FORCED HARMONIC MOTION
Ken Cheney
ABSTRACT
The motion of an object under the influence of a driving force, a restoring force, and a friction
force is investigated using a mass on a spring driven by a variable speed drive. Magnetic damping
provides the friction force. The steady state amplitude as a function of frequency, phase shift,
and the width of the response curve are examined in detail.
INTRODUCTION
GENERAL
PHYSICS INVOLVED
A mass m is acted on by a restoring force ky, a friction force -Rv, and a driving force F cos(ωt).
k is the spring constant, y is the displacement, R is the friction constant, v is the velocity, F
is the amplitude of the driving force, ω, is the frequency of the driving force, and t is the
time.
Including ma gives:
ma + Rv + ky = F cos(ωt)
or
m
dy
d 2y
+ R + ky = F cos(ωt)
2
dt
dt
Equation 1
The motion that results from these forces is best thought of a consisting of two parts. There
is initially a transient part resulting from the left side of the equation as if there was no
driving force. This part decays due to the friction term. After a long enough time the motion
consists of a simple cos motion at the driving frequency. While the motion is dying away
the two components beat with each other and give a very complex effect. This experiment
will only investigate the "steady state" motion after the initial complexities have died away.
Solution of Equation 1.
The motion is given by:
F
cos(ωt − θ)
ωZ
Equation 2
Z =√
R 2 + (ωm − k/ω)2

Equation 3
ysteady =
where
ysteady is the steady state motion of the mass, and θ is the phase difference between the
driving force and the mass.
Combining Equations 2 and 3 and dropping the time dependent part (cos) gives:
F
ysteady max =
Equation 4
ω√
R 2 + (ωm − k/ω)2

where ysteady max is the maximum amplitude reached in the steady state.
LB1BFH.DOC 1.111, May 3, 2002
Ken Cheney
1
also
 ωm − k/ω 
θ = 90o + tan−1


R

Equation 5
Full Width Half Maximum
A plot of the maximum amplitude as a function of driving frequency gives a slightly
unsymmetrical peak with the maximum at the natural frequency ω0
ω0 = √
k/m

For many purposes it is important how wide this "resonance curve" is. If the resonance
is narrow the mass will only respond to a narrow range of frequencies but with a large
amplitude. Conversely if the resonance is broad then the mass will respond to a wide
range of frequencies but only with a small amplitude.
This concept of width is another aspect of the Q of a system.
The Full Width Half Maximum (FWHM) is the width of the resonance curve taken at
an amplitude half of the maximum of the curve. A measure of the width of the curve is
given by:
FWHM
R
=
Equation 6
3
√
ω0
mω0
APPLICATIONS
AMPLITUDE
Applications resulting from Equation 4 are often quite spectacular. Notice what happens
to y when the driving frequency matches the natural frequency and R is zero. y goes to
infinity.
Imagine a bridge, Tacoma Narrows say, with small damping and a wind resonating it at
the natural frequency. Or, perhaps, the wing on your plane has a resonance matching
the frequency of the engine? Or the water in a bay might have a natural frequency
matching the tidal frequency, Bay of Fundy, forty five foot tides. Or apply this to a radio
tuner.
The lack of response to driving frequencies above the natural frequency is very useful
for isolation. Hi-fi turntables are isolated from vibrations and cars isolate the passengers
from bumps in the road this way.
PHASE
It appears strange that anything so esoteric as a phase shift could have spectacular results.
Equation 5 says, in words, that at low frequencies the mass follows the driving force in
phase, at resonance they are 90 degrees out of phase, and well above resonance they are
180 degrees out of phase.
Examples where this does matter are the valves on an internal combustion (car) engine
or the outer part of a hi-fi speaker.
Both examples start out with the mass following the driving force in phase but at a high
enough frequency they may be completely out of phase. This is very unhealthy for the
engine.
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FORCED HARMONIC MOTION Ken Cheney
WIDTH
Finally Equation 6 tells us about the width of a resonance. A small R produces a narrow
resonance, Equation 4 says a big, narrow resonance.
This big, narrow resonance is great for timing and tuning. A timer should respond lots
at the proper frequency and not at all at other frequencies. The lower the resistance
relative to the mass the better. Devices used have been pendulums, tuning forks, quartz
crystals, and atomic or molecular vibrations.
Tuning requires the same properties. A radio tuner should respond to just the desired
station and ignore stations nearby in frequency no matter how strong.
EXPERIMENT
PASCO DRIVEN HARMONIC MOTION ANALYZER
DESCRIPTION AND SETUP
The specific instructions here will refer to the Pasco "Driven Harmonic Motion Analyzer"
model ME-9210A.
OVERVIEW
A variable speed motor drives a relatively large drive wheel through a belt. The drive
wheel has a drive pin offset from its center, this pin pulls one end of a string. The string
oscillates one end of a spring. The other end of the spring drives a transparent mass bar
which is connected to a damping rod.
Photocells measure the amplitude and zero crossing of the mass bar. A LED that rotates
with the drive wheel but is controlled by the zero crossing of the mass bar is displayed
on the front panel and reads out the phase difference between the drive and the response
of the mass bar.
Connections to the game port of an Apple computer allow the program DHMA Plotter
to monitor and record the amplitude, mass-bar frequency, drive-wheel frequency, and
phase.
AMPLITUDE AND FREQUENCY
The driving force originates with a variable speed motor, the shaft of the motor extends
out the back of the control box.
The shaft is connected to a larger drive wheel (approximately 15 cm in diameter) by a
flexible belt. This drive arrangement is common when it is desirable to filter out high
frequencies that might be produced by the motor.
The drive wheel has a drive pin offset from the center by a sliding amplitude bar. The
bar slides along the diameter of the drive wheel and can be adjusted to give the desired
driving amplitude. It is possible to adjust for zero driving amplitude, this leads to rather
dull results.
Finally, the drive cord which will ultimately move the driven mass is attached to the
drive pin. The y motion of the drive pin is given by Rd cos(ωt) where Rd is the radius
from the center of the drive wheel to the drive pin, ω is the driving frequency, and t is
the time. This is just the time dependence we assumed in the theory section for forced
harmonic motion.
FORCED HARMONIC MOTION Ken Cheney
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You want to adjust the amplitude to be as large as possible in order to get easily measured
displacements of the driven mass. On the other hand you do not want the oscillations
to destroy the apparatus. The largest oscillations occur at resonance with the minimum
damping force.
Set the magnetic damping to the minimum possible, set the driving frequency to
resonance, and set the offset of the drive pin, Rd, as large as possible without destroying
the equipment. Be careful with your initial tries, the amplitude can grow very quickly.
The resonant frequency will be quite near the theoretical frequency but probably not
exactly as calculated. You will have to experiment a bit to find the actual resonant
frequency.
DRIVING FORCE
The driving force is produced by the driving amplitude, Rd, stretching the spring which
holds the masses. The force, F, is given by Hook’s law F = −kRd where k is the spring
constant.
k can be obtained by hanging weights from the spring.
ALIGNING THE MASS BAR IN THE GUIDE HOLE
The mass bar must not rub on the guide hole. There are two possible adjustments.
If the bar is twisted relative to the hole rotate the bar relative to the screw it is hanging
from.
If the guide bar is straight but rubbing on the side of the guide hole adjust the levelling
screws at the bottom of the apparatus.
ALIGNING THE MASS BAR WITH THE PHOTOCELLS
The center of the mass bar must be aligned with the photocell in the upper mass guide.
Try moving the mass bar by hand and notice that the phase set LED on the front of
the upper mass guide is on when the center of the mass bar is below center and that
the LED is off when the mass bar is above the center. This is controlled by the dark
bar on the lower half of the mass bar on one side which blocks a photocell in the center
position.
Center the drive wheel at zero degrees. To do this set the mass bar oscillating and
observe the LED on the PHASE scale on the front of the panel. This LED will flash
on and off. Rotate the drive wheel until the phase LED is at the zero degree mark.
Now adjust the length of the string and support so the smallest oscillations of the mass
bar make the phase set LED flash on and off. This means that the center of the mass
bar is centered on the photocell. You will probably only need to adjust the screw at
the top of the support column. If things are really bad gross adjustments can be made
with the plastic clip on the string above the drive wheel.
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FORCED HARMONIC MOTION Ken Cheney
MAGNETIC DAMPING
Under the plastic mass bar is a damping rod made of some non magnetic material. This
damping rod moves between two cylindrical magnets. These magnets can be moved
closer or farther from the damping rod by large plastic screws. The magnets produce
eddy currents in the damping rod which in turn produce damping forces. The forces are
proportional to the velocity of the rod relative to the magnets.
COMPUTER INTERFACE AND PROGRAM
The software to collect and analyze the data with an Apple computer is "DHMA Plotter" by
Vernier Software.
CONNECTIONS
Turn off the computer and DHM apparatus. Connect the two with the nine pin cable
from the back of one to the back of the other.
TURNING ON
Make sure the DHMA Plotter disk is in the left drive of the Apple computer.
Turn on the computer first. If you forget and turn on or reboot the computer with the
DHM apparatus turned on the Apple goes into a self check mode and ignores the outside
world. Turn off the DHM apparatus and reboot the Apple.
To reboot the Apple hold down the CONTROL and the RESET keys at the same time.
This has the same effect as turning the computer off then on but is much easier on the
computer.
USEFUL FACTS
The DHMA Program defines the phase angle as the lag of the mass bar behind the drive
wheel. If the mass bar leads the drive wheel then the phase angle is > 180 degrees or is
negative
The printer is in slot 1.
The printer interface is a Grappler card.
MAIN MENU
The Main Menu is the core of controlling the program. You will select the procedures
from it. The procedures most used are:
Select Dependent Variables to Collect
Mode Y
Collect Data
Mode K
Print Data on Printer
Mode T
Plot Data
Mode P
HINTS
You must select the dependent variables before starting to collect data in Mode K.
When you start Mode K select drive frequency as the independent variable.
FORCED HARMONIC MOTION Ken Cheney
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Mode K displays readings continually on the screen but only reads data into memory
when you tap the space bar. Don’t hold the space bar down or you may collect several
readings.
PROCEDURE
The object is to check equations 4-6.
STEADY STATE AMPLITUDE VERSES FREQUENCY
The object here is to obtain enough data to determine the dependence of steady state
amplitude on the driving frequency. This is to be done for minimum, maximum, and
medium damping forces. i.e. three curves.
A good range is from 0.1 ω0 to 10 ω0if possible.
To determine each curve a number of data points are required. Each point is obtained
by setting the driving frequency and waiting until the motion of the mass settles down
to a constant amplitude. This procedure gives one amplitude-frequency point. You
collect the data just by touching the space bar on the computer.
To get the best data with a minimum number of data points some planing and plotting
is required. The interesting data is centered around the resonant frequency. You need
enough points near the resonant frequency to be sure of the behavior in this region.
The amplitude changes very rapidly in this region so it is easy to miss some vital points.
By far the best way to be sure you have the necessary points is to make a rough plot of
your results as you take the data. The eye is very good as spotting oddities in curves
but can’t do much for you with a data table.
On the other hand you want to investigate frequencies far enough away from resonance
to be sure what the curve looks like at the extremes. If you don’t take larger steps in
frequency in this region you may require weeks to get your data. Again, to get the best
data: PLOT AS YOU GO!
PHASE VERSES DRIVING FREQUENCY
Be sure to record the phase at steady state for each combination of damping and driving
frequency. This is done automatically if you selected both phase and amplitude as
dependent variables.
ANALYSIS EXPECTED
Use FITSALL to compare the resonance and phase curves with theory. Discuss how well they
match, or fail to match theory.
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FORCED HARMONIC MOTION Ken Cheney
RESONANCE CURVE
For FITSALL:
ysteady max =
F
Equation 4
ω√
R 2 + (ωm − k/ω)2

becomes
3000 DEF LONG FNA(X)=A(1)+A(2)/(X*SQR(A(3)**2+
(X*A(4)-k/X)**2))
This all goes on one line.
Don’t type in k but instead put in your value for the spring constant.
Symbol
Meaning
Suggested Guess
X
Driving Frequency ω in
radians
---------------
A(1)
Offset
Zero
A(2)
Driving Force
kRd, Rd is the offset of the drive pin
A(3)
Damping Constant, R
From FWHM
A(4)
Mass
Mass bar + Damping bar + 1/3 spring
It would be good to solve for k also but notice that any number of combinations of m and
k give the same values for (ωm − k/ω). As a result it is only possible to solve for k or m,
not both at once.
PHASE VERSES DRIVING FREQUENCY
For FITSALL:
 ωm − k/ω 
π

θ = + tan−1
2

R

Equation 5
becomes, in radians:
3000 DEF LONG FNA(X)=PIX(0.5)+ATN((X*A(1)-k/X)/A(2))
PIX(0.5) means π *0.5
ATN(X) means tan−1(x)
Just as in the amplitude case we can’s solve simultaneously for m and k.
Symbol
Meaning
Suggested Guess
X
Driving Frequency ω in
radians
---------------
A(1)
Mass
Mass bar + Damping bar + 1/3 spring
A(2)
Damping Constant, R
From FWHM
FORCED HARMONIC MOTION Ken Cheney
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FWHM
Measure the FWHM for each size of damping from the curves obtained above. Compare
with theory.
OVERALL
It is very helpful to see all three resonance curves at once, it is much easier to appreciate the
effect of damping when everything is plotted to the same scale on the same page.
It is easy to write a small BASIC program which will plot your experimental points and the
best fit curves obtained from FITSALL. See the sample below, DATAPLOT.PUB.PHYSCI.
WRITE UP EXPECTED
Formal write-up. No introduction. Enough sketches to explain any measurements. Good
equipment list.
GRADING CONSIDERATIONS
The important part is a thoughtful discussion of how the results compare with theory and what
interesting features there are in your results.
Don’t forget to discuss your computer results including the Standard Errors.
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FORCED HARMONIC MOTION Ken Cheney