On the In-phase Synchronization Time of Coupled Metronomes
Victor Gao
November 2021
1
Abstract
A phenomenon found in various fields of study, synchronization is an intriguing topic with numerous applications. This paper investigated the in-phase synchronization of two coupled metronomes by
analyzing experimental data and conducting a theoretical evaluation.
The two metronomes were coupled through a freely moving board, which allowed the motion of
the pendulums to cause synchronization. The synchronization time was determined by recording the
audio ”ticks” of the metronomes, while the frequency of the metronomes was adjusted by sliding a
weight attached to the metronome’s pendulum. The theoretical analysis was accomplished by modelling
the non-linear dynamical system using Van der Pol oscillators. A set of differential equations were
composed and integrated with the help of Wolfram Mathematica to achieve numerical solutions to the
synchronization time.
The experimental data supported the mathematical model, which indicated an inverse relationship
between synchronization time and frequency. This matches the verdict drawn by various other research
papers and experiments, which suggests the conclusion determined from this experiment is valid.
2
Contents
1 Introduction
2 The
2.1
2.2
2.3
2.4
2.5
Experiment
Setup and Procedure . . . . . . . .
Measurements of the Materials . .
Recording of Values . . . . . . . .
2.3.1 The Frequency . . . . . . .
2.3.2 The Synchronization Time
2.3.3 Control Values . . . . . . .
Results of Experiment . . . . . . .
Evaluation of Data . . . . . . . . .
4
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3 Theoretical Analysis
3.1 Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Nondimensionalizing the Equations of Motion . . . . . . . . .
3.3 Determining Parameter Values . . . . . . . . . . . . . . . . .
3.3.1 Distance from pivot to pendulum center of mass . . .
3.3.2 Moment of Inertia . . . . . . . . . . . . . . . . . . . .
3.3.3 Mass ratio . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Damping coefficients . . . . . . . . . . . . . . . . . . .
3.3.5 Initial angular displacement . . . . . . . . . . . . . . .
3.4 Obtaining Solutions to the Synchronization Time . . . . . . .
3.4.1 Conversion from non-dimensional to actual parameters
3.5 Comparing Experimental and Theoretical Results . . . . . . .
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4 Conclusion
21
4.1 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Acknowledgements
23
6 Concerns Regarding the First Draft
26
3
1
Introduction
As a student who has played numerous instruments as a teenager, I was familiar with the metronome and
its mechanics. Unlike a gravity pendulum, a metronome is able to oscillate for a far longer period of time
due to an escapement, a mechanical linkage that allows a periodic impulse to be transmitted from a spring
to the pendulum [1]. In most mechanical metronomes, potential energy from the spring is supplied by
hand in the form of a rotating crank. This potential energy replaces the kinetic energy of the pendulum
lost to friction, prolonging the metronome’s ability to oscillate. Once the metronome is winded, a small
initial displacement of the pendulum will initiate the escapement mechanism. Oscillations then produces a
sound caused by a secondary mechanism, which is amplified due to the hollow structure of the metronome [1].
The synchronization of coupled metronomes is a common experiment performed to showcase a perplexing
phenomenon in physics. Energy from one metronome is transferred to the other through a common platform, affecting the oscillatory motion of the pendulums. Although extensive research has been conducted on
the state of synchronization itself, few have investigated the effect of initial conditions on the system (such
as the frequency). Moreover, limited research explores a rather trivial parameter, the time of synchronization.
Research Question: How does the common frequency of two coupled metronomes affect the
synchronization time of the oscillators?
Each time the pendulum on the metronome oscillates, it exerts a force on the base of the metronome.
The motion of the base is then transferred to the freely moving board which both metronomes are situated
upon, affecting the motion of the other metronome’s pendulum and eventually synchronizing the two. Since
the frequency of oscillation determines how frequent the pendulums on each metronome affect the motion
of the system, it is reasonable to assume that a higher frequency will result in a shorter synchronization time.
The goal of this investigation was to determine the validity of this hypothesis through the support of
experimental data and mathematical modelling achieved by utilizing the principles of dynamics.
Figure 1: [2] Metronomes in an un-synchronized
state. By Harvard Natural Sciences Lecture
Demonstrations.
Figure 2: [2] Metronomes in a synchronized
state. By Harvard Natural Sciences Lecture
Demonstrations.
4
2
The Experiment
The aim in this chapter is to detail how the experiment was conducted. Due to limitations in the accessibility
of resources and tools, creative solutions were implemented to measure certain parameters, such as the time
of synchronization.
2.1
Setup and Procedure
In this experiment, two Wittner Super-Mini-Taktell (Series 880) metronomes were used for their small size
and mass. The metronomes were placed on a foam board, separated by an equal, fixed distance (6.45 cm)
away from the center of the board. The board with the metronomes were then placed on two aluminum pop
cans oriented so that both would roll in the direction parallel to the length of the foam board. The cans were
to roll on a piece of paper to smoothen the surface and reduce friction between the cans and the ground. A
Lego contraption was then braced onto the top of each metronome to control the initial angular displacement.
A microphone, dangled between the two metronomes, was used to record the audio created by the system.
After recording began, the Lego attachments were taken off to begin the ticking of the metronomes. After a
maximum time of 3 minutes (or whenever the metronomes were both visually and auditory synchronized),
the recording was stopped. The synchronization time was then determined through analysis of the recorded
audio on Audacity. Details on how the synchronization time was determined is explained in section 2.3.2.
Figure 3: Diagram of the experimental setup. The left shows the setup without the microphone, included
for a better view of the system. The right shows the full setup.
2.2
Measurements of the Materials
The values of certain parameters of the metronome were required in the theoretical analysis of this paper
(see section 3.3). The relevant parameters are listed below.
• Mass of the metronome box: Mb = 94 ± 1g
• Mass of the pendulum bob: mbob = 5 ± 1g
• Mass of the pendulum stick: mstick = 23 ± 1g
5
• Mass of the pendulum (bob + stick): m = 28 ± 1g
• Mass of foam board: Mf = 5 ± 1g
• Distance from the center of the base of the metronome to the center of the board: a = 6.45 ± 0.05cm
Figure 4: Labelled parts of the metronome
Figure 4 details the parts of the metronome. The pendulum bob mbob could slide off of the stick, allowing
its measurement. To measure the mass of mstick , the component was disassembled from the box of the
metronome.
2.3
Recording of Values
16 iterations of the experiment was conducted, with each iteration consisting of 5 trials. A different frequency
was utilized for each iteration, and the synchronization time for each of the 5 trials in each iteration were
recorded to produce an error measurement.
2.3.1
The Frequency
Both metronomes were set to the same frequency. The frequency was adjusted by sliding a weight attached
to the pendulum arm. On the pendulum, there were several grooves (see Figure 4) for which the weight
was to be fitted to, with each indent corresponding to an indicated frequency on the metronome. Thus, this
numerical value was taken to be the frequency for each trial of the experiment.
6
2.3.2
The Synchronization Time
To determine the synchronization time, the audio produced by the metronome was analyzed using an audio
visualizer/editing software (Audacity). The sounds the metronomes produced were recorded by an Apple
Earbuds Microphone. Since the ”tick” each metronome produced was much louder than any background
noise, each beat of the metronome could be visualized on Audacity as a large cluster of spikes. To determine
the synchronization time, the following procedure was manually performed for each trial.
For each local produced by the beating of the first metronome, the time (t1 ) at which the highest spike
was would be recorded. A search was then conducted in the interval t1 ± 0.01s1 of the recorded time for
the highest spike from the local audio cluster second metronome (t2 ). If such a spike existed, the times of
the four subsequent clusters from each metronome would then be compared. If, for all four pairs, the times
of the highest spikes were within 0.01s of each other, the synchronization time was recorded as t1 ± 0.005s
(Uncertainty was recorded as half of the smallest scale unit, and the time was visually estimated up to
the thousandth digit). This process was repeated for all spikes produced by the first metronome until the
synchronization time was found.
(a)
(b)
Figure 5: Analysis of the audio recording produced by the 5th trial at a frequency of 208 bpm. (a) shows
a premature check where the condition for identifying synchronization time was not satisfied. (b) shows the
successful identification of synchronization time t1 = 34.890s (all four clusters after also met the condition
|t1 − t2 | <= 0.01s).
1 This
value was estimated to be 0.01s after it was determined that for any two distinct ticks within approximately 0.01s of
each other, the human ear was unable to distinguish the sounds coming from both metronomes.
7
2.3.3
Control Values
To reduce the effects of confounding variables and isolate the relationship between frequency and synchronization time, several parameters were held constant throughout all trials of the experiment. First, the initial
angular displacements of each metronome was kept consistent by using a Lego contraption. This device is
shown in Figures 6 and 7, and was braced on to the metronome box before each trial began. Then, when the
audio recording was to begin, the contraption was swiftly removed from the box to allow the metronomes to
oscillate freely.
Figure 6: Front view of the Lego contraption.
The 1x2 grey beam may be moved along the
black axle to adjust the initial angular displacement of the pendulum (to obtain two different
initial angles for metronomes 1 and 2.
Figure 7: Back view of the Lego contraption.
The brace was built to match the dimensions of
the metronome box exactly to reduce wobble and
ensure the inital angle was constant throughout
all trials.
Furthermore, each metronome was wound to the maximum extent before being placed on the foam board
to ensure energy was supplied from the escapement mechanism throughout the duration of the trial. Finally,
the position of the metronomes were kept constant by measuring the appropriate positions along the foam
board and marking down the outline of each metronome. Then, each metronome was placed onto the board
to match the outline.
2.4
Results of Experiment
Table 1 outlines the data (frequency and synchronization time) collected the experiment.
The synchronization times for each iteration in Table 1 were averaged to reduce uncertainty, and 16 data
points were produced. The frequency was also converted from beats per minute to Hz as follows
fHz =
8
fbpm
60
Synchronization Time (s, ±0.005)
Frequency (bpm)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
208
38.325
37.256
37.986
39.120
34.890
200
40.200
43.024
41.734
40.022
39.832
192
41.945
40.176
45.256
42.344
43.988
184
43.234
39.883
42.875
42.707
42.005
176
41.525
43.211
44.117
45.232
42.163
168
43.146
46.299
47.009
42.667
42.900
160
44.221
44.847
46.013
44.980
45.346
152
46.023
44.368
44.141
46.011
47.023
144
45.454
46.822
45.634
48.341
46.083
132
46.196
50.388
47.243
46.195
49.126
120
47.623
50.232
50.387
46.247
48.095
112
51.105
53.621
50.062
55.565
55.340
104
58.029
55.712
58.792
60.632
62.901
96
60.133
69.023
65.502
64.597
65.002
88
80.144
88.236
95.124s
92.919
99.024
80
128.311
90.133
139.261
103.306
127.522
Table 1: Raw data collected from the experiment
Furthermore, the uncertainty utilized in Figure 8 was calculated by taking half of the range of data points
in each iteration. This random uncertainty was utilized, as the values far exceeded the uncertainty obtained
from audio analysis (0.005s). A sample calculation is shown below for the first iteration of trials.
tmax − tmin
2
39.120s − 34.890s
=
2
∆t = 2s (kept to one significant figure)
∆t =
The data was then graphed using LoggerPro 3.16, as shown in Figure 1.
9
Figure 8: Time of synchronization vs. frequency for two coupled metronomes on a moving base
2.5
Evaluation of Data
The graph in Figure 1 supports the hypothesized inverse relationship. As the frequency increases, the
synchronization time decreases. However, for higher frequencies (f >= 132Hz) the data points seem to
form an unexpected asymptote around t = 45s. For lower frequencies (f <= 88Hz), a significant decrease
in the motion of the foam board was observed in most trials, possibly due to the increased role of friction
between the cans and the paper damping the system. As such, a large range of values for the synchronization
time were obtained, corresponding to large uncertainties in the first two data points in Figure 8.
A unique observation was made in some failed trials. Due to human error, some metronomes were
introduced to initial velocities at time t = 0. For some of these trials, anti-phase synchronization occurred
after a period of time, where the motion of the pendulums were symmetrically opposite to each other.
10
3
Theoretical Analysis
In this section, a mathematical model is presented to simulate the motion of the metronomes on the moving
base. The forces acting on each metronome were analyzed to develop a system of non-linear differential
equations, which were then graphed through using Mathematica and a set of initial conditions. A statistical
approach is then presented to determine the time of synchronization, and theoretical curves were produced
to model the relationship between initial beating frequency and time of synchronization.
3.1
Force Analysis
We shall first consider the forces acting on a single metronome (denoted with subscript 1) secured onto a
moving base. We will first define some parameters of the system.
• M - Mass of the metronome bases and foam board
• m - Mass of the oscillating pendulum bob
• r - Distance from the center of mass of the pendulum to the pivot point
• I - Rotational inertia of the pendulum
• θ1 - Angle formed between the pendulum bob and the vertical for the first metronome
• θ2 - Angle formed between the pendulum bob and the vertical for the second metronome
• xM - Displacement of the center of mass of the board and metronome bases (referred to as the ”base”)
Since the metronome is in an accelerating frame, the pendulum bob will experience an acceleration of the
same magnitude but in the opposite direction of the board’s motion [3]. Using Newton’s second law in
angular form (right will denote the positive direction),
τnet = Iα = I
d2 θ1
dt2
(1)
By analyzing the forces present in Figure 9, we note that both gravity and the fictitious force act to restore
equilibrium to the pendulum’s position. After splitting into components, we obtain,
−mgr sin θ1 − mr
d2 xM
d2 θ1
cos θ1 = I 2
2
dt
dt
(2)
Up until now, we have neither considered the escapement mechanism of the metronome (which supplies
energy to the pendulum in a non-linear fashion) nor the damping (friction) present at the pivot of the
pendulum. We propose a model commonly found in similar literatures studies of metronomes: the usage of
Van der Pol oscillators [4, 5, 6, 7]. In the metronome, once the pendulum surpasses a certain threshold angle
(θ1 > θ0 ), the escapement will act to dampen the motion. However, when the angular displacement of the
pendulum is within (θ1 < θ0 ), the escapement will supply energy to its motion to prolong oscillation [8, 9,
10]. Accounting for friction from c, the linear viscous damping coefficients, this behaviour may be modelled
by the expression
D = c((
θ1 2
dθ1
) − 1)
θ0
dt
(3)
where D represents the damping moment acting on the pendulum [4, 8]. Furthermore, the threshold angle
pc
may be modelled as θ0 =
b , where b is the non-linear damping coefficient [4]Draft 1 Note: The
11
Figure 9: Force diagram for a single metronome on a moving base. The net force acting on the base is
denoted as M ẍM . The cos θ1 component of mẍM is not shown for a clear diagram, but acts in the same
direction as mg sin θ1 to restore equilibrium.
derivation of this expression is not shown, but I intend to include if (if neccesary). Inserting
this term into equation (2) gives us
−c((
θ1 2
d2 xM
d2 θ1
dθ1
− mgr sin θ1 − mr
cos θ1 = I 2
) − 1)
2
θ0
dt
dt
dt
We will now consider both metronomes as part of the system and formulate an expression for
terms of θ1 and θ2 , the angles formed by the pendulums of each metronome respectively.
(4)
d 2 xM
dt2
in
Note that the origin of the coordinate system was chosen to be the midpoint of the two metronomes so
that the variable a may be eliminated [11]. The center of mass of the entire system can now be modelled by
the equation
xCM =
m(xM − a + r sin θ1 ) + m(xM + a + r sin θ2 ) + M xM
m+m+M
(5)
where a denotes the distance between each metronome and the center of the board. To simplify equation
2
xCM
(4) further, we must obtain a relationship between d dt
= 0, θ1 , and θ2 . To achieve this, we will make
2
the assumption that no external forces act on the system. Since the aluminum cans are smooth, its friction
with the ground may be neglected and the assumption is reasonable. Then, the acceleration of the system
12
Figure 10: Diagram denoting the positions of various masses. The reference point (origin) is chosen to be
at the midpoint of the board, O.
is 0. Simplifying,
xM (2m + M ) + mr(sin θ1 + sin θ2 )
2m + M
mr
= xM +
(sin θ1 + sin θ2 )
2m + M
xCM =
Since
d2 xCM
dt2
(6)
= 0, we have
d2 xCM
d2 xM
mr
d2
=
+
(sin θ1 + sin θ2 )
2
2
dt
dt
2m + M dt2
d2 xM
mr
d2
0=
+
(sin θ1 + sin θ2 )
dt2
2m + M dt2
d2 xM
mr
d2
=
−
(sin θ1 + sin θ2 )
dt2
2m + M dt2
(7)
Substituting back into equation (4), we obtain
−c((
θ1 2
dθ1
m 2 r 2 d2
d2 θ1
) − 1)
− mgr sin θ1 +
(sin θ1 + sin θ2 ) = I 2
2
θ0
dt
2m + M dt
dt
Rearranging and dividing by I gives
d2 θ1
c θ1
dθ1
mgr
m
+ (( )2 − 1)
+
sin θ1 −
dt2
I θ0
dt
I
2m + M
d2 θ2
c θ2
dθ2
mgr
m
+ (( )2 − 1)
+
sin θ2 −
dt2
I θ0
dt
I
2m + M
mr2 d2
(sin θ1 + sin θ2 ) = 0
I dt2
mr2 d2
(sin θ1 + sin θ2 ) = 0
I dt2
(8)
(9)
Equations (8) and (9) denote the equations of motion for two coupled metronomes on a moving base. Since
each metronome is subject to the same forces, their equations of motion are identical.
3.2
Nondimensionalizing the Equations of Motion
In this section, we will attempt to simplify the equations of motion derived in section 3.1 by reducing the
number of characteristic parameters. This can be done through the nondimensionalization of the differential
13
equations and replacing physical constants with nondimensional variables. Nondiemsionalization allows us
to simplify the equations of motion and recover characteristic properties of the metronome-board system,
allowing for a streamlined process when numerical integration is required [12, 6]. First, we simplify one of
m
the coefficients of the last term in equation (8) and (9) by using the mass ratio σ = M
.
m
=
2m + M
m
M
2m+M
M
=
σ
1 + 2σ
Also, for small angles, it is well known that sin θ ≈ θ, so using Newton notation (where the dots denote
differentiation with respect to t), equation (8) becomes
c θ1 2
mgr
σ mr2
θ̈1 + [(
− 1)]θ̇1 +
θ−
(θ̈1 + θ̈2 ) = 0
I θ0
I
1 + 2σ I
(10)
p mgr
Now, to nondimensionalize (10), we will make the common algebraic substitution ω0 ≡
I , where ω0
denotes the natural angular velocity of the pendulum in the absence of damping or escapement [12]. Nondimensional time may now be represented as τ = ω0 t. Since
θ̇ =
dθ dτ
dθ
=
and dτ = ω0 dt
dt
dτ dt
dθ
θ̇ = ω0
dτ
Similarly,
θ̈ =
d2 θ
d2 θ d2 τ
= ω02 2
2
2
dτ dt
dτ
Making the above substitutions into (10) gives
ω02
σω02 mr2 d2 θ1
d2 θ1
cω0 θ1 2
dθ
d2 θ2
2
[(
+
ω
θ
−
(
+
)
−
1]
+
)=0
1
0
dτ 2
I θ0
dτ
1 + 2σ I
dτ 2
dτ 2
σ mr2
c θ1 2
[( ) − 1]θ̇1 + θ1 −
(θ̈1 + θ̈2 ) = 0
θ̈1 +
ω0 I θ 0
1 + 2σ I
(11)
where the dots denote differentiation with respect to non-dimensional time τ . We make a final substitution
with non-dimensional parameters ŝ1 = θθ01 and ŝ2 = θθ20 to reduce equation (11) to
1¨
c
1
1
σ mr2 ¨
ŝ1 +
(sˆ1 2 − 1)ŝ˙ 1 + ŝ˙ 1 −
(ŝ1 + ŝ¨2 ) = 0
θ0
θ0 ω0 I
θ0
θ0 1 + 2σ I
Simplifying, we have the equation of motion for the two metronomes in non-dimensional form
ŝ¨1 + α(sˆ1 2 − 1)ŝ˙ 1 + ŝ˙ 1 − β(ŝ¨1 + ŝ¨2 ) = 0
ŝ¨2 + α(sˆ2 2 − 1)ŝ˙ 2 + ŝ˙ 2 − β(ŝ¨1 + ŝ¨2 ) = 0
(12)
(13)
2
σ mr
where α = ωc0 I and β = 1+2σ
I . α and β are the only two parameters that characterize the system.
Equations (12) and (13) are the nondimensionalized variants of equations (8) and (9) respectively.
3.3
Determining Parameter Values
To graph the equations (12) and (13) and obtain numerical solutions, we must determine the numerical
values of the controlled parameters in the system.
14
3.3.1
Distance from pivot to pendulum center of mass
p
First, to convert from non-dimensional values to physical values, we must know the value of ω0 ≡ mgr
I .
Since the independent variable in this investigation is the frequency, it is desirable to formulate a relationship
between the frequency f and distance to center of mass r. The pendulum of the metronome consists of two
parts, the bob and the stick. Thus, the center of mass is given by
r=
mbob rbob + mstick rstick
mbob + mstick
(14)
where rpart denotes the distance from the part’s center of mass to the pivot. mbob and mstick are known
values listed in 2.2. Since the bob is a movable weight that’s length can be adjusted from the pivot to change
the frequency of the metronome, rbob can be measured by using a ruler. To find rstick (distance from center
of mass of stick to pivot), we detach the pendulum from the metronome and the bob from the stick. We then
balance the stick on a razor blade to find rstick = −0.0147 ± 0.0005m. After substituting the appropriate
values, we obtain
r = −0.0121 + 0.179rbob
(15)
For each respective frequency value used in the solving of equations (12) and (13), rbob was measured digitally
using ImageJ analysis. The values of rbob are shown in Table 2. The Mathematica program then used (15)
to determine r, which it used to solve and graph the differential equations. The uncertainty in rb ob was
noted to be 0.005cm after it was determined that one pixel (the smallest unit in ImageJ) corresponded to
0.005cm by means of a scale.
f (Hz)
rbob (cm, ±0.005)
208
3.482
200
3.691
192
3.900
184
4.095
176
4.281
168
4.466
160
4.652
152
4.819
144
4.986
138
5.140
132
5.279
126
5.395
120
5.534
116
5.627
112
5.729
15
f (Hz)
rbob (cm, ±0.005)
108
5.827
104
5.920
100
6.012
96
6.105
92
6.203
88
6.296
84
6.389
80
6.472
76
6.556
72
6.644
69
6.709
66
6.765
63
6.820
60
6.876
56
6.932
52
6.988
48
7.053
44
7.108
40
7.169
Table 2: Values of rbob for different frequencies
3.3.2
Moment of Inertia
The moment of inertia changes with each value of frequency, since the pendulum bob is physically moved to
alter the frequency value. Since the pendulum consists of two parts (bob and stick), its moment of inertia
can be denoted as
I = Istick + Ibob
(16)
where Istick and Ibob denote the moment of inertia about the pivot point for the stick and bob respectively.
Since the bob is relatively uniform in both shape and mass density, it may be modelled as a point mass
2
with moment of inertia mbob rbob
, where rbob is the distance from the pivot to the center of the bob. Due
to limiations in equipment, Istick could not be experimentally determined. Thus, we will take the value
Istick = 1.29 × 10− 5kg · m2 , determined by Martens et al. [5]. After substituting mbob = 5 × 10−3 kg,
16
equation (16) becomes
2
I = 1.29 × 10−5 + 5 × 10−3 rbob
(17)
Values of I for each frequency were then determined using Table 2.
3.3.3
Mass ratio
The mass ratio σ is given by the ratio of the pendulum mass to the mass of the rest of the system. Using
values in 2.2, we have
σ=
m
m
=
M
2Mb + Mf
σ = 0.145 ± 0.007
3.3.4
Damping coefficients
Again, due to limiations in equipment, values of the linear (c) and non-linear damping coefficients (b)
were unable to be determined experimentally. Thus, we will use predetermined values from experiements
that have used the same brand of metronome. Martens et al. determined the linear damping coefficient
for the Wittner Series 880 to be c = 1.8 × 10−6 kg m2 s−1 [5], while Pantaleone experimentally obtained
b = 4.2 × 10−6 kg m2 s−1 [6].
3.3.5
Initial angular displacement
To obtain solutions to differential equations (12) and (13), the initial angles formed by each of the pendulums
with the vertical is required. In this investigation, the initial angles were controlled by a Lego contraption.
By analyzing images of the pendulum with the Lego contraption in ImageJ, the initial angle in degrees may
be determined for each metronome. In all trials of the experiment, the intial angles for metronomes 1 and 2
respectively are θ1 = (30.687 ± 0.001)◦ and θ2 = (37.988 ± 0.001)◦ .
Figure 11: ImageJ analysis to determine the intial anglular displacment of metronome 1 (left) and 2 (right).
The vertical used was the rectangular tempo marking printed on to the front of the metronome.
17
Since equations (12) and (13) utilize non-dimensional parameters ŝ1 and ŝ2 , a conversion was required to
determine ŝ1 (0) and ŝ2 (0) (initial conditions requried to solve the differential equations). The conversion is
shown below:
ŝ1 (0) =
θ1
θ0
r
ŝ2 (0) =
b
c
= 0.818
= θ1
3.4
θ2
θ0
r
= θ2
b
c
= 1.01
Obtaining Solutions to the Synchronization Time
Although the equations of motion for two metronomes were derived in section 3.2, the synchronization time
cannot be algebraically computed without the use of numerous approximation methods. To preserve accuracy, we will implement a graphical solution powered by Wolfram’s Mathematica.
Using Mathematica, we can numerically solve equations (12) and (13) by inputting the set of parameters
determined in section 3.3, as well as the initial conditions ŝ1 (0) = 0, ŝ2 (0) = 0, ŝ˙ 1 (0) = 0, ŝ˙ 2 (0) = 0. Note
the values of ŝ˙ 2 (0) and ŝ˙ 2 (0), as the pendulums are released from rest and have zero initial velocity. To
obtain a numerical result for the synchronization time based on the solved differential equations, we present
the following algorithm.
Since synchronization is defined as the auditory ticks of the metronomes beating in unison and each
metronome will produce a tick when its pendulum bob coincides with the vertical (θ = 0), we can determine
the time at which each metronome ticks by finding the roots to equations (12) and (13). For each root xi
of (12), we will define a margin and look for a root in (13) between the interval x − and x + . If such
root exists, we repeat the same process for the next n2 roots of (12). If all n + 1 tests are satisfied, the time
of synchronization will be xi . If no such root exists or any of the n successive tests fail, we will repeat the
entire process for the next successive root xi+1 .
The above algorithm was implemented in Mathematica using procedural type statements (see Appendix).
First, the equations were numerically solved using NDSolve. Then, using MeshFunctions, the equations were
graphed and the roots of each were appended to a list. Then, the algorithm would use the two lists to
produce the non-dimensional time of synchronization. The program in Mathematica was then compiled for
various iterations of frequency to produce a dataset that was graphed in LoggerPro. Since the value of is arbitrary, multiple iterations of theoretical simulation were performed over the interval ∈ [0.015, 0.005]
(This interval was chosen to include the threshold value of 0.01 used in section 2.3.2.). Four distinct datasets
were found, and each was then graphed and fitted using B-Spline functions powered by Mathematica.
2 Again,
different values of n were tested. It was found that for values n >= 10, the synchronization times produced were
identical. Since n should be as large as possible to maintain accuracy, n was chosen to be 10.
18
Figure 12: Theoretical curves produced by Mathematica. The orange, red, purple, and blue curves correspond to intervals of [0.015, 0.013], [0.013, 0.011], [0.011, 0.008], and [0.008, 0.005]. The bounding for
intervals are only approximate values, as specific values of serve no purpose.
3.4.1
Conversion from non-dimensional to actual parameters
It should be noted that the Mathematica program produced solutions in non-dimensional time. As such, we
must use the parameter ω0 to convert from τ to t (actual time) to produce the curves seen in Figures 12
p
to τ . The numerical values of ω0 was determined by using
and 13. This is done by multiplying ω0 ≡ mgr
I
equations (15) (16), and Table 2.
3.5
Comparing Experimental and Theoretical Results
Due to the arbitrary value, the curve from Figure 12 that most suited the experimental data was chosen,
based on the root mean squared error (RMSE). The purple curve ( = 0.01) resulted in the smallest RMSE
value (14.45), and a comparison with the experimental curve is shown below.
19
Figure 13: Theoretical curves (blue) with = 0.01 with the experimental data points and experimental
curve of best fit (orange).
With the exception of the first two points, all data points were within 10 seconds of the theoretical
synchronization time. However, even with the exclusion of the first two data points, the RMSE value is 6.69,
indicating inaccuracy. Furthermore, when considering the error bars, only one data point fell within the
expected curve. The rest were all above the theoretical values and had relatively small error bars, suggesting
a source of systematic error.
Furthermore, the first two data points are significantly inaccurate, differing with the theoretical values
by over 20 seconds, approximately corresponding to a 25% experimental error. Overall, although the shape
of the two curves are roughly similar, the times produced by the experimental curve are consistently larger
than those modelled by the theoretical analysis. This strongly suggests the possibility of a systemic error
present in the experiment that interfered with the accuracy of the results.
20
4
Conclusion
Despite the low uncertainty of most experimental data points, the most suitable theoretical curve still
failed to accurately model the synchronization time determined in the experiment. Of the sixteen trials
performed, only one produced a value that fell within the theoretical curve. However, both the experimental
and theoretical results share extremely similar shapes and support the hypothesized inverse relationship
between frequency and synchronization time. Furthermore, the theoretical curve closely matches the results
determined by Wang, which suggest the theoretical model is valid [11].
Nonetheless, an unexpected result that can be seen in both the experimental and theoretical curves is
the seemingly asymptotic behaviour the synchronization time exhibits for frequencies larger than approximately 2.25Hz. Further experimentation at higher frequencies would most likely be required to confirm this
behaviour.
4.1
Evaluation
Several systematic and random errors affected the accuracy and precision of the experimental results.
Perhaps the most significant systematic error was due to friction between the aluminum cans and the
ground, which damped the motion of the entire system. Although a piece of paper was used to reduce the
effects of friction, its impact was visible in several trials, especially ones with lower frequencies. For this
reason, much larger times of synchronization were obtained for the first few data points, corresponding to a
significant deviation from the theoretical curve in Figure 13. A simple solution would be to sand the sides
of the aluminum can to reduce bumps and imperfections on the surface, as well as placing the system on a
surface with a smaller friction coefficient (such as ice).
Furthermore, a significant random error observed in nearly all trials was the inconsistency of the initial
angles of the pendulums. Despite the construction of a Lego contraption to control the initial angular
displacements, the removal of the contraption to commence each trial often supplied an unwanted initial
impulse to the pendulum. As a result, depending on the magnitude and direction of the impulse, the
angular velocity at t = 0 would be nonzero, and a larger spread was recorded for the synchronization time.
This error may be reduced by constructing a more elaborate brace that is secured to the ground to increase
precision of experimental results.
4.2
Discussion
This investigation only considered the in-phase synchronization state of two metronomes. In some trials, antiphase synchronization was observed, and synchronization times obtained vastly differed from other trials.
More research may be conducted to investigate the dynamics of the system where anti-phase synchronization
is the final state.
21
References
Articles
[4]
J. Carranza, Michael Brennan, and Bin Tang. “On the synchronization of two metronomes and their
related dynamics”. In: Journal of Physics: Conference Series 744 (Sept. 2016), p. 012133. doi: 10.
1088/1742-6596/744/1/012133.
[5]
Erik Martens et al. “Chimera states in mechanical oscillator networks”. In: Proceedings of the National
Academy of Sciences of the United States of America 110 (June 2013), pp. 10563–10567. doi: 10 .
1073/pnas.1302880110.
[6]
James Pantaleone. “Synchronization of metronomes”. In: American Journal of Physics - AMER J
PHYS 70 (Oct. 2002), pp. 992–1000. doi: 10.1119/1.1501118.
[7]
Jeffrey Tithof et al. “The Time to Synchronization for N Coupled Metronomes”. In: (Dec. 2011), p. 9.
[11]
Xuepeng Wang and Sihui Wang. “In-phase Synchronization of Two Coupled Metronomes”. In: (May
2018), p. 12.
Books
[8]
Steven H. Strogatz. Nonlinear dynamics and Chaos: with Applications in Physics, Biology, Chemistry,
and Engineering. Addison-Wesley Pub., 2000.
Online Sources
[1]
ThePiano.SG. How Does A Metronome Work. 2016. url: https://www.thepiano.sg/piano/read/
how-does-metronome-work (visited on 05/30/2021).
[3]
Seth Popinchalk. Challenge: Metronome and Cart Equations of Motion. Mathworks. 2008. url: https:
//blogs.mathworks.com/simulink/2008/10/09/challenge-metronome-and-cart-equations-ofmotion/ (visited on 08/21/2021).
[9]
Nathan Kutz. The Van der Pol Oscillator. Youtube. 2021. url: https://www.youtube.com/watch?
v=lX4dPij5_WM (visited on 07/30/2021).
[10]
Cornell MAE. MAE5790-10 van der Pol oscillator. Youtube. 2014. url: https://www.youtube.com/
watch?v=O1lQrHemPsw (visited on 08/21/2021).
[12]
Jeffrey Chasnov. Nondimensionalization — Appendix B — Differential Equations for Engineers. Youtube.
2019. url: https://www.youtube.com/watch?v=SMs-40stA10 (visited on 07/30/2021).
Images
[2]
Harvard Natural Sciences Lecture Demonstrations. Metronomes in a synchronized and unsynchronized state. June 2010. url: https://sciencedemonstrations.fas.harvard.edu/presentations/
synchronization-metronomes.
22
5
Acknowledgements
Will be included in the final submission.
23
Appendix
Code utilized in the determination of synchronization time from graphed differential equations.
solve[i_] :=
NDSolve[{a’’[t] + u*(a[t]^2 - 1)*a’[t] + a[t] - B*(a’’[t] + b’’[t]) ==
0,
b’’[t] + u*(b[t]^2 - 1)*b’[t] + b[t] - B*(a’’[t] + b’’[t]) == 0,
(1/o)*c’’[t] - u*(a[t]^2 - 1)*a’[t] - u*(b[t]^2 - 1)*b’[t] a’[t] - b’[t] == 0,
o == 0.145, m == 0.028, z == 0.0000018,
u == z/(wlist[[i]]*ilist[[i]]),
B == (o/(1 + 2 o))*(m*rlist[[i]]^2/ilist[[i]]),
a[0] == 1.34256, b[0] == 1.70058, c[0] == 0,
a’[0] == 0, b’[0] == 0, c’[0] == 0}, {a, b, c}, {t, 0, 10000}]
(*The "solve" method solves the set of differential equations based on a given set of initial
conditions*)
get[i_] := (
f[t_] := Evaluate[a[t] /. solve[i]];
g[t_] := Evaluate[b[t] /. solve[i]];
plot1 =
Plot[f[t], {t, 0, 1000}, Mesh -> {{0}}, MeshFunctions -> {#2 &},
MeshStyle -> PointSize[Medium], PlotRange -> All,
PlotPoints -> 2000];
l1 = Sort@Cases[Normal@plot1, Point[{x_, y_}] -> x, Infinity];
plot2 =
Plot[g[t], {t, 0, 1000}, Mesh -> {{0}}, MeshFunctions -> {#2 &},
MeshStyle -> PointSize[Medium], PlotRange -> All,
PlotPoints -> 2000];
l2 = Sort@Cases[Normal@plot2, Point[{x_, y_}] -> x, Infinity];
compute[l1, l2, e/wlist[[i]], index];
)
(*The "get" method generates the roots of the solved differential equations and stores them in
lists l1 and l2.*)
compute[l1_, l2_, margin_, index_] := (
For[i = 1, i < Length[l1] + 1, i++,
q = 1;
For[j = 1, j < Length[l2] + 1, j++,
q = 1;
If[Abs[l1[[i]] - l2[[j]]] <= (margin),
For[k = 1,
k <= 10 && i + k <= Length[l1] && j + k <= Length[l2], k++,
If[Abs[l1[[i + k]] - l2[[j + k]]] > (margin),
24
q = -1;
Break[]]
];
If[q == 1, AppendTo[listND, l1[[i]]];
AppendTo[list, l1[[i]] *wlist[[index]]];
Break[]], q = -1]
];
If[q == 1, Break[]]
])
(*The method "compute" checks roots within a certain margin and determines whether a root may be
evaluated as the synchronization time.*)
25
6
Concerns Regarding the First Draft
• Level of detail needed for algebraic steps presented in section 3.1 and 3.2, especially the usage of Van
der Pol oscillators to model escapement (more steps need to be shown?)
• Uncertainty
– Since I used values from other papers which did not provide an uncertainty, many of my parameters
and equations do not account for uncertainty (such as in (15) and (17)). I thought of estimating
the literature values by estimating the smallest increment given (ex. 0.0000018 has uncertainty
0.0000001), but I’m not sure how feasible this is
– Even if I do implement uncertainties for parameters and obtain final uncertainties for synchronization time, I can’t really implement into the theoretical curve, since I used 35 data points to
compute the curve (which would crowd the graph). Also, I don’t know if having error bars on a
theoretical curve made from data points serve any meaning
• I’m still unsure if I should use an Appendix or not (Table 2 looks out of place)
• Citation Format: I’m not sure if IB wants the EE to be cited in a certain style, right now I’m just
using a LATEXpackage for citations.
Word Count: approx. 3950
26