Energy Conservation for a Harmonic Oscillator in a Periodic Field

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.

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.

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

(1.1)
x^{··} + ω_{o}² 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_{o}² x²) =
f x^{·} .

The quantity E = ½ (x^{·}² + ω_{o}² 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:

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 case of resonance (ω = ω_{o}) the particular solution (2.1)
must be replaced with

(3.1)
x(t) = ^{A}/_{2ωo} t sin ω_{o}t ,

hence

(3.2)
x^{·}(t) =
^{A}/_{2ωo}
(sin ω_{o}t + ω_{o}t cos ω_{o}t).

This yields the input power

(3.3)
L = f(t) x^{·}(t) =
^{A²}/_{2ωo}
(sin ω_{o}t cos ω_{o}t +
ω_{o}t cos² ω_{o}t) .

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_{o}t cos² ω_{o}t of L is the reason for the growth
~ t² of oscillator energy
E = ½ (x^{·}² + ω_{o}² x²)
on the left hand side of eq. (1.2) in *time average*.

[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/

http://en.wikipedia.org/wiki/Steorn