|MadSci Network: Engineering|
This is a very complex problem from an engineering/design standpoint, but I can see why your teacher wants to do this, as helicopters are cool flying machines. Complete understanding of the physics of helicopter flight is likely much beyond your scope, so I will try to limit my explanations to the parts that are relevant to your vertical only flight. I will also give you some design recommendations, but I won't build the whole thing for you! Many of the terms I will use can be found at the following website:
http://www.copters.com/helo_aero.ht ml and if you have other questions, the site owner (Paul Cantrell, firstname.lastname@example.org) will likely be a great resource. Some of the concepts will probably be over your head but many you will be able to understand and he has great illustrations of what is going on, so I recommend looking at the sight thoroughly.
One of the problems with the earliest helicopter designs was the fact that once they got off the ground, the entire helicopter body began to spin the opposite way of the rotors. This comes from Newton's third law of motion, "to every action there is an equal and opposite reaction" (that is the paraphrased version). If you are spinning the rotors and they have both inertial resistance and wind resistance, you need to apply a torque to the rotors for them to spin. In order to not spin the body, an equal and opposite torque to the body needs to be applied. An experiment to test this would be to stand on one of those rotating platforms (you teacher may have one) and try to spin (like helicopters blades) a heavy umbrella or something similar above your head. See if you can stay motionless on the platform and start and stop the umbrella.
There are two methods for dealing with this spin. One is the tail rotor that nearly all helicopters you see have. The tail rotor serves two purposes, one to steer the aircraft and the other is to counteract the spin from the main rotors. The other method to counteract the spin is to have 2 main rotors that spin in opposite directions. (One rotor is on top of the other and have separate drive shafts, usually a solid one inside for the top rotor and a hollow one outside for the bottom rotor). This is a much rarer configuration but some helicopters do have them. One other method that would work for you, but not in real life, is to put a big fin on the tail and use wind resistance to slow down the spin. This may seem like a pain in the rear to deal with, but I can guarantee that those students who do not deal with the spin issue will have troubles winning, as the spinning saps the power from the rubber band. (If the inertia and wind resistance of the helicopter body is nearly the same as the rotors, it is conceivable that once the helicopter rises from the ground, it will hover in place as the body will begin to spin as fast and in the opposite direction of the rotors, meaning no power goes toward lift).
In order to get lift, there are two ways to get it. One is airfoil lift. If the rotor blade has a cross-section shaped like an airplane wing, there is some lift due to the pressure differences. (An airfoil is flat on the bottom and rounded on top. As air passes over and under the airfoil, the air on top has to travel further than the air on the bottom to reach the same spot they were in before. Since the air has to travel further on the top, its speed is greater, therefore its pressure is less compared to the air on the bottom of the wing. Since there is greater pressure below and less above, this is where your lift comes from. Ask you teacher if you are confused). The other method of lift comes from the angle of attack of the rotors. If you turn the rotor blades at an angle with the front (leading edge) of the blades above the back edge of the blades (see angle of attack section in the website I gave you) you push more air under the blades than over, which also increases pressure below the blades in relation to the pressure above the blades. This gives greater lift. The control in a helicopter for the angle of attack is called the collective. There is likely an ideal angle of attack for maximum lift, but it likely depends on how much torque your engine (in this case the rubber band) can produce. If you angle is too great, the engine can't spin the rotor fast enough for lift off. I would recommend, if you are allowed, just buying a rotor from a hobby store since airfoils are complex and not easy to build.
If you do have to make them, just use something flat and narrow and vary your angle of attack to get your lift. You might need to put an outer ring on the blades for support the way some RC copters do since the blades will see high loads. (Remember, the blades are where all your lift comes from). The ideal angle of lift will come from experiment. If you can't change it, chose something small like 10°-15°. This is likely a good question for Paul Cantrell. Also, the length and area is important. The greater the length and the area of the blade will increase lift, but too long and big and the rotors may break. Also, the rubber band needs to be able to make them go fast enough for lift off. (Greater inertia and resistance that the rubber band needs to overcome when the rotor becomes bigger). The math is a little tricky to figure this out on paper, experimentation may be a better approach to find the best rotor size. Probably whatever you can just fit inside the 50cm box would be ideal!
The last issue is balance. Since you can't steer the helicopter, it needs to be very well balanced, both front to back and side to side in relation to the center of the rotor. If it is not, as the helicopter rises it will tip, and the rotors will pull it off in the direction it is tipping in. It will likely still rise, but horizontal motion is wasted energy. You will need to experiment to get the right balance, probably by adding weight to the light side.
If you want to attempt to build a model with a tail rotor, what you probably should do is anchor the rubber band somewhere in the middle. Then one end would be for the main rotor and the other for the tail rotor (the longer section should be for the main rotor). How you attach the rubber band to get power to the shafts of the rotors is your big design decision! There are a number of options. Hopefully your teacher can help, or if you get really stuck, e-mail me and I'll give you some ideas. Or you can use a big fin on the tail and use air resistance to limit spin. You will lose some energy to spin, but you would be able to use the whole rubber band for the main rotor. The last configuration, the double rotors, is the most complicated but I bet you would get huge design points. You could use half the rubber band for the outer drive shaft and half for the inner shaft. The big trick there is how to support the shafts. You would need a couple of rotating bearings, one of which would have to fit inside of the outer shaft to support the inner shaft! Might be hard to find in such a small scale. If you could figure it out, however, I bet it would be very cool!
Anyway, my last bit of advice is to complete the building of it at least a week in advance so you can experiment and work out any bugs. It is frequently not the best design that wins these competitions, but one that is simple and has had plenty of testing. Good luck and if you need more advice or want some math equations to help describe stuff, e-mail me privately at email@example.com and I will try to help.
Try the links in the MadSci Library for more information on Engineering.