Amplitude Quantization as a Fundamental
Property of Coupled Oscillator Systems
W. J. Wilson
Department of Engineering and Physics
University of Central Oklahoma
Edmond, OK 73034
email: wwilson@uco.edu
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Outline
I. Introduction
II. Argumental Oscillator (Doubochinski
Pendulum)
III. “Theory” of Amplitude Quantization
IV.Oscillator Trap
V. Self-organization Behavior
VI.Implications and Conclusions
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IT’S A TRAP!
Quantum Trap
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Argumentally Coupled Oscillators
x   x  0 x  a( x, t )cos(t )
Introduced by Russian physicists to describe classical
systems where the configuration of an oscillating
system, enters as a variable into the functional
expression for the external, oscillating force acting
upon it
The possibility of self-regulation of energy-exchange is
a general characteristic of argumental oscillations.
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Classical Problems
1. Concept of force implies a rigid, “slave-like” obeisance
of a system to an external “applied force.”
2. A “force” can act, without itself being changed or
being influenced by the system upon which it is acting.
Newton’s third law of action and reaction is not enough
to remedy that flaw, because it assumes a simplistic
form of point-to-point vector action.
3. Attempt to break up the interactions of physical
systems into a sum of supposedly elementary, point-topoint actions.
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Classical Coupled Oscillators
• The idea of an external force, while it may serve as a
•
•
“useful fiction” for the treatment of certain problems
in mechanics, should never be taken as more than
that.
An “external force” is a simplistic approximation, for
an interaction of physical systems
Interacting systems never exist as isolated entities in
the first place, but only as subsystems of the Universe
as a whole, as an organic totality.
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Doubochinski Pendulum
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Doubochinski Pendulum
Doubochinski Pendulum
•
Low Friction Pivot Pendulum
with iron mass (f0 = 1-2 Hz)
•
Alternating Magnetic Field
at base (f = 20 – 3000 Hz)
driven by
V = V0 sin (2π f t)
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Small Amplitude Oscillations
Give familiar
resonance
physics for
Zone 1
oscillations
More interesting
to look at
nonlinear effects
and
f ≈ 10f0 -1000f0
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Yields Quantized Amplitudes
f = 50 Hz, f0 = 2 Hz
•
•
•
Stable amplitudes
are quantized
System “Choice” of
stable mode
determined by i.c.’s
Remarkably stable, large disturbances can cause the
pendulum to “jump” from one stable mode to another
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Period for all Oscillations, ~T0
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Energy Quantized Like Harmonic Oscillator
E = E0 (n + ½)
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Computational Analysis
  0
 A0
A( )  
 0 Otherwise
    0  A( )sin( t )
Numerical integration is surprisingly ineffective.
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Perturbative Schemes More Effective
But require
assuming
oscillates
with ~T0
In this case, since the total number of decelerating half-cycles will
be one less than the number of accelerating half-cycles, after
cancellation of pairs of oppositely acting half-cycles, the net effect
will be equivalent to that of the first half cycle. In this case, the
pendulum will gain energy.
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Phase Dependence
 Changes in the pendulum’s velocity,
and also in the time during which the
pendulum remains in the interaction
zone, as a result of the interaction with
the electromagnet.
 A surprising asymmetry arises in the
process, leading to a situation, in which
the pendulum can draw a net positive
power from the magnet, even without
a tight correlation of phase having
been established.
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Ratio f/f0
101
103
105
107
109
111
Observed
Amplitude
30º
43º
53º
60º
68º
74º
Calculated
Amplitude
23º
39º
50º
59º
66º
72º
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Multiple pendulums with
different natural
frequencies can be driven
by a single
high-frequency
magnetic field
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Trap Oscillator
x   x   x  a0 cos(kx  t )
2
0
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Spatial Analogue
Effective
Size
Point-like
absorber
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Gravitational Segregation
Agitate, f
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Possible Applications
•
Electric motors having a discrete multiplicity
of rotor speeds for one and the same frequency
of the supplied current
•
Vibrational Methods for Sorting
•
Cooling Processes
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Conclusions
• Argumental oscillations can efficiently couple
•
•
•
oscillation processes at frequencies differing by two
or more orders of magnitude
This coupling can be used to transfer energy into or
out of trapped oscillators
Fundamental physics can be investigated using
particle traps and their interactions with oscillatory
fields at much higher frequencies.
Paradoxically one can energize to cool, transmit to
receive, and add kinetic energy to reach lower energy
state.
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References
J. Tennebaum, “Amplitude Quantization as an Elementary Property of Macroscopic
Vibrating Systems”, 21st Century Science & Technology, Vol. 18, No. 4, 50-63
(2006).
[http://www.21stcenturysciencetech.com/2006_articles/Amplitude.W05.pdf]
D.B.Doubochinski, J. Tennenbaum, On the Fundamental Properties of Coupled Oscillating
Systems” (2007). arXiv:0712.2575v1 [physics.gen-ph]
D.B. DoubochinskiI, J. Tennenbaum, “The Macroscopic Quantum Effect in Nonlinear
Oscillating Systems: a Possible Bridge between Classical and Quantum Physics”
(2007). arXiv:0711.4892v1 [physics.gen-ph]
D.B. DoubochinskiI, J. Tennenbaum, “On the General Nature of Physical Objects and their
Interactions as Suggested by the Properties of Argumentally-Coupled Oscillating
Systems” (2008). arXiv:0808.1205v1 [physics.gen-ph]
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