It's true. I sometimes get obsessed with helicopter lift. Perhaps it all started with the S.H.I.E.L.D. Helicarrier. Or maybe it was the human-powered helicopter. I can't remember (but by looking at the dates on the blog posts, it seems the human-powered helicopter came first).

The aerodynamics of a spinning helicopter blade are by no means trivial. However, this has never stopped me from making a simple model before. In this super simple physics model, I am considering that the thrust of a helicopter is due to the change in momentum of the downward moving air. There are essentially two ways you can make a helicopter have enough lift to fly. You can have a small rotor area and push air down at a high speed, or you can have a larger rotor area with a lower air speed.

If the rotor area is *A* and the density of air is ρ, then this would be an expression for the thrust force (magnitude) as a function of air speed.

But what about power? Power is the change in kinetic energy of the air divided by the time interval. Faster air means more kinetic energy and a shorter time interval. If you want the full derivation, check out my human-powered helicopter post. Here is the expression for power.

Since the power is proportional to the cube of the air velocity, you want this velocity to be low for a human powered helicopter. That means you have to make it big. Now go check out a real human helicopter like theUniversity of Maryland Gamera II.

Now, here is perhaps my favorite graph. Since I know my model might be completely bogus, I looked at some real helicopters (data from Wikipedia). From the mass and the rotor size, I can calculate the hovering power (using the possibly bogus model). I can also look at the rated power of the engine. Here is a plot of calculated vs. listed power for some of these helicopters.

I was very surprised to see how linear the data turned out.

## More Helicopter Data

How about something new? A friend of mine is sort of obsessed with the idea of building his own quadcopter drone. He showed me this T-Motor site with tons of different electric motors along with performance data. Here is some of the values they list:

- Prop size
- Voltage
- Current
- Thrust at throttle percent
- Revolution rate (rpm)
- Power – which is just the product of current and voltage

So, what can I do with this? Since I have the rotor size and the thrust, I can calculate the air speed. I can then use this to calculate the theoretical power and compare that to the listed power. Here's what I get.

Boom. Still linear. Honestly, I get a little worried when I go to plot something like this. It seems like a shot in the dark that my bogus-based helicopter lift model would still work on a smaller scale like this. The two plots even have similar slopes: 0.656 and 0.411. What do these slopes mean? Well, this says that my calculated power is about a factor of 2 too low. If I write the power as:

With that formula, the calculated power agrees with the listed power. I'm not sure why there should be that factor of 2 in there. I suspect I made an error in my derivation of the power – probably something about average speed of the air. Just a guess – perhaps I need to think about my derivation a bit more.

As long as I have the data, how about a bonus plot. Here is a plot of my calculated air speed as a function of propeller rotation speed (in rpm).

What does it mean? Well, the faster you spin the propeller blades, the faster the air. Right? I would suspect that there is another variable that is important – the prop pitch (the amount the prop is tilted). I didn't record those values – but that's just my guess.

## Homework

What else could you look at? Go over to the T-Motor site and consider the following questions:

- Use the data from T-Motor to design a flying chair. How many rotors should you use and at what size? How large of a battery will you need (this will have an impact on your number of rotors). Estimate the flight time and range of your awesome flying chair.
- Go back through the data and find motors with constant pitch angles (propeller pitch). Does the idea that faster rotation speeds produce a linearly faster air speed work?
- Calculate the thrust per watt for each motor (they actually list this on the T-Motor site also) and call it the efficiency. Now plot efficiency vs. various other variable to see if you can find the most efficient motor parameters.
- Go back and check my estimations for the Amazon Octocopter delivery drone. Do I need to change any of my estimates?