Partially Assembled Prototype

Generalizing Capacitors And Inductors To Include Spinning Wheels

Let us now consider a wheel spinning in its carriage and a capacitor; their workings and similarities.

Capacitor	Flywheel
A capacitor is a temporary storage device for electrical energy	A flyweel is a temporary storage device for mechanical energy
A given capacitor has a maximum operating voltage beyond which the capacitor will discharge spontaneously	A given flywheel has a maximum operating angular velocity beyond which it will break apart due to stresses with in the material
A discharging capacitor shows a continuous drop in its potential difference	A discharging flywheel shows a continuous drop in its angular velocity
The capacitance is determined by the surface area of plates and the permeability of the medium between the plates	The moment of inertia is determined by the mass distribution of the flywheel
Charge Q = C*V	Angular Momentum L = I*ω
The energy stored in a capacitor is given by E = (1/2)*C*V2	The energy stored in a flywheel is given by E = (1/2)*I*ω2
Charge plays the same role	as Angular Momentum
Capacitance plays the same role	as Moment of Inertia
Potential Difference plays the	same role as angular velocity

Accelerated Spinning Wheels and Inductors

Flywheels can also act in more unusual ways reminiscent of inductors under slightly different conditions. Such an arrangement is shown in the figure above.

When the flywheel is suspended in a carriage and the carriage is offset about a Y-axis as shown in the figure, the inductive condition is invoked as follows: The spinning wheel spins at a fixed angular velocity. The carriage is moved about the Y-axis with a torque that changes in time.

The existence of a significant rate of change of torque is a necessary condition to harness a spinning wheel to transfer energy via the inductive process.

Consider the behavior of an induction coil with a steady current through it. It resists the change of an existing current in the coil. Given that charge is analogous to angular momentum (from our discussion above),

Q(charge) ~ Lm (angular momentum)

We diferentiate this once wrt time to get

dQ/dt (current) ~ dLm/dt

Now, dQ/dt is nothing but current ( rate of change of charge with time).
And since dLm/dt is nothing but rate of change of angular momentum, we can write it as I*dω/dt=> I*α (ang acceleration).

That is, we get a NEW analogy
i ~ α
This adds to the previous three analogies regarding charge, capacitance and potential difference we made just now, i.e.

Current plays the same role as Force (= Moment of Inertia * Acceleration)

The characteristic formula for an inductor is its voltage to current relationship in time. An inductor undergoes self-inductance only in the presence of a varying current, i.e. a second derivative of charge over time i.e., di/dt.

now if i ~ α, in order to find ANOTHER NEW analogy, we differentiate it one more time to get

di/dt ~ dα/dt

The analogy above suggests therefore that the mechanical phenomena equivalent to inductance is given by the second derivative of angular momentum over time.

(Now, the first derivative of angular velocity is angular acceleration. The second derivative is therefore the derivative of angular acceleration).

Thus the amount of counter-angular-velocity and counter-angular-acceleration (since the development of the angular velocity will happen at a rate of acceleration that can be measured by measuring the rate of change of angular acceleration of the applied force) developed by a flywheel depends on the imposed rate of change of angular acceleration.

Just as the characteristic formula for an inductor is its voltage to rate-of-change-of-current relationship in time, the characteristic formula for such a spinning wheel would be its precessive angular velocity to its rate of change of angular acceleration relationship in time (taking into account the Moments of Inertia of the structures involved).

An inductor undergoes self-inductance only in the presence of a varying current, i.e. a second derivative of charge over time i.e., di/dt.

Thus, in analogy with the equation Vcounter(t)= L*di/dt for inductor coils, the equation

ωprecess (t) = (Inductance of Spinning Wheel)*dα/dt

represents the characteristic formula for a spinning wheel freely rotatable about a perpendicular, offset axis. Thus, just as voltage leads current in inductors, the precessive velocity leads the angular force (acceleration) in an inductively suspended spinning wheel.

Further, just as the counter e.m.f causes a counter current to flow through the inductor, so also in the flywheel the counter-precessive-angular-velocity would produce a counter-angular-force (acceleration). Understanding this analogy is the foundation for an alternative explaination of gyroscopic action of flywheels.

In a circuit including an inductor, we apply current and receive voltage.
In a gyro too, we apply torque (gravity pulling down on the wheel) and receive angular velocity (precessive motion), instead of receiving angular acceleration as we should, if we were to apply Newton's Second Law.

All this also implies that as a toy gyro spins and whirls, energy is being given away to the gravitational well at a measured rate as the gyro winds down. (unlike conventional explanations which insist that the energy is simply dissipated via bearing friction - now this transfer rate is very small, which is why it would appear that something minor like bearing friction is involved. However that is simply the easy explanation that got us into all this trouble in the first place.)

Flywheel Assembly

Here is the flywheel assembled into a frame before being mounted into shatter proof casings.

Click here to see the flywheel assembled into the prototype of the Relativistic Machine.

Rotating Inertial Frames

Suppose you are standing at the North Pole on a cloudy night. You happen to have a Foucault Pendulum with you. Having nothing better to do, you set up the Pendulum vertically at midnight and you watch it for 24 hours.

What would you see? You would see the plane of oscillation of the pendulum rotating a full 360 degrees.The plane of the pendulum is precessing i.e., the plane in which the pendulum would appear to be inertially at rest i.e. to NOT precess would be one that is rotating w.r.t. the Earth at the same speed as the precessive velocity of the pendulum.

Now, after the 24 hours are up, lets say the clouds clear away, leaving you with a clear view of the stars in the sky. You would observe that the distant star also rotate, even as the pendulum precesses i.e., the plane of the pendulum is stationary wrt the distant stars.

That is, the 'inertial frame' of the pendulum is definable as the frame containing the distant stars. Thus for practical applications, inertial frames are those frames in mechanics which are at rest w.r.t. the distant stars.

Although Ernst Mach first proposed this principle and it is eminently usable in navigation and other applications, General Relativistically speaking this is not a necessary principle i.e., GR/SR would not be compromised if this principle is discarded in its realm.

In GR/SR, the distant stars are assumed to be a rotating frame of reference i.e., the Universe is assumed to be the rotating entity. The precessing Foucault Pendulum would then be doing the bidding of inertia in rotating with the distant stars as soon as it is positioned.  The Foucault Pendulum's plane of oscillation is then seen as being dragged by the slow rotating massive body (i.e., the Universe). This slow dragging is analyzed using Schwarzchild geometry.

Using this method, 'frame dragging' has been calculated for many rotating objects. or example, the rotating Earth also produces a 'frame drag' worth about 0.3 radians per year. This means that the Foucault Pendulum would show not only the planar precession due to the distant stars but also a 0.3 radians per year change in its plane also due to the Earth's rotation. The gyroscope is then a test body that can be used to observe the dragging of inertial frame.

Introduction To Single Plane Balancing

(The following is the generic introduction cited for single plane balancing in most standard American University Physics textbooks.)

The equation governing the principle behind Single Plane Balancing is simply that the sum of all forces on a moving system, including the d'Alembert intertial forces is zero.

ΣF – ma = 0

Notice that it is essentially a restatement of Newton's Second Law.

The masses generating the inertial forces are assumed to be in or nearly in the same plane.

From turbine blade wheels to automobile tyres, this kind of analysis is valid for all spinning objects whose radius is significantly longer than the width.



The figure above shows a situation where there are two unbalanced masses determined to be at two location w.r.t. the center of rotation. We can model this dynamically as two point masses concentrated at two locations as shown. We can solve for the required amount and location of a third mass that will balance the system.

The system is rotating at a fixed angular velocity. The acceleration of the masses is assumed to be strictly centripetal. The inertial forces will then be strictly centrifugal. Since the system is rotating, the drawing shown is a snapshot of a specific moment.

The above equation reduces to:

-m1*R1*ω2 -m2*R2*ω2 – m3*R3*ω2 = 0

Thus all the forces acting are inertial. Canceling the angular velocity,

-m1*R1 -m2*R2 – m3*R3 = 0

Separating into X and Y components,

-m1*R1x -m2*R2x – m3*R3x = 0
-m1*R1y -m2*R2y – m3*R3y = 0

We can readily solve for m3*R3x and m3*R3y when all other values are known.

New Flywheels Are Ready

Here is the finished flywheel after drying in the sun.
I painted the flywheels blue to contrast with the white reflective strips I will use later on to measure the actual speed of the wheels. It will aid the accurate functioning of the laser photo tachometer.

I also had the flywheels balanced. You can see the 3 dots at the bottom of the wheel where material was taken out after balancing them. Balancing the wheels is important especially for higher speeds because an imbalance in the weight distribution of the wheel will result in anomalously higher centripetal force at the heavy spots.

This particular wheel is a well made one. I can tell that from the data. It was off balance by only 0.955 gram-inch before being balanced. The imbalance was corrected by simple 1-plane balancing. The corrections made ensure that the wheel is safe for 10,000+ rpm.

Click here to see the flywheel suspended in an assembly with housing, bearings and motor.


Click here for a blank drawing template of the flywheel for your use in your own design effort.