8.03 - Lect 3 - Driven Oscillations With Damping, Steady State Solutions, Resonance

we have discussed three oscillations of harmonic oscillators without damping and then we introduce damping but in each of those cases we let the simple home oscillator do its own thing we did not interfere with it today that's going to change today we are going to impose our will onto the simple harmonic oscillator and we can impose our will onto it by driving it with a force and then see what the net result is and let's start with a simple example that I have here a spring we spring constant k and an object Mass n that this be the equilibrium position x equals zero that will be damping I will introduce again B over m equals gamma and Omega 0 squared equals K Over M we've seen this before it is the shorthand notation so now in addition to the fact that when the object is away from equilibrium that there is here a spring Force I am now going to apply on that object a force it may not be easy to do we'll get back to that how we do that but I can apply a force on that maybe through magnetic field maybe through electric fields and I'm going to this force is now going to have the character F0 times cosine Omega t i impose on that system now that frequency Omega and I can choose that anything I want to so now I can write down the differential equation of motion Newton's second law m x double dot equals minus KX nothing new that's the spring Force minus b x dot nothing new that's the damping but now comes this external Force F0 cosine Omega t what I'm going to do now I'm going to move this to the complex plane not that it is absolutely necessary but I'm so used to that so I'm going to write this now in terms of Z and then we take the real part of Z later that gives us back goes back to to X so I'm going to write this now in terms of Z double dot I divide M out we get plus gamma times Z Dot Plus Omega 0 squared times Z and that now becomes F 0 divided by m remember I divided M out and then we get cosine Omega t for which I will write e to the power J Omega T because I work now in the complex plane and through Euler I can always convert that back to cosine my trial function for Z which is a complex notation is some amplitude times e to the power J Omega t minus Delta now crucial is that you understand why this Omega and this Omega are the same this is the Omega of my driving the system that is my will that I impose on that system clearly given enough time in the beginning the system may be unhappy and it may do all kinds of nasty things which we will discuss next lecture but ultimately I will come out to be the winner and ultimately that system is bound to start oscillating with the frequency that I impose on it if I start shaking you in the beginning you may not like that and you may oppose to that but ultimately I will be the winner and I'll make you shakers that frequency Omega so clearly the ultimate solution must have the same omega as the driver what is the meaning of this Delta well it is not at all obvious that the object will be in the same phase as the driver it is possible that when the force is pointing in this direction that the object may be going in the other direction and you will see that that indeed can happen and so this Delta is a phase angle which takes into account the possibility that the driver and the object in their motion are not exactly in face we call this solution a steady state solution studies state that you must wait long enough for the system not to fight you any longer that will be part of my next lecture the fighting issue this is when I ultimately come out to be the winner and when the system follows my will so now I'm going to take the second derivative so I get minus Omega squared J Omega comes out twice so I get minus Omega squared then I get plus gamma times J Omega then I get plus Omega 0 squared and that whole thing multiplied by a to the power e J Omega T minus Delta that now equals F 0 divided by m times e to the power J Omega T so the whole thing is now in the complex plane and you see that e to the power J Omega T cancels on both sides so I lose I lose my e to the power J Omega T and I'm going to multiply both sides by e to the power J plus Delta so I lose my Delta here but it appears then here and so if I make that simple change algebraic change we're going to get minus Omega squared plus gamma J Omega plus Omega 0 squared that multiplied by a must now be equal F 0 divided by m times e to the power J Delta because Omega T is gone and I've moved the Delta to the right side can we live with that and this can be written F 0 divided by n times the cosine of Delta plus J times the sine of Delta I'm still in the complex plane but that's Euler now now let's compare the apples with apples and oranges with oranges this is an apple and this is an apple that means it's real and that means this is an apple but there are also oranges this is an orange it has a j and this is an orange that has a j and so for this equation to always halt at all moments in time the apples must be equal to the apples on this side and the oranges on this side must be equal to the oranges on that side so it looks like one equation but it really is two equations so we now get that minus Omega squared plus Omega 0 squared times a must be equal to F 0 divided by m times the cosine of Delta apples on both sides are equal and now we get the oranges on both sides gamma Omega must be equal to times a I still have this a here so gamma Omega times a equals F 0 divided by m times the sine of Delta two equations with two unknowns a as unknown and Delta as unknown and they're easy to solve if you square them then you get sine squared Delta and cosine squared Delta you add them up that's one and so that immediately gives you then what a is so a is going to be F 0 divided by m upstairs and downstairs you're going to get Omega 0 squared minus Omega squared squared Plus Omega gamma squared so this is the amplitude of the object a real massage that we'll talk about this for at least the next 10 minutes it's very complicated function we want to see through that equation what that actually means and the tangents of Delta is easy to find because you define this equation by that one you get immediately the tangents of Delta H disappear and F0 Over N disappear and so you'll get that the tangent of that angle Delta is Omega gamma divided by Omega 0 squared minus Omega squared we can now return to the real world and if we return to the real world we have to change x z back into X and so our final solution which I will put in color will then be that X as a function of T is this amplitude a times the cosine of Omega T minus Delta and that is the Omega that is my will that I impose on the system notice that there are no two adjustable constants which we were so used to in the past in the past we said well you can start the system at T equals zero you can Define the position you can give it a certain velocity so you always expect that in your solution there are two adjustables in order to meet the initial conditions there are none here and the reason for that is that this is a steady state solution which means the system doesn't even remember anymore what the situation was at T equals zero it had lost all its memory and so a which is the amplitude of that object is not not something that you may choose a follows immediately from this equation which is a complex function of Omega Omega 0 of f 0 and so on and Delta is also non-negotiable Delta has nothing to do with your initial conditions Delta follows from gamma Omega and Omega zero so now we're going to look at a and try to understand the complexity of that amplitude for one thing it is pleasing that F0 is upstairs it is intuitive pleasing that if the force that you apply becomes larger that the amplitude will become larger that's reasonable it is also pleasing to see that there is a gamma here downstairs that means if there is a huge amount of damping you don't expect a to be very large so that's also pleasing that you see a gamma downstairs now I want to evaluate in detail what is hidden in this very difficult equation and let me try out your intuition common sense without looking at my Solutions without looking at differential equations without looking at the equation a just common sense now suppose I apply a force here on this object with the frequency which is near zero so it takes a hundred million years for it to reach a maximum and then it takes another 100 million years for the force to go to zero and so on when that Force has a value f 0. what do you think will be the position of that object if you know that position that may tell you what a is what the amplitude is of that object without any differential equations any one of you able to immediately say of course a has to be this maybe that is a little tougher I see some hands there no you were just doing your hair yeah of course if that Force goes so slowly then at all moments in time there must be equilibrium between the spring Force which is of course KX and the force that you apply which is UF and so if you do it extremely slowly the two must always cancel each other and so I make the prediction now that when Omega goes to zero that a should become F 0 divided by K that is that X now let's look at this equation let's see whether that is true we make omega 0 we make omega 0 so this equation tells us that it is f 0 divided by m and then downstairs we have omega 0 squared but Omega 0 squared is K Over M and you see indeed that that's exactly what you get now without looking at equations can you guess what the phase difference is between the driver and the follower if it takes a hundred million years for that Force to slowly reach its maximum and 100 million years to go back again what do you think will be their phase difference between the two it'll be zero of course there's plenty of time for that object to follow so you expect that Delta becomes zero well if Omega becomes zero this Omega this zero is Omega squared this is so this goes away here's a zero upstairs so you see the tangent of Delta is zero and that indeed is what you see so the two follow each other it's extremely boring the whole thing to watch and the amplitude is exactly what you predict let's now do something more interesting and let us drive it at what we call the resonance frequency we give it that word that is the frequency that the system really would love to oscillate in the absence of any damping and in the absence of my doing this silly Thing by driving it so now we are at what we call a resonance so this term goes away and this term now becomes Omega 0 gamma so you now get that a becomes F0 divided by m and then downstairs you have F Omega 0 gamma well if you remember that we introduced a quality Factor Omega 0 divided by gamma which is a dimensionless number then you can also write this as F 0 divided by K times Q so that's nice to remember that at resonance if you define this as resonance the amplitude of the object is Q Times Higher than what it would be at extremely low frequency interesting to remember so this is the amplitude at very low frequency and when you drive it at resonance it is Q Times Higher and then I will put that here when Omega 2 goes to Infinity everything goes so fast that the object has no time to follow the driver the object goes nuts because of this high frequency it can't do anything and so a will then go to zero and let's check that if Omega goes to Infinity you see the downstairs here goes to Infinity so a goes in D to zero so you have no amplitude at all what is not so obvious that Delta here is pi and what is not so obvious here either that Delta goes to pi over 2 in case of resonance in other words at resonance the driver and the follower are 90 degrees out of face the follower is 90 degrees behind very hard to imagine what that is like but I will demonstrate it you will be able to see it what it means here that at very high frequencies the amplitude of the object goes to zero but what I will be able to show you that if the driver goes in this direction that the object goes in this direction so there are 180 degrees out of phase and that I can show you that's what it means when Delta equals pi so now we can make a a graph a plot of a as a function of Omega so here is omega and here is a and let this be the resonance frequency Omega 0 I'll make it a little straighter so you start at very low frequency if this is zero you start here with f 0 divided by K we all agree that that was obvious and then the amplitude will build up go through a maximum goes down and ultimately goes to zero and at this value Omega 0 this value is q times F 0 divided by K now for those of you who look very carefully you may have noticed that the maximum here that I have drawn is not at Omega equals Omega 0 which may go against your instinct this maximum occurs at a frequency which we will call Omega Max which is always a little bit below Omega 0 but for high Q systems as I will show you shortly it is effectively the same I will come back to this the Delta the face as a function of Omega if this is pi and this is pi over 2 and if this is Omega 0 then that Delta will change in the following way which is way harder to imagine than that what a is doing you are in phase at very low frequencies at resonance precisely at Omega 0 you hit the pi over two 90 degrees out of phase and at very high frequencies you will see that the two are out of phase and I will be able to demonstrate that to you coming back to this mysterious maximum not so mysterious actually where is this at what frequency do we have really the maximum amplitude well to calculate that you would have to take the derivative of that monstrous equation you will have to take the ad Omega and you go you ask that to be zero so that's when the maximum occurs and I will leave you with that exercise may take you a few minutes to do that and you will find then that Omega Max so where the real maximum is located the maximum in terms of amplitude is Omega 0 squared minus gamma squared over two not four but two to the power of one half not so intuitive that it is there and if you like to write that in terms of Q which is often done then Omega Max so that is the frequency at which the amplitude reaches a maximum is Omega 0 times 1 minus 1 over 2 Q squared and then the square root of that whole thing and the reason why this is nice you can immediately if you know Q you can immediately evaluate what the difference is percentage wise between Omega Max and Omega 0. if you want to know what the maximum amplitude itself is so what a Max is so that's really this value it must be very close to Q times F 0 over q but it's a little higher then you can write that in the following form and that just a matter of algebraic manipulation and you get a q here which you expect and then downstairs you get something like this one minus one over four and then you get a q squared to the power one half and so now let's put in some numbers so that you get some feeling for the answers that we have suppose we have an example by Q equals five it's a modest value for Q most pendulums that we have the cues are way higher than five so I take a modest number for Q if I go to this equation here Q squared is 25 2 times 25 is 50 that's two percent but I have to take the square root so it's only one percent off so Omega 0 Omega Max divided by Omega 0 is 0.99 it's only one percent lower there's only one percent below Omega zero and then if you want to know now what Amax is so you would think that a Max is very close to Q times F 0 over k but it is not Q times it is a little larger and so if we Define a Max divided by a0 a0 now is meant to be the amplitude when Omega equals zero that is a shorthand notation this number is not q a little higher is now 5.03 now you can see that if Q is higher then of course these numbers become even closer than Omega Max becomes even closer to Omega 0 and then this maximum a becomes even closer to Q times F 0 over k rarely ever will we be bothered too much with the fact that the resonance frequency which we call Omega 0 is not exactly the frequency whereby the response of the object is a maximum very rarely ever will that become an issue I want to show you now a transparency from your own book so don't take notes this is from French you see here the function a but it is divided by a0 which is that F 0 divided by K so that is the amplitude for zero frequency and so when you start off it's one that ratio is one by definition right because it's a Omega divided by O's a0 and horizontally you see Omega divided by Omega 0 so by definition right here at the one sign is that point that I put Omega 0 there and you see here these various curves for different values of Q and the one that I made green is Q equals 10. and no surprise that the height is at plus 10 because we predicted that it is Q times higher than the amplitude when we have low frequency and you see indeed that the red one is very close to 10. if you look at the one that has a mark Q equals three which is this one if you look very carefully you may see that the maximum a is lifted slightly below the value 1 which is omega Omega is Omega zero but even for Q equals three the difference is insignificantly small and then at the bottom you see the Delta function the phase delay the the object follows the driver at very low frequency precisely Delta is zero at resonance it is precisely pi over 290 degrees very hard to imagine but we'll I will try to show it to you and at very high frequencies they go like this they are 180 degrees out of phase so now comes the question how do you apply a force on a system it's nice to say there is a force but you have to think of a way that you can actually do that and I will just discuss one case with you and then I will try to demonstrate it also if I have a pendulum and I want a force on this object then I can do that as you will see in an indirect Way by starting to move my hand here you will see how that translates into a force on this object by moving my hand I am now the driver my displacement now is in inches not a force but is in inches here is the pendulum length L minus M and this is the equilibrium position of my hand and here is the object but I'm going to move my hand in a way Atta equals Atta zero times cosine Omega t that is the frequency that I decide I impose that frequency on the top of that pendulum and the amplitude of my hand in terms of inches or miles or light years that is linear scale at a zero it's not a force it's not a force it's a displacement well I take a picture at One Moment In Time and what do I see I see that this is what the pendulum looks like this angle is Theta the pendulum is displaced over a distance X from equilibrium and the top is displaced over a distance ATA this is Walter Lewin I am doing that I am there with my hand I can't help it this is where I am and this is where the object is so now I want to put in all the forces that are at work I will move it up a little bit because I want to have a little bit of room for my forces so make the legs a little shorter so here's the object and here's the object there are only two forces on this object and that is gravity which is mg and that is the tension there's nothing else I call this x equals zero and I call this displacement X away from equilibrium for small angles I want to argue that t is very close to mg for one thing if you hold them vertically and you do nothing and there is no motion it's obvious that t is mg the two forces have to cancel each other around that's clear but I can show you that even if the angles are modest that that should also be the case suppose I decompose t in two directions vertical Direction so this is T times the cosine of theta and in horizontal Direction so this is T times the sine of theta if the angles are very small the object is hardly moving at all in this direction the motion is almost exclusively in this direction so there is no acceleration in the y direction or I should say the acceleration in the y direction is negligibly small so that means to high degree of accuracy T cosine Theta is always the same as mg High degree of accuracy but for small angles cosine Theta itself is one therefore T equals mg and so the force that is driving this object back to equilibrium is T sine Theta and so that force is mg sine Theta to a high degree of accuracy I'm going to introduce again that gamma is B over m s order is damping and I'm going to introduce that Omega squared equals g over L Omega 0 squared is G over l g over L being the the square root of G over L being the resonance frequency of a pendulum length l independent of the mass of the object as we have seen before so now I'm going to write down Newton's second law so I get MX double dot and then I get minus b x dot that is the the damping minus b x dot and now comes this Force which is the only one that wants to drive it back to equilibrium it's the restoring Force and so that's Force if you accept my T being mg that is mg times the sine of theta that's the differential equation that I now have to solve that is a driven system now here I had a driven system and boy I saw a force here I don't see anything like that there where on Earth does Walter Lewin come into this picture who is doing something have I overlooked myself perhaps just me I changed nothing would I change anything I've changed nothing but I don't see myself anymore so what's wrong is there anything wrong with this where do I show up in this equation yeah so where in that equation do I show up what is the sine of theta what is the sine of theta what is the sign of this angle X minus ETA that's Walter Lewin x minus ETA divided by L there I am and so I'm going to substitute that in here and I'm going to divide by M well that's not divided by m yet now just say x double dot minus b x dot and now we get minus mg times x over l and now we bring wall to Loop into the other side and so we get plus mg times ETA divided by L and ETA is ETA 0 times cosine Omega T because I am moving my hand that ETA is a function of time let me write down an mg in here and then we'll check this so NG times sine Theta has two terms it has an mg times x over L but it has also an mg times ETA over L and I bring that ATA over L on this side but I know that ETA is changing in time and so you see now Walter Lumen is right there and now I divide by m and I substitute Omega 0 squared in here oh I had an M here too you should have screamed there was an M there I decided not to divide by m remember now I'm going to divide by m so I get X double dot there's an equal sign here you were you should not be sleeping you're not supposed to sleep this is an equal sign minus B x dot right equals minus B yeah we're in business now and so I'm going to this should also you're all sleeping right all of you are sleeping my goodness MX double dot minus BX dot minus MGK X over L and then the minus and the minus becomes Plus all right try not to sleep so X double dot now we get plus gamma times x dot and now we get plus Omega 0 squared times because G over L is Omega 0 squared times x so I divide the M out and now I get equals Omega 0 squared times ETA zero times cosine Omega t I will move this L up a teeny little bit and I'm going to look now at that equation at the bottom and I am now overjoyed happiness because this one looks almost like a carbon copy of the one that I had here with an F 0 cosine Omega T and now instead of an F 0 divided by m cosine Omega T I now have this so this takes the place of my earlier F 0 divided by m f 0 divided by m is an acceleration by the way now it better be an acceleration because this is an acceleration and apples have to be apples so this is an acceleration this is an acceleration and this is also an acceleration multiply Omega squared by a distance then you get distance divided by time squared so you see now how the connection between the two go where originally I got an F 0 over m times cosine Omega T now because of Walter lewin's motion I'm going to get an Omega 0 squared times ETA zero and so you see now how this motion of my hand indeed translates into a force on the object well I have the solution I don't have to do anything all I have to do is change this by Omega 0 squared times at a zero and I'm done differential equations are identical I don't even have to change the tangent of Delta nothing changes this is the only thing that changes so we're done we can now make some predictions the prediction is that if I'm going to shake this pendulum and I'm going to do that very slowly taking one hour to go to the left and talking taking one hour to the right and if my amplitude is at a zero what do you think that the amplitude a the solution of my differential equation will be in other words I'm going to shake very slowly what do you think a will be without looking at the differential equations so I just go with my hand like this amplitude Omega at a zero amplitude at a zero and I do it very slowly what will be the amplitude of a at a zero so you expect that this goes to at a zero well if you don't believe it go to this equation substitute in here zero in here zero you get Omega zero Square each of this Omega 0 squared and you see at the zero it's exactly what that equation predicts but your common sense has the same thing now what do you think Delta is if I'm going to move this pendulum very slowly to the left and to the right of course of course the object will follow me would be ridiculous if I take one week to go from here to here that the object would be there right obviously the object is always here so you also predict that Delta is zero and so now we can make a quick prediction that at resonance you probably get Q times at a zero and then Delta would become pi over 2 and when you go to very high frequency then a would go to zero and then Delta would go to Pi and this is what I want to demonstrate to you so the final solution of this pendulum which I will write down in red is going to be that X equals a times the cosine of Omega T minus Delta just as we had before a is non-negotiable has nothing to do with the initial conditions Delta is non-negotiable has nothing to do with initial conditions this is the steady state solution all right let me take my shoes off because then you can see it better all right here's a pendulum here's a pendulum it's going to be very exciting I'm going to tell you I'm going to move this with Omega very close to zero very exciting I'm doing it right now aren't you thrilled no you're not thrilled but I'm moving and now I'm going to go back do we agree that a the amplitude of that object is exactly the same as the amplitude of my hand do we agree do you see that that is why that a is at a zero and that follows from that rather complicated equation did you see that Delta was Zero did you see that we went hand in hand so to speak no pun implied we're going hand in hand right that one follows exactly my hands so Delta I zero let's now go to high frequency very high frequency way above resonance and what you see now is that the object is not moving very much but if you look very carefully you will see when my hand is here the object tends to go there and where my hand is here the object tends to go there that is that pie ready you see that there's almost no motion a is near zero but can you really see that the phase difference is pi can you see the 180 degrees let's see if the a is exactly zero of course that you cannot tell so I try not to go infinitely fast I go a little slower than anything can you see it okay now comes resonance and now it will be very difficult to see this pi over two that's almost impossible that's not my object objective but my objective is to show you that the enormously small very small at a zero here we'll give an amplitude there which is Q Times Higher so you get a huge swing when my hand is hardly moving at all that's the power of Q so there we go I first get it into it there it is now this is resonance would you agree this is resonance now look at my hands my hand is moving probably no more than with an amplitude of three millimeters no more and yet I see an amplitude there of 60 centimeters that would mean that very roughly this pendulum has a q of 200. namely 60 centimeters divided by three millimeters so this is even a way to make an extremely rough gas admittedly very rough of the Q value you can you cannot even see my hand move we'll be honest you can't even see my hand move but I know I am moving it a little oh you're lying no you will not no you will not okay so you see all the goodies that we have calculated actually can be demonstrated and show up quite dramatically suppose I had a spring system like this and I want a force on that object here well what I can do is just shake it here in a way extremely similar to what I did there and when I shake it there we can make certain predictions we can make predictions now based on the knowledge that we have suppose I shake it with an amplitude at a zero no differential equations know nothing for now but I know that somehow it will come out in terms of a force at the object so I know that when I write down the differential equation of course it shows up exactly this way I get an Omega 0 squared times ETA zero except the Omega 0 squared is now K Over m so what do you think if I shake it as Omega equals zero what is then the amplitude that this object will have relative to my motion at a zero so I move my hand at a zero infinitely long what will this object do we'll just follow it so you get this answer what will be the delta will be zero when I hit resonance what will be the amplitude of that object hanging from the spring will be Q times higher than my at a zero what will be the phase difference 90 degrees when I shake like crazy a will go to zero so with the spring if you shake it like this which is part of your problem set you will see exactly the same results that we have there done for a pendulum and this now I want to independently um demonstrate to you I have here an Air Track I can blow out air so that the object here starts floating so we can make the damping very small by making it float but if we lower the airflow the damping becomes a little higher I have a spring here with springs constant K and I have another spring here with spring constant k they both have spring constant k and now I'm going to drive this here at extremely low frequency over a distance at a zero at maximum at a zero cosine Omega t what do you think the amplitude of this object will be at that very low yeah very very good very good not at a zero but why is it half because we have two Springs so effectively the spring constant is twice that exactly so if I go very slowly you will see that this displacement here will be twice as high as this displacement but what I really want to show you is there in phase this one will go to the right when this one goes to the right now comes the catch I showed you earlier there's a steady state solution in the beginning the system doesn't like me it hates me it fights me it doesn't like that Omega it wants to do something different which is part of next week's lecture and you will see that in the beginning and so we have to be a little patient before my real survives you ready for that so I'm going to start now to drive the system at a frequency which is below resonance I want you to see two things that they go hand in hand and I want you to see that the so you're going to very low frequency I'll give you the amplit here this is twice the amplitude of the driver now it is here the spring and now the spring is here so it's only this much is two at a zero so at the zero isn't it's no more than three quarter of an inch and now we're going to let that object be exposed to this driver and we'll give it a little bit of time to recognize me it takes a little bit of time to reach the steady state solution and next time we will learn how much time it actually takes so if you want to be a little bit patient then you will see if we give it too much damping too little air then of course it starts to get stuck yeah we're close we are close for me close enough now look at it they're going both to the left for me and both to the right for me for you they are going now both to the right and going both to the left going both to the right and both to the uh and both to the left now this was the amplitude twice the amplitude of the driver and when you look carefully here it's less it is that after zero over two that this gentleman immediately noticed because we have two Springs so you see here apart from the factor of two you see the Delta zero and you see that the amplitude indeed is half of the amplitude of the driver because of the two Springs now we're going to Resonance Omega 0 and now nasty things may happen it may break we have to give it time you see what funny things it's doing it's not in steady state yet you have to wait just a little patience give it more time notice also that the remember this is only moving this much and look how much this is moving I may not even be exactly at resonance you know we can only do the best we can here we may not be exactly at resonance oh boy I'm close to Resonance now oh yeah oh yeah oh man look at that oh am I at resonance I think I got it that you see the neither in Phase no outer face now you see the 90 degrees and look at this teeny weeny little displacement here and look what this man is doing that is resonance markable ah Marcos where's Marcos we hit it right on now I will oscillated way over resonance not way but over resonance first we first have a system must first calm down and now I will change the frequency above resonance so that now you will see the phenomenon that I discussed earlier that the amplitude is very small again we have to wait a little look how fast it's going and that time will go 180 degrees out of phase now look this is going this much back and forth and this one is not doing very much now if you can't see it I can it looks like this can you see it that's 180 degrees out of faces very clear five minutes break see you back here in exactly five minutes now so we have discussed today some simple systems pendulum one object Springs one object one resonant frequency but soon in 803 we will discuss systems with more than one object for instance is I if I put three cars on here with four Springs three resonance frequencies if I have a triple pendulum which I will demonstrate next week one pendulum with all the other below the other three resonance frequencies five cars on there five resonance frequencies so simple objects like a dinner plate or just a regular glass has an enormous number of resonance frequencies it can oscillate in many many different ways if you drive your car you will turn around that's certain oscillation a certain period underlying and you may notice at certain speeds that something in your car begins to Rattle very annoying all you have to do is go a little slower or go a little faster and it stops you go off resonance for that object now you may go on residence for another object of course and some cars rattle at any speed you have a radiator in your room which rotates that is also an underlying oscillation and a period that may start to cause resonance in in the frame you may hear some awful noise sometimes unfortunately these fans you cannot change the speed so easily but you can go from state three to two to one and then this terrible noise will go away you take a washing machine or dryer I used to remember a friend of mine in the Netherlands had a dryer and when he started the dryer at the very early phase when it was at a certain frequency the whole dryer would start to walk through the room it would and then at higher frequencies it of course would stop resonance resonances are everywhere and they often occur when you don't expect them you open a faucet you think it's a steady stream of water which I'm sure it is but sometimes you hear an unbelievable sound that drives you almost nuts I'm sure all of you have heard that sometimes if it isn't in dormitory maybe at hotels or at home all you have to do is open the faucet a little more or a little less and it goes away and it's really Extremely Loud and an annoying resonance if you take something as simple as a wine glass which has a tremendous number of resonances then I can make you listen to a well-known resonance which is by rubbing the rim of the glass when I rub the rim of the grass I'm not exciting it at one particular frequency and certainly know that the resonance frequency I'm exciting it at lots and lots of frequencies I dump on it the whole spectrum of frequencies but the glass is mean it just picks out the one which is its resonance that's where it builds up a large value for a it ignores all the others and that's why I can make it resonate at that particular frequency listen to it this is not one frequency what I'm doing and has nothing to do with the time that it's for me to go around it's a very high pitch about it's about 420 Hertz so the rubbing is like dumping a spectrum of frequencies on it and it selects what it's like the most when I was a student I remember we often had an after dinner speaker we had the dinners at the fraternity and we had an after dinner speaker and more often than not we didn't like the after dinner speaker we didn't like the speech and so we made that very clear and the way we did that is all our wine glasses enormous sound in that dining hall and the the speaker very quickly got the message of course that is an enormous sound that you can generate and most of these wine glasses were roughly the same so it was always a tone that was loud and clear and almost one frequency you've seen footage lately of the storms three storms in a row and you must remember sometimes that you saw a traffic sign you hear Paul and then a traffic sign and then even though there's some kind of a crazy wind going the traffic sign goes like this all resonance frequencies it can even break even though the winds appears to be relatively steady the wind then generates in a way a whole spectrum of frequencies and this traffic sign picks out the one that it likes the most and then it goes nuts at a frequency that is a resonance frequency and resonances can become destructive of course if these amplitudes are too high then things can break down you may have noticed I was worried here when the amplitude became so large that the spring might even break or the car might jump off that that track and of course a classic example of this destructive resonance is the destruction of a very famous bridge in this country in 1940 bridge in Washington State was destroyed by wind and this bridge had many different resonance frequencies one like this and some like this and depending upon the wind strengths different resonances were excited at different moments in time but that ultimately led to the destruction of the bridge there are countries where soldiers are not allowed to cross a bridge when they are Marching In step that's the case in the Netherlands my home country it's also the case in many European countries the story has it had in England I think more than 100 years ago when soldiers went over the bridge in Step that the bridge collapsed whether that was the result of the soldiers what we will never know but in any case from those days came the order that soldiers have to go out of Step before they cross the bridge there is a rumor that most of you have heard that there are women who are capable of singing with such a loud voice they can break a wine glass they have to tune exactly at the frequency of the resonance frequency of the wine glass and they come out huge volume and the glass breaks I don't believe it but it's a rumor there was a commercial many years ago some of you may never have seen it for Memorex Memorex was a tape a special tape for audio tape recorders this guy is going to a concert and there is this woman singing loud voice beautiful frequency Bingo glass breaks he comes home and he's tells his wife the story well his wife was smart enough of course not to believe it and he says well it just so happens that I recorded it on my Memorex tape well so he plays the tape at home and at the moment that the woman's voice goes what happens the glasses break at home in his cabinet so much for the physics of Memorex because you can imagine that the resonance frequency of the glasses at home were very different than the glasses in the orchestra so it's all a Swindle but that's of course what commercials are all about what is now what is now the bottom line the bottom line was that if you buy Memorex these tapes then the reproduction is so perfect that you can even take it home and you see the glasses break this brings up now the 64 million dollar question and that is can it be done or can it not be done so any women in my audience who want to give it a try that would be wonderful we've asked ourselves that question can this be done or can this not be done by a person and I think we came to the conclusion that a person alone without all kinds of electronic equipment could not do that and I still believe that today but Professor felt Michael felt here at MIT with one of his graduate students many years ago developed some powerful equipment which you see here which was designed to make an attempt to break a wine glass it doesn't always work but it works often the idea being then that here is the wine glass it's almost a carbon copy of this one when they finally made this to work so it's near 440 Hertz they went to Crate and Barrel and they asked how many of these wine glasses do you have they said we have five thousand and they bought them all five thousand because it's not obvious if you have to change glasses that it will ever work again and so here is one of those glasses here is a loudspeaker the sound comes from this side and what is nice about this arrangement that we can make you see the Distortion of the glass in this resonance mode the glass oscillates like this goes from oval to Circular to oval to Circular and the way we can make you see that is by strobing the glass at a frequency which is a little bit different from the frequency of the sound and think about it two frequencies almost the same give you a beat phenomenon so what that comes down to is you see the motion of the glass very slowly and then if we hit that resonance frequency we will increase the volume and then maybe it will break now I have to warn you the sound will be unbearably high and so for some of you here in the front row you may even want to cover your ears or you may want to move back it's up to you but be careful this is really an enormously strong signal that you're going to hear so or be careful you can move back if you want to I have hearing Hsu have noticed and I have the option that I can turn them off but in spite of that I'm not deaf without them I will still cover my ears so let me first give you the light setting that we want so we make it a little dark and then I will show you uh the glass oh I have to change the setting here there's the glass it's a there's no sound it's not oscillating and now I turn on the speaker so I'm now going to turn my hearing aids off and put this on and I'm going to increase the volume and if it doesn't want to break I will change the sound frequency a little bit to sweep over that resonance curve so that I get onto the maximum because if it's only off the Q is so high of this system that if I'm a little bit off in frequency then the amplitude will be low so let's first see what happens when I increase the volume did I do something wrong yes I change the frequency foreign that's very close getting very close thank you I think I have convinced you that a woman cannot do this I have some last words of wisdom for you and that is falling in love is also a form of resonance and the two can be destructive because it can break your heart so try to remember that next time have a good weekend

Channel: Lectures by Walter Lewin. They will make you ♥ Physics.

Views: 210,022

Keywords: Walter Lewin, Physics, Forced Oscillations, Damping, Steady State Solutions, Amplitude vs Frequency, Resonance, Quality Q, Pendulums, Springs, Air Track, Destructive Resonance - Break Wine Glass, Break Wine Glass, Sound Breaks Glass, Rubbing Wine Glass

Id: Y_DmzZcQR7A

Length: 69min 5sec (4145 seconds)

Published: Tue Feb 10 2015