“I know, let’s put a flat front on this speed boat!” said no one ever.
I am a fairly reserved person, but as an engineer, it is frustrating to see what are considered to be the top quadcopter “racing” frames. Unfortunately for us, there isn’t much of a choice when it comes to a smart design. The main performance destroying design flaw on these frames are the arms.
Quadcopters are strange animals – they behave absolutely nothing like a conventional aircraft. Airfoil shapes work excellent in the case of airplanes since the axis of the airfoil shaped structures (fuselage, wing, etc.) are, in general, parallel with the direction of movement/air flow. If this parallelism deviates much (angle of attack increases too much), then the airfoil goes into a stall which is where turbulent air dominates and causes immense drag. This is why flat/airfoil shaped arms will never work well for quads since our pitch angle varies tremendously while flying. However, airfoil streamlining works great in the case of building a quad in which you plan on staying at a relatively fixed pitch such as speed runs.
Reasons for Using Round Arms:
- Flat arms = air brakes.
- Flat arms under the props = thrust blockers.
- They have nearly half the drag coefficient of a flat plate.
- They are the best way to reduce arm drag regardless of pitch angle.
- They are a great way to rout wires to the motors which further reduces drag.
- The projected area (see concepts) is 1/2 that of an air brake (flat arm) which effectively makes the overall drag force 1/4 that of a flat arm.
Although it is understood that the large face of a flat arm will not always be perpendicular to the direction of movement, flat arms are still extremely inefficient. Actually, as your quadcopter gains speed (and pitch angle increases), the face of the arm gets closer to being perpendicular to the air flow which in turn increases the inefficiency of the flat arm. However, in general, the thrust created by the props is always (theoretically) perpendicular to the arms.
Are Round Arms as Strong as a Flat Arm?
To understand how much deflection an arm will have (a measurement of how stiff the object is), the formula for calculating beam (the arm) deflection can be used. Without getting into detail, the only term that we need to use as a comparison between a tube and rectangular arm is the I term which is called the second moment of inertia.
For a rectangle, the direction of deflection makes a difference in calculating I. There are still many assumptions to be made here, but in general, we are only concerned with deflection along the direction of thrust.
For this calculation, I used a tube that has a 10mm outer diameter and a 8mm inner diameter. Then using algebra, I calculated the thickness a flat arm would have to be to equal the strength of the tube. For the width of the flat arm, I used the average of a typical flat quadcopter arm which is 22mm. Using these numbers, a flat arm would have to be 5.41mm thick to equal the strength of the 10mm tube arm. 5.41mm is very hefty for an arm. However, when comparing arms of the same length, this flat arm has the following drawbacks:
- The flat arm is 4.2 times heavier than the tube
- The flat arm has 2.2 times the projected surface area of the tube (in the direction of thrust)
- The flat arm is obstructing 2.2 times the amount of thrust that the round arm blocks.
- The flat arm has a 66% greater drag coefficient.
- Matters only get worse once the motor wires are run down the flat arm (motor wires for a tube arm are run down the inside).
The same calculation can be carried out for a solid cylindrical arm:
- The solid cylindrical arm is 2.6 times heavier than the tube
- The solid cylindrical arm has .88 times the area of the projected surface area of the tube (in the direction of thrust)
- Although it is .88 times the surface area, this will be thrown out the window (along with the drag coefficient) once the motor wires are run down the arm.
What if I’m Not Going for Speed and my Pitch Angle Stays Low?
At a pitch angle of 0°, the 5.4mm thick arm is a little more than half the projected area of the round 10mm arm. However, as that pitch angle increases, the projected area increases. At around 13.5° of pitch, the projected area equals the 10mm tube and it only gets worse as pitch increases. However, it also gets worse due to the motor wires running across the arm. With the motor wires added, we can only reach a pitch of 7-8° before the projected area equals the 10mm tube.
For a more in depth understanding of drag, please see sections 4 and 5 of the Speed Optimization series.