### Static Thrust Calculation

Calculations of static thrust are needed in order to ensure that the proper propellers and motors have been selected. Static thrust is defined as the amount of thrust produced by a propeller which is located stationary to the earth. This calculation is particularly important for this project because quadrotor helicopters are more likely to perform at low speeds relative to the earth. This low-speed performance ensures that the calculations of static thrust can be applied to a wide range of flight conditions. Also, it is important to note that the final calculations of static thrust are estimates and not actual values.

The first step in calculating static thrust is determining the power transmitted by the motors to the propellers in terms of rpm. Aircraft-world.com has compiled empirical data used to calculate power [1], and the formula used for their datasheet is given in Equation 1.

Where power is in watts and rpm is in thousands. For example, a 6X4 APC propeller has a propeller constant of 0.015 and a power factor of 3.2. Given a rotational speed of 10,000 rpm, the calculation goes as follows: Power=0.015X103.2=24 W.

The next step is to determine the thrust produced by a propeller. Equation 2 gives thrust based on the Momentum Theory.

A commonly used rule is that velocity of the air at the propeller is v=½Δv of the total change in air velocity: Therefore,  and equation 3 is derived.

Equation 4 gives the power that is absorbed by the propeller from the motor. Equation 5 shows the result of solving equation 4 for Δv and substituting it into equation 3. In doing so, Δv is eliminated and torque can be calculated.

Finally, it is advantageous to express the results of equation 5 in terms of mass. Newton’s Law, F=ma, is used to obtain equation 6.

Solving for mass is useful for quadrotor helicopters because it can be directly related to the mass of the aircraft. In particular, a thrust (mass) that equals the mass of the aircraft is needed for hovering. The importance of hovering will be addressed in the following section (DC Motors).

19 Responses to “Static Thrust Calculation”
1. ducvietbgDuc says:

hello, my name is Duc, I am a Vietnamese student. I’m glad to read about quad knowledge on your blog. Can you help me answer some pretty basic questions? (because I am a beginner and have limited language):
-what is proller center hub? (whether it is the outer diameter of the circle between the two wings of the propeller is not?!)
– Powerfactor constant Kp? (I found the constant Kp on bringing in some sources: W = Kp * D ^ 4 * p * ^ 3 Rpm)
thank u!

• jedickey says:

Hi Duc, I’m really glad you’re enjoying my blog.

I can only assume that the propeller center hub refers to the center of the propeller where the two blades meet in the middle. That part of the propeller should have very little to do with thrust and power calculations because it introduces very little rotational inertia and contributes nothing to thrust. The variable of interest (the D in your equation above) is the length from one blade tip to the other blade tip on a two bladed propeller. As you can see in your equation, the prop diameter has a very large affect on power. Increasing the prop diameter by a factor of 2 will increase the power needed to rotate the prop at the same RPM by a factor of 16. WOW! As you can see, small changes in prop size can lead to large changes in the power needed to drive the prop.

As far as the power factor, Kp, the formula that I used in my analysis is from empirical or test data from Aircraft World. Their data comes from multiple test using different diameter props. Basically, they’ve fit a mathematical equation to experimental data. This process is just one way of getting a power equation. I’m sure that the equation that you found from a different source will work just as well. Keep in mind that the Kp that you found and the prop constant that I used are different and not interchangeable.

Send me the link to the site where you found your equation. You mentioned that some people liked my blog. Are they the same people that wrote the equation you’re using? Also, I’m curious about you. Are you working on a quadrotor for school or just for fun? I did my copter for fun. Are you in school? I’m just curious. Thanks again for writing!

2. ducvietbgDuc says:

sorry : W= Kp*D^4*p*Rpm^3

3. ducvietbgDuc says:

Thank u for your answer. I am in university and need to do a software. This software helps select the motor and battery when input mass and flying time. (not talking about speed). So I’m looking for the approximate formula to be used for calculation software. My formula is made ​​from http://www.rcgroups.com/forums/showthread.php?t=393251

• jedickey says:

I like the page you recommended. It’s a little better organized than mine.

I now understand what you are trying to accomplish. Keep in mind that the forum you sent me is for electric planes and not quadrotors. Although most of the information there is relevant (flight time calculations, Kv, etc.) electric plane modelers are not as concerned with static thrust. But us quadrotor enthusiasts understand how importance static thrust is when designing the perfect quadrotor helicopter.

Now for the good news! Your power equation will work in place of the equation I used from Aircraft World. Both equations are power equations as a function of diameter, pitch, and RPM after all. You don’t see pitch and diameter in the equation I used because they are hidden in the data acquired by Aircraft World.

Since I’ve had many viewers ask me about other propellers not listed on Aircraft Worlds webpage, I would like your help in working out the final equation, equation 9, using the formula that you have presented. I shouldn’t be too hard and I as well as many other views of this blog would appreciate it. I’ll post you’re response on this blog and give you full credit for it. What do you say? Are you up for a challenge?

• Bhargav says:

Hey i am designing a small sized VTOL-UAV for my project work
The derivation and formulae given by you are of great use for calculating the thrust required
Thanks a lot for that!
My question is – can u define wat exactly is the POWER FACTOR? and PROPELLER CONSTANT?
I see these are used to find the power – so want to know wat exactly are those parameters and how they can be found..
Thank You

• jedickey says:

The Propeller Constant and Power Factor values are found from experimental/empirical test. Basically, someone put a propeller on a test devise, incrementally increased the power transmitted to the propeller, then recorded the rotational speed of the propeller at each increment. The resulting power vs. rpm data set is then feed into a program like Matlab or Excel to yield a best fit curve. Judging by the the equation (equation 1) given by the experimenters, the best fit curve can be represented by an exponential equation. So, these values are not derived from theory, and you can’t really dig any deeper than the results. I’ve included a link to the values below. Hope this helps!

http://aircraft-world.com/prod_datasheets/hp/emeter/hp-propconstants.htm

4. Hector says:

Hello,
Im working on my senior design project and we are making a quadcopter.
I wanted to show some basic calculations so Im looking all over the interet to find a thrust equation.
What is v? Wouldnt v be zero because the quadcopter is stationary?
What would delta v be?

• jedickey says:

5. Kevin says:

Hi,

For Eq 3 is there a reference for this rule that you used? I’m curious where this rule comes from.

“A commonly used rule is that velocity of the air at the propeller is v=½Δv of the total change in air velocity”

Kevin

6. Rodney says:

Hi Jeremiah,
First of all, I’d like to congratulate you and this website, it is a great resource for undergrad student or general hobbyists.

My question is in regards to propeller constant (Oh no, not another propeller constant questions). I see your .015 Kp value for a 6×4 prop was derived empirically (someone had time on their hands), I tried finding the Kp for my 1045 (10×4.5) but came up with zip. In short is there a formula to compute the propeller constant besides going to the air-craft world website? (the info on the site you provided above no longer works)

Thanks Rod

• jedickey says:

Hey Rob,

Thanks for visiting my site! Yes, the people at Aircraft World who performed that work did all of us a solid! I’m not sure what type of prop you’re using (slow flier, electric, etc.), but it looks like 10X4.5 isn’t on Aircraft World’s list (link below) at all. I recommend extrapolating or interpolating using other 10″ props to get your prop constant. You could also use a finite element fluid model, but I’ve never done that. The empirically determined values in the link below work pretty well. Oh and by the way, my analysis doesn’t consider the actual power requirements of the motor being used. I’m working on that right now. And I’m going to provide videos that support my calculations (or discredit them). So it should be fun!!! Stay tuned. It should be soon…

https://www.aircraft-world.com/Datasheet/en/hp/emeter/hp-propconstants.htm

7. bala says:

Useful

8. Vinh Willard says:

Hi,
My name is Vin, so i am doing my project about weight lifting drone, and your equation is very useful for my project; however, i have a bit concern.
Let say rpm = kv * v
i have 2300 kv motor and 11.4V battery so i have rpm = 34040
take this into power = prop constant * rpm ^ power factor . i have power = 1198 W
put that power into the equation (6). i got about 607 kg; so with 2300 kv motor can lift 607 kg; it seems not right…
Is there any chance you can help me to point out where i did wrong, please ?

I am looking forward to hearing from you

Sincerely,

• jedickey says:

Hi Vin!

You are correct, that thrust is not correct. The rpm you calculated is the no load rpm. That is, it’s the rpm that your motor would achieve without a propeller.

You can make an assumption that your motor, esc, battery and propellers are all functioning at absolute ideal conditions, though. Just assume that your motor is spinning at exactly 1/2 the no load speed. This is the speed at which your motor is producing the most mechanical power. Of course this assumption over predicts performance, so you may want to lower the rpm to about 30 or 40% of your no load depending on how conservative you’d like to be.

Also, double check the numbers you are using. The equation requires rpms to be in the thousands. In my blog example, I used 10 for 10,000 rpm.

I am working on a model that considers the load on the motor. Initial testing of the model has predicted performance within 10% of actual, which is pretty good for a fluids problem. I’m trying to find the time to publish it.

If you would like, give me the details of your setup (motor, propeller, battery and esc) And let me know what the new thrust comes out to.

Regards,
Jeremiah