Characterization and Analysis of a Chaotic Pendulum
H. Potter
(Completed 8 March 2006)
A chaotic pendulum was characterized by determining its natural frequency, its
damping, and its driving torque. These parameters were then used to predict,
measure, and compare resonance curves. Chaotic motion was then observed by
increasing drive torque and decreasing drive frequency. Two Poincare plots were
produced using this data so that the presence of strange attractors could be
observed.
I. Introduction
Periodic phenomena are commonplace, and in an effort to better understand more
complex periodicity it is fruitful to study controlled periodic motion. This experiment
does so by studying the periodic motion of a mechanical chaotic pendulum. A small
mass that is attached to a shaft, as well as all other instruments attached to this shaft, is
the body undergoing periodic motion. An optical shaft encoder-disc that is attached to
the shaft and rotates with the mass transmits data to a computer where data analysis can
be completed. Also attached to the shaft is a ring magnet that, in combination with a
non-rotating copper plate, provides damping through eddy current losses as the apparatus
rotates1. The separation between the copper plate and the ring magnet can be adjusted
accurately using a micrometer that is mechanically linked to the copper plate. A motor
provides a driving force with an adjustable frequency and amplitude. By adjusting the
specifics of the motion, such as the driving frequency, driving amplitude, damping, and
orientation, those factors that affect the motion can be controlled. This makes it possible
to determine the parameters of the pendulum’s motion experimentally.
This experiment has five distinct, but related sections. In the first section the
natural frequency of the pendulum is calculated by measuring the change in its period as
it is released from various initial angles. In the second section the damping associated
with various micrometer spacings is recorded and a calibration curve relating micrometer
spacing to damping is created. In the third section the torque produced by applying
various voltages to the motor is observed for several micrometer spacings, and this data is
used to create conversion curves between applied voltage and resultant torque for these
micrometer spacings. Section four uses the previously acquired data in order to create
expected resonance curves for five different quality factors. Data is then recorded so that
a comparison can be made between the expected and actual resonance curves. The final
section explores the appearance of chaotic motion when the driving frequency is lowered
significantly as the driving amplitude is increased.
II. Section 1: Natural Frequency
The procedure for determining the pendulum’s natural frequency is as follows.
As the initial angular displacement of the pendulum mass increases, the period of its
motion after it is released increases exponentially, assuming that the motion is relatively
undisturbed by damping. Thus the micrometer spacing was increased as much as
possible in order to minimize the effects of damping on the motion. By recording the
period for a number of different initial displacements, a data set can be obtained that is
large and varied enough to make an exponential regression somewhat reliable. By
extrapolating this curve back to a zero initial angular displacement, an experimental value
for the natural period of the pendulum can be obtained. It is critical that several data
points with very large initial angular displacements are included in the data set;
otherwise, it may be concluded incorrectly that the period does not depend on initial
angular displacement because it increases so slowly at first. Including data points with
higher initial angular displacements also helps to distinguish the appropriate exponential
fit from the inappropriate linear fit.
The apparatus used in this experiment records angular position and velocity at
.007ms intervals. The data was exported to Microsoft Excel, and the number of cells
separating the first two angular position zeroes were recorded. This number was then
multiplied by the interval duration and doubled in order to yield a value for the full
pendulum period. The initial angular displacement of each run was also recorded from
the exported data set. These data pairs were then plotted and an exponential regression
was performed in order to determine the pendulum’s natural period. The natural
frequency of the pendulum was then calculated using
0 
2
.
P0
(1)
Pendulum Period
Period (s)
1.0
y = 0.6427e0.1386x
R2 = 0.9844
0.9
0.8
0.7
0.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Initial Angle (rad)
Figure 1: A plot of all data recorded along with the equation of the exponential
regression, and the corresponding r2 value.
2
Data Summary:
Interval Size (s)
0.007
Data:
Run Number
1
2
3
4
5
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
P0 (s)
0.6427
ω0 (s)
9.7762
r2
0.9844
θm (rad) Intervals Passed Period (s)
1.3100
54
0.756
1.3383
55
0.770
1.0643
53
0.742
0.6860
50
0.700
1.0776
53
0.742
1.2488
54
0.756
0.8796
52
0.728
0.7477
51
0.714
0.8196
51
0.714
1.1765
53
0.742
1.2771
54
0.756
1.0425
54
0.756
1.0556
53
0.742
0.9189
52
0.728
0.9833
53
0.742
0.7728
51
0.714
0.9173
52
0.728
1.1272
53
0.742
0.9629
52
0.728
0.6479
52
0.728
0.7524
52
0.728
0.9959
53
0.742
0.9519
52
0.728
1.0524
53
0.742
0.8743
52
0.728
1.0069
52
0.728
0.9550
53
0.742
0.9269
52
0.728
0.9849
53
0.742
1.0085
53
0.742
0.9158
53
0.742
1.0037
53
0.742
0.8969
52
0.728
1.0430
53
0.742
2.5400
66
0.924
2.5180
66
0.924
2.6986
67
0.938
2.5918
67
0.938
2.6405
66
0.924
2.5730
65
0.910
2.6185
65
0.910
2.6012
66
0.924
Table 1: All data recorded that was plotted in order to determine the natural period of the
pendulum, with the exception of runs 6 through 8, which were recognized as being
unreliable due to very small initial angular displacements.
3
III. Section 2: Damping Calibration
The equation of a harmonic oscillator is, in general,
d 2
d
I 2 b
 mgr sin    T ,
dt
dt
(2)
or, in a more convenient form for this experiment,
d 2 b d mgr sin   T


 ,
I
I
dt 2 I dt
(3)
where θ is the angular displacement, I is the moment of inertia of the entire rotating
portion of the apparatus, b is the damping coefficient, m is the mass of the actual
pendulum bob, r is the distance of the bob from the shaft (from the axis of rotation), g is
the acceleration due to gravity, and T is the torque produced by the driving motor.
In order to isolate the damping term in Equation 3, the entire chaotic pendulum
was turned on its side and clamped into place so that the gravitational term no longer
affected the pendulum’s motion, and the driving motor was turned off. This left only the
first two terms in Equation 3 to consider. This portion can be easily solved for angular
velocity to yield
  i e
b
t
I
,
(4)
where ωi is the initial angular velocity given to the pendulum. Thus, by giving the
pendulum an initial angular velocity, the rate at which it slows was observed in order to
determine the damping, b/I. It was most prudent to consider this decay in its logarithmic
form so that a linear regression on the data set yielded a line that had a slope equal to b/I.
Thus the graph of
  b
ln  i   t
  I
(5)
was considered and a linear regression was performed on it in order to determine b/I
directly for each data set. By collecting data for a range of micrometer spacings, a
calibration curve was obtained so that the damping associated with any micrometer
spacing could be determined. This fit was exponential, as is shown in Figure 2.
4
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
ωi (rad/s)
42.225
46.833
39.645
37.485
55.171
55.667
49.403
49.838
54.939
58.826
62.859
56.565
69.250
66.066
74.939
57.075
44.302
μ (mm) b/I (rad/s)
13.00
0.474
13.00
0.4733
13.00
0.4839
13.00
0.4852
12.50
0.5153
12.00
0.6367
11.50
0.7743
11.00
1.0103
10.50
1.1603
10.00
1.4382
9.50
1.8907
9.00
2.3826
8.50
3.3434
8.00
4.3677
7.50
5.7392
7.00
8.009
6.50
10.682
r2
0.999
0.9995
0.9988
0.9988
0.9956
0.9907
0.995
0.9913
0.9976
0.9984
0.9983
0.9998
0.9988
0.9994
0.9992
0.9989
0.9977
Table 2: All data used in determining the exponential calibration curve for damping as a
function of micrometer spacing.
Damping Calibration Curve
Damping, b/I (rad/s)
12
y = 181.09e-0.4661x
R2 = 0.9872
10
8
6
4
2
0
6.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
Micrometer Setting (mm)
Figure 2: A graph of the data collected along with the exponential fit to the data.
Use Curve to Determine Damping:
y=aebx
μ (mm) b/I (rad/s)
a
8.50
3.4458
181.09
9.00
2.7294
r2
b
9.50
2.1620
0.9872
-0.4661
10.00
1.7126
10.50
1.3565
Table 3: The exponential curve is used here to determine the damping for the 5
micrometer spacings that will be used in later sections.
5
IV. Section 3: Voltage to Torque Conversion
In this section another special case of Equation 3 is considered. Rather than
isolate damping, as was done in Section 2, it was desired that torque should be isolated.
In order to accomplish this, the entire apparatus was again clamped in position on its side
so that its motion is entirely perpendicular to the vertical, and thus unaffected by
gravitational forces. In order to render the second derivative term negligible, the torque
was controlled by adjusting an outside direct current voltage supply that was connected to
the motor. For any given voltage, the driving torque should be constant, and thus the
mass will spin around very rapidly, but will approach a terminal velocity very quickly as
well. Once it reached terminal velocity, the second order term vanished. By measuring
this terminal angular velocity, the torque, T/I, was calculated as
T b
 t ,
(6)
I I
where ωt is the terminal angular velocity and b/I is known from Section 2.
As voltage increases, terminal velocity increases linearly for intermediate
voltages; however, nonlinearlity leads to inaccurate extrapolation for values that are
either much lower than those observed, or above a certain threshold voltage, above which
terminal velocity stops increasing. It is highly recommended, therefore, that the
experimenter determine as precisely as possible the voltage range over which the
calibration curve will be used before proceeding.
In this experiment the terminal velocity was taken to be the average of all data
points exported that were not well outside a reasonable range, and thus simply due to
mechanical errors. After sifting through the data for these points, which were relatively
rare, but typically so different from the normal values that they skewed the average
angular velocity, the average of all values recorded for each run was taken to be the
terminal velocity for that run. Voltage was then increased incrementally and another data
set was recorded. This process continued until a voltage of approximately 4 volts was
applied, at which point the micrometer spacing was changed and the process repeated.
It was observed, upon analyzing the data, that although the lower micrometer
spacings had higher maximum terminal velocities, the linear portions of all calibration
curves laid on the same line. Thus, all data points that were in the linear portions of the
various calibration curves were combined in order to yield a single linear calibration
curve. If the voltages involved in Section 4 were in this linear portion of the combined
calibration curve, then a normal linear regression would be appropriate for determining
the torque produced by voltages in this range; however, the voltages involved in Section
4 are well below the voltage range of the data. As a result, the nonlinearity of the
calibration curve near the origin becomes very significant. In order to accommodate this
fact as best as possible given the available data, the combined linear regression was made
to pass through the origin. The fit to the data was still reasonable, but not as good;
however, this change enabled a rough approximation of the torque produced at the much
lower voltages, and was thus quite useful and appropriate. It is this linear fit that is
shown in Figure 3. All other calibration curves are displayed in their entirety for
comparative purposes and qualitative analysis by the reader.
6
Torque, T/I (rad/s 2)
Voltage to Torque Conversion,
Combined Linear Data Set
300
250
200
150
100
50
0
y = 258.81x
R2 = 0.927
0.0
0.2
0.4
0.6
0.8
1.0
Voltage (V)
Figure 3: The regression line was forced through the origin in order to better approximate
torques corresponding to very small voltages, as was necessary for Section 4.
μ (mm)
Voltage to Torque Conversion, 8.50mm
250
Torque, T/I (rad/s 2)
8.50
2
Voltage (V) ωt (rad/s) T/I (rad/s )
0.333
19.088
65.774
0.485
29.718 102.400
0.538
34.351 118.364
0.660
43.703 150.590
0.788
52.881 182.215
0.938
61.485 211.862
1.043
63.983 220.468
1.262
66.145 227.921
1.649
66.503 229.154
1.856
66.285 228.404
2.047
65.416 225.409
2.216
65.781 226.664
2.446
65.115 224.372
2.828
64.366 221.790
3.245
63.845 219.993
3.690
63.252 217.951
4.050
62.407 215.039
4.950
60.866 209.731
200
150
100
50
0
0
1
2
3
4
5
Voltage (V)
Table 3 and Figure 4: Data for 8.50mm micrometer spacing.
7
μ (mm)
Voltage to Torque Conversion, 9.00mm
300
Torque, T/I (rad/s 2)
9.00
2
Voltage (V) ωt (rad/s) T/I (rad/s )
0.351
27.235
74.335
0.463
38.922 106.234
0.594
53.116 144.976
0.660
58.787 160.456
0.742
67.305 183.705
0.838
76.671 209.268
0.956
83.672 228.377
1.123
87.723 239.433
1.431
88.785 242.333
1.638
88.887 242.610
1.949
88.335 241.103
2.145
88.116 240.507
2.373
87.407 238.572
2.580
87.264 238.181
3.141
85.785 234.145
3.528
85.322 232.881
3.827
83.983 229.225
4.230
82.592 225.430
250
200
150
100
50
0
0
1
2
3
4
5
Voltage (V)
Table 4 and Figure 5: Data for 9.00mm micrometer spacing.
μ (mm)
Voltage to Torque Conversion, 9.50mm
300
Torque, T/I (rad/s 2)
9.50
2
Voltage (V) ωt (rad/s) T/I (rad/s )
0.307
34.954
75.571
0.485
59.046 127.658
0.638
79.099 171.013
0.746
93.629 202.429
0.832 103.092 222.887
0.969 108.268 234.078
1.173 110.103 238.045
1.284 110.071 237.976
1.550 110.813 239.580
1.796 110.429 238.750
2.106 110.139 238.123
2.342 110.324 238.523
2.559 109.869 237.539
2.897 109.695 237.163
3.160 109.451 236.636
3.494 108.398 234.359
3.817 108.407 234.378
4.030 108.569 234.729
250
200
150
100
50
0
0
1
2
3
4
5
Voltage (V)
Table 5 and Figure 6: Data for 9.50mm micrometer spacing.
8
μ (mm)
Voltage to Torque Conversion, 10.00mm
300
Torque, T/I (rad/s 2)
10.00
2
Voltage (V) ωt (rad/s) T/I (rad/s )
0.293
43.077
73.771
0.333
51.618
88.399
0.474
68.868 117.941
0.550
86.415 147.992
0.686 109.197 187.007
0.770 113.876 195.020
0.958 147.662 252.881
1.116 152.812 261.701
1.348 154.112 263.927
1.559 155.805 266.827
1.884 154.034 263.794
2.200 153.548 262.961
2.679 151.248 259.023
2.989 150.922 258.464
3.322 149.420 255.892
3.580 148.682 254.628
4.020 148.311 253.993
250
200
150
100
50
0
0
1
2
3
4
5
Voltage (V)
Table 6 and Figure 7: Data for 10.00mm micrometer spacing.
μ (mm)
Voltage to Torque Conversion, 10.50mm
300
Torque, T/I (rad/s 2)
10.50
2
Voltage (V) ωt (rad/s) T/I (rad/s )
0.250
46.881
63.596
0.368
76.850 104.251
0.440
94.375 128.024
0.570 125.112 169.721
0.629 138.826 188.324
0.786 169.006 229.265
0.895 184.777 250.659
0.960 192.297 260.860
1.118 200.416 271.874
1.274 201.369 273.167
1.466 203.966 276.690
1.604 203.269 275.744
1.928 200.748 272.325
2.371 200.735 272.307
2.684 198.036 268.646
3.173 193.384 262.335
3.509 191.456 259.719
3.936 188.512 255.726
250
200
150
100
50
0
0
1
2
3
4
5
Voltage (V)
Table 7 and Figure 8: Data for 10.50mm micrometer spacing.
V. Section 4: Resonance Curves
In order to analyze resonance phenomena, the driving motor was connected to an
alternating current power supply with an adjustable driving frequency. This enabled the
effect of driving frequency on oscillation amplitude to be observed and analyzed. The
responsiveness of amplitude to changes in driving frequency is measured by the quality
factor Q of the particular oscillator. It is defined by
9
Q
0
b
 
I
.
(7)
The reason that this definition is appropriate is directly related to the fact that
m 
T
I

2
0


2 2
2
,
(8)
b
   2
I
where θm is the maximum angular displacement of the oscillator. The larger that b/I is
relative to ω0, the less the value of ω relative to ω0 affects the maximum angular
displacement of the oscillator. A higher quality factor, therefore, corresponds to
significant resonance. Thus, by choosing to observe changes in amplitude due to changes
in drive frequency for relatively weak damping, resonance curves with distinct peaks can
be made by plotting the maximum angular displacement observed for a range of drive
frequencies at a specific micrometer spacing. This was done for 5 different micrometer
spacings, and the expected curves were plotted by using Equation 8 in combination with
the results of Section 1, Section 2, and Section 3. The maximum angular displacement
should be at a maximum when
2
1b
     ,
2I 
2
0
(9)
so it is interesting to note that all of the graphs have distinct peaks at lower drive
frequency than this formula would predict. This suggests that the natural frequency
calculated in Section 1 is somewhat high, and thus that the natural period calculated was
somewhat low. In addition, all of the graphs of expected maximum displacement are
well below those that were observed. This can be attributed to the extreme extrapolation
that was required in order to estimate the torque at such low voltages. More voltagespecific calibration in Section 3 would remedy this problem if a steady source of such
small voltages could be found, and if the alternating voltage supplied by the driving
motor was more consistent. Typically it would oscillate throughout a 10mV range. For
the purposes of determining torque the average of the upper and lower extremes was
taken to be the effective voltage.
10
Expected Largest Max Angular Displacement (rad):
9.468
2
μ (mm) Q
Voltage (V) T/I (rad/s )
8.50
2.84
0.011
2.847
θE (rad)
f (Hz)
ω (rad/s) θm (rad)
0.765
4.807
0.423
0.038
0.829
5.209
0.435
0.040
0.911
5.724
0.447
0.043
1.059
6.654
0.507
0.051
1.161
7.295
0.588
0.058
1.282
8.055
0.664
0.069
1.396
8.771
0.548
0.080
1.447
9.092
0.503
0.084
1.560
9.802
0.459
0.084
1.733
10.889
0.364
0.065
1.920
12.064
0.280
0.044
2.030
12.755
0.247
0.035
2.161
13.578
0.217
0.028
2.300
14.451
0.180
0.023
Table 8: Resonance data for 8.50mm micrometer spacing.
Maximum Angular
Displacement (rad)
Resonance Curve, Q=2.84
0.7
0.6
0.5
0.4
θm (rad)
θE (rad)
0.3
0.2
0.1
0.0
4
6
8
10
12
14
Angular Driving Frequency (rad/s)
Figure 9: Resonance data for 8.50mm micrometer spacing.
11
Expected Largest Max Angular Displacement (rad):
9.584
2
μ (mm) Q
Voltage (V) T/I (rad/s )
9.00
3.58
0.0115
2.976
θE (rad)
f (Hz)
ω (rad/s) θm (rad)
0.770
4.838
0.307
0.041
0.838
5.265
0.322
0.043
0.955
6.000
0.358
0.048
1.083
6.805
0.418
0.057
1.139
7.157
0.439
0.061
1.228
7.716
0.493
0.071
1.392
8.746
0.469
0.097
1.596
10.028
0.344
0.107
1.704
10.707
0.294
0.085
1.881
11.819
0.217
0.054
2.028
12.742
0.181
0.040
2.132
13.396
0.161
0.033
2.282
14.338
0.134
0.025
2.352
14.778
0.126
0.023
Table 9: Resonance data for 9.00mm micrometer spacing.
Resonance Curve, Q=3.58
Maximum Angular
Displacement (rad)
0.6
0.5
0.4
θm (rad)
0.3
θE (rad)
0.2
0.1
0.0
4
6
8
10
12
14
Angular Driving Frequency (rad/s)
Figure 10: Resonance data for 9.00mm micrometer spacing.
12
Expected Largest Max Angular Displacement (rad):
9.656
2
μ (mm) Q
Voltage (V) T/I (rad/s )
9.50
4.52
0.0165
4.271
θE (rad)
f (Hz)
ω (rad/s) θm (rad)
0.734
4.612
0.302
0.057
0.798
5.014
0.338
0.060
0.866
5.441
0.356
0.064
1.071
6.729
0.450
0.082
1.321
8.300
0.615
0.133
1.591
9.997
0.373
0.194
1.741
10.939
0.292
0.127
1.935
12.158
0.204
0.073
2.190
13.760
0.149
0.043
2.321
14.583
0.134
0.035
Table 10: Resonance data for 9.50mm micrometer spacing.
Maximum Angular
Displacement (rad)
Resonance Curve, Q=4.52
0.7
0.6
0.5
0.4
θm (rad)
θE (rad)
0.3
0.2
0.1
0.0
4
6
8
10
12
14
Angular Driving Frequency (rad/s)
Figure 11: Resonance data for 9.50mm micrometer spacing.
13
Expected Largest Max Angular Displacement (rad):
9.701
2
μ (mm) Q
Voltage (V) T/I (rad/s )
10.00
5.71
0.017
4.400
θE (rad)
f (Hz)
ω (rad/s) θm (rad)
0.777
4.882
0.518
0.061
0.875
5.498
0.505
0.067
0.967
6.076
0.539
0.074
1.071
6.729
0.594
0.085
1.226
7.703
0.773
0.114
1.331
8.363
1.032
0.150
1.431
8.991
0.788
0.206
1.612
10.128
0.469
0.235
1.762
11.071
0.384
0.133
1.902
11.951
0.301
0.085
2.034
12.780
0.247
0.062
2.123
13.339
0.224
0.051
2.278
14.313
0.185
0.039
2.345
14.734
0.177
0.035
Table 11: Resonance data for 10.00mm micrometer spacing.
Resonance Curve, Q=5.71
Maximum Angular
Displacement (rad)
1.2
1.0
0.8
θm (rad)
0.6
θE (rad)
0.4
0.2
0.0
4
6
8
10
12
14
Angular Driving Frequency (rad/s)
Figure 12: Resonance data for 10.00mm micrometer spacing.
14
Expected Largest Max Angular Displacement (rad):
9.729
2
μ (mm) Q
Voltage (V) T/I (rad/s )
10.50
7.21
0.017
4.400
θE (rad)
f (Hz)
ω (rad/s) θm (rad)
0.762
4.788
0.544
0.060
0.824
5.177
0.505
0.064
0.922
5.793
0.525
0.070
1.017
6.390
0.568
0.079
1.172
7.364
0.712
0.103
1.255
7.885
0.841
0.125
1.390
8.734
0.672
0.194
1.496
9.400
0.553
0.300
1.670
10.493
0.471
0.216
1.783
11.203
0.368
0.131
1.849
11.618
0.331
0.104
2.042
12.830
0.246
0.062
2.156
13.547
0.225
0.049
2.316
14.552
0.190
0.037
Table 12: Resonance data for 10.50mm micrometer spacing.
Resonance Curve, Q=7.21
Maximum Angular
Displacement (rad)
1.0
0.8
0.6
θm (rad)
0.4
θE (rad)
0.2
0.0
4
6
8
10
12
14
Angular Driving Frequency (rad/s)
Figure 13: Resonance data for 10.50mm micrometer spacing.
VI. Section 5: Chaotic Motion
For sufficiently large drive frequencies and small drive amplitudes the motion of
the chaotic pendulum is periodic, or at least quasi-periodic. When the amplitude is
increased a great deal, however, and drive frequency is kept very small, the resulting
motion of the pendulum becomes chaotic. In order to observe this chaotic motion, these
conditions were put in place and a series of 5000 data points were recorded on a Poincare
plot. Chaotic motion has the characteristic behavior of never repeating its motion
exactly, and yet somehow having a certain tendency towards specific combinations of
angular displacement and angular velocity. This behavior manifests as strange attractors
15
in a Poincare plot of the data: the points have a tendency to group together, but not
coincide. This peculiar behavior can be observed in the two Poincare plots reproduced
below.
Chaos 1
Angular Velocity
60
40
20
0
-4
-2
-20 0
2
4
-40
-60
-80
Angle
Figure 14: Poincare data plot one. Note the grouping, and yet the lack of coincidence.
This type of behavior is known as being drawn to a strange attractor.
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Chaos 2
Angular Velocity
100
50
0
-4
-2
0
2
4
-50
-100
Angle
Figure 15: Poincare data plot two. Note the grouping, and yet the lack of coincidence.
This type of behavior is known as being drawn to a strange attractor.
VII. Conclusion
This experiment involved analyzing and characterizing a chaotic pendulum very
thoroughly. The first three sections involved determining the natural frequency, damping
calibration curve, and torque conversion curve for the specific pendulum. These
instrument-specific data were then used to investigate the more general phenomenon of
resonance. A comparison was made of observed and expected resonance curves, and the
predictions of the theory fell well short of accurately predicting the shapes of the
resonance curves. The resonance peaks also occurred at unexpectedly low frequencies.
These difficulties can be attributed to several causes. The method for determining the
natural frequency of the pendulum left much to be desired in terms of leaving the
experimenter confident in its quantitative accuracy. The process of finding resonance
curves, however, was very reliable, and if more data were taken so that a much more
continuous resonance curve could be plotted, an accurate experimental determination of
the driving frequency that maximized amplitude could be made. Using Equation 9, an
accurate determination of the natural frequency could be made by using the damping data
from Section 2, which seemed to be very reliable. This method would correct the
positions of the predicted resonance curves. In order to correct the heights of the
resonance curves, a very sensitive direct current supply should be used to create a
conversion curve in a more suitable voltage range. A very consistent alternating current
voltage supply should then be connected to the driving motor so that the applied voltage
17
does not vary so greatly. A difficulty with this approach, however, is that the frequency
of the alternating current driving would have to be determined reliably and accurately.
Ultimately, by taking more data with the experience gained from this round of
data collection, and by searching for ways in which to improve the methods of data
collection, the quantitative accuracy of the experimental results could be vastly improved.
Qualitatively, however, this data set is sufficient in order to observe the characteristic
behavior of a damped, driven oscillator, including resonance. Similarly, the two Poincare
plots provided help provide an intuitive understanding of the chaotic behavior at large
drive amplitudes and low drive frequencies, but a more quantitative analysis would
require much more time and effort. If the process of extracting each useful data point
from each Excel Spreadsheet were automated the entire experiment would proceed much
more quickly. Time spent collecting and analyzing data was necessarily significant due
to the data collection procedure.
1
Blackburn, James A. and H. J. T. Smith, Instruction Manual for EM-50 Chaotic
Pendulum. (Daedalon Corporation, Salem, MA 1998).
(The source was used extensively throughout the lab report. It was only cited explicitly
once, when a nearly direct quotation was used.)
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