I am trying to better understand the dynamics of forward flight in multirotors.

Assuming I have a quadcopter with 4 motor/propeller combinations capable (each) of a propeller pitch speed of, say, SpeedMax= 100 mph.

In forward horizontal flight, the quadcopter will pitch down at a certain angle, let's say AlphaP, from horizontal. If AlphaP is, say, 45 degrees, and drag is neglected, wouldn't the quadcopter be capable of a max theoretical speed of sin (45)* SpeedMax ~ 70Mph?

Also, seems to me AlphaP cannot go all the way to 90 degree (quadcopter flying like a plane), as at that point the propellers would not produce any upward thrust to maintain the copter aloft given there is no wing loading as available in a plane. If drag was to be neglected, what factors would the optimum AlphaP be depended on, and what would that angle be, for maximum speed?

• Where does your 100mph number come from? At any rate, what determines the maximum pitch is exctly what you said, being able to maintain the altitude against gravity... the thrust vector's component pointing up. – Guy Sirton Mar 29 '14 at 6:45
• Slightly off topic, but an important point: even if you stay at 0 degrees of tilt, there is still a maximum forward speed that you can achieve. Helicopters have a lot of trouble breaking the $\mu$-1 barrier. – Ian May 27 '14 at 16:24
• Motors generate acceleration, not speed. If the drag (and relativity) are neglected, the maximum theoretical speed is infinite! – marcv81 Oct 13 '14 at 23:35

As you said, as AlphaP increases, the amount of upward thrust decreases, and when AlphaP = Forget about speed. This idea that each propeller is capable of 100mph isn't that helpful. The problem is that the thrust produced by the propellers will be a function of speed. The faster they are traveling through the air, the less thrust they'll produce. Think instead about thrust. How much thrust are the propellers capable of?

90º, there's no upward thrust at all. Your tilt angle, AplhaP will be limited by the upward thrust needed to keep the quadcopter aloft. the amount of upward thrust produced is proportional to cos(AlphaP).

If all 4 propellers together produce 20N, and your Quadcopter weighs 10N, then to stay aloft:

• cos(AlphaP)*20N >= 10N
• cos(AlphaP) >= 0.5
• Alpha <= invcos(0.5)
• Alpha <= 60º

Now the forward thrust produced will be sin(60º)*20N = 0.866*20N = 17.3N. What speed does this relate to? You can't neglect drag here. The quadcopter will accelerate until the drag equals 17.3N.

But of course, it's not that simple. What's confounding this is that the thrust produced at speed will be less than that produced when stationary. A very rough first approximation is to think of the power output of the propellers.

• power = force * speed

Double the speed means half the thrust. But it's not that simple either. Generally, when you do these simple calculations, I would divide the results by 4 to get a more realistic estimate of the maximum speed.

 F*cos(maxAlphaP) = x*g     //cos or sine alphaP, depending upon how you measure the angle.