Oscillations, Resonance & Normal Modes
(Suzanne Amador Kane 10/2008)
Introduction
Physics 213 and many of our experimental phenomena in Physics 211 lab rely on an
understanding of oscillatory motion. In this lab, you will be able to explore a mechanical
analog of the electrical oscillations and resonant behavior you saw in the earlier RLC
circuits lab. You will use a torsional oscillator first: a mechanical device in which a
circular disk attached to a stiff wire can undergo rotational oscillations. Second, you will
use a series of masses and springs arranged on an air track to explore the phenomenon of
coupled oscillations and normal modes.
Oscillations and resonance in a torsional oscillator
Background: What’s a torsional oscillator? Why torsional oscillators?
You will explore mechanical oscillations and resonance using a torsional pendulum. (Fig.
1) These devices are important for experimenters measuring a variety of phenomena
because they can convert a tiny signal (such as a very small force) into a large angular
displacement that is more easily measured and calibrated. In fact, many fundamental
experiments on gravity have been done using torsional pendula and oscillators.
Cavendish and Eotvos used these devices in their early explorations to determine the 1/R2
dependence of the gravitational force, to measure the gravitational constant, and to
determine its independence on the exact material makeup of the masses used. Even
today, groups like University of Washington’s EotWash research team use torsional
oscillators to probe for departures from these classical results, searching for experimental
signatures that allow them to look for prediction of string theory and other theories which
combine gravity with particle physics. (In a less happy outcome, the famous Tacoma
Narrows bridge disaster had an entire suspension bridge undergoing resonant oscillations
that included a strong torsional oscillatory component! Structural engineers must worry
about these types of oscillations in designing structures to withstand earthquakes,
windshear and other environmental driving forces.)
Our TeachSpin torsional oscillator consists of a very strong steel wire that runs vertically
through the device and is connected to a copper rotor disk. (Fig. 1 & 2) The copper rotor
gives the device a considerable moment of inertia, I, which plays the same role as mass in
a spring-mass-dashpot system. You can change the moment of inertia by adding
additional masses that are stored in a small cardboard box. The stiff steel wire responds
to any twisting motion that rotates the rotor disk from equilibrium by generating a
restoring torque, analogous to a spring’s restoring force. (Even an unstretched wire
would generate this torque, but our wire is also stretched so it is under tension. The
higher the tension, the greater the restoring force. The natural frequency of our torsional
oscillator is relatively independent of tension since the steel of the wire is so stiff to begin
with.) Just as you can characterize a spring by its spring constant, k, the steel wire has a
rotational spring constant, . Together, they determine the natural (or resonant)
frequency of the torsional oscillator via:
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o 

I
,
the rotational analogy to o 
Resonance & Normal Modes
(1)
k
for a spring-mass system
m
Figure 1: The Teachspin Torsional oscillator.
The rotor also is attached to an angular position
transducer, which allows you to directly read out
the angular position, , as a function of time via a
BNC connector on the front panel. You can read
out the angle in radians using a calibrated scale
glued to the rim of the copper rotor; a clear
plastic guide allows you to read out position.
(Fig. 3)
Your device can also measure angular velocity
directly using a clever magnetic system. Also
attached to the rotor disk is a set of strong
permanent magnets. These have a magnetic
dipole moment oriented out of the page in Fig. 2.
On either side of these magnets are a set of
“Helmholz coils” located just under the copper
rotor. Helmholz coils are two sets of circularly
wound coils with a common central axis. Our
coils are wired so they are both in series and
wound in the same orientation (i.e., both are
either clockwise or counterclockwise, depending
on your perspective. When the entire device
rotates, this relative motion between the
permanent magnets’ dipole and the Helmholz
coils creates an induced EMF in the Helmholz
coils that is proportional to the angular velocity
of the moving rotor. (They do not interact with
the nonmagnetic copper of the rotor however.)
By setting a front panel switch, you can measure angular velocity directly by measuring
this signal, which is available on the main panel via BNC connections.
Our device has been constructed so it can either oscillate freely, or else be driven by an
external forcing signal. This external torque is applied using the very same Helmholz
coils and permanent magnets described above. You can switch the Helmholz coils so
they are connected to two gray BNC connectors on the front panel. A function generator
can then be used to apply a driving voltage to these connectors. This results in a timevarying current through the Helmholz coils, which generates a time-varying magnetic
field, B, that creates a time-varying torque, d , on the permanent magnets attached to the
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rotor: d =   B. This plays the same role as the driving force, Fd, in lecture and your
textbook. The driving waveform can be sinusoidal, with a varying amplitude and
frequency f d =  d /(2, set by the function generator.
Figure 2: Detailed view of the central rotor and angular position transducer section.
In your Physics 213 lecture and textbook, you have encountered the equations for
oscillatory motion using the spring-mass-dashpot system. In our device, we have a
similar equation (see your text’s Chapters 3 and 4 for more background). Fundamentally,
these equations are just special cases of Newton’s 2nd law for angular motion:
net = I = I d 2 / dt 2







where  = torque,  = angle and  = angular acceleration. For our system, the net torque,
net , has two components. The first is the restoring torque that results from rotating the
rotor from equilibrium, which obeys Hooke’s law for rotational motion:
spring  








where  = angular spring constant. The second is a contribution due to the inevitable
frictional or damping forces in the environment. The damping in your wire + rotor
device is itself very small. However, there are two sets of magnetic dampers installed on
either side of the rotor disk. (Fig. 3) These consist of permanent magnets mounted on
small translation stages that can either be fully retracted, for essentially no magnetic
damping, or translated so as to hover over the copper rotor disk. When the magnetic
damping is engaged, the relative motion between the magnetic dampers and the rotating
rotor disk generates eddy currents in the copper. This creates a damping torque that is
proportional to angular velocity:
damping ddt
(4)
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Figure 3: Magnetic dampers
The resulting differential equation is:

     o2  o cos d t
(5)
I
Where we have introduced a damping constant /I and we have assumed a sinusoidal
driving torque. You have encountered the solutions to the resulting equations of motion
in Physics 213.
Prelab Exercise 1:
Find in your textbook and notes the following relationships to use in your experiment
and data analysis. Mathematically these will be the same relationships as you’ve
studied before, but you will need to understand how the parameters correspond to
make sense of your results in lab.
a) The relationship between amplitude and phase vs. driving frequency for a
driven torsional harmonic oscillator.
b) The phase difference between the driving force and the angular displacement at
resonance and at frequencies far above and below resonance.
c) The definition of quality factor, Q, and its relationship to the equations and
parameters in part a)
d) The equation of motion for a damped, undriven oscillator. Your answer should
give (t) as a function of time, in tersm of the parameters described above.
Experiment 1: Getting acquainted with the torsional
oscillator
Your first activity will be to get used to the operation of your torsional oscillator by
setting it into torsional oscillations. Read through the background section above quickly
identifying each of the elements described. First, you will observe free, undriven
oscillations with minimal damping. Using the thumbscrews on either side of the rotor,
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retract the magnetic damping magnets as far as possible. (Fig. 3). Then, use the small
clamp in the bottom half of the steel wire to rotate the rotor slightly by hand and release
it. (Do not exceed 1 radian of motion in your experiments, so as to avoid damaging the
apparatus.) Your oscillator should undergo slow (~ 1Hz) oscillations that are easy to
see by eye. Now, connect your oscilloscope’s Channel 1 input to the Angular Position
BNC output from the front panel. (Fig. 4) You should be able to get a display on your
oscilloscope of the sinusoidal output signal, although you may have to use a very long
time per division setting (1 sec/div or thereabouts.) This means each sweep of the signal
may take a long time to refresh on the oscilloscope screen, so be patient if you do not see
a signal for a moment.
Figure 4: Front panel connections & switches.
You can zero this Angular Position signal so it is centered on GND using the small black
knob labeled “Zero Adjust” on the left middle panel. Do so now if it is not centered.
In the absence of magnetic damping, you should observe free oscillations with a
magnitude that varies little with time. Record your natural frequency, fo, for later
reference. Vary the natural frequency by adding additional weights as shown in Fig. 5.
Each quadrant weight adds I = kg/m2, where
M  R12  R22 
I 
(6)
2
Record how the resonant frequency changes as you add each mass, for several different
masses. Measure the mass of each with the scale provided and the two radii, R1 and R2
(see Fig. 5) so you can compute I.
After you have finished the experiments below, return to this point and use this data to
extract the rotational inertia of the rotor disk (without added masses) and the wire’s
rotational spring constant. Do this using a plot that utilizes your data for resonant
frequency as a function of the number of added masses (or, equivalently, extra I.) Then
use a fitting program to deduce Io, the rotor disk’s rotational inertia, and the torsional
spring constant, . (For example, think of how you could use a simple linear plot—of
what vs. what?-- to extract these parameters.) What are your units and error bars for each
quantity?
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Figure 5: Brass quadrant masses for varying rotational inertia
Experiment 2: Free oscillations & damping
Now, connect Channel 2 of your oscilloscope to the rightmost Angular Velocity BNC
output on the Torsional Oscillator’s front panel (Fig. 4). Once again, retract your
magnetic dampers fully so you see undamped oscillations. Observe and sketch the time
behavior of your Angular Position and Angular Velocity. What is the phase relationship
between the angular position and angular velocity? Explain why this does or does not
make sense mathematically for simple harmonic motion.
Now, use the XY feature of your oscilloscope to make a phase space plot of angular
position (X on Ch1) vs. angular velocity (Y on Ch2). Sketch this relationship for
undamped (magnetic dampers fully retracted) and for significant damping (magnetic
dampers in an intermediate state). Explain why each case looks the way it does. Be sure
to sketch the full time behavior, not just one representative point!
Now that you are familiar with the basic operation of the experiment, let’s compare its
behavior to theoretical predictions. In your RLC laboratory, you were unable to observe
the full range of behavior of free oscillations: the response of the system without a
driving force. We will do so now and compare the results to theory.
In this part of the experiment, you will use the LabPro interface with a voltage probe.
Connect this probe to your Angular Position output and to the DIG 1 signal input on the
LabPro. Log onto the Logger Pro software package and make sure you are seeing a
voltage vs. time plot. Adjust the magnetic damping and the time range on your
LoggerPro plots so you can see many oscillations and also see the full effect of damping.
(Your plot should look something like Fig. 3-13 in your 213 text, but including both more
oscillations and a longer time to see the full effect of damping.) Capture a good set of
data, then cut and paste it into Origin to analyze. In Origin, make a plot of your voltage
vs. time, then do a fit using your functional form for damped free oscillations. You will
be most happy to learn that this function is available (sort of) as part of Origin’s
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collection of fitting functions! (Sort of, because you will have to relate Origin’s
parameters to your own. Be sure to write down the fitting function carefully and its
parameters names, save the values you fit to, and later figure out how they correspond to
your own parameter choices!) Before you do the fit, it’s a good idea to subtract off any
nonzero y-offset. That’s because the voltage from the Angular Position transducer is
hard to zero exactly. Do this by selecting the column with Angular Position in Origin,
then choosing Statistics/Descriptive Statistics/Statistics on Columns. This computes the
mean, which you can subtract off your data by selecting the appropriate column, rightclicking on it and using the Set Column Values option to subtract off the nonzero offset.
To do the fit, go to Analyze/Nonlinear Curve Fit/Advanced Fitting Tool/Waveform/Sine
damp. That function (Sine damp) is the one you wish to use. Again, write down the
function and its parameter names. Perform a fit to your actual experimental data and
record the best fit parameters; plot out your best fit along with the experimental data all
on one plot. (You may have to make good guesses at the initial amplitude and other
parameters to get the fit to resemble your data, so be sure to record all your work. Ask
your instructor for help if you have a hard time with this part!) In your final report,
explain the meaning of your fitting parameters by relating them to those used in your
textbook’s description of damped free oscillations, such as Eq. 3-34(c).
See if you can also capture critically damped oscillations by adjusting the magnetic
damping. (See your text pp. 52-53.) Don’t spend too much time getting it perfect! Print
out your best result.
Experiment 3: Forced oscillations & resonance
Now you will use a function generator to drive oscillations. Be sure you have noted the
natural frequency of resonance of your oscillator (without any additional masses) before
you move on to this part. Start with the magnetic damping retracted so that you have the
dampers just clearing the rotor, for light damping. (If you fully retract the dampers, we
have found that you will excite both torsional oscillations plus a guitar-string like
vibration of the wire. Your angular position vs. time signal will then exhibit beating as
energy feeds between these modes with very similar resonant frequencies. This was
discovered by two students, Jennifer Campbell and Alex Cahill in 2008!)
Connect one of the newer (gray not black) Wavetek function generators so its LO voltage
output is connected to the Coil Drive via a BNC cable to banana plugs. (See Fig. 4) (You
will not want to use the HI output, since it will drive your oscillators’ oscillations too
strongly! You may instead need to use the attenuation feature rather than choosing the
LO vs. HI output. The gray Wavetek function generators have frequency select dials
with a digital readout, for better resolution in finding  and hence Q.) Set the Torsional
Oscillator’s toggle switch to its left side to connect the Helmholz coils to the drive
circuitry. Use a BNC tee connector so you can monitor the driving voltage on an
oscilloscope on Channel 2. Continue to monitor the Angular Displacement signal on
Channel 1 of your oscilloscope. Again, you will want to use a time/division of about 1
second or so.
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Drive the oscillations with a sinusoidal signal, with an initial frequency close to the
natural frequency of your torsional oscillator and starting at about half the full amplitude.
In the RLC Circuit Lab, we had you make a detailed plot of the dependence of amplitude
and phase on driving frequency. Here, you will spot check that the same behavior holds,
using your Prelab Exercise Results as a guide. Hint: Make sure to stop the vibrations in
between each frequency measurement, or otherwise it will take too long!
1) Drive the system at as close to the natural frequency as possible, with minimal
magnetic damping. Then, find & record the frequency at which the system gives
a maximum oscillatory amplitude. Record the phase difference between the
driving signal and the oscillation. (For this part, make sure your drive signal has
the BNC-to-banana plug oriented so the banana plug with the grouncing notch is
uppermost. This ensures the right orientation of the voltage so your oscilloscope
display of driving voltage is in phase with the driving torque.) Which signal leads
and by how much in radians or degrees? Does this agree with your expectations?
Refer to your plot from the PreLabs.
2) Now go to higher and lower frequencies and record the phase shift at each
extreme, recording all your values. Again, compare to your expectations.
3) Estimate Q by changing the drive frequency to see the correct (approximate!)
drop in angular position amplitude to 1/2 the peak value. (Recall from the RLC
circuits lab, Q = o/.) Don’t spend too much time on this part—it’s a very
narrow frequency range! You will have to stop the oscillator by hand at each
frequency after exciting it at the resonant frequency, rather than allowing it to
slowly decay to equilibrium at the new frequency off-resonance.
4) To see the effect of damping on Q as the magnetic damping is varied, measure Q
as in part 3 for a moderately damped oscillator. Discuss how your data compares
with expectations.
Troubleshooting:
1. Is your frequency range set to 1 Hz?
2. Did you remember to flip the switch on the torsional oscillator panel to its left
side? If it’s not set properly, you haven’t connected the function generator to the
drive torque circuitry.
3. Did you connect the Angular Position output from the oscillator back to the
oscilloscope (in the last experiment, you connected it to the LabPro cable)?
Experiment 4: Coupled Oscillations and Normal Modes
One very important class of oscillatory behavior occurs when two or more objects are
coupled together by springs (or by forces that act approximately spring-like for small
displacements—such as atomic vibrating about their chemical bonds!) You have seen in
lecture and your textbook that this case can be analyzed to yield predictions for normal
modes: collective motions of the system as a whole that can be related to the spring
constants and masses of the individual objects. In lab, you have an airtrack that blows
forced air from its length. Several small sleds can freely slide along its length. Two such
sleds are mounted such that they have springs connecting them, and each is connected by
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a spring to a fixed mount on each end of the rail. (It looks like this photo from:
https://wiki.brown.edu/confluence/display/PhysicsLabs/Experiment+120 )
Prelab Exercise 2:
Look up in your notes or your textbook for Physics 213 the predictions for the
frequencies of oscillations of the normal modes of two identical masses, m, coupled by
identical springs with spring constant k, in terms of k and m. If you do not know how
to do this, either consult the similar derivation for two coupled pendula, or else consult
with
your instructor
forishow
to derive
this.
The
driving
mechanism
a rotating
shaft
connected to a rod that wags back and forth as
the shaft rotates. This shakes one of the end springs, driving the motion of the coupled
masses. A DC motor creates the rotational motion; it is driven by a DC power supply.
You can adjust the driving frequency by changing the DC voltage output. The two
normal mode frequencies correspond to approximately 5.7 and 8.7 volts. (But, don’t
change the drive voltage too fast! Remember, a motor is also an inductor, so you wish to
change the voltage across it slowly. Also, always reduce the power supply voltage to
zero before turning off the supply! The spike of voltage across the (inductive) motor
could burn out the power supply otherwise. Also, do not use a voltage greater than 12V
with the motor, even though your supply can go higher than this. Since the voltage
increases abruptly after 12V, be extra careful about this!) You can measure the
frequency of the driving force using the photogate system and its output box. (See figure
below from Pasco™. This system works by using a black U-shaped piece mounted close
to the rotating driving arm. As the arm rotates, it blocks a light beam that shines from
one end of the U to the other. This is detected by the photogate box, which records the
period. You will need to Select Measurement: Time, then Select Mode: Pendulum
mode, then press Start/Stop to get this reading. (The power switch is inconspicuously
located on the left side!) Record both the driving voltages and a few values of the period
to get the frequencies of each normal mode and the associated uncertainties.
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In your writeup, compare the approximate ratios of your two normal mode frequencies
with your theoretical estimate from Prelab Exercise 2. Sketch the motion of the two
masses in each normal mode, indicating which has the higher frequency. The frequency
comparison will be qualitative, but you should be able to check the approximate
correspondence.