Brad I'm sure you must have considered 4 large rotors with a low disc loading and high inertia for the bulk of the lifting and 4 smaller ones with lower inertia for the stabilization control. All fixed pitch. I'm curious what you may have concluded about that configuration. I really enjoy and benefit from your posts.

Brad Hughey > Chris MilnerJanuary 9, 2016 at 1:47pm

Thank you for your kind words, Chris. Yes, I have thought about this.

Control authority is all about how much of the total thrust you can change and how fast. Consider the case of a conventional cyclic-pitch-controlled helicopter as an example.

A typical production helicopter has a rotor speed of 500 RPM. That translates into about 8 "control modification cycles per second", so to speak. A helicopter rotor's pitch angle, and therefore the blade section coefficient of lift and thrust, can be varied over its entire range from zero (or even negative) to maximum in only one quarter of a rotational cycle. It can do this 32 times per second, yielding a theoretical control response time of 31 milliseconds. Actually doing that in a flying, full-scale helicopter would produce a very violent control reaction, but the fact is: it CAN be done. When you see those crazy-agile R/C helicopter videos, remember their head speeds are about 1500 RPM, so the on-ship control response is three times faster than at full scale. They can and do vary the whole thrust vector in about 10 milliseconds (not counting obvious control system delays).

If you push the cyclic stick forward in a regular full-size helicopter, it will be changing the pitch of the blades from about 3 degrees in the front to 6 degrees in the back every 62 milliseconds. To match that, a typical electric quadcopter would have to have fixed-pitch blades that go from roughly half speed or 1/8th of total thrust to full speed or 1/4 total thrust in that same 1/16th of a second. And that's for a quad, where one blade is changing 25% of the total thrust that fast. The Dr. Jonathan How/Mark Cutler team at MIT was reporting responses in the 270 millisecond range for 9-inch diameter fixed-pitch blades. My own testing of various propeller and motor combinations at this scale has generally confirmed a very wide thrust transient performance gap.

In light of this, it's amazing that small quads actually fly as well as they do. This is due in no small part to the fact that the control dynamics of a typical quad are quite different than an overhead-rotor helicopter. Having a center of mass near, or even slightly above, the lifting plane makes for a much more stable craft.

Smaller propellers would certainly be able to change thrust faster than longer ones, but their percentage of the total would also be proportionally smaller.

Another direction is to increase the number of propellers. Why not go with an Octo or even with 16 props? Less prop momentum issues, more reliabililty etc.

No, seriously, I have been waiting for someone to ask just such a question so we could walk through the math. There's a caveat, though, that theory will only take you so far. If you've set out to design a new aircraft, you'd best have the budget for some failed parts and the time to experiment.

First of all, I assume you don't want to build your own propellers. Since 30 lbs is rather large, you'd want to pick some larger props. There is a compromise here, because the larger the diameter of the props, the more efficient it will be (which translates to longer flight durations). Unfortunately, larger props have more rotational inertia, which translates directly into more sluggish control responses. As a starting point, we could chose 24"

I am also assuming that you say 30 lbs of thrust because that will be the gross weight target of the aircraft. Let's add at least 30% to that for control headroom and round it all off to 40 lbs total just for convenience. That means each 24" prop must produce 10 lbs of thrust.

Using momentum theory as a starting point, that means a disk area of 3.14 square feet. The formula for induced velocity is the square root of (thrust in pounds divided by (2 times the density of air times the disk area)) or sqrt (10/2 * 0.00238 * 3.14). The answer is 25.87 feet/second ^-1. The ideal power at the rotor disk is the original thrust (10) times the induced velocity or 258.7 pound feet/sec ^-1. Simply divide this number by 550 to get an ideal power of 0.47 horsepower. To convert to watts, 746 * 0.47 = 351 watts. But this is the ideal power, assuming the propeller is 100% efficient, which of course, no propeller is.

Based on testing, we know that model airplane propellers are somewhere between 25 and 50 percent efficient. Assuming worst case (and allowing for some margin), then your motor must be capable of 351/.25 or 1,404 watts at the shaft. Since it can be assumed that a brushless DC motor with controller is about 80% efficient, you'll need to supply 1,755 watts of power from your battery. Under load, a lithium polymer pack usually sustains about 3.4 volts per cell, so that's a 6 cell pack and 86 amps approximately. You would do well to upsize the ESC and motor a bit, because VTOL aircraft stay at nearly full throttle for much longer durations than airplanes do.

Now you have some shopping to do. :-)

The tricky part here, of course, is the KV rating on the motor. Ideally, the full maximum battery voltage times the KV rating should not exceed the maximum recommended RPM of the propeller. So, 6 X 4.2 X KV /=/ > RPM Max. Also, besides obviously needing a pair each of matched regular and counter-rotating props, you need to find them with a relatively low pitch, lest the static thrust conditions (no inflow) exceed the propeller section stall angle. Perhaps something along the lines of a 24 X 8 through 24 X 12 ought to do the trick. You might want to get a static thrust RPM recommendation from the prop manufacturer and back into your motor KV spec that way.

## Replies

Brad I'm sure you must have considered 4 large rotors with a low disc loading and high inertia for the bulk of the lifting and 4 smaller ones with lower inertia for the stabilization control. All fixed pitch. I'm curious what you may have concluded about that configuration. I really enjoy and benefit from your posts.

Thank you for your kind words, Chris. Yes, I have thought about this.

Control authority is all about how much of the total thrust you can change and how fast. Consider the case of a conventional cyclic-pitch-controlled helicopter as an example.

A typical production helicopter has a rotor speed of 500 RPM. That translates into about 8 "control modification cycles per second", so to speak. A helicopter rotor's pitch angle, and therefore the blade section coefficient of lift and thrust, can be varied over its entire range from zero (or even negative) to maximum in only one quarter of a rotational cycle. It can do this 32 times per second, yielding a theoretical control response time of 31 milliseconds. Actually doing that in a flying, full-scale helicopter would produce a very violent control reaction, but the fact is: it CAN be done. When you see those crazy-agile R/C helicopter videos, remember their head speeds are about 1500 RPM, so the on-ship control response is three times faster than at full scale. They can and do vary the whole thrust vector in about 10 milliseconds (not counting obvious control system delays).

If you push the cyclic stick forward in a regular full-size helicopter, it will be changing the pitch of the blades from about 3 degrees in the front to 6 degrees in the back every 62 milliseconds. To match that, a typical electric quadcopter would have to have fixed-pitch blades that go from roughly half speed or 1/8th of total thrust to full speed or 1/4 total thrust in that same 1/16th of a second. And that's for a quad, where one blade is changing 25% of the total thrust that fast. The Dr. Jonathan How/Mark Cutler team at MIT was reporting responses in the 270 millisecond range for 9-inch diameter fixed-pitch blades. My own testing of various propeller and motor combinations at this scale has generally confirmed a very wide thrust transient performance gap.

In light of this, it's amazing that small quads actually fly as well as they do. This is due in no small part to the fact that the control dynamics of a typical quad are quite different than an overhead-rotor helicopter. Having a center of mass near, or even slightly above, the lifting plane makes for a much more stable craft.

Smaller propellers would certainly be able to change thrust faster than longer ones, but their percentage of the total would also be proportionally smaller.

This is easy.

These blades:

http://shop.spinblades.com/index.php?option=com_virtuemart&page...

This motor:

http://www.hobbyking.com/hobbyking/store/__21869__Turnigy_HeliDrive...

You'll need to use something like this as a gearbox:

http://helidaily.com/sneak-pics-trex-800-trekker/

:)

Regarding big, I have some practical numbers for a 24" setup I did on giant scale R/C airplane.

Setup: Turnigy 80-100 180kv motor, Jeti Spin 200A ESC, 12S Lipo and APC-E 24x12 propeller.

Result: 6200W, 37lbs static thrust at 6500rpm.

Another direction is to increase the number of propellers. Why not go with an Octo or even with 16 props? Less prop momentum issues, more reliabililty etc.

Big.

No, seriously, I have been waiting for someone to ask just such a question so we could walk through the math. There's a caveat, though, that theory will only take you so far. If you've set out to design a new aircraft, you'd best have the budget for some failed parts and the time to experiment.

First of all, I assume you don't want to build your own propellers. Since 30 lbs is rather large, you'd want to pick some larger props. There is a compromise here, because the larger the diameter of the props, the more efficient it will be (which translates to longer flight durations). Unfortunately, larger props have more rotational inertia, which translates directly into more sluggish control responses. As a starting point, we could chose 24"

I am also assuming that you say 30 lbs of thrust because that will be the gross weight target of the aircraft. Let's add at least 30% to that for control headroom and round it all off to 40 lbs total just for convenience. That means each 24" prop must produce 10 lbs of thrust.

Using momentum theory as a starting point, that means a disk area of 3.14 square feet. The formula for induced velocity is the square root of (thrust in pounds divided by (2 times the density of air times the disk area)) or sqrt (10/2 * 0.00238 * 3.14). The answer is 25.87 feet/second ^-1. The ideal power at the rotor disk is the original thrust (10) times the induced velocity or 258.7 pound feet/sec ^-1. Simply divide this number by 550 to get an ideal power of 0.47 horsepower. To convert to watts, 746 * 0.47 = 351 watts. But this is the ideal power, assuming the propeller is 100% efficient, which of course, no propeller is.

Based on testing, we know that model airplane propellers are somewhere between 25 and 50 percent efficient. Assuming worst case (and allowing for some margin), then your motor must be capable of 351/.25 or 1,404 watts at the shaft. Since it can be assumed that a brushless DC motor with controller is about 80% efficient, you'll need to supply 1,755 watts of power from your battery. Under load, a lithium polymer pack usually sustains about 3.4 volts per cell, so that's a 6 cell pack and 86 amps approximately. You would do well to upsize the ESC and motor a bit, because VTOL aircraft stay at nearly full throttle for much longer durations than airplanes do.

Now you have some shopping to do. :-)

The tricky part here, of course, is the KV rating on the motor. Ideally, the full maximum battery voltage times the KV rating should not exceed the maximum recommended RPM of the propeller. So, 6 X 4.2 X KV /=/ > RPM Max. Also, besides obviously needing a pair each of matched regular and counter-rotating props, you need to find them with a relatively low pitch, lest the static thrust conditions (no inflow) exceed the propeller section stall angle. Perhaps something along the lines of a 24 X 8 through 24 X 12 ought to do the trick. You might want to get a static thrust RPM recommendation from the prop manufacturer and back into your motor KV spec that way.

Keep us posted!