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    1 Phys15c: Lab2, Fall 2009

    Lab 2: The Coupled Pendulum and a bit on the Chaotic Double Pendulum

1 Introduction

    This lab looks at coupled harmonic oscillators (one of the oldest systems of interest in physics), the double pendulum and the chaotic double pendulum (a system from one of the newest fields of physics). The lab requires the use of a digital oscilloscope, in order to detect the frequency spectrum of the coupled pendula. The ―’scope‖ (normally

    referred to as just a ―scope‖), which presents its options under menus, is relatively easy to useor, at least, perhaps

    less formidable than an analog scope.

    In preparation for the lab you may want to look at any of several treatments of coupled pendula and normal modes: for example, Chapter 3 in Georgi or in Pain or (very helpful on the particular case:) pp. 32-36 in F. Crawford, Waves (Berkeley Physics Course, volume 3 (1968)). We will try to review the basic properties of the coupled harmonic oscillator at the start of the lab session.

Reminder: ―f‖ and ω: while angular frequency, ω, is convenient in the expressions describing the behavior of

    harmonic oscillators, what you will observe is not angular frequency, but oscillations per second or Hertz. As you

    know, ω=2πf.

2 The Coupled Pendulum

    The coupled harmonic oscillator may be the most important classical system that you will study. The basic behavior of such systems comes up over and over again in every branch of physics. (The quantum mechanical analog differs little from the classical system.) We will study the behavior of coupled pendula in detail.

    Figure 1: Coupled Pendulum

    2

    2.1 Coupled Pendula, in a Nutshell

     Modes: The behavior of many systemspendulums, vibrations on strings, musical tones,

    energy levels of atoms, for examplecan be explained in terms of normal modes. For our

    purposes normal modes can be thought of as ―stable‖ or preferred modes (or, to use more

    ordinary language,―preferred patterns‖) of oscillation. If we start our pendulum in one of its

    normal modes it will stay in that mode of oscillation. The normal modes of vibration of a pendulum have frequencies associated with them. A single pendulum is capable of only one mode of oscillation; so, for small displacements it will always swing with a particular frequency. The coupled pendulum shows more complex behavior; it has two normal modes, each with its own associated frequency.

     Linearity: The ―normal modes‖ description of behavior is useful because, for linear systems, all of the behavior of the system is simply a linear combination of the normal modes. This means, for example, that we can describe the frequency of oscillation of either one of our compound pendulums as

    1+Bsinf2 Asinf

     Fundamental Frequencies: simple pendulum

For a simple pendulum, force balance gives:

     22 F = m dx / dt = - mgx / L

and

     22dx / dt + (g/L)x = 0

    The fundamental frequency ωsimple is equal to the square root of the coefficient of the x term or ωsimple= ?(g/L)

     Fundamental Frequencies: coupled pendula

Coupled pendula exhibit two fundamental frequencies which correspond to a parallel mode of oscillation

    and a symmetric mode of oscillation.

    Parallel Mode: The resonant frequency of the parallel mode is the same as that for the simple pendulum. Describe why.

    Symmetric Mode: In the anti-symmetric mode the force equation contains one more term, describing the effect of the coupling elementin our case, a spring:

     22 F = m dx / dt = - mgx / L 2kx

     (―2kx‖ because anti-symmetric displacement of x for one mass causes a stretching of the spring by twice that amount). And

     22dx / dt + (g/L + 2k/m) x = 0

    This results in a modified fundamental frequency for thesymmetric mode, of the form

    ω = ?(g/L + 2k/m) symmetric

     Superposition and ωc and ωe

    cos A + cos B = 2 cos( {A-B}/2 ) cos ( {A+B}/2 )

    The oscillation of either one of our coupled pendulums will look like two different sine waves added

    together. The two superposed sine waves look like a high frequency wave that is being modulated in

    amplitude at a lower frequency. The higher frequency feature is called the carrier wave, ωc, and the

    lower frequency modulation is called the envelope, ωe. These two frequencies are linear combinations

    of the fundamental frequencies. In particular:

    ω = (1/2) (ω + ω) carrierparallelsymmetric

    and

    ω = (1/2) (ω ω) envelopeparallelsymmetric

2.2 Setting up the Apparatus

Adjusting pot and scope

On each shaft on which a pendulum will be hung, a ―potentiometer‖ is attached. A potentiometer (or ―pot‖) is a

    device whose out voltage varies as the ―slider‖ moves, making contact with a fixed resistor that runs between the two endpoint voltages. Thus, the pot produces an output voltage that varies as the shaft turns. It’s a very simple device,

    no doubt familiar to many of you:

    Figure 2: A potentiometer lets one pick off a ―potential‖ between its extremes

    1In the lab, a pendulum is placed on the shaft of each pot, thus:

Figure 3: Pendulum hung from pot lets V out read pendulum angle

    1In addition, series resistors on the ends of the pot restrict the range of voltages available atVOUT to much less than the whole applied voltage range.

Phys15c: Lab2, Fall 2009

You will drive the V+ and V- ends of the pot with the positive and negative outputs from a ―split‖ power supply—

    one that provides both positive and negative voltages, relative to its ―COMMON‖ terminal. Here’s what the power

    supply front panel looks like:

Figure 4: ―Split‖ power supply powers the two ends of each pot

We use a split supply in order to place V close to zero volts, and to keep its swings approximately symmetric OUT

    about zero.

Watch V with channel ONE of the digital scope. The scope ground (the black alligator clip) must be connected OUT

    to the power supply’s COMMON terminal. Now move the pendulum left and right by hand. Adjust scope gain so

    as to get a few divisions movement of the scope trace, as the pendulum swings.

    If the scope shows a voltage centered close to ground (zero volts), fine; if it does not, then loosen the clamp that holds the potentiometer in place and rotate the pot until V is close to zero when the pendulum is vertical. OUT

Couple the Pendulums

When you have adjusted both pots, and are watching both pots on channels one and two of the scope, it’s time to

    couple the two pendulums.

If the pendulums were left in the parallel configuration you will need to move them to the side-by-side configuration.

    Undo the bolt, move the pots so that the pendulums swing side-by-side as in figure 3 and fig. 5, below. There is a hole in the side of the aluminum beam so that you can put on the nut. Tape the middle joint of the double pendulums so that you have two simple pendulums. Then join the two pendulums with a length of spring.

Figure 5: Two pendulums coupled, again

2.3 Useful Information for Taking Data and Using the Scope

    Please note that this section is not the procedure section. The exact procedure is outlined later, however you need to read and understand this section in order to take good data later.

    You can then play with the coupled pair: if you start them swinging in unison, then they will behave like twin single pendulumsfor a while, at least. In contrast, if you start them with opposite phases (one to left, the other to the right) then you will see the other, quicker ―mode,‖ the oscillation whose period is affected by the spring. A third

5

Phys15c: Lab2, Fall 2008

interesting case occurs if you start one pendulum only: you will see it swing for a while, then gradually transfer its

    energy to the other, whence it will be transferred back, and so on . . . .

Once you’ve seen this qualitative behavior, take a look at what the two pots are showing on the scope screen. One

    of the pots showed us, for example, the following trace:

Figure 6: Scope image of one pendulum’s behavior, when coupled

This image is of some value. From it, for example, you could get a good estimate of the frequency of the ―envelope,‖

    at least. This envelope frequency we met earlier, as

ωenvelope = (ω ω) / 2 parallelsymmetric

But the scope can do much more to help us interpret this waveform. In particular, it can show us a frequency spectrum

    of the waveform we just looked at: it will do a ―fast Fourier transform‖ and show us the result. Here’s the FFT shown

    below the trace we displayed in figure 6, above. Here are the two traces: the original (―time-domain‖) and the FFT

    (―frequency-domain‖):

Figure 7: FFT shows frequency analysis of the original waveform

    The two peaks show the frequencies of the two oscillation modesthe parallel (the lower frequency), and the symmetric. We can expand the FFT to get more detailed information:

    Figure 8: Expanded FFT delivers more information about the coupled oscillation

6

Phys15c: Lab2, Fall 2009

    How to expand the FFT Step one: press the MATH button; and then increase the ―sweep rate,‖ using the knob labelled HORIZONTAL. You will notice that the region displayed narrows in frequency, and you can see that on the bar at the top of the scope display. In order to see the interesting part of the FFT, in this case very-low-frequency peaks, you must display the range of frequencies close to zero. To achieve this, use the HORIZONTAL POSITION knob to move the region that you want to watch in detail to the leftmost end of the display bar (detail below):

Figure 9: Use horizontal controls (sweep and position) to watch low frequencies

In the figures above (figs. 9 and 8) we have also taken advantage of the scope’s vertical ―cursors‖ to find the

    frequencies of the two peaksand the difference between these two frequencies, here, 182mHz. From this difference you could predict the ―envelope‖ frequency.

Phys15c: Lab2, Fall 2009

Patience is Rewarded

You will discover that you can get better resolution if you take your data more slowly. Here, we slowed the scope’s

    ―sweep rate‖ by a factor of five, from 2 seconds/division (―2 s/div‖) to 10 s/div. Gathering the data took more than 1

    1/2 minutes. But the frequency peaks are better defined, as you can see.

    Figure 10: Slower acquisition rewards you with better frequency resolution

    In fact, no new information appears; what was dimly-visible before is now just a lot prettier. From these screen images you should extract the frequencies of parallel and symmetric oscillation, and the

    difference between these two, from which you can predict envelope frequency. Don’t forget that the cursors can

    help you to find all these values. Then compare your envelope prediction against what you observe, as we have said above.

Procedure: Coupled Pendulums

     Excite the 2 normal modes and find the 2 normal mode frequencies by:

     Using the time waveform and measuring 1/T.

     using the FFT and zooming in on the peak.

     Excite a non-normal mode and do the following:

    o Set one of the pendulums in motion. Optimize the system (as is described earlier) so that you

    have good coupling data on your scope screen. Print out the time data and tape or stable it into

    your notebook.

    o Use an FFT to find the 2 peaks of the normal modes together and print them out as well. Label the

    normal modes with their frequencies. Verify that these frequencies correspond to the frequencies

    of the peaks found in Part 1.

    o Measure the frequency of the carrier and envelope wave from the time waveform then very the

    relationships: ω = (ω + ω) / 2 and ω = (ω ω) / 2 carrierparallelsymmetricenvelopeparallelsymmetric

     Change the magnitude of coupling by changing the spring constant K

    8 Phys15c: Lab2, Fall 2009

    Do this by putting springs in parallel and in series and measuring the normal modes and the envelope fre-

    quency. Comment on the effect on the two normal-mode frequencies. Also note and describe the effect of the

    spring constant upon the rate of energy transfer between the two pendulums. Check your observations against

    what the equations predict and comment.

3 The Compound Pendulum or Double Pendulum

3.1 The Compound Pendulum, in a Nutshell

    Figure 1: Pendulum, Double Pendulum and Phase Space

    Consider the case of the double pendulum shown in figure 2. It turns out that for small angles we can describe the motion of the pendulum with 2 coupled equations resulting in 2 normal modes and 2 fundamental frequencies. For larger angles, however, the equations of motion become non-linear. These equations can be solved numerically on

    a computer. When this is done the behavior seems chaotic. What is chaotic?

    A definition of chaos is that nearby trajectories in phase space tend to diverge exponentially in time.

What is phase space? Consider the simple pendulum. As we have seen for small angles sinθ ? θ and

The last equation is that of a circle in dθ/dt, θ?(g/l) space. The trajectory that the particle takes in phase space is along

    the circle. In the case of a damped harmonic oscillator the trajectory is a spiral. Consider a trajectory that is started with the initial condition θ (dθ/dt) and a trajectory started with almost exactly the same initial conditions θ (dθ/dt). , , 0011

    These two trajectories are (at least initially) nearby trajectories. Now the above definition of chaos should make sense. λtIf the distance (s) between nearby trajectories is phase space increases as an exponential function of time (s ; e)

    then the system is chaotic. (It turns out that in order for these trajectories to really diverge exponentially, the phase space variables must be chosen properly. For our system the simple variables chosen are not the right ones, butthe idea is the same.) Don’t let this discussion throw you: we’re really saying something very simple in somewhat high-fallutin terms!

    9 Phys15c: Lab2, Fall 2009

    Figure 2: Divergence in Phase Space

3.2 Procedure for the Compound Pendulum

    The double pendulums we have set up are chaotic for large angle motion. In this section of the lab we will measure angle vs. time of two ―identically‖ prepared double pendulums (each pendulum is jointed). Qualitative aspects of

    chaos will be demonstrated.

    1) Set up the pendulums in the parallel configuration. Make sure that the pots are set up so

    that voltage is proportional to angle over our range of interest. Begin with just one compound

    pendulum.

     Excite just one of its fundamental modes. Describe the mode and sketch it in your

    lab book. Measure the frequency of this mode using the FFT.

     Exite the other fundamental mode, describe it, sketch it and measure its frequency.

     Excite a low amplitude oscillation that involves both modes. Print out the FFT, label the

    two frequencies and verifty that they are the same as the frequencies that were measured

    earlier.

    2) To study chaos prepate the two pendulums, as carefully as possible, with identical initial

    conditions. Release them at the same time. Watch the two pots with two scope channels.

    Print out the time data. How long did it take the behavior of the two pendula to significantly

    diverge? Look at the FFT and print it out. Describe how the FFT for the ―chaotic‖ pendulum

    differs from the FFT for the small amplitude compound pendulum. Explain what the difference

    has to do with the concept of linearity.

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