# What is the equation for calculating static thrust?

I'm building a quadcopter as my bachelor's degree thesis. I'm having trouble finding a reliable equation for calculating the static thrust for a quadcopter. The equations I have found give me ridiculous results. I know that it's difficult to derive such an equation and I know that there are online calculators (like this). But how do these calculators calculate static thrust? There has to be an approximate equation somewhere. This is what I'm asking. I don't mind the equation(s) that lead me to there, I would just like to know them.

The following snapshot is taken from Mathcad software. It's various equations i have gathered from supposed trustworthy websites. These equations lead me to the calculation of static thrust. I think the equations are pretty self explanatory. Thank you. Components of the quadcopter: Battery, ESC's, Motors, Propellers, frame

Various useful websites that i've found most equations: 1, 2, 3, 4

• Please show what you have - in the end, we will give you the equations which do not work for you. You would also post sample code in a question on SE, right? – Peter Kämpf Aug 28 '15 at 14:55
• Did you check your units? Quadcopter with 3 meter diameter rotors? – DeltaLima Aug 28 '15 at 15:32
• Thanks. It seems i've made some careless mistakes. 3 is in inches and one equation is supposed to be in inches but the other is in meters. I've also made a mistake with the RPM. I will correct them all and let you know soon. – Nikos Aug 28 '15 at 15:43
• No the RPM wasn't in thousands. Forget that. I have uploaded a new picture after applying various fixes. Things now have definitely improved. Still though that 15 kilograms in the end is A LOT. In essence since the quadcopter's weight is 500kg, i've found from another equation that the total thrust generated should be from 2 to 3 times the weight of the quadcopter. So between 1 and 1.5 kilos. So there must be further improvements to be made. I've also added links to the critical components of the quadcopter and various useful websites in which i've found most of these equations. – Nikos Aug 28 '15 at 16:30

By now you should have found this answer on static thrust. It explains how to arrive at this equation for static thrust $T_0$: $$T_0 = \sqrt{P^2\cdot\eta_{Prop}^2\cdot\eta_{el}^2\cdot\pi\cdot \frac{d_P^2}{2}\cdot\rho}$$ Your P is at most 11.1 V × 4.5 A = 49.95 W. Your prop efficiency $\eta_{Prop}$ must be guessed; let's take 60%. The electric efficiency should be higher; let's use 95% for now. The prop diameter $d_P$ is 3 or 5 inches (depends if your comments or the picture in your question is more reliable), so I use 0.127 m for now. Air density is given with 1.18 kg/m³.

When I plug in the numbers, I get 2.894 N. One propeller can lift approximately 295 g of quadcopter, and all four will produce thrust equivalent to the weight of 1.18 kg in Earth's gravitation. Looks about right for a quadcopter of 500 g.

Don't assume that the motor delivers all the power the propeller can absorb. Also, don't assume that the RPM of the isolated motor can be maintained when it is driving a propeller.

• Your answer made me look a lot more into the theory of aviation and try to understand more about all these concepts. I realized i didn't even know the basics. Also this SE website helped me a lot in very little time, in finding answers to my questions. Of course, i'm only getting started. Thank you very, very much! – Nikos Aug 29 '15 at 16:28
• @RestlessC0bra: You best thank me by contributing back, and you did already. I reworked the numbers, and now they should be even more right. – Peter Kämpf Aug 29 '15 at 18:12
• I believe it should be a square root instead of the cube root (the same mistake is in the original answer as well). It's a nice approximation, however, I think $\eta_{Prop}$ is a function of wind speed so you should be sure to use an appropriate value. – Chris Jul 11 '16 at 12:56
• @Chris: I don't think so. Please look at the original answer: The thrust is on the left side and in a square root on the right. First you need to square both sides and then bring thrust over to the left. This gives you an expression for thrust cubed. I know I made mistakes in other answers but try to correct them as much as I can. Please prove me wrong! – Peter Kämpf Jul 13 '16 at 8:12
• @PeterKämpf Yes, you are right I missed the $T_0$ under the square root. Sorry about that. – Chris Jul 13 '16 at 9:16

There are a couple of examples on this site now (here and here for instance) for static propeller thrust calculation, based on existing considerations for helicopter hover. The theory can be found in books like Principles of Helicopter Aerodynamics by J. Gordon Leishman; Helicopter Performance, Stability, and Control by Raymond Prouty. The books start their explanations based on impulse theory for lifting disks, then expand with Blade Element Theory for a comprehensive explanation of rotating thrust blades.

The theory presented in the books is generally for open rotors, in Leishman there is some impulse based discussion on influence of ducting. Also, measurement based data in small ducted rotors in a masters thesis here.

All above is for relatively open rotors, with a low solidity ratio, up to 0.1. If power is not the limiting factor but diameter is, a fan with a solidity ratio of close to 1 as found here can be used, and scaled up or down according to blade area.