 Today, we are going to discuss about detection of cracks in shafts, because you know predominantly you would have seen if cracks occur materials and structures fail, but we need to monitor the cracks. In fact, to monitor the system so that the cracks can be detected at early stage because invariably you know cracks can occur in any system be it a shaft be it a plate cracks can occur because of many reasons. Cracks are sometimes visible if they are on the surface and they are if they are invisible they are inside the body. Particularly there are methods you know once there are cracks are occurring in a system what happens this is a shaft which is rotating. Obviously, its strength is going to weaken strength is going to decrease and eventually this crack goes undetected the component of the material is going to fail and then we will have catastrophic consequences because of such and failure. Imagine a case of a railway axle where we have this wheels going on this axles and suppose this cracks occur and then the systems fail then we are going to have a problem because there will be derailment accidents etcetera. So, cracks have to be detected at an early stage. Now, if you will notice here this crack a very generic crack is going to weaken the structure its strength is going to decrease and once the shafts are rotating systems are rotating this becomes still a very complicated issues things are rotating and if say imagine if there are weights being carried on the shaft because they are rotating this cracks are subjected to varying loads there will be fatigue loads on the crack and a material subjected to fatigue load loads is going to fail much quickly compared to a static load. Imagine if I have this as a shaft and this is rotating and there is a small crack and every time this rotating this crack is going to open close open close and then eventually because of fatigue loading it is going to fail much quickly. This is something which we need not we should not do. This is one type of cracks and we in this class we are going to focus on such cracks on shafts which are visible from the outside and how we can reduce them. But cracks do occur in structures like I was telling you inside the structure outside the structure if it is inside the structures it is very difficult to visually see it. So, people do what is known as the ultrasonic inspection and we are going to study about this later on ultrasonic inspection or what is known as the NDT type of tests. But one thing you will notice here if a crack has occurred the machine or the local area around the shaft is going to be not that stiff it is going to have stiffness is going to reduce. Stiffness is going to reduce. Reduce is a pretty strong word I would say stiffness is going to change if a crack has occurred in a system. And you will recall the natural frequency of any system is root over k by m. So, if there is a change in the stiffness obviously there is a change in the natural frequency. So, with this you will see there are methods by doing the Eigen value analysis of a system we can find out the change in natural frequency and then say may be a crack has occurred. The systems transfer function is going to change if you all recall the mobility of a system is defined by force by velocity is not it. And then we have the impedance is inverse of mobility is not it ok. So, by giving an excitation to the system some force by measuring some velocity I can find out the systems impedance ok. The systems impedance in fact this is impedance yeah impedance is force by velocity and impedance is inverse of mobility. So, I by changing giving an excitation to a system I will measure its response. If a crack has occurred this mobility is going to change that is also another way of detecting a crack of a system has undergone or not. Before we go into the discussions on the methods to find out cracks let me tell you what are the two types of cracks which can occurred in a shaft ok. So, in this class as I was telling we are going to focus only on shafts which have cracks and cracks which occur inside structures will be dealt when we discuss about ultrasonics. So, a load on a shaft could be because of bending loads. So, after supported on bearings these are bearings there could be a crack and there could be some bending load because of a weight and then we are going to have this kind of a crack which is usually a transverse almost in this line these are transverse crack and if I exaggerate it in the drawing. So, each type the crack is the shaft is rotating at once it comes to the bottom because of the loads it is going to close. So, this is a load here strong load here it is going to open and close once it goes to the top this crack is going to close and again it comes down it will sorry once it comes down it will close goes up it will open. So, this opening and closing of cracks occur in a rotating shaft and this is known as the crack breathing. Now, to our naked eyes at times this will be not visible at all and another parameter by which the cracks are defined is actually if this is the diameter of the shaft as D and the crack depth is A is a non dimensionalized parameter A by D crack depth diameter and usually of course, you know A by D is 0.5 it is almost till the diameter I mean no system would go till even 0.5 because the structure would have become so weak it would have failed usually 0.1 about 10 percent of the D A is equal to it is a limit I mean beyond which we cannot have any structure sustain loads beyond crack depth of 10 percent more than the shaft diameter. So, we have to keep a track on this is the parameter by which this crack is defined and this is usually negligible it is like an hairline you know if you are talking about a crack of a shaft of about 12 mm diameter even this dimension could be about 0.5 mm very thin hairline crack in our laboratory I will show you we have done experiments wherein we use a hacksaw to cut a slot and make a crack also and that is a massive crack I mean having a hacksaw cutting a slot and a shaft. But usually this cracks are hairline cracks this is a type of crack and which is known as the transverse crack because of the flexural loading on the shaft. Another important thing happens actually as you know the shafts usually carry torsional loads they transmit power. So, they carry lot of torque and usually because of the torque you know the maximum shear stress occurs in the 45 degree lines if you are transmitting torque usually the cracks are at this 45 degree axis and somewhere here and these are the slant cracks. And if you see cracks oriented on a shaft in these lines you can know this cracks are because of the torsional loading on the shaft and not because of the flexural loading. Flexural loading as I just told you they will be transfer signature. So, essentially in shafts either a transverse crack occurs a slant crack occurs or a combination of them both of them occur because of the type of loading which is there on the shaft which is easily a flexural loading because of the bending forces or the torsional loading because of the torsional moment which the shafts are carrying. And so what are the symptoms or science by which we can detect whether a crack has occurred to vibration monitoring. Number one is the stiffness will change. So, we can monitor the Eigen values or the natural frequency. Number two the mobility inverse of the impedance mobility will change. So, we have to do through impedance measurements. Of course, once we are discussing through vibration monitoring there are techniques beyond vibration monitoring by which cracks are detected and these are usually stationary cracks on structures. One is through the NDT techniques of dye penetrant test particularly to find out you know hairline cracks on castings particularly given examples. So, for example, we cast an engine cylinder block because there are. So, this is some view section of a cylinder block and a crack has occurred a very thin hairline crack. Now, if you see a cylinder block there are many things which happens. There are in a cylinder block there are lot of water passages which are made in the casting itself. So, that the cooling water goes all around the cylinder block. Now, they are under pressure. So, if there is a crack what is going to happen this water is going to ooze out. So, how do you detect such kind of cracks? So, that is why they do what is known as a dye penetrant test after the cracks are after the castings are done they smear it with a dye and then rub it off. And if the crack is there this dye after you have smeared the surface the dye and this dye will make a mark ooze out and clearly show that crack has occurred just looking at the dye. And this dye could be not visible in the naked eye it could have a fluorescent lamp to where the dye is going to glow and then you can clearly see crack has occurred that is one way which we will discuss later on in the ultrasonic in the NDT test section. And another crack detection technique which they do for shafts which are rotating particularly imagine in railway bogies or wagons rolling mills, paper mills, cement plants. Here we have a axle of the train which is rotating if the crack is inside it will have consequence that the structure is going to weaken it will fail. Rolling mills lot of rolls you know are there which run continuously if a roll breaks the steel plant the cold rolling mill or the hot rolling mill cannot produce. Paper mills there are about 100 200 mills rolling rolls the rolls crack the paper mill is going to get shut down. Cement mill the rotary kiln like a big shell about 2 3 2 3 meters in diameter which rotate same there is a crack eventually the system is going to weaken and fail. So, these are actually tested by an NDT technique of ultrasonics. We will discuss about the dye penetrant and the ultrasonics in the later class, but again in this class I wanted to focus on the dynamics of cracks in a shaft and how do we monitor it. So, we know that we can measure the mechanical impedance of such a system. We can measure the natural frequencies of a system because if cracks occur the stiffness is going to change and so the natural frequency is going to change and then let us see how we can do that. This is a setup here which we have in the laboratory and if you can see here we have a crack and if you will see there are two disc of aluminum two flanges basically and these flanges are held together by four bolts and eventually if you will see here this is actually a crack shaft is totally crack and to simulate cracks what to do is tighten these four bolts. Luson may be the top two or any two or any one so that at one end of the crack may be if I draw it here see here we have put two flanges there are actually two crack shafts already and then if you bring them together and then put bolts we join them kind of and then if you loosen this and tighten this cracks may be this shafts may be could be like this. To stimulate crack we do this in the laboratory and what we do here is again if when we have a crack shaft we always measure at the bearing locations as I told you beforehand in condition monitoring easily when you monitor rotating machines this is the only good place we have to keep the transducers and here also we have to keep the transducers on these two bearing locations and this black disc here is actually a loader the heavy weight which is rotating on along the side this cracks so that this is responsible for giving that load bending load which is coming down and which will be responsible for opening and closing of the crack and create this breathing mode. So, we have got the vibration measurements from this system and I will show it to you later on and then we will another set of which we do is to find out the mechanical impedance like I told you the techniques for detecting shaft cracks or mechanical impedance and then the Eigen value analysis. This is an experiment which we did in the laboratory and this is a shaft wherein and if you will see this is about a 12 mm shaft and which we where we cut a saw tooth crack onto the shaft by just by using a hacksaw. Now, because of this we rotated at a particular speed and then we tried to see what is the change in the mobility actually these two graphs summarize the results. Let me explain you what we did here what we did essentially is we measured the mobility. Now, to do that what we did is we have this system wherein you see this shaft where is a heavy disc and this particular heavy disc here and then where we introduced a crack very close to the disc. So, this disc is responsible for opening and closing the shaft and we tried to measure the impedance of the system at any one bearing locations. So, what we had is an exciter you see this black one here is actually an exciter and this exciter we are giving a force in signal. So, this exciter is exciting the structure and here what we have is an impedance head impedance head is a transistor I told you which has a force gauge and accelerometer together. So, that I will measure the acceleration of this system and measure the force which is being applied this acceleration will be double single integrated to get the velocity. So, I know the velocity I know the force to the system and then thus I know the mobility. So, mobility of the system is known for uncracked system when we did two measurements with a good system wherein the crack the shaft had no crack and next we replace that shaft with a shaft which had crack. So, we have these two ratios you know the response to M crack with M original mobility with crack system to mobility with the uncracked system this two ratios and then you will see here this is at this x axis is the crack depth alpha which I told as A by D you know we have gone as high as 0.4 pretty high A by D is 0.4 and we have run the system at two different frequencies one was 22.75 hertz and one is at 40 hertz and you will see how with crack depth suddenly this mobility changes and this is a good indication that by monitoring the mobility of a system I can tell that the system has undergone a crack. Just by monitoring vibration I would not be able to tell whether a shaft's crack has occurred and this is actually published in one of our journal papers. So, to summarize you know this for different various crack depth and different mobility mobility is very sensitive at and these are at the rotational speeds 22.75 40 hertz 22.75 at 120 hertz and so on. At different speeds we have seen the mobility. So, in a system if we want to say for example let us talk about a gas turbine. So, if you want to place a system in a gas turbine to monitor whether cracks have occurred or not we can do this. Gas turbine has long long rotors with sets of compressor blades low pass low pressure stage high pressure stage combustion the turbine etcetera all these are sets of vanes compressor turbine etcetera. So, what we have to do is do through a modal testing through experimental modal analysis. Well what is experimental modal analysis basically we found out the transfer function of a system response to the excitation in our case mobility is nothing but velocity by force and dependence is force by velocity. So, to a unknown system where we have to do crack monitoring all we have to do is give a certain known force at the frequency of operation suppose a gas turbine rotates predominantly at say may be may be 24000 rpm to may be 30000 rpm and this will correspond to 24000 by 60 400 to 500 hertz. So, from a range of 400 to 500 hertz or few of its multiple you excite this system through a force. How do you excite the system through a exciter usually because it is a large gas turbine we can use heavy electro dynamic exciters I give a known force F from the frequency may be 400 to 2000 hertz and the same time at the bearing locations obviously, but in the turbines because they are supported with many bearings at any of the bearing locations you can measure the force and then with the suitable transducer you can mount an accelerometer etcetera measure the velocity measure the force and plot a mobility curve V by F as a function of frequency and then now all you have to do is continuously keep on monitoring V and F by continuously I mean you know you need not do the force give the force excitation every time, but momentarily now every day once you measure V by F every day once you measure V by F suddenly you will see this V by F if you do the ratio of the so this is my original and which I say as m naught and then you will have m nu m nu is V nu by F nu. So, you will plot m naught by as a function of frequency and you will see that with a crack occurring suddenly you know they should have short off like you go back to the slide here I am plotting m nu to m naught as a function of crack depth, but at particular frequency I can monitor them and this frequencies are the operating speeds and I will see that with particular crack depth it is increasing. If the crack depth has increased this value is going to increase because you see this is crack depth if crack has increased this is going to increase. So, at a particular frequency I can I can decide on the frequencies, frequencies usually will be the operating frequencies operating frequencies and this this is out of our research which we did about you know 10 years ago and just measuring the mechanical mobility of system to monitor cracks occurring in a system. You can get a reference to this paper at my website this has come out in JASA in year 2002 volume 112 number 6 pages to 2825 to 2830 the details about this technique are given in that paper of ours wherein just by monitoring the mobility of a system we can tell whether a crack has occurred in a system or not. Now, let us so you know in a nutshell to monitor the occurrence of cracks in large systems you have to measure the mobility of such a system periodically and if the mobility value increases you will get an indication that the crack has developed or crack is growing and once happens you know crack also takes time to grow and that is what is saving us if the cracks occur suddenly without no reasons and they did not grow and they just fail it will be disastrous. So, crack takes time to grow and this helps us in the prognostics you have the you know the famous Perris equation for the crack growth rate that depends on the material property the crack dimensions etcetera. So, it takes time to grow and eventually crack will grow to a such an extent the material would have become weak and it would fail, but through a mobility measurement initially when the crack has occurred and it has just grown to few a's you know point where a I define as the crack depth by d the crack has increased we will see that the mobility value suddenly shoots up. So, mobility is sensitive to crack other than just monitoring the velocity of vibration. By velocity of vibration it would be very difficult for us to monitor whether crack has occurred, but by doing the mobility measurements we can surely say the crack has occurred. This is what actually happens is another view the better drawing of that is this what I meant by this crack occurring at a distance crack can have a maximum distance of l at a location next from one of the bearings. So, is a rotor here the shaft of diameter d in a is the crack depth which you defined as and because of these there will be forces coming in the bearings the stiffnesses are going to change displacements are going to happen and so on. And when the crack shaft rotates this is how the stiffness varies this is what is the breathing model for the transverse crack in a complete rotation this is how the shaft undergoes. The flex the stiffness changes the flexibility changes because the cracks are opening closing opening closing at any location the flexibility which I measure at the bearing locations are going to change the stiffness is going to vary. Now, let us see what happens in the case of the breathing behavior for a slant crack here I am subjected to them to a torsional loading and the cracks have occurred in the 45 degree axis. So, cracks will once open and once close this is also the breathing behavior of a crack, but it is slant, but then the question would be how do I distinguish between whether a crack has occurred because of a torsional load or because of a bending load because once you have to do the diagnostics you will see eventually a crack has occurred in the 45 degree axis. But if you are measuring the vibrations you would not know perhaps you know whether it is breathing mode because of slant crack or a transverse crack and how do you because I need to know the cause of the crack was it a transverse load which created the crack was it torsional load or another fundamental question which you would be wondering is a designer would have taken care of the loads. So, why crack is occurring to answer that question there are many ways these materials which are used to make the case of a shaft. Say for example, let me take the case of a pulley by pulley I mean a pulley of a conveyor system by let me it is not the conventional pulley which we use for power transmission and in our you know may be small motors or pumps, but I am talking about conveyors where and these are actually supported on bearings and this length could be about 2 meters just give you feel of the dimensions and this diameter could be about 0.5 meter and this thickness of this it is a hollow drum and then there are some end plates here on to which we put a shaft and this thickness is about 20 mm. Now, the shaft is solid shaft is solid solid shaft on to which we have the end plates this is the shell usually on the shell they usually put a rubber material to protect the shell this is known as the lagging rubber material and in a conveyor there could be you know series of such pulleys and there would be this conveyor which is going on this belts and this could be a big ship here carrying may be coal and then there is a crane here at the port which will scoop and then drop it at a bend and then this is getting transmitted transferred and this comes to the place where there will be railway wagons carrying the coal and out and this could be as long as may be 1 kilometer and each of these you know these are there may be you know 5 meters apart or 2 meters 5 meters apart we have put this conveyor pulleys and they could be carrying coal at the rate of may be 5000 tons an hour. This is the typical scenario of if you go to any port either the loading circuit or the unloading circuit either in our country in all of the ports we do unloading on the loading of the iron ore we send out the good iron ore and get the good coal from outside because we have not good coals in our country or we get the coals in and the iron ore goes out in the conveyor systems and in such a system if a crack occurs in a pulley of the conveyor the conveyor is not going to work. Now question is a designer would have done a good design to ensure that the cracks or this conveyor system does not fail but there has been instances I have seen some of the pulleys. The material of the pulley is supposed to be isotropic steel of a particular blade usually it is low carbon steel with particular you know this pulleys are made out of plates these plates are produced by hot rolling and then they are welded at one location they are welded at one location and imagine welding a 20 mm plate there will be lot of heat concentration at one zone around the weld and that is known as the H S Z heat affected zone. So usually after welding you have to do good amount of post weld heat treatment post weld heat treatment otherwise what happens you look at the microstructure of such systems such plates around this you will see that the grain sizes are not uniform and there could be banding in it and this could be you know ferrites and this could be perlites and this banding give rise to alternate hard and soft areas around the weld. For a designer when I design it I when I take a value of stress I assume that the stress is uniform everywhere hardness is uniform everywhere that is fine but when I manufacture it when I have done welding my heat treatment was not proper I have artificially created high zones of soft material high zones of hard material and you are rotating a soft and hard material together ok one zone is soft one zone is hard and this is how because of one inherent defect this crack is going to originate and then if one crack goes unnoticed it is weakening the structure few more cracks will come up and then once cracks start to propagate with time because you are rotating it cracks will propagate and eventually system will fail and this is what actually happens and that is why you will see all those you know bridges we see in our country a harder bridge the new bridge why do they never weld those structures do you know why they do not weld because of these problems that is why they rivet they make a hole and rivet and bolt because heat treatment cannot be done at such a larger scale institute in a large place where you are going to weld things ok. So, post weld heat treatment is very very essential to prevent failures in mechanical systems many mechanical structures systems are failed because of crack because of fatigue only because the post weld heat treatment was not done correctly to ensure that I have uniform grain size. So, this otherwise the material is going to have hard zone soft zones the temperature time cooling curve after welding is very very important if you have studied iron carbon diagram you see how this the carbon compound if you quickly quickly cool it will have hard material there will be lot of martensite formation where to slowly cool it they have to be a lot of pearlite formation. So, heat treatment becomes essential, but as a maintenance engineer we need not worry about it we assume that the guys who have designed it who have manufactured it would have taken care of these, but when we go to site a system has failed they will all point to the bearing because that is why we have where we have put the transducer they will say because the bearing has failed the whole system has failed no that is not true because the person who had manufactured it did not do a good weld job perhaps did not do a good post weld heat treatment after the welding. So, such variations in the microstructure occurred such hard variations in hardness occurred and thus the crack initiated ok. This is a one case study you all should remember in particular and usually I should tell you usually when you are doing and welding actually when you weld you know you have to go to the melting point temperature usually for steel for plate is 20 mm thick immediately after the welding you have to hold at that weld temperature that material which have been fused for about 1 hour that means you should not cool it suddenly if you quench it in a cold bath there will be there will be brittle surface on the top it will harden and then it will crack ok. So, we have to always hold it many times the fabricators at site do not do this because to hold it at the temperature you have to insulate it usually and asbestos cloth is put over the weld and even before you weld it suddenly you cannot bring for a room material from a room temperature and suddenly start to heat it they are slowly heated up to bring to a temperature and then the welding is done held for 1 hour at that temperature if it is 20 mm thick and then slowly it is cooled and that is usually done by covering the weld with an asbestos cloth and if you do these things you can avoid cracks in systems and shafts ok. Another reason of course cracks occur because of casting defects blow holes, blow holes in castings which go undetected poor casting and then of course these blow holes etcetera can be checked by alder sonic attesting ok. Now, is another thing why the last few slides before this we had shown you about mobility here we will show you about the change in Eigen frequencies ok or the natural frequencies once the crack occurs this again the normalized Eigen frequencies the first three Eigen frequencies with crack depth increasing you will see at some particular frequencies they are very very sensitive to the change in the Eigen frequencies. So, Eigen frequencies mobility are two indicators that a cracking has happened if you just monitor one Eigen frequency you may not be able to see you have to normalize them with respect to the original natural frequency how much is the variation with respect to the original mobility how much is the variation just by doing an absolute mobility measurement or an Eigen value measurement you will not be able to know what the natural what the change has occurred because of cracking, but if you take the ratios this clearly show that this kind of changes do occur ok. Now, lot of signal processing techniques have been used to detect track all also particularly though in this class we are not going to discuss about the details of non stationary signals I will just tell you briefly particularly when you start up a shifting system or which is known as a coast up or coast down and if you do something what is known as an wavelet analysis and if I have a shaft some velocity presence of sub harmonic lobes indicates correct cracks and this can only be done when we excuse me when we coast up or coast down a system by coasting up I mean changing the speed and ramping up the speed from 0 to of some value when you are ramping up because the signals are not at constant speed this is a non stationary signal then we can see this kind of lobes happening once you do an wavelet analysis though in this course we are not going to discuss about wavelet analysis. So, through advanced signal processing technique we can also detect cracks earlier you know people were never monitoring cracks through such vibration monitoring they only did an NDT test either through ultrasonics or dipenetrant to locate cracks, but now a days with the improvement and the understanding of the science and then these algorithms have come up people can find out the occurrence of such cracks in such systems. Now, to do the impedance analysis of the cracks in the laboratory these are some of the data which we use the length of the shaft was about 50 centimeter diameter was 2 centimeter and the density etcetera and then the first Eigen value was 24.27 hertz a rotor was rotating at 9.55 hertz with this kind of a simulation we did the analysis and this last table here gives an indication of how do you compare comparison of the transfer crack with slant crack. So, when there is a breathing opening and closing of crack cracks breathe with a frequency equal to rotation of shaft speed in the transfer crack and in the case of a slant crack excuse me the breathe with the frequency equal to the torsional frequency. Here the in the transfer crack the Eigen values reduce significantly compared to that in the case of the slant crack. In the steady state response the FFT shows the characteristics 1 x 2 x 3 x, but in the slant crack they show the subharmonic and superharmonic of the rotational speed by sub means less than the rotational speed super means more than the rotational speed and transient analysis we can use the wavelets to detect transient crack like I told you and then the in the case of the response to impulse when I give an impulse force the normalized mechanical impedance is highly sensitive to crack depth in the case of the transfer shaft and in the case of the slant crack it is relatively less. So, to summarize what we discussed today cracks can occur in structures cracks can occur in rotating shafts to monitor the cracks in rotating systems we can use the method of Eigen value analysis or the mobility analysis to find out whether cracks have occurred to find out cracks in structure we can do NDT testing of structures by ultrasonics or dipenetrant and this cracks rotating cracks sorry cracks in rotating shafts could be either transfers or slant and the transfer cracks are created because of the flexural or the bending loads and the slant cracks are created because of the torsional loads and you have seen the relative sensitivity of the one crack to the other towards mobility towards Eigen values and which could be used as indicators for detecting cracks in rotating shafts and of course, cracks do take time to grow and that is the breathing room we have till we take remedial measurements and cracks why they occur because of defects in casting defects in welding post weld heel treatment was not appropriate the raw material during rolling was not appropriate hot rolling was not done properly and so on. Thank you.