Where Does the Resonance Energy Come From?
Energy Conservation for a Harmonic Oscillator in a Periodic Field

Gerhard W. Bruhn, Darmstadt University of Technology

07.02.2008

Modern inventors of PMMs feel attracted by resonance effects of oscillatory systems under exterior fields. Their hope is that energy that appears in resonance phenomena stems from a hidden miraculous source which is set free by resonance: They claim that the energy comes from "spacetime" or from "zero point energy" or something like that. [1], [2], [3], [4], [5].

We must destroy all their hopes: The energy of resonance effects is merely delivered by the exterior forces working on the oscillatory system as will be shown here by a simple example.

All resonance energy is taken from the exterior field.
The energy must be fed in from outside to maintain the exterior field.

The following consideration applies to mechanical and electrical oscillators as well and can easily be generalized to more complicated systems.

1. The conservation of energy

We consider a one-dimensional harmonic oscillator in a time periodic field, described by the differential equation

(1.1)                 x^·· + ω_o² x = f(t) = A cos ωt .         (ω_o and ω positive constants)

Multiplication of eq.(1.1) by x^· yields

(1.2)                 ^d/_dt ½ (x^·² + ω_o² x²) = f x^· .

The quantity E = ½ (x^·² + ω_o² x²), sum of kinetic and potential energy of the oscillator is the total energy of the oscillator. Eq.(1.2) means that the change of the oscillator energy E during dt is just equal to the work that is done by the force f along the distance x^·dt . That's the energy conservation law:

The change of the oscillator energy is just given by the work of the exterior field.

2. The case of non-resonance

In case of no resonance (ω ≠ ω_o) the de. (1) has a particular solution

(2.1)                 x(t) = ^A/_{ωo² − ω²} cos ωt ,

hence

(2.2)                 x^·(t) = − ^Aω/_{ωo² − ω²} sin ωt .

Therefore we obtain the input power of the exterior force f(t) = A cos ωt to be

(2.3)                 L = f(t) x^·(t) = − ^A²ω/_{ωo² − ω²} sin ωt cos ωt .

This result shows that the input power L is time periodical and its time average is vanishing.

In non-resonance case in time average no energy is transmitted from the oscillating exterior field to the oscillator.

3. The resonance case

In case of resonance (ω = ω_o) the particular solution (2.1) must be replaced with

(3.1)                 x(t) = ^A/_2ωo t sin ω_ot ,

hence

(3.2)                 x^·(t) = ^A/_2ωo (sin ω_ot + ω_ot cos ω_ot).

This yields the input power

(3.3)                 L = f(t) x^·(t) = ^A²/_2ωo (sin ω_ot cos ω_ot + ω_ot cos² ω_ot) .

Here the first term in (...) is periodical in t and hence does not contribute to the time average of L. However, the second term yields an increasing contribution to the time average of L which grows ~t . This part ^A²/_2ωo ω_ot cos² ω_ot of L is the reason for the growth ~ t² of oscillator energy E = ½ (x^·² + ω_o² x²) on the left hand side of eq. (1.2) in time average.

In case of resonance the growth ~ t² of oscillator energy E in time average stems from the oscillating exterior field.

References

[1] M.W. Evans, 94th Paper now the lead paper “SCR Applied to the Bedini Device”
      http://www.atomicprecision.com/blog/2008/02/07/94th-paper-now-the-lead-paper-scr-applied-to-the-bedini-device-2/

[2] Wikipedia, on John Bedini
      http://en.wikipedia.org/wiki/John_Bedini

[3] M.W. Evans, (blog on Steorn), Simplest Type of Resonance Circuit,
      http://www.atomicprecision.com/blog/2007/08/06/simplest-type-of-resonance-circuit/

[4] Steorn, Website
      http://www.steorn.com/

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