Why Change Designs?

The Problem: My original design was tested in the first prototype and produced at maximum 1.21 V. This low voltage output forced me to reconsider my calculations, which had predicted much greater voltage. I discovered that my initial calculations had not converted Teslas (flux density) to Webers (total flux). Because of this error, I had used Faraday's Law to predict far greater power output than was actually possible in my design.

Considering the alternatives: Returning to basic electrodynamics, I realised that my initial design produced no force vectors opposed to the direction of the wheel's motion. I understood how the Lorentz force is opposite the direction of motion in an eddy current brake, but had dismissed the design after the brainstorming phase. I rejected this design because the electromagnets required too much weight and a complicated design. But I now realised that I may be able to generate an eddy current with permanent magnets instead, reducing complexity and weight. The concept was to place magnets on typical bicycle brake pads, thus when pulled close to the rim the Bfield would be perpendicular and through the aluminum bicycle wheel rim. This would induce eddy currents in the rim whose interaction with the Bfield would slow the wheel. But first I had to test the practicality of the idea.

Second Prototype: Eddy Current Test rig.

My second prototype has an eddy current brake (an aluminium disc about the size of three stacked CDs) on a shaft connected to a motor. A photodiode mounted to the rig measures angular speed in revolutions per second (PASCO scientific) by counting the amount of time that passed between holes that let the IR beam through. A Magnet holder provides a ~.15T Bfield across the disc.

A top View of the Test rig.

A Side View. The 8 holes halfway between the axis and the rim allow IR light through to the photodiode.

A video of the Rig in Action.

Based on the mass of the disc I calculated its moment of inertia:

I=.106kg*.06²m/2=1.91E-4

Having measured the spin down rate, average frictional torque could be calculated:

Tavg=I*Δω/Δt=1.91E-4*(21-0)/(0-1.31)=0.003Nm

Due to Newton's first law, the rubber band-drive must be providing about this much torque to overcome the bearing's friction and keep the disc rotating at a constant speed.

The same calculation was preformed with the data gained when the disc decelerated due to the introduction of a magnetic field across the disc. The torque generated was T=0.012Nm. Power: P=T*2π/t=0.012Nm*2*π/0.357=0.211W

Acting at the radius of the disc the force exerted was calculated: T=F*r   0.012Nm=F*0.06m      F=0.15N

Since the eddycurrent brake generates a Lorentz force: F is proportional to V assuming negligible E. Thus I could proportionally scale my results to see if an eddy current induced in the rim of a moving bicycle could produce enough power to significantly slow the wheel. The rim of the brake in my test rig was moving at 2.8r/s*2*π=17.6 rad/s, prior to the introduction of the magnetic field.

Since V=ω*r the speed at the rim is V=17.6rad/s*0.06m=1.06m/s.

The design parameter was 15MPH=6.7m/s thus the same system would produce F=0.15N*(6.7/1.06)=0.95N.

Allowing for 5 magnet pairs, and the increased rim size of the bike wheels this still only amounted to: T=5*(0.95N*0.29m)=1.38Nm. Power would be just P=1.38Nm*2*π/0.225s=38.5W

This estimated maximum power was still well short of my design goal, as well as being less than the power generated by a lightly pedaling rider. This dispiriting fact caused me to reconsider my design yet again.

Final Design

The essential problem was that the bicycle wheel rim moved too slowly to produce a substantial Lorenz force at any normal riding speed. This led to the obvious conclusion that if only I could increase the speed of the eddy current brake relative to the wheel speed I could produce a substantial braking force. This condition necessitates a drivetrain by definition, a system that adds complexity and weight to the system, two things I had been trying to avoid. More importantly it undermined my goal of having a contactless braking system. By scaling up my result from the previous test rig however, it appeared that substantial power could be generated. 

For example the bike wheel would spin at about 3 revolutions/second at 15MPH, but a 1 inch diameter shaft driven off of that wheel's tire would spin at 82r/s. Scaling proportionally the result from my second prototype, my eddy current brake's rim would be traveling at 30.9m/s and would produce 4.4N of force, 0.26Nm of torque, and dissipate a power of 134 W. Multiplied by a couple sets of magnets, I would quickly be at my 500W goal. Finally I had a mathematically promising design.

The Final Design I decided on would have a shaft connected to a brake disc surrounded by several magnet pairs, when brought into contact with the wheel this shaft would spin at a great enough angular velocity to produce substantial braking forces in the disc. I envisioned this shaft being held by bearings on the end of rods that could pivot on frame-mounts, allowing the brake to be selectively applied. A spiral torsion spring in the frame mount would hold the system off of the bike wheel during normal riding, a Bowden cable would pivot the brake shaft down into contact with the tire.

One downside of this design is that when the stationary shaft is brought into contact with the spinning tire it will slip, wearing both parts, particular at high speed. However for my purpose of keeping a moderate speed down a hill this would not be such a problem as the brake would be applied prior to descending the hill, when the wheel is not spinning that fast.

Unfortunately at this point in the semester there was not enough time remaining to fully design, construct, install and test such a system. Instead I decided to build a proof of concept prototype which would be simple enough to quickly build and test. I reduced the design to it's most essential components: a bearing, a shaft, a brake disc and a magnet holder. This prototype would demonstrate that using a shaft could increase the speed of the brake disc and produce sizable braking power, but because it would have no selective application it would be breaking all the time and thus render the bicycle impractical.

Proof of Concept (Prototype 3)

Despite the fact that most breaking occurs at the front wheel due to weight transfer under deceleration, especially when riding downhill, the easiest mounting point on my bicycle was the rear frame just below the seat, so this is where I put the braking system in this prototype. For expedience and simplicity a single bearing is used. A steel shaft 5/8" in diameter was machined to interface with the brake disc, this removed the majority of the perpendicular load on the screw that held the disc to the shaft. This perpendicular load had caused many screws to break on prototype 2. The proof of concept prototype installed on the test bicycle can be seen below.

The steel rectangle is the magnet holder. Below one can see the radial arrangment of the magnets in the holder and the brake disc (out of focus).

Testing

Now I had to test this system to asses its efficacy. I did so by riding the bike down a slope of known angle and legnth, and timing the trip. The angle and legnth told me how much higher the start line was than the finish line. Knowing this my potential energy could be calculated  

By dividing the potential energy lost over the course by the time of the course I could estimate the rate of energy conversion, Power.

The test was preformed on Wriston quadrangle, and runs were repeated three times so that the average transit time could be taken. After the first test the magnet holder was removed, after the second test the entire eddy current braking system was removed.

The course had a slope of 3° and a legnth of 46.3m, thus the height h=46.3m*SIN(3°)=2.42m.

The test bicycle and myself together have a mass of 82kg. ΔU=mgh=82kg*9.81m/s²*2.42m=1947J

Thus in this test the braking system developed a total power of 75W, of which more than 2/3 was from the friction of the bearing and the shaft-tire interface. 

Subjective test results

My first attempt to collect a useful dataset failed because the bicycle traveled faster with the full eddy current braking system than without the magnet holder. This cause of this was the observed slipping between the tire and the shaft, this had not occurred without the magnet holder installed because in that case the shaft was unloaded.  I readjusted the bearing mount to force the shaft to press harder into the tire to try and eliminate slippage. This adjustment resulted in the successful data set seen above, although intermittent slipping still did occur. This slipping should be less of a problem in my final design because the force pushing the shaft onto the tire can be modulated by the rider (via the bowden cable) to prevent slipping.

The proof of concept prototype was effective at drastically reducing acceleration when riding downhill, with the system installed my run over the course was effectively at constant speed, with only slight accelerations during moments of slippage. Without the system the bicycle accelerated during almost the entire downhill run. In this way the proof of concept prototype was effective at carrying a constant moderate speed downhill.

Final Results

Analysis of my design goals

The proof of concept prototype showed that the eddy current braking system of my final design is effective at maintaining a safe downhill speed. Slipping between the shaft and tire will always be a problem with this design, however intelligently applied Bowden cable pressure can limit this problem. The largest compromise in my final design is that it wears both the braking shaft and tire when in use, thereby creating servicing needs. The minimization of maintenence was one of the motivations to undertake this design project. Future work could implement my final design to test its real-world braking and wear.