 In this class, we will be discussing about the basics of machinery vibration. As you recall in the few first few classes, I had told to do CBM or condition based maintenance, we need to monitor the vibration signals from machinery. And once we monitor the signal in the sense, acquire the signal, analyze the signal and then we could say something about the machine's condition. So, it is very important that we understand the basics of machinery vibration. In this class, I will be introducing you to few basic terminologies, concepts of vibration. Though as you know, machinery vibration itself is a full fledged 40 hours course and I am trying to squeeze in all that information in about 1 to 2 hours. And I would also refer you to good standard textbooks on machinery vibration to know more in details about the vibration. However, in this course, I will be or rather in this lecture, I will be focusing mostly on the important aspects of machinery vibration which will be used for condition monitoring. To begin with, you know, how do you define vibration? Well, vibration is an oscillation about a mean position that you would have recalled in your high school physics and that is still the definition which we will use. It is nothing but a motion may be of a pendulum about its mean position. This could constitute what I mean as vibration is the mean position and there are two extremes of this pendulum of certain length l and mass m. So, motion about a mean position or I will still say it is an oscillatory motion about a mean position. So, this is what I mean by vibration. Well, this motion whether motion can have direction. So, if I take a body say of mass m in space, this body can have six independent directions of motion. So, in a three dimensional plane, if there could be rotations over them, so we have six translations and sorry three translations and three rotations altogether, six independent directions of motion and such independent directions are known as the degrees of freedom of this body. Now, imagine if this body is constrained in five directions and only allowed to move in a particular direction say x t, I may consider this body to be a body having a single degree of freedom or S, D, O, F. So, if we look into this diagram over here, this body has a mass m and though I have shown the direction x coming down, this body could very well be supported on a spring having stiffness k and a damper having a coefficient of damping as c and suppose it has a motion in this coming down and I give a force given by F 0 sin omega t. So, if I was to write the equation of motion of this body, I could very well write it as m x double dot plus c x dot plus k x is equal to F t. In this case, it happens to be sin omega t. So, this is the equation of motion of this body in this one degree of freedom or single degree of freedom. Now, vibration is nothing but essentially these responses which I have written as x, x double dot or x dot and x double dot. So, if x is the displacement, then x dot is the velocity, x double dot is the acceleration. So, what is vibration? Well, vibration is nothing but this motion represented either as displacement or velocity or acceleration. If I assume that x is of the form, small x is of the form x sin omega t, I can very well find out the displacement or the velocity as and the acceleration as. So, the amplitude of displacement is x, the amplitude of velocity is omega x and amplitude of acceleration is omega square x. So, these three terms x omega x and omega square x as you will see, they are the velocity and acceleration that dependant on the frequency of motion and vibration and vibration means representing this motion either in the form of displacement, velocity or acceleration. Now, you will see certain interesting relationships here in the sense, why where are we studying this vibration and how does it help us in CBM. Now, if you take a machinery, these forces could be many which I am writing as you know summation of say phi sin omega i t, there could be many such forces which could be because of defects, defects of the form of misalignment in the system, unbalance, cracks etcetera. So, you will see later on that these defects which happen in a machinery give rise to certain forces and these forces are actually responsible for generating the response either in the form of x, x dot or x double dot. So, we try to understand about the defects in this machinery based on our measurements of the vibration response and this constitute what is the signal from the machine. So, we have transducers or sensors which will measure this and by trying to analyze these signals, we will get a clue as to what could possibly be the forces or we would have had knowledge beforehand. If the responses are of the form measured as x, x double dot or x dot, we know that these responses are created by faults which are characteristic like misalignment, unbalance, cracks and so on. So, you want such knowledge basis available or signature is known to us, we can then very well find out default and diagnose the fault in a machinery. So, if from in that sense, the study of vibration is very important for us. The question is if I go back to the model which I was describing, a single degree of freedom mass having a spring stiffness k and a damping coefficient c, it has certain response because of certain force have 0 sin omega t. Now, to the equation of motion, I have told that x sin omega t is a solution and this k is known as the stiffness coefficient and its units is usually Newton per meter and this is the damping coefficient c and its unit is Newton meters per second or Newton seconds per meter. So, it is force by velocity and now force here the unit is Newton. So, the way this equation of motions are derived is there is a lump mass given as m, if I have a forcing function f naught sin omega t, I will have a spring force which is nothing but as a linear spring force which is nothing but proportional to the displacement and a viscous damping force proportional to the velocity. So, from the Newton's second law using the equation, we will come up with this equation of motion m x double dot plus c x dot plus k x is equal to f naught sin omega t. Here also I should, so this is the equation of motion for a single degree of freedom system. Here I have described x as the linear motion where the body is only having translational motion, but mostly in the machines you will see that we have shafts which are supported on bearings and this shafts undergo certain rotations because of an applied torque or a defective torque. So, how do I write the equation of motion for such a torsional system? So, this will be nothing but the mass moment of inertia times the angular acceleration plus the damping coefficient in the rotational domain plus the torsional stiffness is equal to the applied torque which I am writing as t as a function of t capital T is the torque. So, this is the equation of motion in a rotational we can call it as torsional vibration as theta, theta dot, theta double dot and in the previous example I had described about linear or translational vibration wherein we could measure x, x dot and x double dot, but traditionally we try to understand this rotational torsional vibration by measuring x, x double dot, x dot and x double dot because of the simple reason that earlier days such torsional vibration measurements directly measure theta or theta dot or theta double dot where not easy. But as you all know the phenomena is actually a torsional vibration if there is a defect in a softening system or a machinery carrying a pulley supported on bearings if there is a defect occurring in this machine in the shaft it is actually a torsional motion and this torsional motion could be very easily understood if there was system to measure this torsional vibrations. Instead what we did is we put linear sensors or accelerometers to measure the vibration only at the stationary bearing locations. So, in effect to understand or to measure theta and theta dot and theta or double dot we use to measure x, x dot and x double dot. However, lately with the use of laser torsional vibrometers with use of optical encoders it has been very easy to measure theta, theta dot, theta double dot and then diagnose default. And this is a very important and upcoming area in CBM as to measure torsional vibrations directly by using either laser based systems or even optical encoders. And we had had a Kharagpur particularly in our lab we have been very successful in using optical encoders for directly measuring torsional vibrations of machinery systems. And even we are able to diagnose faults in gear boxes where there could be defects in the gears there could be defects in the misalignment in the shaft and so on. And this is to say that you know we not to use x or x dot or x double dot, but these were measured by accelerometers which are essentially contact type and have to be mounted to the casing wherein the vibration is occurring unlike the laser torsional vibrometer or the optical encoders which are non contact in nature and they can be very easily used and adapted to measure torsional vibrations. Another important relationships which I had told that you know people always ask in fact this question I get asked every time I teach a short course is what parameter of vibration should we measure? Is it displacement? Is it velocity? Is it acceleration? Well to answer this question if I look at x it is capital X the magnitude of the displacement magnitude of the velocity is omega x and magnitude of acceleration is omega square x where omega is the rotational frequency in radiance per second where omega is equal to 0. 2 pi f, f is the frequency given either in cycles per second or in hertz which stands for but you will see at high frequencies because of this omega square term there is a large quantity at high frequencies. So, it is always advisable to measure any high frequency vibration as acceleration and low frequency like you know shaft rotating at less than 1 rpm, 2 rpm just to measure the static run outs or dynamic eccentricity etcetera we can measure displacement and intermediate rule of thumb you know sometimes 0 less than 10. So, 10 and any frequency greater than this is for x this is for omega x and this is for omega square x there are of course you know standards which also say whether you should measure displacement or velocity or acceleration this is something you have to keep in mind as to what kind of response has to be measured and has to be determined from mechanical systems which are vibrating. So, this equation of motion is actually a second order differential equation but I should tell you something more about this equation in the sense there are certain terms which we can denote here. This is the right hand side is what is known as the forcing function and you will notice this omega here omega is nothing but the right forcing frequency. So, if a machine is running at say n rpm is its operating speed then this omega is equal to 2 pi f and this f is nothing but n by 60 rotations per second or hertz. So, usually this is how this omega is related to the speed of the machine and in condition monitoring literature people usually denote this as 1 x 1 x frequency nothing but n by 60. Harmonics of it will be 2 times 3 times and so on. So, this is the response now this response x which I will measure would be of this form sin omega t minus certain field difference. So, this is the equation to the response and this is a sequence of the steady state response because you would appreciate that this system which has been represented by this equation will have a response but where is the energy coming to generate this response it obviously has to come from this forcing function. So, that means if this forcing function was not there if the right hand side was 0 I would not have a steady state response but I would if for some instance I if the right hand side was 0 that means there is nothing to excited and if there was a small disturbance to the system it would actually have certain oscillations and which would damp out because of the damping present in the system and such a response is known as the transient response as opposed to the steady state response which I have written here. But in condition based maintenance we are actually interested in the steady state response because we would like to see how the machine behaves when it is being running at certain operational speed and what happens to the response. But then there are certain characteristics of the machine which are present whether you have force or you do not have force and this is actually depending on depends on the coefficient and one such very important is the rotational natural frequency given by over k by m or frequency the natural frequency is 1 by 2 pi whatever k by m and this is known as the natural frequency of the system or the system which is vibrating and this is an inherent characteristics of the machine or the system and which is not going to change no matter what your forcing frequency is. But there are many consequences of this for example, in one case we have the forcing frequency omega when it is equal to omega n. So, we will have a strong case of resonance wherein there will be large motions of x or the displacement. So, this have to be avoided while designing the machines. So, machine designers designers and machine avoid resonance conditions during operation of a machine. So, we are supposed to not operate the machine at its resonance frequency. So, people always you know designers tell the safe operating speed of a machine such that the resonance conditions are avoided because once we have resonances there will be large motions and which will lead to large fatigue failure and finally, a catastrophic failure of the machine. And if I was to there are many ways I can represent this forcing function one method is you know if I look here this form of excitation could be very general in the sense you know f t is equal to f e to the power j omega t and then similarly I will have a next response x t is nothing but x e to the power j omega t. So, by substituting such an equation in the equation of motion I will have this form x by f is equal to 1 by minus m omega square plus k plus j omega c this is one form of the response to the force is given by this where m k and c are the system characteristics. So, if I was to find out the magnitude of the quantity is a complex quantity. So, the and this is known as the frequency response function and the magnitude x by f is given by 1 by k minus m omega square whole square plus c omega square the square root of it and there will be a phase difference between the response and the input and thus could be given as the phase difference nothing but tan inverse of c omega by k minus m omega square. Now, this frequency response function which I had just denoted in a manner as x by f that means I have measured the response as a displacement and the forcing function as f and then there are many ways I could also have represented them for example, it could have been x dot by f could have been x double dot by f and then there are some names associated with it. For example, you all know f by x is nothing but stiffness then f by x dot is the damping f by alerted as small x dot and the third one is the nothing but the mass and the inverse of this x by f is nothing but the compliance and then we have x dot by f as the mobility x double dot by f as the impedance. So, these are the same things represented as different ratios depending on the convenience which we have. So, to this equation of motion if I have if I divide both sides by m I will line up with and then I will have if I denote new terms k by m as omega n square where omega n is the circular natural frequency and zeta is the damping ratio. So, this equation is the what x by f could also be represented as 1 by 1 minus omega by omega n square whole square plus twice zeta omega by omega n square n to the power 1 by 2. If I again denote omega by omega n as r. So, the frequency response function is given by 2 zeta r whole square and this is known as the dynamic magnification and looking at this equation there will be x by f by k actually. In fact, I take it back I just did a typo here. So, make a correction it is a by k f by k. So, in fact I was to write it again. So, x by x naught is given by 1 by 1 minus r square plus twice zeta r whole square and this is what is known as the dynamic magnification factor where x naught is the static displacement and x is the dynamic displacement. So, if omega is equal to omega n resonating conditions will have r equal to 1. So, this is going to disappear and I will have a x by x naught as 1 by twice zeta. So, you see this displacement actually depends on the forcing frequency depends on the damping. So, typical plot of the vibrating response of a single degree of freedom system I denote this as r and then increasing damping what happens this as I increase the damping the response reduces at sweat resonance the amplitude of motion can be controlled by damping. In fact, if you look at this plot here this is the magnification factor and the frequency response with very low damping this amplitude shoots up and also to the phase the phase angle undergoes a change of 90 degree and it changes with damping. So, this is a typical response of a single degree of freedom system subjected to a forcing function. So, just to summarize what we have studied in the single degree of freedom system we studied about the degree of freedom system. Though I have not talked about the multiple degrees of freedom system yet then we talked about the effect of stiffness damping and then natural frequency which is a phenomena or the system. System characteristic and then we have the forcing frequency sometimes in some literature people call it as the driving frequency which is nothing, but the operational speed of the machine and then we saw the behavior of the system as given by the dynamic magnification factor. The fact that the frequency response function can be measured by the few terms like flexibility or the mobility, inheritance, stiffness, impedance, mass and we have the mobility, compliance and the inheritance. So, one is the reciprocal of the other. So, if I was going to write the first one this will be forced by displacement, this will be forced by velocity and this will be forced by acceleration and the inverse of it is mobility. So, impedance is actually forced by velocity, mobility is velocity by force, stiffness is forced by displacement, compliance is displacement by force, mass is forced by acceleration, inheritance is acceleration by force. So, these are the six quantities which are there and then this can be measured, estimated and so on. Another important characteristics which you have studied is the transient response and the steady state response. Now, this we have discussed with relationship to a linear single degree of freedom system having translational or linear motion, but the same principles hold true also for a torsional system wherein all the x, x dot, x double dot can be replaced with theta, theta dot, theta double dot and the mass can be represented or replaced by the mass moment of inertia, the damping by the torsional damping and the linear stiffness by a torsional stiffness and the same system will hold true also for a torsional system. And these are few plots of how this six different transfer functions look like as it related to the frequency. So, this is the frequency and this is the acceptance, mobility and inheritance. Now, question is this transfer function how do we determine? Of course, one we can solve the differential equations and other is experimentally we can determine them. Question is our focus of attention is how do we implement CBM in a machinery? Of course, for a machinery we do not know its M, we do not know its C, we do not know its K and we do not know how many degrees of freedom are there and how they are related. So, obviously it is a challenge to mathematically model an unknown system. Suppose today I gave you a gearbox and asked you to find out its system of equation of motion, it would be little difficult for a beginner. Of course, there are now mathematical tools, numerical tools by which we can find out at least not exactly, but very close to the systems actual response. And then find out this equation of motion of such systems, but if my objective is to find out the natural frequency, if my objective is to finding out the damping present in the system, I can also experimentally determine such parameters by experimentally exciting the system and that is what is actually done in experimental modal analysis. Wherein I excite the system, if this is my system, if I given certain input f t, I will get certain response, excitation is in my mechanical system and this is my response. As I was telling this response could be velocity, could be displacement, could be acceleration, excitation is usually a force and then of course, you know they are all in real time. So, I have denoted them as a function of time and then looking at the x t by f t response, I can what is known as the impulse response function or in the frequency domain the f r f, which we know the six forms of f r f by now, we can find out the damping present in the system and the natural frequency of the systems. So, as a designer or as an condition based engineer for example, if I measure the response of a system and try to obtain the frequency distribution, though we will discuss this later on, I get a response like this. So, happen some of these peaks could be related to the forcing function may be omega, some could be may be multiples of omega 2 omega etcetera, but some could be certain frequencies omega star, which are and if there is a provision of changing omega, I would may be get another plot and is a different ink here may be. If I change omega may be reduce the speed of the machine, I got something so, this is my new omega and then I have this 2 omega here, but you see here the red curve the previous red curve and the green curve they did not move and this omega star could be a constant frequency, which could perhaps be the natural frequency. Of course, you know designers always design machines and equipments such that the natural frequencies are never in the operating frequency zone, but because of certain retrofits, if something is happen you all look at the complaint from the shop floor that no matter what we do the maintenance we know we had a nice overall, we had put new bearings, we do a regular check on the machines, but still there is a strong vibration at a particular frequency may so happen that this frequency happens to be the natural frequency of the component, which was retrofitted or some new component, which was totally overlooked and its natural frequency totally matched with the operating speed and then a condition of resonance occurs and then you have large motion. So, such cases are to be avoided by having a proper design. In fact, a designer in machinery vibration or in equipment design, which is subjected to a lot of dynamic loads they do what is known as a modal alignment chart. I will give this example to you from an say an automobile engine. Somebody asked you to design an automobile engine such that the condition of resonance between various engine components to be avoided. Well, we in an automobile engine say for example, an gasoline engine four cylinder gasoline or a petrol engine we may say that the operating speed will be from 0 to say 6000 rpm. If there are four cylinders the firing frequency would be 4 times 600 6000 by 60 that will be 4 times 100 that will be 400 hertz this is the maximum speed. So, maximum operating speed so maybe you know we should not have any resonating components in this band or maybe just to be on the safer side maybe twice of this. So, 0 to 800 hertz. So, this is to be avoided while designing a component such that in the operating band of 0 to 800 hertz no two components should have the same natural frequencies and this could be done by finding out the individual natural frequencies either through mathematical model or through experimental modal analysis and as a designer we avoid this frequencies. So, to summarize the important characteristics of response are for a harmonic forcing function the response is harmonic by harmonic I mean the forcing function is a sinusoidal or cosine function. The system always responds only at its force frequency and the phase angle is always related to the forcing function. Well, for the multi degree of freedom systems this only gets complicated that we instead of one differential equation I am going to have n number of differential equations where n is the number of degrees of freedom and then of course, they have to be solved simultaneously. And then we will have a case in the next class wherein I will be telling you about the case for the application of such machinery vibrations to multi degree freedom systems. How does the vibration get transmitted to different components? How do rotating systems behave to different excitations and unbalanced forces and so on. Thank you.