Some More Model Plane Math!

While making a design for a flying wing recently, I came across some interesting things that can be very useful when designing an RC plane.

Lift Equation 

Lift = 0.5 * Wing area (in square feet) * speed squared (in feet per second) * air density * lift coefficient of the airfoil

If the number for lift is equal to the weight of the plane in pounds, it will fly. For example, say I have an aerobatic 3D RC plane with a symmetrical airfoil that has a lift coefficient of 0.5, a wing area of 5.5 square feet, and a total weight of 3 pounds. We think it should be able  to fly at a speed of 25 feet per second (27 kilometres per hour).

An example of a 3D (aerobatic sport) RC plane. 

Lift = 0.5 * Wing area * Speed squared * Air density * Lift coefficient       

       = 0.5 * 5.5 * (25 ft per second * 25 ft per second) * 0.0027 * 0.5
       = 0.5 * 5.5 * 625 * 0.0027 * 0.5
       = 4.64 pounds 

Because the lift generated by the wing (4.64 pounds) is more than the weight (3 pounds), assuming that the centre of gravity and everything else was correct, the RC plane would fly. The great thing about the lift equation is that not only can you find out if the lift generated by the plane is enough to overcome the weight at a given speed, you can also find the minimum speed that the plane would take off at (the point where the lift is equal to the weight). Let's go back to that example.

Lift = 0.5 * Wing area * Speed squared * Air density * Lift coefficient
       = 0.5 * 5.5 * (21 ft per second  * 21 ft per second) * 0.0027 * 0.5
       =  0.5 * 5.5 * 441 * 0.0027 * 0.5
       = 3.27 pounds 

We eventually find that the RC plane would need to be moving at a minimum of just under 21 feet per second (23 km/h) to takeoff, (remembering that we had 3.27 pounds of lift and the weight of the plane was 3 pounds).

Calculating Centre Of Gravity

Hi everyone. In one of my earlier posts, I talked about calculating flight speed and how a larger diameter of propeller offers more thrust and more pitch will increase the pitch speed because the propeller is taking a larger bite of air. One thing I want to add is that at the point where the pitch speed of the propeller is equal to the speed that the plane is moving through the air, there is no thrust.

CENTRE OF GRAVITY 

My Dad and I are trying to design an RC plane right now that will have a flight speed of about 90 km/h and a thrust to weight ratio of at least 1.2 for vertical climbs. We decided that since we know what our possibilities are for motors and batteries etc. we decided to estimate the centre of gravity to see if our design was any good in terms of balance. Here is a drawing I made of a design we made on the computer:

Of course, the vertical stabilizer is too small and the plane will be much more streamlined, but this is the best drawing that I could do. The big line going through the airfoil is where we want the centre of gravity to be. We estimated the c of g on a spreadsheet but I will show you as best as I can what we did. The arm is the distance of these different main parts from the centre of gravity and the motion comes from multiplying the weight by the arm. So, if the motion is roughly the same for behind the centre of gravity and in front of it, we will know that the plane is fairly well balanced. 

                          Front Of Plane                                    Back Of Plane 

              Weight     Arm  Motion                          Motion     Weight      Arm 

      Battery: 65 grams 7 cm  455                          1350 Motor: 54 grams 25cm

      Gear:     80 grams 20cm  1600                       315  ESC:   35 grams 9cm 

                                                                             354   Gear:  118 grams 3cm 

         Total Front Of Plane Motion:               Total For Back Of Plane:

                         2055                                                         2019

I should probably say that I don't remember exactly what the different arms where from the actual spreadsheet for this plane, but I think that most of the weights of the different parts are accurate, and these motions were very close to the ones that we got from the spreadsheet. On the actual spreadsheet, the difference between the front C of G and back C of G was only around100 which as far as I know is not very significant considering that the motions were around 2500. 

Estimating Stall Speed For A Trainer

I have been very interested in finding the calculation for stall speed. I have found it, but it is way to complicated for me to understand. So, I tried to make my own simplified version. I thought: Okay, what effects stall speed.

• Air density
• Maximum lift coefficient of the airfoil
• Wing loading

When I tried to go on this without much success, I decided to focus just on the relationship between wing loading and stall speed for a given lift coefficient and air density. Since the weather is not very good here in the wintertime, I went on the simulator and got the E-flite Apprentice with SAFE technology. Here is a graph I made with the relationship between wing loading and stall speed. 

The wing loading is the vertical (y) and the stall speed is the horizontal (x). What is basically happening is that the up until 61.8 grams/sq. decimetre of wing loading, for every 10 grams/sq dm. of wing loading I added, the stall speed or (more useful) landing speed increased by five kilometres per hour each time. After 61.8 grams per square decimetre, the landing speed was increasing by 3 - 4 kilometres per hour instead. I may repeat this test again just to be sure of these results. 

All of this means that my estimate for stall speed, with the airfoil of the Apprentice (which I'm guessing has a lift coefficient of 0.9 on landing) is: wing loading / 1.35 = stall speed. As far as I know, this would become less accurate after 60 grams/sq dm. or so because the relationship changes. 

Many people say that knowing the stall speed for an RC plane isn't useful because all you have is your eye which can't tell you exactly what speed the RC plane is flying at. This is true. 

But my suggestion is to set out some cones that your plane will have to fly over before it lands. By finding how far apart your cones are, and estimating your landing speed, you can say that if the RC plane went over those cones in x number of seconds, it's going at the right speed. 

Designing An RC Plane - Design Process

Now that we know some of the basics of RC plane design (different wing shapes, aspect ratio, wing loading, power systems and estimating maximum speed of a model), I thought it would be a good time to talk about a process that you could use to design your own RC plane. Airfield Models is a great sight to visit that really simplifies RC plane design and some of what I write is inspired by that website.

Step 1: Define The Purpose Of Your RC Plane

Here are some examples of things to consider when deciding on a purpose

- Maneuverability (aerobatics it should be capable of, maybe nothing) 

- Airspeed envelope (minimum and maximum flight speeds)

- Type of Plane (trainer, cargo RC plane, swept-wing jet etc.)
- Thrust To Weight Ratio

An airplane made by the youtube channel Aplane.  Has a low aspect ratio, rectangular wing, built for carrying lots of weight.

Step 2: Be More Specific! Decide On Some Specifications 


Things such as:

-Wingspan
-Chord or average chord (the width or distance across a wing. You can find average chord by adding up the chord at the part of the wing closest to the fuselage and the chord at the tip, then dividing by 2.)
-Weight (includes the weight of the servos, the receiver, battery, ESC, motor and airframe weight)
-Motor and prop combination (think about speed and thrust)
- Airspeed envelope (the minimum and maximum airspeeds that your plane should be able to fly at)




Step 3: Decide On A Target Wing Loading


Now that you know the weight of your electronics and have an estimate for airframe weight, add those two weights together. Then, you can decide on the target wing loading and play around with different wing areas until you find the wing loading that you are aiming for. Because of all things, wing loading should not be the surprise result after you have built the RC plane.

Highly loaded wing of an E-flite Habu 32

Step 4: Create A Plan

It doesn't matter whether its on a computer or on paper. Be specific with the dimensions of all the parts. 

Step 5: Build The Plane!

If you decide to change your specifications while building, remember to go back a step and see how this will affect the performance of your RC plane. Also, if the thing just won't fly or does not fly well, don't despair because this is almost exactly what happened with the TundraTrike from the previous post.

1. What was the problem

2. Why did it happen

3. How can you change the design and start flying again

Also, don't decide to stop any design because you feel that you don't know enough to make it fly properly. Just use your imagination and build what you have in your head. And again, if it doesn't fly, just analyze what went wrong, why it went wrong and redesign the plane. By doing it this way, you can get closer and closer to the RC plane that you really want, learn lots and make a great-flying airplane.  

Our TundraTrike RC Plane!

Hi again everybody! It hasn't been very long, and I know that I wrote only yesterday, but I recently put a new battery in my camera, and decided that I should write about some more personal things, as oppose to just information. So I took some photos of the foam board RC plane that my Dad and I built a couple of years ago. Here's what the plane looks like:

The underside of the plane 

A view of the plane from a back view as it was hanging on the ceiling

We decided to call it The TundraTrike because of the utility/bush plane type appearance from the
two-and-a-quarter inch front wheel and the two-and-a-half inch back wheels. This size of wheels works fairly well on a 580 gram plane on the average grass but we recommend three-inch wheels instead because the takeoffs were sometimes very . . . sketchy. Also more clearance between the prop and the ground. When my Dad designed it, he gave the wing a low aspect ratio of five so that the wing could carry all of the plane's bulkiness along with making it a reasonably maneuverable trainer RC plane.  It has an E-flite Park 370 motor with a 7-inch pitch propeller that (I think) has a diameter of 8 inches which gives means that the plane can fly at about sixty km/h and it has a thrust-to-weight ratio of almost one, because it comes close to hovering vertically for a few seconds when its in a vertical position. Even though I had already been flying in the simulator for a few years, it is always good to start with a very easy-to-fly RC plane when you begin flying in real life because it is a very different experience.

I will try to find some videos of this plane flying that blogger will accept because the ones that I have tried to upload so far have had to much content for blogger.

Receiver, ESC, Motor, Battery and Propeller

Hello everyone! In terms of power systems, there are five essential ones in RC planes. As you saw in the title, they are the receiver, ESC or engine speed control, motor, battery and propeller. Let's talk about them all.

Receiver

I like to think of the receiver as the central piece of electronics on an RC plane. The engine speed control or ESC and all of the servos connect to the receiver. When you make a certain control input on your transmitter, the transmitter sends that signal to the receiver. Since all the servos and engine speed control are connected to this receiver, it allows your plane to function according to your control inputs. 

ESC or Engine Speed Control

For what I know, the ESC is just what it sounds like. A way for the motor to vary its speed. There are many different types of speed controls, so it is important to get the right one based on the amount of current that that motor/propeller combination is using. A great site for this is www.flybrushless.com.

Battery

This one is quite obvious: to provide power for the motor, but there is something important here that I learnt from my Dad. If you multiply the mAh of the battery by its maximum continuos discharge rate, you will get the maximum amount of current that can be drawn from that battery. To make things simple, say that its the type of battery that I use in the foam board plane that my Dad and I built. An E-flite 800 mAh battery with a 20C maximum discharge rate. Multiply 800 mAh by 20C and you get 16000 mAh or 16 amps as the maximum current. This is important because it ties into the whole motor/prop combination thing. If your chosen motor and propeller combination is using say 18 amps of current and you have a battery that has a maximum current of 16, then you need a bigger battery or a different motor/prop combination. Don't worry if your motor/prop combo is using the absolute highest amount of current that the battery can handle being drawn from it. Getting the maximum performance is a good thing. 

Propeller

The propeller is what makes all of this stuff amount to your plane soaring through the skies. So, it is really important to choose the right one! Propellers have a twist in them called pitch. The difference that pitch makes is in the speed or thrust that the RC plane will have. Pitch is the limiting factor for speed because the RC plane gets to a point where the pitch of the propeller is sucking the air in and pushing it out the back as fast as it possibly can, which limits the top speed in level flight. A larger pitch will have a higher speed because it is taking bigger bites of air but lower thrust, and less pitch will have lower speed and a propeller with a larger diameter will have higher thrust. Again, flybrushless.com is a great website to visit for this. There is a way to calculate the maximum pitch speed of the propeller. This could also be considered the top speed of the model, assuming it had no drag which is not true. Here it is:

Pitch speed = (motor rpm * propeller pitch in inches)  * 60 * 0.0000254
(example) Pitch speed = (15096 rpm * 7 inch pitch) * 60 * 0.0000254
                 Pitch speed = 161.04 km/h 

Disclamer first: Just remember that this is what I think and may not take some things into account. But I have compared this against several online pitch speed calculators and an experienced person's website so I think that it is correct. 

Okay now that I have that out of the way, let me explain it. If a propeller has a pitch of 7 inches, then, every time it makes a full revolution or turn, the plane will move 7 inches forward through the air (at full rpm). Then, if you multiply the pitch of the propeller by the amount of revolutions it makes per minute (RPM), then you will get the amount of inches that the RC plane moves forward through the air every minute. If you multiply that by 60, then that is the amount of inches that the plane moves forward in an hour. Multiplying that by 0.0000254 will convert inches per hour to kilometres per hour.

Of course, this cannot be the maximum speed of the model. The main reason that I know of is drag. Drag slows the plane down. So, how much is the question. After doing some research on the e-flite website, testing RC planes out on the flight simulator and typing some things into my spreadsheet, I have made an estimate for the actual top speed of a model. For a high drag, box shaped plane by comparison such as the E-flite Apprentice or Carbon Z Cub,  http://www.e-fliterc.com/Products/Default.aspx?ProdID=EFL3100 the actual estimated top speed should be about 60% of the calculated pitch speed. You can do this easily on a spreadsheet by multiplying the cell with the value of pitch speed by sixty percent. For a medium drag type plane such as the HobbyZone Sportsman S, you could expect a higher top speed closer to 65% of the pitch speed. For a low drag plane such as the E-flite Rare Bear, the max should be about 70% the pitch speed on a good day. This is a great thing to add to a spreadsheet if you decide to make one.



Motor

The motor is what turns the propeller of an RC plane. We already talked about the interconnectedness of the motor, battery and propeller so I will just end by giving you a useful tip for motors: The motor KV multiplied by the voltage of the battery connected to it will give you the maximum potential RPM of the motor (e.g 1800 KV * 11.1 volts = 19980 RPM). Remember, this will change depending on the propeller that you use. I'm assuming that a propeller with a large pitch taking big bites of air will slow down the motor RPM, which would then affect the pitch speed of the propeller slightly, since pitch speed takes motor RPM into consideration. You could then expect the pitch speed to be slightly lower than your calculation depending on the type of propeller you use. 

The E-flite Park 370 Motor. Motor on the left, spinner in the centre and motor mount on the right. 

Designing an RC Plane Using a Spreadsheet

If you start trying to design an RC plane, just about everything affects other things.

Say that you are designing a 3D airplane. You have completed most of your design, but then you remember that if you want your plane to be able to climb vertically, you will need a thrust to weight ratio of at least slightly more than one. You decide that you then need a more powerful motor to give you the thrust that you are looking for. This motor happens to weigh an extra 25 grams than the previous one that you were initially planning to use. Assuming that you did not change the weight of the plane, since wing loading = weight / wing area, you have just increased the wing loading. 

Another example is finding out that the wing loading is to high, then you increase the wing area to lower the wing loading and decrease the plane's stall speed. 

However, There is a way to design the best possible RC plane, plus keeping track of all of the changes that are taking place in as many areas of the design as you want. Use a spreadsheet!

They look really boring, but my Dad showed me how useful and fun they could be really recently when we were designing an RC plane (not the foam board one in the about me page). 

This is a copy of the spreadsheet that I made a few days ago.

Blogger does let you enter spreadsheets very easily so you will have to scroll around for a while to see where everything is and what information is there, sorry. Pay most attention to the aspect ratio, wing loading, wing area and weight. Unfortunately, it doesn't allow you to edit. These are the formulas I used. Remember that the boxes that I use in the formulas (e.g L39) change depending on which boxes you actually use for the different values. The brackets are not part of the formulas. You don't have to do it this way, this is just the way I did it. Use whatever units of measurement you like, just remember to be consistent

• Wing Loading : = (weight in grams) i18 / (wing area in sq. dm) K18
• Aspect Ratio: = (length in cm ) K21 / (width or chord in cm) K23
• Weight: = All the parts added up (Engine speed control, motor, battery, airframe, servos) = J32 + J34 + J36 + J38 + J40 (weight in grams if you used grams before)
• Wing Area: = (wingspan in centimetres ) K21 * (multiplied by) K23 (width or chord in centimetres) 

If you decide to design an RC plane and want to use a spreadsheet. Great! Because they are really helpful. Something that I did not do in this version of the spreadsheet that I recommend is putting the units in the boxes below the values because it gets hard to remember after a while.