Non linear dynamics of a driven pendulum in a replusive

magnetic �eld

Ibrahim Shehzad

2014-10-0008

May 13, 2013

1 Abstract

The dynamics of a driven pendulum in the presence of a replusive magnetic �eld is studied based

on [1]. The equations in the model proposed in [1] are solved numerically and the results obtained

are then compared with the experimental results given in [1] with one objective being to verify how

the model proposed in [1] matches experimental results. The numerical solutions are also studied

to identify the periodic and chaotic regions as well as to realize how sensitive a chaotic system is to

the choice of inital conditions.

2 Introduction

In Physics, we are used to approximating most oscillating systems with just simple harmonic os-

cillators. These happen to be linear systems. What this means is that the force here has a linear

dependance on displacement. Most oscillating mechanical systems that we come across in real life

however are governed by non-linear equations. These systems are what we call chaotic systems.

The drawback with these equations is that they can, in general, not be solved exactly and we can

not obtain nice, closed form solutions for them. They can only, in general, be solved numerically.

The magnetic pendulum is a well known example of such a system. A simple con�ugration for such

a system in shown below:

Figure 1: Our system of interest

1

3 The theoretical model

Maxwell's equations are not completely symmetric in electricity and magnetism. What this means is

that the divergence of of an electric �eld may be non zero but that of a magnetic �eld is always zero.

Stated otherwise, maxwell's equations do not allow any magnetic charges or magnetic monopoles

whereas electric charges are known to exist. This apparent asymmetry led British Physicist Paul

Dirac to postulate the existence of magnetic monopoles. The outcome of that is well known, it leads

to charge quantisation and we know from Milikan's oil drop experiment that electric charge really

is quantised. Therefore, eventhough magnetic charges have not yet been observed, we may have

reason to believe that they exist. The aim of this short digression was to motivate the following:

The replusive magnetic �eld in the paper under consideration is modelled by something that is similar

to the Coulomb's law for electric charges. Here m1 and m2 represent magnetic charges :

Fmagnetic =�0

4�

m1m2

r2r̂ (1)

In addition, the other equations, all of which can be easily derived just from the geometry of the

above �gure are:

r� =√(L sin �)2 + h2 (2)

And :

h� = d + L(1� cos �) (3)

Here L is the length of the pendulum and d is the shortest distance between the magnets. We

now know the forces involved in our system. In addition to the magnetic replusion, there is a driving

force on the pivot of the pendulum whose angular frequency is taken to = 2�s�1, a damping

force and gravity. We know that the analogue of Newton's second law for rotational motion is given

by the following:

Id2�

dt2=

∑i

�i (4)

Where:

∑i

�i = �grav ity + �dr iver + �damping + �magnetic (5)

It can be shown that the system, as a results of these, follows the following equations of motion:

ML2

3

dw

dt=

�L

2Mg sin � + �dr iver sin�� ! +

j�j

�L�0

4�

m1m2

r2cos(j�j+ arctan(�j

h

L sin �j) (6)

d�

dt= ! (7)

d�

dt= (8)

We can rewrite equation(6) in a slightly more useful form:

dw

dt= �A sin � + B sin�� C! +

j�j

�

D

r2cos(j�j+ arctan(�j

h

L sin �j) (9)

Where the constants, A,B,C and D depend on the physical parameters of our system. The

following section presents the analytic solutions of these equations.

2

4 The solutions and corresponding plots

The equations were solved in matlab using the RK-4 algorithm.

The parameters used were as follows:

A = 110s�1

B = 0:01s�2

C = 0:001s�1

D = 0:2m2s�2

The initial conditions are varied. For each set of initial conditions d is varied and the respective

phase plots and time series plots are obtained. Then the initial conditions are varied slightly and

the same process is repeated.

I started with the following initial conditions:

�0 = 0:100rad

�0 = 0:00rad

!0 = 0:00rads�1

Initially d was set at 72mm:

Figure 2: The phase plot for d = 72mm, !0 = 0:0rads�1, �0 = 0:1rad

3

Figure 3: The time series plot for d = 72mm, !0 = 0:0rads�1, �0 = 0:1rad

Clearly, we see periodic motion, that is to say, chaos has not set in as yet because the maximum

replusvie magnetic force is not too strong.

Now, as we decrease the distance to d = 71:8mm, the motion is still periodic but a little more

complicated:

Figure 4: The phase plot for d = 71:8mm, !0 = 0:0rads�1, �0 = 0:1rad

4

Figure 5: The time series plot for d = 71:8mm, !0 = 0:0rads�1, �0 = 0:1rad

Now, if we decrease d to d = 71:3mm, we see chaos setting in:

Figure 6: The time series for d = 71:3mm, !0 = 0:0rads�1, �0 = 0:1rad

5

Figure 7: The phase plot for d = 71:3mm, !0 = 0:0rads�1, �0 = 0:1rad

At d = 71mm, we see further proof of this:

Figure 8: The time series plot for d = 71mm, !0 = 0:0rads�1, �0 = 0:1rad

6

Figure 9: The phase plot for d = 71mm, !0 = 0:0rads�1, �0 = 0:1rad

If we reduce the distance any further, i.e set d = 70:8mm, the repulsive force is so strong that

the pendulum becomes restricted to only the portion where � is positive. It can not overcome the

repulsive magnetic force and therefore fails to make it to the other side:

Figure 10: The phase plot for d = 70:8mm, !0 = 0:0rads�1, �0 = 0:1rad

7

Figure 11: The time series plot for d = 70:8mm, !0 = 0:0rads�1, �0 = 0:1rad

4.1 Changing the initial conditions by changing �0

The next step is to change the value of �0. We set �0 = 0:1002rad We set d = 71:3mm:

Figure 12: The time series plot for d = 71:3mm,!0 = 0:0rads�1, �0 = 0:1002rads

It is clear that only a slight change in initial conditions has changed the phase plot. If we compare

it to the corresponding plot for �0 = 0:10rad ,that is, to �gures (6) and (7), we see that there is

a higher degree of chaos in this case. In that plot, the pendulum is starting to get con�ned to the

positive-theta region whereas in this particular case, that stage has not arrived yet. This signi�cant

8

Figure 13: Thephase plot for d = 71:3mm, !0 = 0:0rads�1, �0 = 0:1002rad

change in phase plots corresponding to only a slight change in initial conditons is a characterisitc

feature of chaotic systems. Notice that we changed �0 by just 0:2percent.

9

4.2 Changing the initial conditions by changing !0

The next step is to repeat this process, this time, by changing !0. This is supposed to provide further

proof of the sensitivity of our system to initial conditons. We change it to !0 = 0:10rads�1. Here,

�0 = 0:00 and �0 = 0:1. Therefore, the comparison here is being made with the �rst case. The

results obtained are as follows:

Figure 14: The phase plot for d = 71mm, !0 = 0:1rads�1, �0 = 0:1rad ,

Figure 15: The time series plot for d=71mm, !0 = 0:1rads�1, �0 = 0:1rad ,

d was here set equal to 71mm. Now, if we compare this to the �rst case, that is, to �gures (8)

10

and (9), we see a clear di�erence. With !0 = 0:00, the corresponding plot showed chaos. This plot

shows that the pendulum has already passed that stage and has entered the stage wherewith its

motion is restricted to only positive values of �, or in other words, it is restricted to one side only.

11

5 The experimental results

:

It is clear that for the following initial conditons:

�0 = 0:100rad

�0 = 0:00rad

!0 = 0:00rads�1

We see a transition to chaotic behavior at about 71:8mm

The experimental plots obtained by [1] for the same initial conditions are as follows:

Figure 16: The experimental results

12

Figure 17: The experimental results

6 Conclusions: A comparison of the experimental and numeri-

call results

The chaotic behavior of the pendulum and its sensitivity to initial conditions was studied using a

particular model which is often called the Gilbert model. From our calculation, we see that the

transition to chaotic behavior occurs at roughly d = 71:8mm However, according to the exper-

imental results, it does not occur until d = 55:60mm: Moreover, the transition to the behavior

wherewith the pendulum becomes restricted to one side, that is, to positive values of � does not

occur anywhere near the region in which it occurs according to our calculations. To reconcile this,

we conclude that the equations we used in our model, in particular, that for the repulsive magnetic,

is not accurate. Using that model, we can roughly identify where the periodic and chaotic regions

lie but for higher-precision computations, that model is not suitable.

13

7 References

[1] A. Siahmakoun, V.A. French, J. Patterson

Nonlinear dynamics of a sinusoidally driven pendulum in a repulsive magnetic �eld,

1997 American Association of Physics Teachers

May 1997

[2] Mudassir Moosa

Choatic Behavior in a Repulsive Magnetic Pendulum

May 2012

[3] R. Douglas Gregory

Classical Mechanics

2007

14