The Pendulum and Floquet Theory

[Music] so we want to pick up on floquet theory and consider this classic problem that comes out of physics and this is related to the pendulum so i'm calling this this lecture the pendulum and flow k theory so we want to understand is uh the stability of the pendulum the pendulum has periodic orbits right we can set the thing swinging and obviously it's it's very easy for us to understand that the pendulum in the downward position can be stable but there's another interesting physics phenomena i want to highlight today which is what we want to do is take the pendulum and here's the picture of the pin so we're going to drive this thing in particular the way we're going to drive it is we're going to take the support here and oscillate it back and forth let's say with some amplitude epsilon and cosine omega t so omega is going to be the driving frequency epsilon is going to be the oscill the amplitude of those uh oscillations and so we're going to just take this pendulum on a support and then oscillate it and then the question we're going to ask it's going to be related to what happens in particular one of the things that we're going to address is can you with such a perturbation to the system stabilize the pendulum in the upright position in other words is the inverted pendulum stable by driving it with this kind of forcing now generically what we'd really like to do is drive it like this just with some sinusoidal oscillations right smooth continuous and what we're going to do analytically is going to approximate this smooth forcing by some piecewise forcing because then we can actually work out all the algebra explicitly so in other words the support is going to be up down up down instantaneously just like in this picture here okay this is a little fake but it's going to allow us to do the computation and calculation about the stability of the pendulum in the upright inverted position now the governing equations in this point are given by here this is fully non-linear it's the second derivative f equals m a right so it's second derivative and here the pendulum has sine x remember that's the nonlinear pendulum has its x double prime plus sine x or minus sine x here i'm going to say that there's usually a constant front of it delta but now i'm adding to that epsilon cosine omega t which represents the forcing from this amplitude fluctuation here there's two parameters here there's the delta and the epsilon so epsilon remember again is actually three parameters but delta and epsilon are the magnitudes sort of this is the strength of your standard pendulum in terms of mg right and then over here this epsilon is the amplitude of the forcing and omega is the driving frequencies so we're going to do is explore the pendulum as a function of these three parameters and start to try to understand what happens to the physics of this pendulum under this kind of driving forcing that we have here okay so let's look at this um and here's what i'm gonna do i'm gonna look at the pendulum near the inverted and the downward position when you're downwards you're near x equals zero so sine x you know can be approximated by x minus x cubed over three factorial so forth normally what we do is we do small amplitude uh in the small amplitude limit we place sine x by x so that's going to allow that would allow us to address the stability of the pendulum in the down position now we are that's kind of a boring case we already know what happens there right the downwards pendulum is stable in that position the inverted pendulum is when x is equal to pi so now you're pi up you're facing straight upwards so this sine of x plus pi that's this here you just use your trig identity to show that's like negative sine x which when you do the taylor series expansion of this for small amplitude fluctuations near the top is minus x plus x cubed notice the sign change that's the big difference when you're in the upright position you get this minus x now in the linear linea remember that this is x double prime is equal to minus x here which is sines and cosine solutions this would be x double prime equals x which is exponential solutions it's like saddle you have one growing one decaying mode so you walk away from the top you say it's unstable but now we've introduced a perturbation to the system which is this modulation of where the bracket holds this thing into place this equation here is called matu's equation then and depending upon when you're if you're in the upper inverted position this is a plus or minus sign there and so we're going to study matu's equation and we're going to study stability of periodic solutions driven by this epsilon cosine omega t term and we're going to ask does anything really change in the physics i am in upright position you imagine you're always going to get pulled over and you're in down position you're always going to stay there because there's nothing to make you go away from that gravity is holding you in place okay so this is the question we want to address this is a mathematical construct we're just going to ask the question can you invert can you stabilize this pendulum in the inverted position okay fair question and of course at least at first maybe you haven't thought about it maybe you say like i don't know probably not right gravity would just pull this thing down you know whenever you put it up it might stay there for a bit but it's going to fall over but now the question is what's happening with this term here this driving term that i have that's oscillating that bracket so what i'm going to do is break this up into two components remember i'm going to assume that this thing here is going to be piecewise so it's going to be plus and then minus and then plus and then minus i'm going to approximate that sign by two constant pieces okay so here it is piecewise oscillations are like this here's the governing equations from 0 to half the period t over two and here's the governing equations from t over two to t the big difference here is the sign so this it's plus epsilon minus epsilon that's the only difference between the two okay so that's going to give us in the up or down position and remember this is an approximation if i come back to here it's saying i'm going to approximate this here this forcing sign by piecewise constant forcing here and the reason i'm going to do this is because i can work out my flow k discriminant exactly analytically in this case if i have this piecewise constant form where i can't do it in the case where it's actually sinusoidal and continuous okay so that's what i'm going to do i'm going to take these are my governing equations and it shifts from time 0 to t over 2 it's this 2 over 2 to t it's this and then i keep going back and then it's here and then here just keep going back every half period so remember for the flo k discriminate what i have to compute this is from the last lecture is this quantity here gamma this flow k discriminant is equal to the first fundamental solution evaluated at t the second fundamental solution eval this derivative valued at t and remember x1 satisfies the initial condition that it is one its value is one and its derivative is zero at time zero and x two satisfies that the solution is zero and its derivative is one at time t equals zero so i've to find these solutions with these two different initial conditions and once i find x1 and x2 i could construct this flow k discriminant for this problem okay so let's go compute these things so first of all what are the solutions well your fundamental solutions between zero to t over two is given by here this is just here's your solution they're exponentials remember we're near that saddle so one's a growing exponential one looks like a decaying exponential okay and that's also the case when you're here in the regime t over two to t remember you're near the saddle so you say i got these two solutions they have saddle type structure inside of this square root there's a plus epsilon in the first part of it and a minus epsilon in the second part of it and then so those are the two solutions and for me to construct a solution from time zero all the way to capital t the initial condition of this i run it from time 0 to time t over 2 and that's the initial condition now for the second part which runs from t over 2 to capital t okay so you can do a lot of algebra here so let's just call this the algebra health slide i've done it all for you you can work this all out i'm not going to go through it in detail but this is just taking those two solutions and now what you're going to do with these two solutions you're going to say my fundamental solution x1 satisfies that at time t equals 0 the solution is 1 its derivative is 0. i am put those initial conditions in work this thing all the way through those are the two constraints i have that need to be satisfied okay my second fundamental solution x2 it's initially its solution is 0 the derivative is 1. and so i can put those in solve and here is what i get x1 is this solution right here written all out notice that i have hyperbolic cosines here and here hyperbolic sines here and there same here now a mixing of hyperbolic sines and cosines hyperbolic sines and cosines so i have x1 i have x2 and what i need to do then is take x1 evaluated at capital t i need to take the derivative of x2 and evaluate it at capital t so even more algebra comes up which is i have to do those calculations evaluate them once i have them i add them together to form my floquet discriminant and there is the flowcase discriminant okay so this is actually a quite an instructive problem right it's it's the pendulum it's about as simple a problem as you can come up with that still seems that still has like this physical nice physical interpretation and it's a lot of work to get this but there you are this is your flow k discriminant as a function of this delta and a function of this epsilon and we have it all worked out here with the period variable t directly in there so a couple things to notice these are complex complicated expressions but i can just plot them on you know matlab or any other kind of software tool you want but i do want to highlight this first if you take omega to be big in fact this is one of the calculations you can show us for large frequencies what's going to happen with this thing here it's going to be stable because in fact this flow k discriminant is going to go if you take it into this asymptotic limit you can actually show that this thing collapses to 2 cosine square root of epsilon minus delta pi over omega okay remember that the frequency is related to this capital t here the frequency of the forcing omega that's what i'm actually looking at here is directly rated to the period t it determines the period t so for very high frequencies in other words a very short period i can actually find that gamma goes to this value here cosine is bounded between 1 and -1 so this thing here is going to a value of 2 and stability was determined if the flow k discriminant was 2 or less so it's 2 or less so for high frequency oscillations of that thing when i'm forcing at very high frequencies what it tells you is the flow k discriminant satisfies in fact the stability properties that we talked about in the last lecture which is the flow okay discriminant is less than two that's interesting right it tells you at least for high frequency oscillations i should be able to stabilize that pendulum in the right position in fact i can actually compute that flow k discriminant right i can just plot it and these are for different regimes there's a lot of plots here but the force panels up here are for delta being bigger than epsilon remember that delta is just the standard pendulum parameter like x double dot equals delta x so that's the delta and epsilon is the amplitude of the forcing so if delta's bigger than epsilon here are your flow k discriminants as a function of forcing frequency omega and if delta's less than epsilon again flow k discriminant as a function of four cons for forcing frequency omega and for each of these i want to point some things out so when you're in the downward pendulum the two the dotted lines on all these is where the flow okay discriminant is two or minus two and so everything is stable if i'm in this between the dotted lines so the downward pendulum notice in either case whether it's a null in your pendulum or a linear pendulum and by the way the nonlinear pendulum here i actually just computed this thing for the full nonlinear pendulum what i worked out in theory was here for the linear pendulum but this computation of the flow k discriminant can just simply be a numerical simulation right you can just say i want to run through a period with the initial con find one fundamental solution with with initial conditions one zero and the other solution with zero one is the initial conditions run them construct to flow okay discriminant and that's what you get here so the downward pendulum is always stable in this case it's also uh sorry and here in an inverted pendulum it's not stable until you get to a sufficiently high frequency where this thing now drops to the value of two i can do the same calculation here for the for instance the inverted pendulum again now it has this structure here where it drops down to two and it's stable here it's stable over here and up here even in the downward pendulum here for this case here epsilon can be pretty big and it can actually destabilize your pendulum until you force it a high enough frequency notice the amplitude fluctuations are much bigger than the natural than the then the frequency than the actual pendulum dynamics itself for gravity it's not a small perturbation it's a very large it's a larger one so then you do destabilize the pendulum until you get to a critical value and right here of frequency forcing here and here so this gives you a summary of this and what it's telling you is something really interesting you can invert that pendulum just by oscillating it on its support at a sufficiently high frequency every single one of these plots show you that if you go past this critical value of that frequency of forcing you can stabilize the pendulum so now here comes the question do you believe that is this just this mathematical result where we've neglected some physics this certainly can't be true that you could stabilize the pendulum in an upright position just by oscillating it so this is where you can go to youtube and actually get some videos so we're going to do now is like almost an inception scene i'm going to be on i'm on youtube and then we're going to watch youtube within youtube like and if i did one more layer i'd be like three layers down in the youtube in the youtube stream but here we're just stopping at two layers just to show that it can be done and here we go this is a youtube and video and this is a guy named inverted pendulum starring alan all right here he goes what he's got is a jigsaw there with this pendulum he's attached it to the arm he's just attached this arm to a jigsaw and you can see it's just a pendulum right and you can see how it's just you know it's exact acts exactly like depends on what you expect it to do now he's going to do is he turns on this jigsaw which basically is oscillating the bracket here very rapidly and he just stabilized the pendulum in the upright position this is not magic this is all just physics kind of awesome right so you can find lots of videos like this on youtube or maybe not lots but you can certainly find some like for instance alan here who shows you and does the actual experiment showing you that in fact you can stabilize the pendulum in their upright position with a bracket that oscillates here and here the epsilon is smaller than delta so it's a very small oscillation but sufficiently high frequency and everything tells you that if you make a sufficiently high frequency you will stabilize it and this video shows you exactly that and there you go we were two youtube levels in and now we're back out to just one youtube level and now this is the last lecture that concludes this course on advanced differential equations but i i like i like some of the results here that you've seen hopefully you've enjoyed them as well and this is all about these ideas of stability and here we've now addressed it directly for periodic systems using floquet theory and even close it off with a nice little experiment not by me but by others thank you very much [Music]

Channel: Nathan Kutz

Views: 1,884

Rating: 4.878788 out of 5

Keywords: pendulum, inverted pendulum, Floquet theory, kutz, Matthieu equation

Id: 3TbVWzRY2hU

Length: 18min 21sec (1101 seconds)

Published: Mon Jan 18 2021