Physics 76 Spring 1997 Chaotic Pendulum

Transcription

1 Physics 76 Spring 1997 Chaotic Pendulum A violent order is disorder; and A great disorder is an order. These Two things are one. Wallace Stevens, Connoisseur of Chaos, Reading: Prior to doing this experiment, you should read the first two chapters ofchaotic Dynamics by Baker and Gollub (QA/862/.P4/B35/1990 in Kresge Library) and skim the rest of the book. I. Introduction In this experiment, you will investigate a simple mechanical oscillator that has very complicated behavior. Usually, unpredictability in physics is associated with randomness, as in statistical and quantum mechanics. In either case, there are systems (the state of a radioactive atom, the velocity of a molecule) that can only be described in probabilistic terms: there is so much of chance that the atom has decayed, or there is some probability that the gas molecule has a certain velocity. In classical dynamics--the science of weights and pulleys, and orbits and planets--on the other hand, the equations of motion are completely deterministic. If the position and velocity of a particle at time t=0 is known, then the position and velocity at some later time t. can be calculated exactly. In almost all cases that you ordinarily encounter in a physics course (as opposed to in "real life") the equations of motion are linear. For example, in simple harmonic motion, the restoring force depends linearly on the displacement, and explicitly nonlinear systems, such as the simple pendulum, are only treated in a small angle limit where the dynamical equations can be linearized. In this laboratory experiment, you will investigate the dynamics of an inverted pendulum with a torsional restoring force. Because the equation of motion can be approximated by a nonlinear equation known as the Duffing equation, this sort of pendulum is known as a Duffing oscillator. As you will discover, although the system is very simple (a flexible steel ruler with a little weight on top--sort of like a metronome) the motion has regimes where the behavior is neither regular, nor predictable: knowing the position and velocity at some point in time is of no use in predicting the posiiton and velocity at some later time. This unpredictability has nothing 1

2 to do with randomness, however, as can be readily shown by simulating the motion on a computer. The version of the Duffing equation that describes the inverted pendulum is a nonlinear, second order, ordinary differential equation, and can be written in dimensionless form as (see Question 1) θ = θ bθ c θ + f cosωt (1) where q is the angle the pendulum makes with the vertical, and the unit of time is taken to be l / g. The chaotic behavior of the system arises from the θ 3 term. Without it (and with b negative) Eq. (1) would describe a damped, driven harmonic oscillator. The full equation can exhibit periodic motion, but also considerably more complicated behavior. The full equation-- like most other nonlinear second order ODE s--also has no closed form solutions. That s why you mostly encounter linear equations: the answer can usually be written down as, for example, a sum of cosines or Bessel functions. Nonlinear equations on the other hand, must (almost always) be solved numerically, as you will do in this lab. Equation (1) assumes, among other things, that the pendulum arm is a rigid body. Our actual mechanical oscillator is made from an ordinary steel machinist s ruler and a small brass weight. The bending of the ruler provides the torsional restoring force. The bending also violates the rigid body assumption of Eq. (1). Nevertheless, you will find that the behavior you observe in the experiment is very much the same as that predicted by the above equation of motion. In other words, there are aspects of the behavior that are universal in the sense that they depend only on some general characteristics of the system rather than on the exact details. In this case, the universality arises from the shape of the potential in which the system moves. II. Characterizing Chaos Given that the behavior of a chaotic system is very complicated, how are we to describe it? By far the most useful description is through the use ofphase space. Technically, phase space is a multi-dimensional space with one dimension for each of a system s generalized coordinates and generalized velocites. (See Classical Mechanics by Marion, Ch. 7, for a discussion of generalized coordinates and velocities) For example, the phase space of a particle moving in three dimensions is 6-dimensional: one dimension for each of three coordinates, and one dimension for each of three velocity 2

3 components. Our inverted pendulum has a much smaller 2-dimensional phase space specified by q and θ. To understand what makes chaos different from other behaviors, let's start with a couple of simple examples. Consider a damped simple pendulum. Start it at some initial angle, and let it go. It will oscillate back and forth in smaller and smaller swings, and eventually just hang straight down. In phase space, it will follow a spiral path toward the point (0,0). If we start it with both an initial angle and and initial velocity, it will follow a different spiral, but still wind up in the same final state. The phase space path that the motion settles into is called an attractor. In this case the attractor has a very boring shape: it s just a single point, called a fixed point. Now consider a perfect pendulum with no damping. If we set it swinging at some small angle, the motion will be simple harmonic, and it will swing forever (no friction). You should take a moment to convince yourself that the path in phase space is now a circle. Since the phase space motion just continually retraces itself, the circle is also an attractor, and in this case is known as a limit cycle. Now consider our Duffing oscillator. It has friction, and is being driven at some frequency w. This system can also display chaos, but for the moment let's only consider drive amplitudes that are small, and drive frequencies that are far from the natural frequency of the pendulum. In this case, after the transients associated with the initial conditions have died away, the motion is again periodic, with a limit cycle attractor. The pendulum just moves back and forth about its equilibrium positon. To get ready for chaos, we need one more visual tool. Let us introduce the variable φ = ωt, and make it periodic. That is, when enough time has gone by so that f has gotten up to 2p, we reset it back to 0. Now get ready to think in three-dimensions. We ll take the θ θ plane as being vertical and displace it from the origin by some large distance. The variable f is the angle that this plane makes with a fixed vertical plane, so that as f swings through 2p, the θ θ plane makes one cycle about the origin. The circular shape of the attractor in the θ θ plane will be swept along the surface of a toroid (doughnut). The actual attractor is not the entire surface of the toroid, but only a spiral confined to the surface. As f goes through 2p, the pendulum makes one circle around the origin and one circle around the minor diameter of the toroid. The full path is a spiral that winds around the origin and closes up on itself. 3

4 This last step may be a bit hard to visualize, so Fig. 1 may help. The solid line is the attractor for a simple harmonic oscillator (not the Duffing equation). In this particular case, the winding variable f has been chosen to be φ = ωt / 5, so that the oscillator makes 5 complete orbits around the circle in the θ θ plane while f goes through 2p. This factor of 5 was added only to make the attractor wind more times around the torus and show more clearly what is going on. Because this attractor describes simple harmonic motion, it is periodic, and closes on itself. Fig. 1 If we now drive the oscillator a little harder, and a little closer to its natural frequency, then the fact that the motion is not really simple harmonic will start to be evident. The cross section of the toroid will no longer be circular, but the attractor is still a limit cycle, i.e., still a closed spiral around the origin. If we drive the pendulum just right, it will become chaotic. As you will see, the cross-section of the toroid becomes dumbell shaped. In addition, the path of the system is no longer confined to exactly the surface of the toroid: it can be inside or outside at any particular instant as it spirals about the origin. Finally, and most importantly, when f has gone through 2p, the curve does not 4

5 close on itself. In fact, as time goes on, the path of the system never intersects itself. In each oscillation of the drive, the state of the system makes one orbit around the origin, and at the same time moves around within this dumbell shaped region of the θ θ plane, but it never returns to any of the (θ, θ, φ ) points that it has already visited. This dense, three-dimensional, non-intersecting tangle is the attractor for the chaotic system, and is known as a strange attractor. A three dimensional doughnut shaped tangle of paths in phase space is difficult to draw, and difficult to read. As an example, Fig. 2 shows a portion of the strange attractor for the Duffing oscillator. In this case, the attractor goes once around the origin for each single cycle of the drive (not once around for each five drive cycles as in the last figure). You can see that the paths sort of cluster around a doughnut shaped region of phase space, but you can t tell much else! Fig. 2 In order to try and get a better understanding of the attractor, there are two ways of looking at it that are usually used. The first is to draw only the θ θ plane and ignore f. 5

6 In this view, the paths in phase space appear to intersect, but that is only because they have all been compressed together along the time direction. The second is to draw only a single cross-section of the toroidial attractor. This is known as a Poincaré section. Computer programs for exploring both of these views are part of this experiment. III. Equipment The equipment is shown schematically in in Fig. 3. The inverted pendulum is a 12" machinist's rule to which a movable brass weight is attached: the position of this weight determines the relatives sizes of the q 3 and q terms in the Duffing equation. The drive is provided by two electromagnets acting on a magnet made of neodymium-iron-boron (a very "hard" permanent magnet material: a hard magnet is one which retains its magnetization even when subjected to strong external demagnetizing fields) attached to the ruler. The whole system is mounted on a table which can be accurately levelled, and enclosed in a box to exclude drafts. The angle of the pendulum relativet o the vertical (i.e. q ) is measured by two strain gauges glued to each side of the ruler. A strain gauge is a resistor whose resistance changes when the gauge is stretched or compressed. These gauges are wired to form a "Wheatstone Bridge", as shown in Fig 4. When the ruler is straight, the resistances of gauges A and C and of B and D are equal in pairs, and there is no output voltage (see Question 3): the bridge is then said to be balanced. When the ruler bends, two of the resistances (say A and B) increase and the other two (C and D) decrease: this unbalances the bridge and produces a voltage. This voltage is amplified, and its derivative taken by the circuit shown in Fig. 5. The outputs of this circuit are proportional to q and to θ respecitvely, and are fed to two channels of a digitizing oscilloscope. IV Procedure A. Alignment 1. The brass wieght should not be on the ruler. If it is, take it off. WHEN YOU ARE DONE FOR THE DAY, YOU MUST REMOVE THE WEIGHT!!! This means that data taken on different days will not be intercomparable. In other words, plan to spend half a day checking the system out and seeing what it does, and then a nice long day taking all your final data. If you leave the weight on the ruler, you will 6

7 permanently bend it, and then have to replace it, which will take you much more time!! 2. Level the base along the width using a small bubble level and the two leveling screws. 3. Using a plumb line (any piece of string with a weight on the end) hung from the ceiling, adjust the third leveling screw until the ruler lines up with the string. (Trick: to keep the plumb line from swinging about wildly, put a cup of water so the weight hangs into it.) 4. Replace the brass weight on the ruler. Adjust the position of this weight so that its top is 0.14 from the top end of the ruler -- just high enough to make the ruler flop over to one side. Make sure that the ruler doesn t want to lean more to one side than the other. If it does, make very small adjustments to the leveling screw until it doesn t. 5. Check that the q output from the electronics box is connected to channel 1 on the oscilloscope, and that the θ output is connected to channel 2. Put the scope into x-y mode and set both channels to read 500 mv per division. Turn on the electronics box. If everything is set up correctly, there should be a spot on the oscilloscope screen that wanders around as the pendulum moves back and forth. If you hit the Auto Store button in the upper right corner of the scope front panel, the spot will trace out a line as it moves around. You can use the Erase button to do what it says. Check the DC offsets on the scope inputs by grounding both channels and adjusting the position knobs so that the spot is in the center of the screen. 6. You may have to balance the electromagnetic drive. With the drive off (just disconnect the cable from the front panel of the signal generator) set the pendulum into a large amplitude oscillation, and measure its frequency. Now set the signal generator to that frequency and set the drive amplitude to 10 V. The trace on the oscilloscope should be a sort of dumbell, and it should not favor one side of zero or another. If it seems to be far off, you can use the two horizontal adjustment screws to shift the magnets and even out the shape of the curve. If you have to make a really big adjustment, you probably didn t have the ruler precisely vertical. In that case, you should repeat the entire alignment procedure. 7

8 B. Experiment 1. With the signal generator off, use a stop watch to measure the natural frequency of the oscillator in one of the two potential wells. Your expression for this frequency is only valid near the very bottom of the well, so be sure to use very small oscillations. Our equation for the motion uses the quantity l / g as the unit of time, so you should measure l and convert your frequency to these units (don't forget to also express it as an angular frequency in radians per unit time). 2. Still with the signal generator off, measure the rate at which these oscillations decay. There are many ways to do this. One possibility is to use the scope in storage mode, and time n oscillations of the pendulum. You can use the cursors on the scope to measure the initial and final amplitude of oscillation. The coefficient c in Eq. (1) is then given by c = 2ln(A f / A i ), T where T is your measured time (in units of l / g ). 3. Find regions of chaotic and periodic behavior. This will take considerable patience. Here is one possible procedure. Set the drive frequency to 0.20 Hz (this is close to your measurement in part 1?) Set the drive amplitude to 2V and start the pendulum from a known initial condition. The nicest way to do this is to disconnect the drive cable from the front of the signal generator, and manually stop the pendulum so that it sits with (almost) no oscillations in one of the equilibrium positions. Then reconnect the drive. (You should do this procedure every time you change the drive level or frequency.) Determine whether the limiting behavior is periodic or chaotic. You should be prepared to wait 15 or 20 minutes: something that looks chaotic might eventually settle down into a periodic orbit (and vice versa). You should keep nearby windows closed and the plexiglass cover over the oscillator. Air currents can easily kick the oscillator into what is known locally as a shoe orbit from which it will not recover without manual intervention. Disturbances will also make periodic orbits noisy and blur the boundary between chaos an periodicity. Try not to bump the table, and be especially careful when hitting the Erase button on the oscilloscope. 8

9 If the behavior was periodic, make a big leap up in drive amplitude (double it, or so) to find chaotic behavior (and vice versa if it was chaotic). By continually splitting the difference between chaotic and periodic drive levels, you should be able to localize the boundary between the two regimes to.001 V in the drive level. 4. Once you have established the boundary between periodic and chaotic motion as a function of drive level at fixed frequency, set the drive to be just inside the periodic region, and find the boundary as a function of frequency. You should look in both directions: both higher and lower frequencies. Localize these boundaries as well as you can. 5. Hard copy. There is a very fancy Hewlett-Packard x-y plotter on a cart in the lab. Use it to make some hard copy pictures of the chaotic behavior that you ve found. 6. What if I can t find chaotic behavior? Try moving the weight farther up the ruler. Try halving or doubling your drive frequency. Chaos is out there. This device should do more chaos than just about anything else. 7. When you are done, TAKE THE BRASS WEIGHT OFF THE RULER! 8. TAKE THE BRASS WEIGHT OFF THE RULER!! 9. By the way, did you remember to take the brass weight off the ruler? 9

10 Questions 1. Consider a simple pendulum composed of a massless shaft of length l and a bob of mass m. In addition to gravity, a torsional restoring force of magnitude aq acts on the pendulum, where q is the angle the shaft makes with the vertical. a. Show that for sufficiently small q, the equation of motion for the pendulum, is given by ml 2 θ = (mgl + α)θ where g is the acceleration due to gravity. (Q1) b. If dissipative effects such as friction are included, the above equation becomes ml 2 θ = (mgl + α)θ k θ, (Q2) where k is a (small) damping coefficient. Assume that at t = 0 the pendulum has zero angular velocity and is at some angle θ 1. Find an equation for the subsequent motion of the pendulum. You may assume weak damping, i.e. k 2 /ml 2 <<(mgl+a). c. Now consider the same pendulum, but in an inverted configuration: the bob is at the top of the shaft, and the fixed point about which it rotates is at the bottom. Find the new equation of motion using the same small angle approximation as in part (a). Find the value of m (for a fixed l) for which the inverted configuration is unstable and the pendulum will flop over rather than remain upright. d. If the pendulum is unstable, then as it flops over, the small angle approximation will clearly be violated. Show that to the next order in q the equation of motion can be written in the following dimensionless form θ = θ bθ c θ, (Q3) where the unit of time has been taken as the natural period of the non-inverted pendulum. In other words, parameters m, l, and k. l / g = 1. Expressb and c in terms of the orginal 10

11 e. Show that if the vertical configuration is unstable, then b is always positive and there are now two equilibrium values θ 0 for the position of the pendulum. Express θ 0 in terms of b f. Equation (Q3) can be thought of as the force on a particle. If the non-conservative damping term is neglected, this force can be derived from a one-dimensional potential. Find an equation for that potential. Use Mathematica or the equivalent (or even a calculator and some graph paper) to plot this potential and compare it to the potential you would get if you did not expand the sine function in the equation of motion. Do this for b =.2,.1, and.05. g. Show that for sufficiently small deviations dq from equilibrium, the motion is again simple harmonic, and find the resonant frequency in terms of the parameter b. (Take c = 0 for this calculation.) 2. With the addition of a driving term, Eq. (Q3) becomes a form of the Duffing equation, which is written most generally as: θ = aθ 3 + bθ cθ + f cosωt. (Q4) For the inverted pendulum, a = 1/6. Because of the particular nonlinear form of this equation, there are no neat, closed form solutions for θ(t) as there are in the case of Eq. (Q2). Instead, solutions for a particular set of parameters and intitial conditions must be found by numerically integrating the equation on a computer. (For a discussion of analytic techniques for solving the Duffing equation, see Harold T. Davis Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962), pages This book can be found in the Mathematics library: QA/303/D268.) In the Physics 49 folder on the Public file server, you will find a pair of True BASIC programs entitled duffing and poincare which you should download to your own Mac. If you don t have True BASIC installed already, you should download that too. a. Start with the program duffing, which numerically integrates the Duffing equation using a Runge-Kutta technique with an adaptive step size. It makes a plot of q vs. θ, with a few additional bells and whistles. The program is self- 11

12 documenting, and you should use it to go exploring. Try running it right off the bat, just to get the flavor of what it does. b. The following parameters have been chosen to approximately mimic the actual apparatus: B=.03, C =.02, and W =.2. Use initial conditions XINT = -.42, VINT = 0. This starts the oscillator at rest in one of it s minima. Explore the behavior for F in the range from to How many steps in the period doubling cascade can you observe? Do all the periodic orbits have a power-of-2 periodicity? c. Explore widely. Report on anything interesting you find. Some hints on using the program. Set up your output window so that it is almost entirely hidden by the listing window. While the program is running, you can then click on the listing window and click back on the output window, which has the effect of erasing the window. This is very useful for seeing if the program has settled down into a periodic orbit. While looking for interesting behavior, it is probably best to set ANIMFLAG = 0. Then, when you need to investigate the periodicity of an orbit, you can set it equal to 1. By adjusting XOFFSET, VOFFSET, XSCALE, and VSCALE, you can enlarge or shrink the output, as well as move it around on the screen. You can take a snapshot of the screen by pressing COMMAND+SHIFT+3 and then use a graphics program to clip out only the part you want and print it or paste into a Word document. d. You can use the program poincare to make a Poincaré section of an attractor. You should run this for a long time: 10,000 drive cycles or so. The program writes to the screen rather than to a file, so be sure to turn off any screen saver program you may have installed, or you ll lose all your output. Set the program running for a very large number of cycles just before you go to bed, and you ll have very nice results in the morning. You can also use this program to verify the periodicity of the limit cycles you found using the duffing program. 3. a. Show that the Wheatstone bridge of Fig. 4 is balanced (i.e. V out = 0) if A/D = C/B. 12

13 b. Assume that in the absence of strain A=B=C=D. The effect of a given strain is to increase A and B by a fraction e and to decrease C and D by the same amount (e << 1). Find V out / V in.for this strain. c. Explain why is it desirable to use the strain gauges in pairs, with one member of the pair on each side of the ruler, rather than a single gauge on one side. (Hint: consider the effect of a uniform change of temperature on the balance of the bridge). Fig. 6 Poincaré diagram for b = , c = 0.033, f = 0.06, and w = The attractor was sampled half-way through the drive cycle. 13