeds of the rotating shafts because of excited resonances and stabilities and varying loads and this is what is responsible for vibration and sound generation in machines. Instabilities could be because of the operational difficulties because of the dynamic conditions loads could be varying because depending on the applications and some of the resonances in the structure or in the machine could be excited while you are doing a coast up or coast down. So, we will see how such frequencies can be identified by this very convenient technique of order analysis. So, to summarize you know you recall order analysis is a technique by which instead of looking at the frequency domain in the x axis we are looking in the orders or in how many rotations the shaft has undergone and one rotation means you know one complete order and so on. So, the what is the principle behind this order analysis it is the key to sound and vibration analysis on rotating machines because the rotational speed of the machine is simultaneously measured with the measurement of sound and vibration. I just gave you an example how the key or the key phasor signal was measuring the rotational position or rotational speed and simultaneously it was triggering the accelerometer kept on the bearing to measure the vibration. So, this is combinedly measured and when the analysis is related to rotational speed it is called as order analysis. So, rotational speed given by rpm is nothing but revolutions per minute and that is the first order. So, for example, if the machine is running at 1000 rpm the first order is 1000 rpm its harmonic would be 2000 rpm and so on. So, well what is the what are the methods of order analysis? We will be in this class discussing on FFT based order analysis, but then there are certain two other techniques time signal recording and short time Fourier transform and world Kalman order tracking filter technique. Now, these are techniques which we will discuss in the advanced level courses on either on signal processing or machinery fault diagnostics, but this suffices to say for example, in a multi stage gearbox or a turbine if there are just not one single shaft, but you know there are multiple shafts one turbine many rotating shafts if this was a rotating at n rpm this could be 48 times n and so on. And suppose from the vibration monitoring of this machines I just want to know what is the condition of this shaft. So, I know this shaft is occurring at 48 times the rotational speed. So, I would like to see the vibration spectrum of the 48th order. If I capture the order here by a key phasor signal by certain estimation techniques like the world Kalman filtering technique I can find out how in the total vibration spectrum how is the 48th order changing and then I can find out the only the vibration response of the or the this will be in orders of the 48 order into something like this. So, I can find out what is the variation of this we are not going to discuss this in detail, but we are going just focus our attention to just a single shaft how by doing a rotational speed analysis I we can find out the order. So, I will take you to an example which we do in the lab. So, let me first explain you this setup here what we have is you know this is a rotating shaft which is supported on two bearings bearing 1 bearing 2 on top of bearing 1 I have put an axle or emitter which measures the vibration in the vertical direction and this disc has the provision to we we can introduce unbalance by just putting in a bolt here and this is there is a motor driving the shaft motors behind this panel and we can set up the speed of the motor in terms of whether you want to run up run at a constant speed or you can by varying this potential meter you can do a run up or run down of this machine. Now, to measure the very important rotational speed this is a photoelectric tachometer as in there is a optically reflective reflective strip here. So, a light will be focused or pointed at this shaft and this light is going to reflect back. So, every rotation of this shaft I will get a one light pulse back. So, this is how this tachopulse will will measure that one rotation of the shaft is complete and simultaneously it is going to through the equation process capture this actual emitter signal or the vibration signal and then it is going to plot it. So, there are many ways of representing this order case one is this is the normal condition of the shaft wherein there was no unbalance and the top one this is the y axis here is the amplitude of vibration the x axis is the frequency and this is up to 3.2 kilohertz and the z axis is the rotational speed in rpm of the shaft. And the such a kind of plot is known as waterfall plot for example, this takes time to build up. So, there is one spectrum behind which is another spectrum another spectrum and so on and all these spectrums are stacked behind each other. And this looks like a waterfall plot for what happens you will see here there is it is very difficult to distinguish the frequency because the frequencies have got kind of smeared. And then rather if I looked at the order here this is the same time axis or the rpm axis this is the amplitude and this is the order. I see a peak in the third order in the sixth order and so on. And so this means that the vibration here is related to the rotational speed is what I can very clearly see this is the first spectrum behind it is the second spectrum third spectrum and so on. So, all these spectrums are stacked. So, I can see all these frequencies which you see in this plot here they are related to the rotational speed. Unfortunately in this experiment because this experiment was this setup is designed. So, that none of the natural frequencies are there in the operating zone I do not see a fixed frequency sticking up here. But you will see something very interesting happening once we go to the water to the unbalanced case. If you look at this disc yellow disc in the middle this yellow disc in the middle there are many holes ok radially placed. And if I put one if I physically put one bolt and this is made to rotate. So, I will get an unbalanced force m omega square r where m is the mass of this unbalanced. So, this unbalanced force is going to give me force every rotation. So, here you look on the third rotation and sixth rotation and so on and some higher rotations I am getting a lot of vibration levels and that could be something related to the other phenomena in the rig. But if I go to the another plot of this and this is nothing but the same thing as on a contour plot where you can see the because you know that was a little skewed. And you see this is the color is the amplitude band and that is the amplitude and this is the frequency and this is the order. You see looking at here particularly in the third order you can see this red strip the amplitudes of vibration are high. This is related to the third rotational speed. Now what we did was we introduce unbalance onto the system. Once we introduce unbalance into the system and what we did was in all these cases even in the previous cases what we did was while this data was been collected we physically move these rotated the potentiometer. So, that the speed went from 0 hertz to 4000 rpm or 0 rpm to 4000 rpm while the data was been collected. So, we have sweep the entire frequency spectrum and this is the collection of such a frequency spectrum because had I just an simple FFT and all the frequencies will get represented and you will perhaps not see here, but there is a small because this is only until 160 hertz this lines are not straight and there are little bend. But if I sorry if I go to the case of the unbalance and there is frequency smearing, but the most important thing is in the very first order the frequencies have shown up the amplitudes are high in the first order because they are related to the first rotational 1 x components. So, this becomes very important to find out because in this frequency spectrum it is very difficult to know whether my 1 x is the most offending frequency or 2 x or 3 x, but if I do an order analysis you see things which are related to the rotational speed they show up. And then you see these frequencies now the same waterfall plot I mean just to. So, waterfall plot is a 3 dimensional plot this is your frequency this is your amplitude and this is your time. You all are familiar with the single frequency amplitude and that was nothing but the spectrum, but with time with time if you stack up all the spectrums right. And the frequency was constant if the rotational speed was constant what would happen in this line you will get a peak like this agreed. But if the rotational speed was changing you will get something like this if you stack up and if I just picked up the peaks and if you. So, this is if I this was just for a single spectrum if I average all of these together I would get a frequencies varying, but if I just stack them up in terms of the orders I will see the phenomena which are related to the rotational speed. So, waterfall plot is a 3 dimensional plot instead of frequency I could also be having orders. In orders in this example you see for the unbalanced condition there is lot of smearing occurring. If you look at this red curve here you cannot see whether this is because of a rotational speed or something else, but if you look at the orders here you will see distinct peaks coming out. Frequency smearing has occurred here with the top axis this is frequency and the top plot this is frequency in the bottom plot this is order. Here frequency smearing has occurred, but in this case nice distinct orders have come out and the first order is related to the unbalanced condition. So, waterfall plot is a 3 dimensional plot which gives us an instant recognition of the change in frequency in the spectrum and then if you look at the contour plot this is another way of representing that same 3 dimensional plot you see the first order is very strong here case of the unbalanced. And again if you cannot perhaps see this in the black curve here because we had done a ramp up operation there is a speeding up or the change in the frequency. Now, once you have talked about the waterfall plot there is another plot I will just explain you or rather tell you camp bell diagram. This is used in our rotodynamics where as a function of rpm the rpm are the with time frequency and rpm they will at different rpm's you will generate the kind of the amplitudes and wherever they kind of meet you can find out this is to find out I will suffice to say that to determine resonances resonance or critical speeds shafts need not have only one critical speed. Particularly when we design say for example, an electric motor say an electric motors or the gear boxes etcetera we run it is at 1440 rpm. As a designer I can very well design the shaft depending on the mass and stiffness you would have studied that in vibration so that the rotational or the critical speed of this shaft or the resonant frequency of the shaft is well beyond 1440 rpm. If you take it 1500, 1560 is about 30 hertz so I can design a shaft that its resonating frequency is beyond 30 hertz right no problem. But say for example, I have a gas turbine gas turbine and 100 megawatt gas turbine. So, it rotates at 20,000 or may be 30,000 rpm 30,000 rpm corresponds to 30,000 by 60 500 hertz. So, its operational speed is 500 hertz and it would be almost impossible to have a resonant frequency is only above 500 hertz a designer would have a tough time designing such a system. Whose resonating frequency is only beyond 500 hertz so that would require for a that would require a very very thin shaft and obviously, a 100 megawatt gas turbine cannot be so thin that its resonating frequency is beyond 500 hertz. Well as a designer why do you require it beyond 500 hertz because it should not excite or it should not get excited by the operational speeds. But that is not the case there will be maybe 3 or 4 frequencies resonating frequencies below 500 hertz. So, it suffices to say that there are shafts whose resonating frequencies are within the operating zone right. So, I have to find out what these resonating frequencies are and only during operations I can make sure that I do not excite this resonating frequencies. So, cambell diagram is used by designers by a maintenance engineers to diagnose and find out rotating speeds critical rotating speeds of large rotating machines or machines operating at high speed high speeds 500 hertz and so on. For electric motors where we are talking about single speeds we do not talk about cambell diagrams, but when there are flexible rotors of lot of rotational critical speeds we will talk about cambell diagrams. So, right now I will not focus more on the cambell diagrams, but let us see the importance of this key phasor. You will see in condition based maintenance where vibration is an important parameter to be measured. Rotational speed is the speed is also a very very important parameter which is to be measured because you just saw an example wherein many of the frequencies which we see in the frequency spectrum they are related to the rotational speed of the machine. So, it is, but important for us that we must measure the rotational speed. So, the key phasor technique is used as a very convenient means to measure the rotational speed wherein it just gives a voltage signal and this could be through inverse of this because inverse of this time period is the rotational speed. If the time period changes I do not care the rotational speed would be measured. The rotational speed could be changing, but there are again algorithms if I will get a pulse strain square pulse out of such certain voltage. The average spacing between these will be my rotational speed. So, there could be you know there could be a digital indicator of the rotational speed just by measuring it is nothing, but a frequency counter. Just by putting a frequency counter digital frequency counter I can get the rotational speed, but mind you this rotational speed you know if the digital display or the your sampling was very quick you will see these numbers jumping down and that happens in many of the machines which will go out to the plant and see that the numbers will be jumping around may be 1239, 1241, 1230 etcetera. These numbers jump around because this speed is fluctuating. Imagine while the speed is fluctuating if you are doing an average FFT you will have disastrous results. Results will have no meaning. So, that is the reason why we synchronize our acquisition process by just the triggering signal. So, order tracking or order analysis requires two very very important measurements. One is the rotational speed and other is the parameter. It could be the vibration. It could be noise etcetera also and then we will see anything which is related to the rotational speed will show up as you know 1 x, 2 x, 3 x and so on. One very important application of order analysis is the engine firing frequency. Let us take the case of an engine firing frequency. 4 cylinder, 4 stroke say this was running at 600 rpm idling at 600 rpm. The question to you is what is the engine's firing frequency? By firing frequency means at what frequency the combustion is occurring? Imagine for a 4 stroke engine means for every two revolutions I will have one combustion, is not it? Right? So, in 600 rpm that means 600 revolutions per minute or 600 by 60 hertz divide by 1 by 2 is the firing times 4 is the firing frequency because there are 4 cylinders. Now, if this was a 2 stroke engine this would have been 1 because for every rotation I was getting a firing. So, this is what is the engine firing frequency and that will come down to 10 to 20 hertz. I do not know if you have experienced this, but you know if you have sat on the jeep you do not see them in the modern SUVs or MPVs etcetera, but if you are talking about those old jeeps or the taxes which come from the railway station very ill kept taxes. If the driver was idling you will see you will kind of feel this rattling noise and that is actually at the engine firing frequency and if you measure that you will see this 20 hertz component and you will be perhaps being sometimes even see the steering column shaking because the steering column which is fixed to the firewall and next to it is the engine with the engine is running once you are idling the engine is running at 20 hertz. So, there will be instances where the steering columns natural frequency could be close to 20 hertz and there will be the steering column would be rattling and how many of you have observed that particularly in old jeeps and cars particularly in the case of diesel engines where the forces are high you will see that the steering column of the vehicle gets excited at a particular speed particularly in idle. What is the source of excitation here it is the engine firing frequency because engine by engine firing I mean the engines combustion cycle. There will be a certain built up of pressure and this will come down with time and this is like a burst every cycle one burst is happening and this burst is an excessive pressure wave. So, this pressure wave is giving the forcing function and this forcing function is coming at 20 hertz and then it may so happen that the steering column is having a resonant frequency. In fact, you know we are I was doing some work for a tractor company they had a tractor as soon as the tractor was switched on there was a lot of shake on the steering wheel and imagine if a driver is going to you know hold on to the steering wheel and the steering wheel was having lot of vibrations it would not be very convenient or comfortable for the driver to hold. So, we find out that steering column or the steering wheel in fact steering wheel actually it has a steel structure inside it steering wheel had a natural frequency of 29 hertz and somehow that engine firing frequency was very close to that 29 hertz during idling at a particular RPM. RPM was about 780 was the RPM engine idling RPM. In this case example we took at 600 RPM I vaguely remember it was around 750 RPM was the idling speed and the way to do this was you know we could not do anything with the engine engine was made by somebody else. So, engine had its idling frequency at about firing frequency at about 30 hertz. So, what we did is by changing the mass and stiffness of the steering wheel we could shift its frequency natural frequency. So, that henceforth once this engine was running it the steering wheel was no longer vibrating. So, this is how we can diagnose you know this was not in case of a diagnosis or mostly a case of a design, but by such techniques we can find out the natural frequencies and how to avoid them in design. I was telling the example of gas turbines we cannot possibly avoid them, but at least if we know these natural frequencies we will not operate at those natural frequencies, but for the case of a tractor I mean you can always idle the engine and still, but this idling frequency if it is close to the resonant frequency of the steering wheel it is going to either have large motions of the steering wheel. So, that is to be avoided. So, order analysis is one technique by which we can find out the frequencies which are related to the rotational speed of the machines. So, if in the spectrum say if some of the of course in this examples here some of this frequencies are not related to the rotational speed if they were related to the not related to the rotational speed they could be lying in between these. And perhaps if you go to the higher frequencies you will see that some of the frequencies do show up and they are beyond the operating zone of the rotational of the test ring. Thank you.