Post on 26-Aug-2020
Problem 11. Azimuthal-
Radial PendulumBy: Aarij Atiq
1
Problem Statement
Fix one end of a horizontal elastic rod to a rigid
stand. Support the other end of the rod with a
taut string to avoid vertical deflection and
suspend a bob from it on another string. In the
resulting pendulum the radial oscillations
(parallel to the rod) can spontaneously convert
into azimuthal oscillations (perpendicular to the
rod) and vice versa. Investigate the
phenomenon.
2
Outline of report
1• Preliminary Observations
2• Theory
3
• Experiments
• Experimental Set-up, and Results
• Conclusion
3
Construction of system
Elastic
material
(constant
(k)Length (l)
Taut string
Mass (m)
4
Diagram – Top View5
Diagram Top View6
Observation Video7
Radial oscillations Azimuthal oscillations
The pendulum behaves like a
conical pendulum with an
oscillating fulcrum
Investigation- Conditions of the
Experiment
8
The pattern isn’t observed when
disturbed in the phi direction
Investigation-Conditions 9
Behaves like a simple
conical pendulum when
disturbed in the 𝛼 direction
Continued…10
Behaves like a simple
conical pendulum when
disturbed in the 𝛼 direction
Effect achieved due to the
contraction of the string
Theory-Assumptions of the Model11
Ignoring the minimal deflection of
the rod in the z-direction
Assuming that 𝑙2 remains constant
(no contraction of the string)
Assuming that the rod traces a
circular motion 𝛼
𝛼
𝜙L
𝑙𝑦
𝑧
Defining the generalized co-ordinates12
𝛼
𝜙L
𝑙𝑀𝑦
𝑥𝑧
𝑥 = 𝐿𝑐𝑜𝑠 𝛼 + 𝑙𝑠𝑖𝑛 𝜃 cos 𝜃𝑦 = 𝐿𝑐𝑜𝑠 𝛼 + 𝑙𝑠𝑖𝑛 𝜃 sin 𝜙
𝑧 = −𝐿𝑐𝑜𝑠 𝜃
ሶ𝑥 = −𝐿𝑠𝑖𝑛 𝛼 ሶ𝛼 + 𝑙(− sin 𝜃 sin(𝜙) ሶ𝜙+ cos Φ cos(Θ) ሶΘ
ሶ𝑦 = 𝐿𝑐𝑜𝑠 𝛼 ሶ𝛼 + 𝑙 sin Θ cos Θ ሶΦ +
sin Φ cos(Θ) ሶΘ)
ሶ𝑧 = 𝑙𝑠𝑖𝑛(Θ) ሶΘ
Lagrangian of the System13
Kinetic Energy:
1
2𝑀 ሶ𝑥2 + ሶ𝑦2 + ሶ𝑧2 +
1
2
1
3𝑀𝐿2 ሶ𝛼
Potential Energy:
−𝑀𝑔𝐿𝑐𝑜𝑠 θ +1
2𝑘𝛼2
Lagrangian= K.E-P.E
Velocity of
the bob
Velocity of
the rod
Potential
Energy of
bob
Elastic potential
of the rod
Euler-Lagrange Equations14
𝑑
𝑑𝑡(𝜕𝐿
𝜕 ሶθ)=
𝜕𝐿
𝜕θ𝑑
𝑑𝑡(𝜕𝐿
𝜕 ሶ𝜙)=
𝜕𝐿
𝜕ϕ𝑑
𝑑𝑡(𝜕𝐿
𝜕 ሶ𝛼)=
𝜕𝐿
𝜕𝛼
(1)
(2)
(3)
𝑀𝐿 ሷ(𝑦 cos 𝛼 − ሷ𝑥sin(𝛼)) = −𝑘𝛼 +1
3𝑀𝐿2 ሷ𝛼
ሷ𝑦 = ሷ𝑥 tan ϕ
𝑀𝑙 ሷ𝑥 cos ϕ cos Θ +𝑀𝑙 ሷ𝑦 sin ϕ cos θ+ ሷ𝑧𝑠𝑖𝑛𝑛 θ = −𝑀𝑔𝑙𝑠𝑖𝑛(θ)
Governing EquationsCan be written as 2nd order
differential equations describing
the accelereation in the
(alpha),(theta) and (phi)
directions
Experimental Set-up
Rigid stand
Strings
Bobs of variable masses
Camera
Release mechanism for
bob
Camera
15
Experimental Set-up16
Displacement-Time Graphs17
-300
-200
-100
0
100
200
300
0 10 20 30 40 50 60 70 80 90
X-D
isp
lac
em
en
t
Time (s)-300
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10 20 30 40 50 60 70 80 90
Y-D
isp
lac
em
en
t
Time (s)
Azimuthal
oscillationsOscillations decay
and become
indistinguishable
Fourier Analysis18
0
200
400
600
800
1000
1200
1400
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
Am
plit
ud
e(c
m)
Frequency(Hz)
Two bands corresponding to the
Azimuthal and Radial Oscillations
Comparing the Amplitudes19
Amplitude Ratio= 𝑀𝑎𝑥 𝑎𝑚𝑝𝑙𝑡𝑢𝑑𝑒 𝑖𝑛 𝑋 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
𝑀𝑎𝑥 𝑎𝑚𝑝𝑙𝑡𝑢𝑑𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑌 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
-200
-150
-100
-50
0
50
100
150
200
1 7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
139
145
151
157
163
169
175
181
187
193Tim
e45°
x y
Obtained using the Max
function in Excel
Phase Portrait20
-300.00
-200.00
-100.00
0.00
100.00
200.00
300.00
-1500 -1000 -500 0 500 1000 1500
X-v
elo
city
X-Position
Phase Portrait
Shows non-
chaotic, periodic
motion for the
general case
Experiment- release angle21
30°
90°
-300
-200
-100
0
100
200
300
-600 -500 -400 -300 -200 -100 0 100
Y-P
osi
tio
n
X-position-300
-250
-200
-150
-100
-50
0
50
100
150
200
250
-300 -200 -100 0 100 200 300
Y-A
xis
X-Axis
60°
Bigger angles => higher transitions
rate
60°
Initial Angles22
-150.00
-100.00
-50.00
0.00
50.00
100.00
150.00
-250.00 -200.00 -150.00 -100.00 -50.00 0.00 50.00 100.00 150.00 200.00
Y-P
ost
ion
X-Position
45°
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
-12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 8.00 10.00
Y-P
ost
ion
X-Position
30°
45° 30°
Smaller angles => Lower transitions rate
Change in Mass (light)23
30g
-300
-250
-200
-150
-100
-50
0
50
100
150
200
250
-300 -200 -100 0 100 200 300
Y-P
osi
tio
n
X-Position
30g
-20
-15
-10
-5
0
5
10
15
20
-25 -20 -15 -10 -5 0 5 10 15 20 25
Y-P
osi
tio
n
X-Position
15g
15g
Change in Mass (heavy)24
-60
-40
-20
0
20
40
60
80
100
-400 -300 -200 -100 0 100 200 300 400 500
Y-P
osi
tio
n
X-Postion
100g
100g
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-30 -20 -10 0 10 20 30
Y-P
osi
tio
n
X-Position
200g
200g
Effect of Mass on the Elasticity of rod25
-60
-40
-20
0
20
40
60
80
100
-400 -300 -200 -100 0 100 200 300 400 500
Y-P
osi
tio
n
X-Postion
100g
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500
-400 -300 -200 -100 0 100 200 300 400 500
Phase Portrait
Phase Portrait show Chaos in
the system during the
Transitional Period
Effect of mass on the elasticity of rod26
*
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-30 -20 -10 0 10 20 30
Y-P
osi
tio
n
X-Position
200g
The heavy mass limits the
movement of rod, leaving an
almost rigid rod that can
rotate(but not bend)
https://www.youtube.com/watch?v=1BhCXEd1zM4
-150.00
-100.00
-50.00
0.00
50.00
100.00
150.00
-30.00 -20.00 -10.00 0.00 10.00 20.00 30.00
Phase Portrait
Harmony and periodicity
returns to the system when the
rod is completely restricted
Length of the String27
-200.00
-150.00
-100.00
-50.00
0.00
50.00
100.00
150.00
200.00
-250.00 -200.00 -150.00 -100.00 -50.00 0.00 50.00 100.00 150.00 200.00X-A
xis
Y-Axis
10 cm
-1000.00
-800.00
-600.00
-400.00
-200.00
0.00
200.00
400.00
600.00
800.00
-1000.00 -800.00 -600.00 -400.00 -200.00 0.00 200.00 400.00 600.00 800.00 1000.00
X-A
xis
(pix
els
)
Y-Axis
30cm
Conclusion-Preliminary28
Pendulum only
works when
disturbed in
the Radial
Direction
Succesfully described the Initial Conditions
of the pendulum system
Conclusion-Theory29
The bob has 3
degrees of
Freedom which
complicate the
modelSuccesfully Modeled and defined
the System Mathematically
Conclusion-Experiments(Describing
Frequencies)30
-300
-200
-100
0
100
200
300
0 10 20 30 40 50 60 70 80 90
X-D
isp
lac
em
en
t
Time (s)-300
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 10 20 30 40 50 60 70 80 90
Y-D
isp
lac
em
en
t
Time (s)Azimuthal
oscillations
Oscillations decay
and become
indistinguishable
0
200
400
600
800
1000
1200
1400
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
Am
plit
ud
e(c
m)
Frequency(Hz)
Verified through Fourier
Transform
Conclusion-Experiments31
-300
-200
-100
0
100
200
300
-600 -500 -400 -300 -200 -100 0 100
-300.000
-250.000
-200.000
-150.000
-100.000
-50.000
0.000
50.000
100.000
150.000
200.000
250.000
-300.000 -200.000 -100.000 0.000 100.000 200.000 300.000
X-A
xis
Y-Axis
-150.00
-100.00
-50.00
0.00
50.00
100.00
150.00
-300.00 -200.00 -100.00 0.00 100.00 200.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
-15.00 -10.00 -5.00 0.00 5.00 10.00
Investigated
and Explained
relations of the
system to 3
critical
parameters
Thankyou!
32
Relevant Parameters
Length of the string
Elasticity of the rod
Weight of the bob
Initial Angles
33
Theory- Conservation of energy
The radial oscillations of the bob disturb the equilibrium of the elastic rod causing to move in the direction of the bob’s motion.
Since the rod is fixed on one end, it traces a circular path (azimuthal oscillations)
The restoring force of the rod acts against the motion of the bob.
This sets up a chain of energy transfer where the motion of the bob is transferred into the azimuthal perturbations of the rod
34
35
-25
-20
-15
-10
-5
0
5
10
15
20
25
0 10 20 30 40 50
Dis
pla
ce
me
nt(
cm
)
Time(s)
X-displacement
36