Physics 321A: Experiment # 12
Harmonic Motion
Mike Ward
0230227
Introduction:
The objective of this experiment is to study the variation of the amplitude and the phase
angle of the forced harmonic motion with frequency for various damping conditions.
This is accomplished by studying the motion of a spring under various conditions: in air
and in water, driven and un-driven, and with different masses hanging from the end of it.
The basic equation we use while dealing with simple springs is known as Hooke’s Law:
the restoring force of the spring, F, is equal to the displacement of the spring from its
equilibrium, x, times the spring constant, k. In a more compact form, this equation is
F  Kx (1). A system that obeys Hooke’s Law will undergo Simple Harmonic Motion.
The frequency, f 0 , of this motion is related to the spring constant and the mass of the
1 K
(2), where K is the spring constant and
2 M
M is the mass of the system. The mass of the system in this case is defined to be onethird the mass of the spring plus the mass hanging from the spring.
system by the following equation: f 0 
When a damping force is applied to the system, it changes the equations of motion.
Hooke’s Law now includes a damping term, and this has consequences in other aspects
of the motion as well. The amplitude of the oscillations will now decay until it reaches
zero. The ratio of the amplitude of one maximum to the next can be written in terms of
x 
b
this equation:
T  ln  1  (3), where b is the damping constant and x1 and x2 are the
2M
 x2 
successive maxima; M and T were introduced previously.
In the case where a sinusoidal driving force is applied to the top of the spring, the motion
undergoes another dramatic change. Transient solutions will appear and disappear, and
the amplitude of the resulting steady-state solution will depend on the frequency of the
driving force. The resulting amplitude of the system can be determined from the
F0 / 4 2 M
following equation: A f  
(4), where A is the amplitude of
2
2
f 02  f d2  bf d / 2M 
the resulting motion as a function of frequency, F0 is the amplitude of the driving force,
f 0 is the natural frequency of the system were it not driven, f d is the frequency of the
driving force. M and b have already been introduced. Furthermore, the phase angle of
the motion relative to the driving force, δ, is related to the driving frequency by the
bf / 2M
equation tan   d2
(5).
f 0  f d2


The procedure followed in this experiment began with disassembling the apparatus and
weighing its components: the spring, mass bar and rod, along with the damping vanes.
The apparatus was then reassembled and calibrated so that the system oscillated
completely in the vertical plane, and not in the horizontal plane. This was verified by
making sure that the mass bar descended completely through the mass guide without
touching its sides.
The spring constant was determined via two different methods. Firstly, a measure of the
length of the spring in its neutral position was recorded, and then various masses were
attached to the bottom of the spring and a measure of the spring’s displacement from
equilibrium was taken. Secondly, the Digital Harmonic Motion Analyzer was set to
record the period of the oscillation, and the system was set in motion.
The damping vanes were then attached to the mass bar in order to add a damping factor
to the system. Another measurement of the period of the damped system was then taken.
Nine measurements of amplitude were then taken, one every ten periods, in order to
determine the decay of the amplitude of the oscillations.
The driver was then turned on, and it was verified that the amplitude bar was in the
horizontal position. The driving frequency was set to be 0.1 Hz, and then the system was
calibrated using the fine-adjust screw so that the phase LED flashed at 0˚ on the display.
The driving frequency was then increased to 0.3 Hz, and the transient solutions were left
to die out. The amplitude and the phase angle were then recorded, and the driving
frequency was then increased in 0.2 Hz increments until it reached 3.0 Hz, with
measurements being taken at each step, after the transient solution disappeared.
The plastic cylinder was then filled with water and the same measurements as in the air
damped case were repeated.
Apparatus:
Observations:
The following is a graph of the observed values for amplitude versus frequency for both
air and water. Consequently, the graph for phase angle versus frequency is presented.
The data table used for both graphs follows the last graph.
By using equation (1), and inserting some measured values for F and x, we can determine
K, the spring constant. First though, the mass of the system was determined as such:
M sys  M barvanes  M spring / 3  51.6 g  0.05 g  2.7 g  0.05 g  / 3  53g  0.07 g . For a
force of (11.3g  0.05 g )  9.8  110 N  0.4% , the spring constant is thus
0.110  0.4%
K  F/x
 6.875N / m  1.5% , where 0.108m is the
0.124m  1%  0.108m  1%
equilibrium length of the spring.
Similarly, for different values of F, the table below gives values for K:
Mass
Force
Length of spring Displacement Spring Constant
0.0213
0.209
0.142
0.034
6.147
0.0313
0.307
0.166
0.058
5.293
0.0413
0.405
0.172
0.064
6.328
The average of these values is K = 6.162 N/m ± 0.658 N/m (where 0.658 is the standard
deviation of the four values of K).
We can solve equation (2) for K as such:
4 2 M 4 2  0.053kg  0.07 g
K

 6.55 N / m  1.25% . This is definitely consistent
T2
0.565s  0.005s 2
with the other method of measuring the spring constant.
It is easy to calculate the ratio of successive amplitudes for the case of damped motion
x
171  0.5
through both air and water. For example: 1 
 1.41  0.5% . These results
x2 121  0.5
are summarized in the following table:
Amplitudes for air (mm +- 0.5)
171
121
92
70
54
40
27
11
1
x1/x2
1.41
1.32
1.32
1.35
1.35
1.48
2.45
11.00
Average 1.38+-0.07
Amplitudes for Water (mm +- 0.5)
63
58
39
27
22
14
12
2
x1/x2
1.09
1.09
1.45
1.23
1.57
1.17
6.00
Average 1.27+-0.20
The last values can be safely ignored, as they are clearly erroneous (including the last two
values for the air case). The values for the damping constants can now be determined
 x  2M
using equation (3). b  ln  1  * , where T* is 10 times the period for air, and T for
 x2  T
water. Thus, for air,
 x  2M
2  0.053kg  0.07 g
bair  ln  1  *  ln 1.38
 6.04 x10 3 kg / s  0.9%
x
5
.
65
s

0
.
9
%
T
 2
 x  2M
2  0.053kg  0.07 g
And bwater  ln  1  *  ln 1.27 
 0.0458kg / s  0.9%
x
0
.
565
s

0
.
9
%
T
 2
It is notable that the damping constant for the water case is almost an order of magnitude
higher than that for air.
Discussion of Results:
The degree of damping in this experiment has relatively small effect on the graphs of
amplitude and phase angle versus frequency. For the phase angle graph, the higher
damping factor for water acted to smooth out the transition from ~0˚ to ~180˚. In the plot
of amplitude versus frequency, the higher damping constant acted to reduce the
singularity at the resonant frequency to a lesser, more manageable value.
According to equation (4), the maximum amplitude should occur when the driving
frequency is exactly equal to the natural frequency – this is called resonance. This is
confirmed by the results of the experiment. The experimental resonant frequency for the
case of water damping was 1.70 Hz, while for air damping it was 1.78 Hz – both were
found by extrapolating the peak of the curve from the graph of amplitude versus
frequency. These are both very close to the theoretical resonant frequency of 1.77 Hz.
For very low frequencies, equation (4) predicts that as f0 goes to 0 the  f 02  f d2  term
goes to (fd)4, and this will limit the amplitude of oscillation to
F0 / 4 2 M
F0 / 4 2 M
A f  

 0 , and this is confirmed by the
2
2
2
2 2


f

bf
/
2

M
f 0  bf d / 2M 
0
d
graph above. As f0 goes to infinity, the denominator of A(f) will go to infinity, and thus
the amplitude will go to zero. This is also seen in the experimental results (and graph).
2
 
bf d / 2M
 0,
f 02
because for very low frequencies, fd = 0. At high frequencies fd will go to infinity, and so
bf / 2M

tan   d2
 2  0 , but it will be one half revolution after the case where fd = 0.
2
f0  fd

Thus, δ = 180˚ = -180˚. This is by the experimental results, as shown in the graph above.
For very low frequencies, equation (5) predicts a phase angle of tan  
Some sources of error in this lab include the inability to completely zero the phase angle
at a frequency of 0.1Hz. The LED that flashed to indicate the phase angle was very, very
dim, and only flashed once every few revolutions when the frequency was low. I am led
to believe that this piece of equipment is beginning to malfunction. Another source of
error is the fact that I was unable to get the system to undergo ten full oscillations in the
air damped case, whereas the lab manual asks for ten. The ratio of x1/x2 was averaged
over only six values, because of the fewer number of periods recorded, and the fact that
the last value for the amplitude was extraneous in both cases. Still, the values obtained
are consistent and within the range expected.
Conclusions:
In this experiment the variation of amplitude and phase angle of forced harmonic motion
was successfully studied under both air damping and water damping. All data gathered
and all calculations made with it conform to the expected results within experimental
uncertainty. This data, and graphs made based on it, were all shown previously.