This is my invention as well as the physical representation of my theory. My theory is simple and it actually works. I present all of this as open source data, that you are welcome to COPY and USE in anyway!
I am going to derive how energy is calculated mathematically in most force induced systems such as a windmill or hydroelectric turbine in order to show a simplified method for harnessing energy from mechanical vibration.
Typically when we go to generate energy in a force induced system what is required is an outside input force, where: Force = Mass x Acceleration. (Units
Work is then defined as Force x Distance, and the units are in (Newton x Meter = Joule).
All that is required to create energy is Force x Distance.
Power then is measured in Watts = Joules per second. It is the rate at which Energy (Joules) is being created through time (Seconds).
Consumption, or what you pay the utility is measured in kiloWatt-Hours. If you notice the units are the same as Joules, just on a different time scale. If you take Power and multiply it by hours, you get kiloWatt hours, that is to say 1 kiloJoule x 3600 = 1 kiloWatt-Hour, because there are 3600 seconds in an hour.
So in my version I did something slightly different. In the first formula F=MA, I added a sinusoidal component. Planet gear systems will create a sinusoidally oscillating centripetal acceleration on whatever is placed in the planet gear so long as it isn't place at the center of the planet gear.
Gravity and centripetal acceleration have the same units, namely Meters per Second Squared, (m/s^2).
So instead of having a steady push force on a mass across a distance, which would give you Work in Joules, I have decided to take a different approach.
The centripetal acceleration at the edge of a spinning disk is defined as A=V^2/R, where V = the instantaneous velocity of a point on the edge of a disk and R = the radius of the disk.
If A=V^2/R and for disk 2 it is A2=V2^2/R2, then the net acceleration is the sum except that to add them you need to know the angle, because you are adding two vectors.
Without going into it too deeply, you can conclude that at some point a tooth from the planet gear will face the sun gear and at another point in time it will be 180 degrees opposed to the sun gear, call this angle x. Therefore the sinusoidal acceleration would look something like this A(t) = A1 + A2 sin(x). Where t is time, and A1 and A2 are the magnitude of the acceleration force for each gear at that RPM.
So in summary, we can conclude that in a typical system:
F = MA
W = FD, or MAD
P = W/S or MAD/S
We can also conclude that in our system:
F = M(A1 + A2 sin(x))
W= M(A1 + A2 sin(x))D
P = M(A1 + A2 sin(x))D/s
F=Force, M=Mass, A=Acceleration
W=Work, F=Force, D= Distance
P=Power, S=Seconds, for D and W it is the same as above
In short what all of this means is that the power input to the system for very small systems would be much more than the power output, because induction is proportional to the size of the magnet being used, but for larger systems where the volume of the magnet has increased to a significant size, the output energy should be significantly impressive, it is much much more than the power input.
The input motor torque is opposed by three forces: air resistance, gear tooth friction and bearing resistance, it doesn't actually have a mechanical load attached to it and the inertia doesn't change after the gears have accelerated to their full RPM.
The output energy on the other hand is proportional to Mass x Acceleration x Distance which is equal to Work.
Final notes:
1) You can add mass to a magnet by encasing it in steel.
2) Theoretically if the RPM of a system could achieve extremely high speeds then the power output could also be extremely high, as acceleration is proportional to the velocity squared. So as the RPM climbs the power output which is proportional to velocity squared would as climb as the square of the velocity.
3) As RPM goes up, Power density (kW/m^3) increases, but material strength must also increase in order to prevent structural failure. So as we engineer motors with very high RPMs we will witness very high power densities but the bottle neck on power density will be decided by the strength of the material.
4) Magnets that are a minimum of 1 inch cubed or greater are a good starting point for your research, if you make the magnets too small your output amperage will be hard to measure as it is miniscule.
5) You can get the electricity out of your system by using conductive gears. You'll need 2 conductive plates for every magnet pair.
Here is a link to high speed motors: http://www.celeroton.com/en/products/motors.html
Some of the motors can achieve speeds of 1 million RPM.
