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Amplitude Quantization as a Fundamental Property of Coupled Oscillator Systems

W. J. WilsonDepartment of Engineering and Physics

University of Central OklahomaEdmond, 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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Quantum Trap

IT’S A TRAP!

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Argumentally Coupled Oscillators

0 ( , )cos( )x x x a x t 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 Problems1. 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-to-point 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 Fieldat base (f = 20 – 3000 Hz) driven by V = V0 sin (2π f t)

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Small Amplitude OscillationsGive 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 ( )sin( )A t

0 0( )0 Otherwise

AA

Numerical integration is surprisingly ineffective.

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Perturbative Schemes More Effective

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.

But require assuming oscillateswith ~T0

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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

20 0 cos( )x x x a kx t

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Spatial Analogue

Point-like absorber

Effective Size

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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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ReferencesJ. 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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