Recently, a Canadian team won the $250,000 Sikorsky Human Powered Helicopter Prize - details here. The prize is for the first human powered helicopter that can hover for 1 minute and reach an altitude of 10 feet while staying in a 10 foot by 10 foot area.
If you look at their flying machine or if you take a look at the University of Maryland's Gamera II, you might notice that both have HUGE rotors. The Atlas helicopter in the video has a rotor radius of 10.2 meters. Why do they make these things so big?
How Does It Hover?
Ok, I am going to skip over many of the bigger details. But in short, how do you get a helicopter to hover? Sure, you could talk about the rotors as though they were wings with lift if that made you happy. For me, I prefer a more fundamental approach. Let's say that you are sitting on an ice rink with a heavy medicine ball. Why? Why not. Now you take that ball and throw it horizontally.
In order to throw the ball, you have to push on it for some time. This force changes the momentum of the ball and it moves across the ice. But don't forget - forces are an interaction between two objects. In this case, the two objects are you and the ball. So, if you push on the ball with some force F, then the ball pushes back on you with the same force (but in the opposite direction).
If that force changes the momentum of the ball, it will also change your momentum by the same amount. Yes, you have a larger mass and thus for the same change in momentum, you will have a smaller change in velocity. You throw the ball, and you recoil in the other way. It's just plain basic conservation of momentum.
If you threw the medicine ball straight down, it would push straight up on you. In the crazy case that you could throw this ball super fast, the force it pushes on you could be as large as the gravitational force pulling down. Would this mean you could fly? No, you only have one ball.
Of course there is a way to solve this problem. Get many many balls. Or maybe you could use air. Air is kind of like tiny balls. So you take above you and throw it down. This means that you push on the air and it pushes back up on you. This air force depends on two things: how many air-balls you throw and how fast you throw them.
But Why Is Bigger Better?
Suppose we have two human powered helicopters. Both are hovering and both have the same mass so that both have the same thrust force from pushing air down. One of these humacopters has a smaller rotor size. This means that it will be "throwing down" fewer air balls. In order to compensate for the lower number of balls, each ball has to be thrown faster.
Both hover, but which is better? Yes, you already know that the bigger one is better - but why? Let's consider a short amount of time for hovering. Both humancopters push air with the same momentum. But suppose that helicopter 1 pushes half the amount of air during this time because of the smaller rotors. This means that it will have to push that less air with twice the velocity in order to have the same momentum.
Great. But which set of air will have a larger kinetic energy?
The smaller rotor produces air with the same momentum but twice the kinetic energy. What about power? In this case, power can be defined as:
If the change in kinetic energy is twice as much for the smaller rotor, the power would be twice as much. Wrong. It's actually more than 2 times greater for the smaller rotor. Why? Time - that's why. If you have a greater air speed, you are going to have to push it in less time. With this factored in, I actually get the following expression for the power of a hovering helicopter (full derivation here)
In this expression, ρ is the density of the air and A is the area that the rotors sweep out. So there you have it. If double the area, you can decrease the air speed and thus reduce the power.
What kind of power would the Atlas take to fly? It has a mass of 55 kg (plus a person of say 60 kg). The density of air is 1.2 kg/m3 with a total rotor area of 1307 m2. In order to hover, it would need to push the air to a speed of 0.848 m/s. This requires a power of 239 watts. But really it would take even more since the above calculation assumes everything is 100% efficient.
But wait! Aeronautical engineering isn't as simple as this. I made some crazy assumptions for something that is extremely complicated. I completely agree with this statement. However, what if I looked at the actual power of actual helicopters? If I know the rotor size and the mass, I can also calculate my theoretical power. Here is a plot of calculated vs. reported power for a few helicopters I found on Wikipedia.
A linear function seems to fit pretty well. You can argue with my basic assumptions, but you can't argue with a line.
