 Good morning, today we will discuss about Rotodynamics as you know in any machinery rotating shaft is a very critical component. So, in this lecture on Rotodynamics I will be just covering the basics of Rotodynamics from a condition monitoring point of view. In fact, Rotodynamics itself is a full-fledged 40 hours course and in this lecture of one hour we are just trying to discuss about the important aspects of Rotodynamics and how actually the problems of unbalance in a rotor misalignment and the effect of the support stiffnesses play a role in the dynamics of the rotating shaft system. And in practice we will be finding many rotor systems comprising of turbines, compressors, pumps, impellers, sets of impellers or fans with sets of blots and so on. So, Rotodynamics is very important that we understand the physics of such rotating machines and then how do we try to control the unbalance forces, control the Rotodynamic stability etcetera, calculate the critical speeds and so on. So, the basic objectives of Rotodynamics is you know predictorical speeds and so on. But, well let me explain to you what this what I mean by a rotor system. I have a long rotor which is essentially supported in some sort of a bearing. This is undergoing a speed of rotation omega, it is rotating at a certain speed and the span of the rotor. This bearings have certain stiffness say k 1, k 2 and there could be a heavy mass m on the rotor. So, if this rotor stiffnesses are constant and stiff from the bearing supports are rigid or soft. When we have rigid stiffness on the supports and the simple disc which is rotating this is basically a disc. We have the simple Jeff-Cott rotor. Many physics is understood by understanding the Jeff-Cott rotor where we have a rigid support stiffness and then we have a certain disc which is rotating. So, when they are rotating we will have what is known as the critical speed a certain speed at which the speed is equal to the natural frequency of the system and then we are going to have the condition of resonance. So, this is very very important in the rotor systems. I have shown here one disc there could be multiple discs. For example, in in in actual conditions think of an aero engine or aircraft engine which is essentially a gas turbine engine. So, we have a long system where in we have the first stage this may be a compressor set and then we have the high pressure turbine followed by another low pressure turbine. We have the air coming in inlet air may be some sort of a fan could be here fan and this compressor there will be there are lot of set of vanes and same is true in the case of turbines and then we have the exhaust which essentially gives the thrust to the engine for propulsion. Of course, in between we have the combustion system of course, I am not going to do go into details of the gas turbine, but essentially our domain of interest is this drawn here in red and then this portion here and this is this is the long shaft which are essentially these are all supported on may be bearings intermediate bearings. So, a typical gas turbine engine has this kind of a configuration where in on a long shaft I have a compressor followed by a set of couple of high pressure turbines low pressure turbines and each of these system rotates at a very very high rotational speed and typically of the order of 30,000, 40,000 rpm. So, you can imagine we have such a engineering complex system where in on a shaft I have sets of vanes or blades which are mounted on desks there could be multiple desks and they are rotating at a very very high rpm. Now, think of a scenario there could be many issues one is what if the rotating speed it has a speed which equals to the natural frequency of the system. Then there will be condition of resonance. So, condition of resonance occurs when this critical speed is equal to the natural frequency of the system. So, when a aircraft is rotating or sorry not the engine is rotating and it is powering the aircraft and there could be conditions of resonance and because of this what would what could happen is there could be large amplitudes and large motions and this shaft because of resonance if it is having if it is going to have large motions it means so happen if this was the casing and then this blade sets may touch the casing. So, there could be little rubs. So, this can lead to lot of things high amplitudes high amplitudes this could further lead to rubs between rotor system and the stator in this case it is the casing. So, lot of wear and tear will occur materials may fail components may fail and then this is one instances. So, by studying rotor dynamics if you can physically model such a system through techniques of the analytical formulation or through the techniques of our finite element method. We can try to estimate the critical speeds and avoid them during the operations. So, this is what is one of the one of the important objectives of studying rotor dynamics. The next is determine design modifications to change critical speeds. Now, we can change the do certain design modifications to change the critical speeds through these mathematical models. We can change the mass stiffness of the appropriate system and change the critical speeds you know omega n is equal to root over k by m. So, this we can play around as a designer we can play around with the values of k's and m's and change the critical speeds. We have in one case wherein there is lot of rotating systems many rotating systems subsystems in one unit that could be for example, a motor driving a gear box driving a set of blades. So, and they are all rotating the shafts are rotating motor driving a gear box driving a set of blowers. All of them have rotary natures and then they have the corresponding the rotary displacements etcetera. So, such subsystems can have the and then you have the natural frequencies like in the case of a linear system we had multiple degrees of freedoms and every degree of freedom has a corresponding natural frequency. When you have the torsional rotating systems every subsystem subsystem 1 subsystem 2 subsystem 3 each of the subsystems can also be having natural frequencies and these are basically torsional natural frequencies and if we can predict them and then know them beforehand we can as a designer or an operator ensure that we do not run the system at its natural frequencies. So, estimating the critical speeds doing design modifications and then estimating the natural frequencies of torsional vibration is very important in the study of aerodynamics. And the next would be in a rotating system if you think the case of an like the gas turbine we just talked about the small amount of unbalance in one of the vanes or the blades. And once they rotate at high speeds we will be having very very high unbalance forces and these forces are radial in nature and because of this unbalance forces we will be having forces at the bearing supports and sometimes these forces could be very very high and then we have to ensure how to reduce these forces otherwise the bearings do not come up the supports or the bearings unable to withstand such high forces. So, as a good designer we have to ensure that the unbalance mass rotates at its center of rotation or everything is centered around the rotation and not away. Suppose I have an unbalance mass m at a distance r I am always going to have an unbalance force m omega square r and this is going to be harmful to my bearings if I put the bearings here they will be harmful. So, this has to be calculate the we can do the balance correction and locate a location from the measure virus in data will when you talk about balancing in the few classes down the road we will see how we can balance such unbalance forces either in single plane or in multiple planes and give correcting unbalance masses. So, that the net effect is reduced for example, in this case if I have an unbalance mass at a distance r on this direction I can give another unbalance mass of the same quantity at distance r opposite you know if I if I draw this plane if I have an unbalance mass if I have unbalance mass here I can give a correction mass here I can reduce this forces the net forces. Then another problem is you know we can also predict the amplitudes of this vibration caused by such rotor unbalance. Now, I am having this two words synchronous. So, if when a system is rotating at a particular r p m we denoted say omega as 1 times x which is the rotational speed and any vibrations at this frequency is actually known as the synchronous vibration. So, any frequency less than omega is sub synchronous and any frequency less than omega is super synchronous. Later on you will see when this rotating shaft is rotating at a particular speed having disc there will be speeds at which there will be lot of dynamic instability and in fact, this is one of the serious problems in the limitations of rotor speeds. In particular if you if you plot it as omega may be the amplitude at certain speeds you know it will happen. These are the which in this case if you increase the speed there is a critical speed beyond which we should not operate or if we have to operate it how this can be controlled and the I will just tell you right now by controlling the damping at the supports we can control this dynamic instability in rotor systems and there are many ways to suppress this dynamic instability. This is essentially by designing or incorporating changing damping at supports and this has been possible by if you think of a by having what is known as a squeeze film damper in the case of journal bearings and sometimes in rolling element bearings which are essentially very high stiffness bearings almost rigid supports. In some way by having an SFD squeeze film dampers in the outer race some amount of damping control can be done of course, you know people have used magnetic bearings to control dynamic instability in rotors. And that is still right now in still in a research stage I would say or very few practical applications have come out of magnetic bearings to control the dynamic instability in small systems they have been successful but in large systems there are other issues with using magnetic bearings. But traditionally squeeze film dampers have been used in the case of journal bearings to suppress this dynamic instability and though we use rolling element bearings some amount of SFD particularly in aircraft engines this has been done to support the dynamic suppress the dynamic instability. So, to summarize the objectives of rotor dynamic analysis as I have listed here predict the critical speeds, determine the design modifications to change the critical speeds, predict natural frequencies of torsional vibration, calculate the balance correction masses and location from measured vibration data, predict amplitudes of synchronous vibration caused by rotor imbalance, predict threshold speeds and vibration frequencies for dynamic instability, determine design modifications to suppress dynamic instabilities. Now, there is a lot of things happen when we are rotating a shaft at its at a particular omega. So, there will be a whirling of the shaft and this is the bow of the shaft. First assumption is we have considered the shaft is flexible so that it can bow. So, this whirling amplitude may be if I denote it as u is the or objective is to reduce this whirling amplitude because what happens if it wills sometimes it may fall with the casing in a system about this is my casing. So, once it touches here these are the regions where rubs are going to occur and this is going to create a wear and tear, see force and then material may fail as I was telling you to reduce the synchronous whirl amplitudes I have to balance the rotor. So, that this unbalance mass is a minimum. So, the does not fly out or avoid rotating at that particular speed omega try to change the speed and if nothing is possible at this bearings we can use SFD so that the support forces on to the bearings is going to reduce. Now, if I think of a rotor system suppose let me just draw this the ratio of the dynamic force to the forces on the supports when the supports have rigid stiffness and this is u and now if I increase the damping where f infinite is the force at the support in the case of rigid supports example our rolling element bearings. So, if I have a shaft which is supported on rolling elements bearing and if this is going to have an amplitude u the f infinite any location will be half m omega square u and you will see for large machines this force is very high when the m is high or at high speeds the forces are very high in case of large rotating systems when you are talking about say steam turbine which is essentially used in say power plants if they were supported on rigid bearings like rolling element bearings ball and ball bearings etcetera the forces on the support would be very high and we have to and there is no way no mechanism no physics by which we can reduce this forces which are coming to the supports. There are another way by which we can reduce this forces and that is where which is known as the journal bearings, but I should just give you an example in journal bearings essentially what happens I will come to the journal bearing first suppose I have a shaft which is rotating and this is a filled with oil and because of the eccentricity between the center of the journal and the center of the shaft I will have a converging diverging section and this shaft is rotating like this and the journal is fixed because of this converging diverging section there will be a fluid pressure this hydrodynamic force which happens because of the converging section on the fluid viscosity the load of the shaft load which is coming mg can be supported by the bearing forces because this will give a lift force. So, when I rotate a shaft at an omega in a journal wherein it is filled with a viscous fluid because of the eccentricity E this fluid will develop try to build up a pressure and this pressure is going to act in the upward direction and support the weight which is coming at the support. So, in the journal bearings essentially the fluid viscosity comes into the play and we can introduce damping and thus reduce the force which is coming on to the bearing if I go back to this plot here this is because of the journal bearing and this is because of the rolling element bearing. So, I can reduce the loads coming to the supports by introducing damping which is only possible when a journal bearing. So, in any system like in the case of the steam turbines power plants if I have a large system or large shaft or long shaft carrying a lot of disc of the compressors turbines etcetera and if they were supported on rolling element bearings what would happen is lot of forces would come on to the supports. Instead if I use a journal bearing and then in journal bearing is because the fluid viscosity and the eccentricity of the shaft from the center of rotation I will build up generate a fluid pressure, pressure acting in the upward direction and then this is able to support the loads coming on to the supports and reduce the forces and this can be very easily controlled by varying the amount of fluid flow the viscosity etcetera this can be changed and then I can reduce the forces. So, invariably in many of the stationary plants you will see that the large plants which are on earth on ground rather is a better word to use we can have journal bearings which will try to reduce the forces coming on to them and on this journal bearings if I have a squeeze film dampers basically I have a system where in I am squeezing and having a vertical motion radially I can put a squeeze film dampers and basically and this is made to oscillate along the outer race and then this basically this I can introduce damping and if you have seen by introducing damping at the supports the load which is carried at the supports does reduce and this is so damping has a very very important role in roller dynamics as to it can reduce the support forces it can control instability. So, if I have to go back the effect of the support bearing stiffness bearing support flexibility can greatly reduce the dynamic load transmitted through the bearings like we just discussed by having the support flexibility properly selected I can reduce the dynamic load which is transmitted through the bearings bearing support damping increases the dynamic load transmitted through the bearings at high speeds as is also obvious from that plot bearing support stiffness may be necessary to keep the transmitted load within acceptable limits while traversing the vertical speed improperly chosen support parameters can produce dynamic loads in excess of the rigid support values in a imagine in the suppose the I have chosen certain parameters of damping such that the forces are very very high then we have to have a failure of the supports and this will not be accepted or allowed. So, how we can reduce that now what do I mean by rotor dynamic instability super synchronous vibrations due to soft alignment misalignment sub synchronous and super synchronous vibrations due to cyclic variations of parameters mainly caused by loose bearings or shaft rubs non synchronous rotor wheeling that becomes unstable. So, these are the conditions wherein we have to be careful to avoid such conditions. So, we need to obviously have a shaft which is perfectly aligned we need to should not have any loose components which will give rise to shaft rubs. So, these are the conditions which we can avoid. So, that the rotor dynamic instability does not happen and with this I will tell you there is one suppose I have this rpm curve and these are the first natural frequency second natural frequency. So, there will be some frequencies and these are basically known as the Campbell diagram. So, there are regions on which you know you would like to avoid operating at these zones wherein your runoff speed should be such that in your operating at a speed at which you are below this natural frequencies or operating at a speed you are in between these natural frequencies. So, this kind of rpm which is natural frequencies plots are to be generated for multiple rotor systems or rotor large rotor systems. So, that the conditions of resonances because of the critical speeds are avoided. So, with this basic understanding about the importance of support bearing stiffness and how support bearing stiffnesses contribute to the overall forces which are coming under the supports. When we design a turbo machinery these are some of the things which one needs to keep in mind. Avoid critical speeds if possible if you have to operate through the critical speeds you know traverse it very quickly. Like in suppose a gas turbine you know we have to rotate at 30,000 rpm once you go from start to 30,000 rpm I am sure there will be number of resonances you have to pass through. So, quickly you need to change over to the resonance quickly you need to move into the operating speed of 30,000 we cannot be moving or rotating constantly at the same natural frequency the. So, the ramp up or the speed up has to be very very high. Minimizing dynamic response at resonance if critical speeds must be traversed is just what we discussed we need to move up the speed increasing mechanism very quickly. Minimize vibration and dynamic loads transmitted to the machine structure throughout the operating speed range. Avoid turbine or compressor blade tip or seal rubs avoid rotor dynamic instability avoid torsional vibration resonance or torsional end stability of the drivetrain system. So, as you can imagine a turbo machinery is a very very complex mechanism other than the you know the fluid mechanics operations the fluid machines part of it wherein we have a certain energy coming in certain energy going out and then we get a thrust. But beyond that there is a lot of engineering into a designing a perfect turbo machinery as to its dynamics is concerned in terms of not rotating at its critical speed, how to avoid critical speeds, how to traverse up high speeds, how to avoid instability, how to avoid rotor rubs between the stator and the rotors. So, these are and how to avoid multiple resonances and torsional systems. So, these are the complex issues of turbo machinery we can talk of our aircraft engines, steam turbine even a large set of pumps. But these have to be taken into account you know this some of them are vertical you know in if you go to many of the power plants particularly the hydropower plants you will see lot of the vertical you know maybe the kaplan turbine etcetera. Vertical turbines the shafts are not horizontal, but vertical we have the effect of the gravity loads all together on the bearings you know how to select the bearings the thrust bearing the pivot bearings the pad bearings etcetera. Now, how do you take care of these issues? So, a turbo machinery from a dynamics point is a very very complex thing as a designer people do take care of it, but then our goal in this class is how do we maintain and monitor the health of such a large complex turbo machineries. So, just to recap now we have already discussed about the bearings you know. So, bearings essentially are the of the two types the rolling element bearing or the journal bearing fluid film bearing. And as you know in the rolling element bearing we have the outer race the inner race and the rolling elements which could be either a roller or a ball and then they are supported by what is known as a retainer or a cage which ensures that they know two rolling elements come in contact with each other. And basically they are used for rigid supports. So, in essence rolling elements are good where the rigid the forces in the supports are less, because when the forces are very very high because of the physical dimensions of the bearings they may not be able to support very very high loads as in the case of the turbines which we discussed about. But the other kind of bearing is the journal bearing or the fluid film bearing. So, basically in the journal I have a shaft and because of this film and because of this converging diverging section and because of the eccentricity of the shaft there is a pressure builder and this pressure is actually supporting the shaft. So, in no point when during rotation there is any rigid contact no rigid contact between the shaft and the journal only may be at rest this is in contact. So, every time you start or stop a machine the entire rotor system comes and sits on this. So, there is lot of wear and tear in the journal, wear and tear in the journal that is why this journals are a minute of soft materials soft bearing material like Babitz etcetera which can be withstand a high temperature as well and which should be able to. So, because of the physics of the journal bearing they can support large loads. So, basically all the steam turbines etcetera or the gas turbines which are land based we have journal bearings and basically this film lubrication is the hydrodynamic lubrication which is occurring. But in this journal bearings the pressure I was actually build up because of the eccentricity of the shaft and the journal and because and thus creating a converging diverging section. But if I have a source of externally pressurizing the fluid and applying it to the journal I could be supporting the large loads which are coming at the bearing supports and such are the hydrostatic bearings and of course, you know we are not going to go into details of the bearings. But as opposed to hydrodynamic bearings or the journal bearings there is never any contact between the shaft and the journal. So, very precision instruments or equipment have such hydrostatic bearings where in we apply externalized pressurized oil or lubrication into the journal and the shaft and then. But in our case we have the journal bearings or the rolling element bearings. So, in rolling element bearings if I in the outer race if I have some provision of adding a squeeze film damping arrangement where in I can change the damping arrangements which is there actually in the aircraft engines other than the rigid rotating element bearings they also have a squeeze film dampers on the periphery which is which will not rotate, but only oscillate and thus create this damping. And as you know because of the damping the forces on the bearing support comes down. But when you talk about the journal bearing we can have the cases where in the because of the fluid wedge conditions the oil is lifted off or the oil gives a pressure and the support shaft is picked up is held at a position. But then in rotor dynamics also we do lot of tests on the rotor dynamics systems to understand from a condition monitoring point of view or from a dynamic analysis point of view. These are some of the tests which are done on the rotor dynamic systems. One is the begin with static stiffness test or now what is known as the impedance measurements of the system. So, at the bearing support stiffness at one location I can give a force and measure the deflection by having a transducer this deflection could be measured by an LVDT or even you can put an accelerometer and then integrate it to get the deflection. So, just to know the static stiffness k delta I can use such an arrangement wherein we give in a force measure deflection and find out the static stiffness. Many a times to understand the cases of the resonances we can do a co-stop test wherein we increase the rpm at a particular rate or come down from high speed course down. Particularly in the case of the journal bearing a very important thing happens like in this journal bearing you would think here basically the forces here at the if you enlarge this one is at rest and other is moving at a speed of omega v by r so linear speed. So, there will be a point wherein the entire film in the middle will be having a speed of omega by 2. So, this fluid in the wedge is approximately theoretically moving at a frequency of 0.5 omega or the fluid is churning fluid is churning is the or whirling at a speed of 0.5 omega, but usually because of the frictional losses this frequency is around 0.42 to 0.48 of f where f is the rotational speed. So, any time I am having a shaft rotating at a frequency of f I will see a predominant frequency somewhere around 0.4 to 2.48 f. So, when we do a co-stop and co-stop analysis you will be very easily seeing the effect of whirling and this can be removed by changing the damping etcetera. In the laboratory you know we have a rotor rigs wherein at one end we have a journal bearing wherein we have a particular type of lubricant and then say for example, in oil and next time we have a kerosene you will see the effect of whirling pretty much there. Many times we do lot of constant speed measurements in rotor systems and of course, the last one is the resonance test which we do on the rotor systems to find out the natural frequencies of such systems so that we can avoid rotating at the natural frequencies. So, some of the common faults which occurs in turbo machinery are the cases of imbalance, misalignment, load variations, mechanical loosenesses, critical speed or resonance, excessive clearance in the fluid film bearings, rotor rub, oil whip and oil whirling. Another important thing happens as an oil whip for example, we are going at a rotating at a particular frequency, but this frequency happens to be an oil whirling has occurred. Oil whirling occurs at point 422.48 f is this oil whirling occurs what happens it will there will be some multiples at which this will be equal to the natural frequency of the system in such a case the and then this is going to whip around at that frequency if it is going to bow and then it will be latching on to this frequency and that is known as whipping frequency. This we will discuss when we talk about the case of unbalanced responses and rotor systems where this whipping occurs. So, to summarize in this class we discussed about the requirements on what we do in rotor dynamic analysis how basically the bearing support stiffness and in particular the damping plays a role in the support forces which come to the supports and then how we can select the kind of bearing the rolling element bearings or the hydrodynamic bearings to take care of the loads and then what are the effects which could be there in rotor dynamic systems and then how we avoid. Thank you.