Magnetic Pendulum (2.12)
Introduction:
This experiment is aimed at exploring the idea of chaotic dynamics using a magnetic
pendulum. There will be certain objectives discovered in this experiment as followed:
•
•
•
•
•
•
•
How simple systems can be highly non-linear and exhibit complex behaviour under
certain condition,
How non-linearity can be made more prominent using simple methods and
hardware modification,
Discovering the simplistic richness of the mathematical and physical structure of
dynamic systems.
The conditions and consequences of the notion of super-sensitivity and its
relationship with chaos,
How to tell chaos apart from statistical indeterminancy
Constructing and interpreting phase portraits and Poincare Maps for different types
of responses of a system
How fractals are associated with attractors and are manifest in the graphical data of
such systems.
Simple Equation of a pendulum is as followed:
𝜏 = 𝐼𝛼 => −𝑚𝑔𝑆𝑖𝑛𝜃𝐿 = 𝑚𝐿2
𝑑2𝜃 𝑔
+ 𝑆𝑖𝑛𝜃 = 0
𝑑𝑡 2 𝐿
𝑑2𝜃
𝑑𝑡 2
𝑆𝑖𝑛𝜃 ≈ 0
𝑑2𝜃 𝑔
+ 𝜃=0
𝑑𝑡 2 𝐿
Non linearity may exist in the equation due to the presence of 𝑆𝑖𝑛𝜃 but if take small
amplitudes in considerations that 𝑆𝑖𝑛𝜃 maybe eliminated and linearity in equation may
exist.
Apparatus and Method: Student Manual. [1]
Background Theory:
Determinism: This concept states that for everything that will happen, there are conditions
given, without them this process may not happen. A deterministic system is whereby no
randomness is involved in the development of future states of the system. [2]
Chaos Theory: Usually chaos is used to define an aperiodic time behaviour of a system in
terms of a nonlinear dynamical system and is sometimes reffered as “noisy” or “random”
behavior. Chaotic behavior occurs in a system that is free from noise and has few degrees of
freedom. This tells us that chaos will occur in a system where the system is sensitive to
initial conditions, it is topologically mixing and the periodic orbits are dense. Small changes
in initial conditions lead to chaos in a system even if the system is deterministic, which
means that the future behaviour is fully determined by the initial conditions and no
randomness is involved.[3]
Phase Potrait: It is a graphical representations of trajectories in a dynamical system in the
phase plane. The initial conditions are represented with different curve or a point. Phase
portrait is a graphical representation of the trajectories of a dynamical system in a phase
plane. It is the plot of Angular Speed against Phase angle. This plot consists of typical
trajectories in the state space. We may find out information an attractor or repeller being
present in the chosen parameter value. [4]
Poincare Map: It is the intersection of a periodic orbit in the state space of a continuous
dynamical system with a certain lower dimensional subspace, called the Poincaré section,
transversal to the flow of the system. It involves considering a periodic orbit with some
initial conditions within a section of space, which leaves the section afterwards, and
observes when the orbit returning to a point in that section. Then a map is formed to send
first point to second. Poincaré maps can be interpreted as a discrete dynamical system. The
stability of a periodic orbit of the original system is closely related to the stability of the
fixed point of the corresponding Poincaré map. [5]
Attractors: An attractor is a set towards which a dynamical system evolves over time.
Points thats close to this set remain close to this even if it is disturbed. It maybe a point, a
curve, a manifold or even a complicated set with a fractal structure known as a “strange
attractor”. [6] A fractal is "a rough or fragmented geometric shape that can be split into
parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a
property called self-similarity. [7]
Self-similarity: A self –similar object maybe described as exactly or approximately similar to
a part of itself. [8]
Resonance: Resonance is the tendency of an object to resonate with a larger amplitude at
some frequencies as compared to others.
Results and Discussion:
The advantage of using a flywheel in the apparatus is that it maintains energy in the system
as rotational energy. Also it was necessary to break the connecting rod into two to allow
movement of the pulley and flywheel system. If we had used a smaller magnet instead of a
large ring magnet, the magnetic field will be smaller and chaos could not be seen as
prominently.
4.1: Measuring Nonlinearities
•
Increasing the amplitude and measuring the Time Period.
Angle of
displacement
(o)
(approximate)
10
15
25
30
45
•
No. of oscillations
5
10
15
15
15
Time (s)
4.18
7.38
11.35
11.47
10.75
Time Period (s)
0.836
0.738
0.756667
0.764667
0.716667
Placing ring magnet and measuring Time Period for different amplitude.
Angle of
displacement
(o)
(approximate)
10
15
25
30
45
No of oscillations
8
8
4
10
15
Time (s)
6.62
6.81
3.09
9.53
15.09
Time Period (s)
0.8275
0.85125
0.7725
0.953
1.006
It can be seen when the amplitude is increased, the time period shows a general trend of
increase without the ring magnet being placed under the pendulum. However, when the
ring magnet is placed, the time period is seen to generally increase but the increase is
greater as compared to when there was no ring magnet placed. The reading show when the
amplitude of oscillations is small, the trend of time period is linear as 𝑆𝑖𝑛𝜃 maybe ignored.
However, when oscillation amplitude is increased, non-linearity can be observed as the
𝑆𝑖𝑛𝜃 may not be neglected in the equation anymore.
Following graphs show graphs (time series, phase plots and fourier spectra) collected from
MATLAB for different amplitude oscillations:
• Small amplitude
16
14
12
10
Phase Angle
8
6
4
2
0
-2
-4
30
20
10
0
40
Time
80
70
60
50
80
60
Angular Velocity
40
20
0
-20
-40
-60
-80
-4
-2
0
2
4
40
60
80
8
6
Phase Angle
10
12
140
160
14
16
4
9
x 10
8
7
Power
6
5
4
3
2
1
0
0
20
120
100
Frequency
180
200
Larger amplitude
50
45
40
Phase Angle
35
30
25
20
15
10
5
0
0
10
20
30
40
50
Time
60
70
80
10
15
20
25
30
Phase Angle
35
40
90
100
200
150
Angular Velocity
100
50
0
-50
-100
-150
-200
0
5
45
50
5
10
x 10
9
8
7
6
Power
•
5
4
3
2
1
0
0
50
150
100
Frequency
200
250
Magnet placed
70
60
50
Phase Angle
40
30
20
10
0
-10
50
0
100
150
Time
300
250
200
300
200
Angular Velocity
100
0
-100
-200
-300
-10
0
10
20
100
200
300
30
40
Phase Angle
50
60
70
500
600
700
800
6
2.5
x 10
2
1.5
Power
•
1
0.5
0
0
400
Frequency
Increased non-linearity can be observed in the graphical data when the frequency of
the time series starts to differ and does not remain constant. When the frequency
changes considerably, degree of non-linearity is higher.
4.2: Stepping into Chaos
Placing ring at 7cm, 0.9 cycles/s
9
8
7
Phase Angle
6
5
4
3
2
1
0
-1
80
60
40
20
0
140
120
100
Time
80
60
40
Angular Velocity
•
20
0
-20
-40
-60
-80
-1
0
1
2
3
4
5
Phase Angle
6
7
8
9
12000
10000
Power
8000
6000
4000
2000
0
50
100
150
200
Frequency
250
300
350
Ring at 4 cm, 1.5 cyles/s
8
6
Phase Angle
4
2
0
-2
-4
0
10
20
30
40
50
60
70
80
90
Time
80
60
40
Angular Velocity
•
0
20
0
-20
-40
-60
-80
-4
-2
0
2
Phase Angle
4
6
8
18000
16000
14000
Power
12000
10000
8000
6000
4000
2000
0
0
50
100
150
200
250
Frequency
Ring at 4 cm, 1.8 cycles/s
0
-1
-2
Phase Angle
•
-3
-4
-5
-6
-7
-8
0
10
20
30
40
Time
50
60
70
80
80
60
Angular Velocity
40
20
0
-20
-40
-60
-80
-8
-7
-6
-5
40
60
-3
-4
Phase Angle
-1
-2
0
12000
10000
Power
8000
6000
4000
2000
0
0
20
80
100
120
Frequency
140
160
180
200
For a chaotic response, the fourier spectra observed shows that there are higher
energy level at higher frequencies as compared to the fourier spectra of a periodic
response. The time series and the phase portraits of a chaotic response are more
random and disturbed as compared to the plots of a periodic response. It has been
observed that for a lesser nonlinear response, the frequency remains almost the
same but for a greater nonlinear response, it has been observed that the frequency
changes a considerable amount.
Sampling time=1.5 samples/s, speed=1.5 cycles/s
4
3
2
Angular Velocity
•
1
0
-1
-2
-3
-4
0
1
2
3
Phase Angle
4
5
6
When the sampling time is made same as the speed of the pulley, the graph becomes
more random and non-linear which proves that chaos has occurred.
If the pattern of the graph diverges from its normal pattern and becomes non-linear,
it can be said the system has reached a state of chaos. We can identify a chaotic
phase portrait by looking at the shape of an attractor. If the shape of an attractor
looks random or strange, it can be said that system has reached a chaotic state.
Fractals can be observed in this phase portrait, as the attractor becomes “strange”:
80
60
Angular Velocity
40
20
0
-20
-40
-60
-80
-8
Conclusion:
-7
-6
-5
-3
-4
Phase Angle
-2
-1
0
To conclude, it can be said that the experiment successfully showed us chaotic
dynamics using a magnetic pendulum. Time series, phase portraits and fourier
spectra helped us gain an insight to the chaotic behavior of this system and identify
different aspects of a phase portrait.
Bibliography:
[1]- Student Manual
[2]- Wikipedia
[3]-wikipediaHasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a
Panorama of Recent Developments. Cambridge University Press. ISBN 0521587506.]
[4]-Jordan, D. W.; Smith, P. (2007). Nonlinear Ordinary Differential Equations (fourth ed.).
Oxford Univeresity Press. ISBN 978-0-19-9208241. Chapter 1.
[5]- Shivakumar JOLAD, Poincare Map and its application to 'Spinning Magnet' problem,
(2005) Retrieved from "http://en.wikipedia.org/wiki/Poincar%C3%A9_map"
[6]- Attractor at Scholarpedia, curated by John Milnor.]
[7]- Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company..
ISBN 0-7167-1186-9.]
[8]- Benoît Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and
Fractional Dimension]