1
Magnetic pendulum
Vitaly Matiunin
The problem
Make a light pendulum with a small magnet at
the free end. An adjacent electromagnet
connected to an AC power source of a much
higher frequency than the natural frequency of
the pendulum can lead to undamped
oscillations with various amplitudes. Study and
explain the phenomenon.
2
3
First observations
Experimental setup
4
Rotation sensor
Light wooden
plank
Frequency ≈ 1
Hz
Neodymium
magnet
Electromagnet
Undamped oscillations
5
Plan of the report
Magnets interactions
Conditions for undamped oscillations
Experimental detection of undamped oscillations
Orders of resonance
Cycles of undamped oscillations
6
Main goal of our investigation
Prediction of
undamped oscillations
Computer simulation
Theoretical model
Poincare map
7
8
Qualitative
explanation
Region of magnets’ interaction
9
A magnet on the pendulum interacts with
the electromagnet only flying over it.
The span happens with almost constant velocity.
10°
10°
Limitations of amplitudes

10    180

Characteristic times and frequencies
v  2 gl  2.5 m/s
2d

 0.04 s
v
1
f   25 Hz

10
DC solenoid
11
If the current in solenoid is direct, the change in
energy of the pendulum is equal to zero.
is accelerated
is braked
AC solenoid
12
The energy of the pendulum can either increase…
is accelerated
is accelerated
AC solenoid
13
…and decrease.
Energy transfer depends
on the phase of the current
is braked
is braked
Dependence on the phase
14
We denote the time when the pendulum passes its
lowest point as t = 0.
Current oscillations in the coil:
I (t )  I 0 sin(t  )
I (t )   I 0 cos    sin t   I 0 sin    cos t
Energy transfer
No energy transfer
Impact of the electromagnet
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Angular velocity
Angle
Undamped oscillations (1 period)
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Angular velocity
Angle
Undamped oscillations (2 periods)
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Angular velocity
Angle
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Experiments
First attempt
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130
120
110
100
Amplitude, °
90
80
70
60
50
?
40
30
20
10
0
0
5
10
15
20
25
AC frequency, Hz
30
35
40
45
Experiment #1
20
130
120
110
100
Amplitude, °
90
80
70
60
50
40
30
20
10
0
0
5
10
15
20
25
AC frequency, Hz
30
35
40
45
Undamped oscillations
21
130
120
110
100
Amplitude, °
90
80
70
60
50
40
30
20
10
0
0
5
10
15
20
25
AC frequency, Hz
30
35
40
45
Order of resonance
In this test 30 current oscillations occur during
1 pendulum oscillation. Order of resonance = 30.
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Orders of resonance
23
130
28
120
24
110
36
32
44
40
20
48
100
Amplitude, °
90
12
16
80
70
8
60
50
46
50
42
40
38
34
30
20
22
6
10
26
30
18
10
14
0
0
5
10
15
20
25
AC frequency, Hz
30
35
40
45
Why all orders are even?
Attraction of magnets
Repulsion of magnets
Half of a cycle
N  2  2 k
24
25
Cycles of
undamped
oscillations
Experiment #2
• AC frequency is fixed (25 Hz in the main
series).
• Some undamped oscillation of kicked
pendulum is found (resonance order = 30,
amplitude ≈ 80° in the main series).
• Slowly shifting the voltage we monitor
changes of the fine structure of the phase
portrait.
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Period doubling
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Appearance of the cycle = 1.7 V
Period doubling = 6.9 V
Destruction of the cycle= 9.5 V
3-periodic cycle
This 3-periodic cycle exists
in the range from 4.5 V to 6.0 V
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The fine structure of the resonance
6.9 V
2V
4.5 V
6V
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9.5 V
Why the cycles are destroyed?
• At low currents: kicks are too weak to compensate
the energy dissipation.
• At high currents: phase adjustment can’t exactly
compensate too strong too strong kicks. This
eventually leads to destruction of the cycle.
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31
Computer
simulation
Limitations of natural experiment
• There are limitations on the voltage and
current in power supply.
• Fine effects are difficult to observe because of
vibrations and external disturbances.
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Model of magnets interaction
33
 x 
F ( x)  F0 sin  
 d 
–d
0
d
Computer simulation
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Limitations of the computer model
• We do not know which initial start conditions
lead to the establishment of undamped
oscillations.
• In each test, we need to wait 10 minutes to
establish the stationary regime.

To conduct all the necessary measurements it will
take us more then decade of continuous work!
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36
Poincare map
Poincare map
The pendulum interacts with an
electromagnet in the narrow angle range.
This allows to relate values (v, φ) on two
consecutive spans.
 vn 
 
 n
 vn 
 vn1   vn1 
Theoretical
    model
   
 n
 n1   n1 
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Change in velocity for half-period
The electromagnet is switched off:
vn1  (1  )  vn
The electromagnet is switched on:
 d 
 d 
2 
 sin 


vn 
vn 
F0


vn1  (1  )  vn 

 cos n
2
m





d
 2  



vn  



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Oscillations with large amplitude
f ()  f 0  
/2
 2  cos   cos   d 
0
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Change in phase for half-period
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Dependence of the oscillation period on the velocity.
  1 2  v 2   1  3 2  v 2 2  1  3  5  2  v 2 3

T (v)  T0  1    


 
 
 
2  
2 
2 
2
2U   2  4   2U   2  4  6   2U 
   

Our calculation takes into account
first six expansion terms.
T (vn1 )
n1  n  
2
U 2  2 gl
Final formulas
 d 
 d 
2 
 sin 


v
v
F0
 n   cos 
vn1  (1  )  vn 
  n 
n
2
m


 d 
2
 



vn  



T (vn1 )
n1  n  
2
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Poincare mapping
1,5
Velocity, m/s
2,0
2,5
3,0
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3,5
Phase φ
0
π/2
π
3π/2
Velocity
Phase
2π
2.92
0.31
2.73
1.57
3.22
3.06
2-cycle
2.92
0.31
3- cycle
2.73
1.57
4- cycle
3.22
3.06
1-cycle
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Poincare map
1,5
44
2,0
Velocity, m/s
2,5
3,0
3,5
0
Phase φ
π/2
π
F/m = 13 N/kg
3π/2
1-cycle
2-cycle
3- cycle
4- cycle
2π
Increasing the strength (video)
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Behavior of the fixed points
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135
135
120
120
Angle,
Angle,
Angle, °°°
105
105
90
90
75
75
60
60
45
45
00
55
10
10
15
15
Force/mass
Force/mass (N/kg)
(N/kg)
20
20
25
47
Summary
Conclusions
48
References
• П.С.Ланда, Я.Б.Дубошинский (1989) “Автоколебательные
системы с высокочастотными источниками энергии”. УФН,
158, 729–742.
• V.Damgov, I.Popov (2000) “Discrete oscillations and multiple
attractors in kick-excited systems”. Discrete Dynamics in
Nature and Society, 4, 99–124.
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