Modelling Pendulum
Experiment
Terms of reference
Modelling 334 Technical report
Authors
H Barnard 25244337
2023
Department of Mechanical and Mechatronic Engineering
Departement Meganiese en Megatroniese Ingenieurswese
Privaat Sak X1, Private Bag X1, Matieland, 7602
Tel: +27 21 808 4204 | www.eng.sun.ac.za
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Signature:
Name:
Henri Barnard ..........................
Date:
10/05/2023 .............................
i
Student no:25244337 ................
Abstract
This report has taken a look into the feasibility of using mathematical modelling to
model complex mechanical systems. The report focussed on assessing the model
of a simple pendulum with the formula:
1
𝐽𝜃̈ + 𝐵𝜃̇ + 2 𝑚𝑔𝐿𝜃 = 0.
The report decided on this model since it takes the damping coefficient into
consideration unlike various other reports. After an experiment was conducted
the data was analysed and this report confidently concludes that this particular
model does a good job of describing the pendulum’s motion and that it is
possible to model a physical system using a mathematical model but suggests
that further investigations be done to support the findings of this report.
ii
Table of contents
Page
Plagiarism declaration ....................................................................................... i
Abstract ........................................................................................................... ii
List of figures .................................................................................................. iv
List of tables .................................................................................................... v
List of symbols ................................................................................................ vi
Introduction..................................................................................................... 1
1
Methods.................................................................................................... 2
1.1 Equipment required ................................................................................ 2
1.2 Experiment set up ................................................................................... 2
1.3 Conducting the experiment .................................................................... 3
1.4 Data extraction ........................................................................................ 3
2
Results and Analysis .................................................................................. 5
2.1 Data analysis ............................................................................................ 5
2.1.1 Obtaining theoretical system parameters .................................. 5
2.1.2 Calculating damping coefficient and mass moment of inertia
from experimental data .............................................................. 6
2.2 Comparing Experimental data and Theoretical data .............................. 8
2.2.1 Theoretical Data .......................................................................... 9
2.2.2 Experimental data ..................................................................... 10
2.2.3 Comparison................................................................................ 12
3
Discussion ............................................................................................... 13
4
Conclusions ............................................................................................. 15
5
References .............................................................................................. 16
Appendix A
17
Appendix B
18
iii
List of figures
Page
Figure 1 Snapshot of pendulum befor it drops ....................................................... 2
Figure 2 Snapshot of the pendulum being tracked in Tracker ................................ 4
Figure 3 Screenshot of Inventor program showing the modelled Pendulum bar
and its mass moment of inertia. ........................................................... 5
Figure 4 Screenshot of calculations done in SMath. ............................................... 6
Figure 5 Excerpt of Matlab code showing how J and B are calculated using
backslash ............................................................................................... 7
Figure 6 Graph of Theoretical plot of angular displacement over time ................ 10
Figure 7 Excerpt of Matlab code for plotting angular displacement over time
graph ................................................................................................... 11
Figure 8 Graph of experimental angular displacement over time ........................ 11
Figure 9 Both angular displacement graphs on the same axis .............................. 12
Figure 10 Graph of angular velocity over time in Tracker ..................................... 13
Figure 11 Graph of angular acceleration over time in Tracker .............................. 13
Figure 12 Graph of Angular displacement over time ............................................ 17
Figure 13 Graph of Angular velocity over time...................................................... 17
Figure 14 Graph of Angular acceleration over time .............................................. 17
iv
List of tables
Page
Table 1 Variables and their types ............................................................................ 2
Table 2 Measurements of Pendulum: ..................................................................... 5
v
List of symbols
a0
coefficient of 𝜃̈
a2
coefficient of θ
B
Damping coefficient
g
Gravitational acceleration
J
Mass moment of inertia
L
Length of pendulum
m
Mass of pendulum
r
characteristic equation roots.
T
Period of swingmen pendulum
t
Thickness of pendulum
w
width of pendulum
α
Alpha
β
Beta
ζ
damping ratio
θ
Angular displacement of pendulum
𝜃̇
Angular velocity of pendulum
𝜃̈
Angular acceleration of pendulum
ωd
Natural frequency of pendulum
ωn
Damped frequency of pendulum
vi
Introduction
Modelling enables us to represent various complex systems using simpler
mathematical equations, which can be easily solved and analysed. This capability
allows companies to save time and money. Moreover, modelling facilitates the
study and enhanced understanding of the behaviour exhibited by complex
mechanical systems without the need for conducting physical tests.
The Modelling 334 course requires students to conduct experiments aimed at
determining model parameters. This evaluation assesses whether students have
attained ECSA Graduate attribute 4, which pertains to their proficiency in
conducting experiments, investigations, and data analysis.
The focus of this report is do conduct an investigation of a mathematical model of
a simple pendulum in the form of:
1
𝐽𝜃̈ + 𝐵𝜃̇ + 2 𝑚𝑔𝐿𝜃 = 0.
The objectives of the report are to determine the system parameters and to
evaluate the model’s accuracy in describing a simple pendulum system.
This report will explain the procedures used to conduct the experiment as well as
how all the data was handled and processed. Furthermore, this report will do a
thorough analyses of the data and explain how the different system parameters
were calculated and/or measured using experimental data. Near the end of the
report some important point regarding the report and the integrity of the report
will be discussed.
Most reports found online today do experiments with pendulums in the form of a
point mass at the end of some string whereas this report focuses on a pendulum
made of a bar. This has many more practical use cases in the real world as very
few pendulums consist of solely a point mass at the end of a string. Furthermore,
this report also takes the damping coefficient into account where it is normal seen
as negligible in reports involving a simplified version of a pendulum.
1
1 Methods
Pendulum
1.1 Equipment required
•
Pendulum
•
Pendulum stand
•
Scale
•
Calliper
•
Tape measurer
•
Video recorder (Phone Camera)
•
Bright coloured stickers
•
Note pad
•
Pencil
Bright stickers
Pendulum stand
Figure 1 Snapshot of pendulum
befor it drops
Table 1 Variables and their types
Variable
Variable type
Mass of pendulum
Control
Length of pendulum
Control
Width of pendulum
Control
Thickness of pendulum walls
Control
Initial angular displacement
Independent
Angular displacement
Dependant
Angular velocity
Dependant
Angular acceleration
Dependant
Period
Dependant
1.2 Experiment set up
1.
The length of the pendulum bar was measured using the tape measurer.
2.
The width of the pendulum and the thickness of the thin walls were
measured using the calliper.
2
3.
The diameter of the bar connected to the top of the pendulum was
measured using the calliper.
4.
The weight of the pendulum was measured by using the scale and
weighing the pendulum 5 times and taking the average of the result to get
a more accurate value.
5.
All the values were recorded by writing them done in a note pad.
6.
Two bright orange stickers per placed 25 mm away from one another on
the pendulum to serve as tracking points for tracker software and to set
up various calibration points.
7.
The pendulum was placed on the stand securely.
8.
The stickers were ensured to be visible to the camera.
1.3 Conducting the experiment
1. The pendulum was raised to approximately 45 degrees and then let go to
swing freely.
2. The freely swinging bar was then filmed using a phone camera.
3. Although the phone was attempted to be help still, after viewing the video
it is apparent that the camera did shift while filming.
1.4 Data extraction
1. The video of the pendulum swinging was imported to Tracker, a program
designed to track objects in a 2-dimensional plane.
2. The bright stickers were used to set up a calibration stick to give Tracker a
scale of the pendulum.
3. A “point mass” was placed at both bright stickers. Tracker’s Autotrack
feature was used to track the position of each sticker in each frame to
create a mapping of the path that each sticker follows while the pendulum
swings.
4. Two additional calibration points were placed at the top and the bottom
of the pendulum stand to track and account for the drifting of the camera
while filming.
3
5. Tables and graphs of all the relevant data were set up in Tracker and can
be found in Appendix A.
Origin axis
Calibration stick
Point mass tracking points
Figure 2 Snapshot of the
pendulum being tracked in
Tracker
4
2 Results and Analysis
2.1 Data analysis
2.1.1 Obtaining theoretical system parameters
2.1.1.1 Moment of inertia
Below shows a table of all the measurements that were taken of the pendulum.
Table 2 Measurements of Pendulum:
Variable
Measurement
L
0.5 m
w
0.0195 m
t
0.002 m
m
0.595 kg
d
0.01 m
These measurements were used to recreate and model the pendulum in Inventor
to get an accurate theoretical value for the mass moment of inertia of J = 0.050
kgm2.
Figure 3 Screenshot of Inventor program showing
the modelled Pendulum bar and its mass moment of
inertia.
5
2.1.1.2 Damping coefficient
The damping coefficient was calculated with the help of SMath Studio and
equations found in (Kluever, 2015). It is important to note that gravitational
acceleration, g, is assumed to be 9.81 at low altitude.
Kluever(2015:214)
Kluever(2015:221)
Kluever(2015:221)
Figure 4 Screenshot of calculations done in SMath.
2.1.2 Calculating damping coefficient and mass moment of inertia from
experimental data
2.1.2.1 Data processing
Tracker was used to set up tables with the values for angular displacement,
angular velocity and angular acceleration. These tables were then copied and
6
pasted to an Excel spreadsheet. Then Matlab’s “Import Data" feature was used to
import the data from Excel by selecting the relevant data and giving them
appropriate variable names and storing each one of them in Matlab as 983x1
columns vectors. A 983x2 coefficient matrix was set up called A, existing of a
column containing every 𝜃̈ value and every 𝜃̇ value, respectively. Next a second
matrix, the solution matrix, called B is created. The Matlab’s backslash operator is
used to determine that the mass moment of inertia, J = 0.0499 kgm2 and damping
Figure 5 Excerpt of Matlab code showing how J and B are calculated using backslash
coefficient, B = 0.0114 kg3/2m3.
The figure above is an excerpt of Matlab code used to calculate the mass moment
of inertia and the damping coefficient of the pendulum system. The full Matlab
code can be found in Apendix B.
The backslash operator (\) in Matlab performs matrix division or solves systems of
linear equations. It provides a concise and efficient way to find solutions to linear
equations by leveraging numerical algorithms and matrix factorizations.
When you have a system of linear equations of the form AX = B, where A is an mby-n matrix, X is an n-by-p matrix of unknowns, and B is an m-by-p matrix of
constants, the backslash operator computes the solution X such that AX = B.
The backslash operator uses various numerical methods to solve the system
depending on the properties of the input matrices A and B. The most common
techniques employed by Matlab are LU decomposition and QR factorization.
1.
LU Decomposition:
•
If matrix A is square and non-singular (i.e., invertible), Matlab performs LU
decomposition to solve the system exactly.
•
LU decomposition factorizes matrix A into two matrices: L (lower triangular)
and U (upper triangular). It can be written as A = LU.
•
Once A is decomposed, Matlab solves the system by solving two simpler
equations: LY = B and UX = Y, where Y and X are intermediate matrices.
7
•
2.
The LU decomposition method is efficient and provides an exact solution
when A is non-singular.
QR Factorization:
•
If matrix A is rectangular, singular, or the system is overdetermined (more
equations than unknowns), Matlab uses QR factorization to find the leastsquares solution.
•
QR factorization decomposes matrix A into two matrices: Q (orthogonal) and
R (upper triangular). It can be written as A = QR.
•
The QR decomposition allows Matlab to solve the system by transforming it
into a simpler form, known as the least-squares problem.
•
The least-squares solution minimizes the sum of the squares of the residuals
between AX and B, providing the best approximate solution when an exact
solution does not exist.
The backslash operator automatically determines the most appropriate method
based on the characteristics of the input matrices. It selects LU decomposition for
square, non-singular matrices and QR factorization for rectangular or singular
matrices.
Additionally, the backslash operator in Matlab has other useful features:
•
It can manage sparse matrices efficiently, taking advantage of their sparsity
to optimize computations.
•
It can solve multiple systems of equations simultaneously by providing
matrices B and X with appropriate dimensions.
Matlab’s backslash operator is a very powerful tool and is extremely helpful and
efficient when solving linear systems.
2.2 Comparing Experimental data and Theoretical
data
Matlab was used to first set up and plot both theoretical and experimental data
and creating graphs.
8
2.2.1 Theoretical Data
An equation for the angular displacement of the pendulum arm was created by
applying theory in the following way.
1. First the initial mathematical model is considered:
1
𝐽𝜃̈ + 𝐵𝜃̇ + 2 𝑚𝑔𝐿𝜃 = 0.
2. Then the characteristic equation is determined to be:
1
𝐽𝑟 2 + 𝐵𝑟 + 2 𝑚𝑔𝐿 = 0.
3. Now the values for mass moment of inertia and damping coefficient of the
pendulum that were determined earlier in the report on pages 5 and 6 can
be substituted in and the following equation is created:
0.05𝑟 2 + 0.0112𝑟 + 1.4592 = 0.
4. There is a variety of different ways to calculate the roots of this equation
and in this report, Matlab’s built in “roots” function is used and 2 complex
conjugate roots are found being:
𝑟1,2 = −0.1118 ± 5.4011𝑖.
5. These values are used to set up the final equation that will be used in
Matlab to create the graph of the theoretical angular displacement. The
final equation is:
𝑦(𝑡) = 𝑒 𝛼𝑡 [𝑐1 cos 𝛽𝑡 + 𝑐2 sin 𝛽𝑡].
In this equation α is the real part of the root conjugate pair and β is the
complex part of the conjugate pair. c1 is the initial angular displacement of
the pendulum arm, which in this case is c1 = 35.4°, but is important to
𝜋
convert this into radians by multiplying it by 180 which is equal to
0.6178rad. c2 is the angular displacement of pendulum arm as time tends
to infinity. For this damped system it is known that the angular
displacement tends to 0 as time tends to infinity. Thus c2 = 0 rad and the
𝑐2 sin 𝛽𝑡 term can be discarded.
6. When substituting in all the knowns we are left with:
𝑦(𝑡) = 𝑒 −0.1118𝑡 [0.6178 cos(5.4011𝑡)].
9
7. An array of t values is created using Matlab’s “linspace” function which
creates a row vector of points spaced out linearly between a given start
and end value, in this case from 0 to 17.
8. Matlab’s plot function is used to plot a graph of the expected angular
displacement and we receive the following graph:
Figure 6 Graph of Theoretical plot of angular displacement over time
2.2.2 Experimental data
1. Data from Tracker is exported to Matlab using the process described under
point 4.1.2.1, but this time the time data is also exported to Matlab.
2. Matlab’s plot function is used to create a graph by plotting every angular
displacement measurement made by Tracker with their respective time
value.
10
3. The code for this procedure looks like this:
Figure 7 Excerpt of Matlab code for plotting angular displacement over time
graph
4. The following graph is produced:
Figure 8 Graph of experimental angular displacement over time
11
2.2.3 Comparison
When comparing the two results it is easy to start by comparing the two graphs.
Matlab is used to plot these to graphs on the same axis:
Figure 9 Both angular displacement graphs on the same axis
From this graph one can clearly see that the experimental data does tend to follow
the model closely in some sections. However, the theoretical plot seems to be less
affected by damping, this could be due to the theoretical and experimental values
for the damping coefficient and mass moment of inertia differs slightly.
When comparing the theoretical and experimental system parameters there is
some evidence that indicates that the mathematical model does a good job in
accurately describing the model as both the mass moment of inertia and the
damping coefficient are very similar with Jtheoretical = 0.05 kgm2 and Jexperimental =
0.0499 kgm2 and Btheoretical = 0.0112 kgm2/s and Bexperemental = 0.0114 kgm2/s.
12
3 Discussion
There were a few results that did stand out as they were not expected. For
instance, when plotting the data gathered by Tracker some abnormalities came to
light like these last few points in the angular velocity graph created in Tracker
which don’t follow the same smooth path as the rest of the data.
Figure 10 Graph of angular velocity over time in Tracker
The problem is even more exaggerated when examining the angular acceleration
graph:
Figure 11 Graph of angular acceleration over time in Tracker
This discrepancy in of the data at these points in the graphs is most likely due to
the manually placed points placed by the user when Tracker’s Autotracker could
not find the point on its own. These abnormalities could have been lessened by
keeping the camera more stable when filming the swinging pendulum, which
would have reduced the reliability on the calibration point to track the moment of
the camera.
To try and lessen the impact of these abnormalities only data from between the
time stamps 2.999 and 13.993 were used which don’t include any extreme outliers
to what we expect to see.
13
Another surprise was the striking difference in the rate of decay in the angular
displacement, where the theoretical model initially decays much faster that the
experimental data does and then slows down where the experimental data seems
much more linear. This could be due to the model assuming that the magnitude
of the damping is dependent on how fast the pendulum is swinging due to the 𝜃̇
term. However, in actuality friction from the connection between the supports of
the pendulum stand and the rod connected to the pendulum is the main force
effecting the damping coefficient. Since friction does not vary with the speed an
object travels at this results in a much more linear decay in the angular
displacement.
14
4 Conclusions
With how important the mathematical modelling of a system into a simpler
mathematical equation is in an industry filled with complex mechanical systems,
it is vital to determine how effective mathematical models are at describing how
a system would react and behave in specific conditions. In this report an
investigation was launched with the objective of determining how trustworthy a
model can be in describing the behaviour of a mechanical system.
It has been determined that the model :
1
𝐽𝜃̈ + 𝐵𝜃̇ + 𝑚𝑔𝐿𝜃 = 0
2
describes the physical system fairly well but not extremely accurate. Mathematical
models can be used describe a mechanical system, but it is recommended that
these models be used merely for approximations as the results in this case are not
reliable enough to base important decisions on as the theoretical data does not
accurately represent the experimental data.
Further investigations can be done on more complex systems to determine
whether these findings correlate to other systems as well, but the findings in this
report are satisfactory for the objective of this report.
15
5 References
Kluever, C.A. 2015. Dynamics Systems Modelling, Simulation, and Control 2nd
addition. USA: Wiley.
16
Appendix A
A.1
Tracker graphs
Figure 12 Graph of Angular displacement over time
Figure 13 Graph of Angular velocity over time
Figure 14 Graph of Angular acceleration over time
17
Appendix B
% Theoretical data.
L = 0.5;
m = 0.595;
g = 9.81;
w = 0.0195;
d = 0.01;
J = 0.05;
t1 = 11.777;
t2 = 12.360;
T = t2-t1;
thetaInitial = 35.4;
a0 = 0.5*m*g*L
a0 = 1.4592
a2 = J
a2 = 0.0500
NatFrequency = sqrt(a0/a2)
NatFrequency = 5.4023
DampFrequency = 2*(pi/T)
DampFrequency = 10.7773
zeta = a2/(2*sqrt(a0))
zeta = 0.0207
B = 2*zeta*NatFrequency*a2
B = 0.0112
p = [J B 0.5*m*g*L]
p = 1×3
0.0500
p1 = 1×3
1.0000
0.0112
1.4592
0.2236
29.1847
18
Roots = roots(p)
Roots = 2×1 complex
-0.1118 + 5.4011i
-0.1118 - 5.4011i
r = 2×1 complex
-0.1118 + 5.4011i
-0.1118 - 5.4011i
Alpha = real(Roots(1))
Alpha = -0.1118
Beta = imag(Roots(1))
Beta = 5.4011
td = linspace(0,20,1500)
td = 1×1500
0
0.0133
0.0267
0.0400
0.0534
0.0667
0.0801 ⋯
thetatheoretical =
exp(1).^(Alpha.*td).*(thetaInitial*(pi/180).*cos(2*Beta.*td))
thetatheoretical = 1×1500
0.6178
0.3973 ⋯
0.6105
0.5906
0.5585
0.5149
0.4608
plot(td,thetatheoretical)
title("Theoretical plot of angular displacement of the Pendulum")
xlabel("time (s)")
ylabel("Angular displacement (rad)")
xlim([0 17])
19
% Experemental Data
Theta;
ThetaDot;
ThetaDoubleDot;
t;
Theta1 = Theta*(pi/180);
ThetaDot1 = ThetaDot*(pi/180);
ThetaDoubleDot1 = ThetaDoubleDot*(pi/180);
figure()
plot(t,Theta1)
xlim([0 16.358])
title("Experemental plot of angular displacement of the Pendulum")
xlabel("time (s)")
ylabel("Angular displacement (rad)")
20
figure()
plot(t,ThetaDot1)
xlim([0 16.358])
figure()
plot(t,ThetaDoubleDot1)
xlim([0 16.358])
21
% Calculating J and B from experimental Data
% only using data between t = 2.999 and t = 13.993
A = [ThetaDoubleDot1(181:841) ThetaDot1(181:841)]
A = 661×2
-49.3465
-49.9734
-41.4602
-41.1018
-25.2212
-9.7338
-18.6052
-15.1464
3.9911
28.3584
⋮
-1.4099
-2.2215
-3.0841
-3.6440
-4.3478
-4.6654
-4.5046
-5.2543
-5.2733
-4.8212
B = -0.5*m*g*L*Theta1(181:841)
B = 661×1
-0.6665
-0.6230
-0.5585
-0.4731
-0.3813
-0.2617
-0.1545
-0.0427
0.1010
0.2137
⋮
22
sol = A\B
sol = 2×1
0.0125
0.0029
Jexperemental = sol(1,1)*4
Jexperemental = 0.0499
Bexperemental = sol(2,1)*4
Bexperemental = 0.0114
% Comparing Theoretical data to Experemental data
figure()
plot(t,Theta1)
hold on
plot(td,thetatheoretical)
hold off
legend("Experemental", "Theoretical")
title("Plot of angular displacement of the Pendulum")
xlabel("time (s)")
ylabel("Angular displacement (rad)")
xlim([0 17])
23