Smart exercise bike computer. Part #3: Power measurement

Good day!

This week I want to continue my mathematical calculations. Last time I made the assumption that the force needed to pedal the exercise bike is constant and equivalent to that of a real bike riding straight on High Gear. This statement is valid only if you ride an exercise bike always with the same tension.

Today I plan to do the following:

• determine the actual force (or rather the torque) created by the exercise bike, depending on the position of the tension knob;
• calculate the power that creates a rider rotating the pedal;
• compare the received power with the power that is consumed when the real bicycle moves.

Let's start the experiments!

Experimental research

Initially, I wanted to conduct an experiment, spinning the exercise bike up to a certain speed using an electric drill, and then measure the power consumed by the drill. However, this did not work in my case for several reasons:

1. The bike comes with a flywheel and in order to quickly spin it up just a huge torque is needed, which literally breaks the attachment between the drill and the pedal assembly of the exercise bike.

2. I do not know exactly what power factor my drill has, but in my home workshop I can only measure the RMS values of current and voltage, so even if I could perform this experiment, I would not speak about any accuracy.

3. It turned out that the tension of my exercise bike depends not only on the position of the tension knob, but also on how quickly one or another "gear" was switched on and even of what "gear" was turned first, for example:

• switching 6 – 7 – 8 – 7 gives the torque Te(A);
• switching 6 – 7 gives the torque Te(B);
• according to sensations, Te(A) > Te(B).

Disclaimer: further calculation uses an obviously simplified model of the dynamics of the exercise bike, the conclusions obtained as a result of this calculation do not pretend to be scientific truth!

I was saddened by the current situation and even thought not to do the power calculation at all, but I still managed to find one solution: if we assume that the torque at the central axis of the pedal assembly of the exercise bike does not depend on the speed of rotation of the pedals, we can conduct the following experiment:

• set the desired "gear" (G) with the help of tension knob;
• put an empty cylindrical jar on the pedal;
• fill the jar with water until the pedals begin to rotate;
• determine the weight of water (m) in the jar;
• determine the torque Te(G) by the formula:

Te(G) = 10 * m / (S1 / 2000)

where S1 is the distance between the pedal axes of the exercise bike (we already know that S1 = 480mm).

For the experiment, I used a cylindrical jar with a diameter d = 85mm, knowing that 1 liter of water weighs 1 kg, the mass of water in the jar can be determined knowing the water level l by the formula:

m = 3,14 * (d  / 1000/ 2)² * l

As a result of the experiment, I obtained the following data table:

As I said, and as it can be seen from the table, it is not possible to define Te(G) precisely, for this reason I needed to approximate the value obtained. I performed the approximation with the help of my favorite program [Curve Expert], a polynomial of the second order. As a result of the approximation, the following model was obtained:

Te_a(G) = 5,49 – 0,26*G + 0,13*G²

Power calculation

Power, as you surely know well, is the product of torque at speed. If everything is clear with the torque, then what would be the speed?

In the course of my training I found the optimal speed for myself – It is about 32 km/h or 20 mph. As for the rationale for choosing this speed, I propose to postpone it for one of the following articles, but for now let's try to understand what is that 32 km/h?

As I wrote in my [last blog], my old bicycle computer interprets one cycle of pedals of exercise bike as D3 = 6.67m. Knowing D3 and speed Vn = 32 km/h it is easy to calculate Wn – the speed of rotation of the pedals in RPM:

Wn = (Vn * 1000  / 60) / D3

Wn = (32 * 1000 / 60) / 6,67 = 80,0 rpm

Great, we have the angular velocity Wn! Now we can calculate the power

P(G, W = Wn) according to the following formula (power in watts):

P(G, W) = Te_a(G) * W / 9,549

In the highest gear G = 8, we get:

P(G=8, W=Wn) = (5,49-0,26*8+0,13*64) * 80,0 / 9,549 = 98,27 W

On the Internet, I managed to find a very interesting [link] following which you can find the table "Watts required to maintain a speed of 20 mph". If we make the assumption that we are really driving at a speed of 20 mph, the closest thing to our result is the Low racer bicycle. Let's take this as a reality.

But how can we estimate the figure obtained? To do this, you need to determine the calorie consumption.

Calculation and correlation of power

It is known that 1 kcal/h is equal to 1.16 W. However, one very important parameter is not taken into account here – the coefficient of efficiency (u) of rider. In our calculations, I am guided by the data from [the next article], which says that when riding a bike u = 0.25. Thus, rider’s power can be calculated by the formula:

Pc(G, W) = P(G, W) * 1,16 / 0,25

As a result, when riding a bicycle at the 8th "gear" at a speed of 32 km/h, we have:

Pc(G=8, W=Wn) = 98,27 * 1,16 / 0,25 = 455,97 kcal/h

My old bicycle computer counts Pc(G, W) according to the following formula:

Pc3(G=any, W=Wn) = 835,6 / 55,71 * 32,0 = 479,97 ckal/h

The values turned out to be very close (difference of ~ 5%, taking into account all the assumptions made in the article, it is possible to write off the error of the method), so we take the following: when G = 8, the speed of my old bicycle computer (V3) – is true.

Based on the foregoing, I compiled the formula for the dependence of the speed of the exercise bike (V) on the "gear" (G) and on the speed of rotation of the pedals (W):

V(G, W) = Te_a(G) / Te_a(8) * 32 * W / Wn

By analogy, you can determine the distance (here N – is a count of rotations):

D(G, N) = Te_a(G) / Te_a(8) * N * D3 / 1000

I plan to use these formulas in my new exercise bike computer.

Conclusion

I believe that I managed to achieve good results! I finally managed to connect the power (P) with the gear number (G)! The results obtained by me do not pretend to be scientific truth, however, for home use – this is quite enough (especially at a G=6..8). The main thing that I was able to learn (and this result is quite scientific) is the non-linearity of the tension variation when changing “gears”, which I previously only vaguely suspected. Now I just can not wait to bring the formulas I received into a new bicycle computer and to conduct the first real tests!

Oh yeah I still do not have an exercise bike computer Unfortunately I have not received my  FRDM-KW41ZFRDM-KW41Z board yet and I do not even know when to expect it Well if the board does not come this week I will have nothing to do but start working on another board that is close in functionality  FRDM-K64FFRDM-K64F which I got in one of the previous [Road Test]. I hope you will support me in this decision!

Thanks for reading and have a nice day!