 ## 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. , , , , .

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·· + ωoý x = f(t) = A cos ωt .         (ωo and ω positive constants)

Multiplication of eq.(1.1) by x· yields

(1.2)                 d/dt ½ (x·ý + ωoý xý) = f x· .

The quantity E = ½ (x·ý + ωoý xý), 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:

### 2. The case of non-resonance

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

(2.1)                 x(t) = A/ωoý − ωý cos ωt ,

hence

(2.2)                 x·(t) = − /ωoý − ωý 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) = − Aýω/ωoý − ωý sin ωt cos ωt .

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

### 3. The resonance case

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

(3.1)                 x(t) = A/o t sin ωot ,

hence

(3.2)                 x·(t) = A/o (sin ωot + ωot cos ωot).

This yields the input power

(3.3)                 L = f(t) x·(t) = /o (sin ωot cos ωot + ωot cosý ωot) .

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 /o ωot cosý ωot of L is the reason for the growth ~ tý of oscillator energy E = ½ (x·ý + ωoý xý) on the left hand side of eq. (1.2) in time average.

### References

 M.W. Evans, 94th Paper now the lead paper SCR Applied to the Bedini Device

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

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

 Steorn, Website
http://www.steorn.com/

 Wikipedia, on Steorn
http://en.wikipedia.org/wiki/Steorn