Derivation of Resonance Frequency

Consider the figure below of a machine made up of an assembly with an outerframe supporting a spinning wheel in a carriage, with the entire carriage being made suitable to spin about the main Y axis of the machine. The spinning motion of the wheel provides the coupling force (inductive) and the non-spinning part of the carriage and the outerframe provide the inertial force (capacitive).



This is identical to the electrical situation below.



By analogy with electrical circuits, the resonance frequency would equal
ωR = 1/sqrt(LC) ~ ωR = 1/sqrt(Inductance of wheel * Capacitance of carriage and frame)

Recall that since by analogy with Faraday’s Law,

ωXprecess (t) = (Inductance of Spinning Wheel)*dαY/dt

=> Inductance of Spinning Wheel = ωXprecess (t)/dαY/dt

Capacitance of carriage and frame is a function of the design and the moments of inertia and is taken to be

IFX* IFY

where the subscript F indicates that its the frame (and w indicates that its the wheel) we are referring to.

Then, ωR = 1/sqrt(ωXprecess (t)* IFX* IFY/ (dαY/dt) )

Also for a spinning wheel, ωXprecess (t) = Torque/AngularMomentum = τwy/(Iwz*ωz)

where ωz is the angular velocity of the spinning wheel.

Substituting this, we get:

ωR = 1/sqrt(τy * IFX* IFY/( (Iwz*ωz)*(dαY/dt) ))

or

ωR = sqrt( ( Iwz*ωz*dαY/dt)/(τwy*IFX* IFY))


Now,

Iwz*dαY/dt = dτFY/dt (since torque to the wheel about the m/c Y axis can only be given by pushing against the frame)

So substituting that into the above equation, we get

ωR = sqrt( ωz*(dτFY/dt)/(τwy * IFX* IFY))

Note that since the frame and the wheels are reacting against each other,

(dτFY/dt)/τwy = (Iz/ sqrt(IFx*IFy))* ωv, i.e. a fixed frequency signifying how many times the torque produced by the vertical motor must change per second.

Substituting that further simplifies the above equation to:

ωR = sqrt( ωz*ωv*Iwz/sqrt(IFx*IFy)*sqrt (IFX* IFY))
Now, suppose we set ωv equal to the resonance frequency, ωR

ωR = sqrt( ωs*ωR*Iwzsquare/(IFx* IFy)

ωR = ωz*Iwz/sqrt(IFx* IFy)

The above formula represents the frequency at which the vertical motor must vary the carriage’s torque in order to emulate resonance conditions for the mechanical LC circuit. At resonance, the torque, angular momentum and angular displacement of the wheel’s center of mass are related as shown in the graph below (only one full cycle of a wave of variable torque is shown).



At resonance, there would be a conversion of internal energy into external energy, resulting in motion of the entire frame under certain conditions. When resonance occurs under such conditions the entire carriage assembly will acquire kinetic energy(’fly’) equal to an amount calculable from the rotational velocity of the flywheel and the moments of inertia of the component parts per every cycle of the variable torque.