Free Introductory UAV Guide

Receive your free introductory guide on UAV/Small Aircraft Design.

Thrusting Forward

Issue 4, 8 August 2022

By: Anthony O. Ives

\[\ \]

Thrust is important in a fixed wing aircraft to overcome drag and move the aircraft forward and generate airflow over its wings. In the case of a rotary wing aircraft or helicopter the thrust provides both forces to overcome drag in forward flight and produce the lifting force to keep the aircraft airborne. Rotors and propellers use the same principle as wings to produce lift.

Propellers and rotors must constantly rotate to generate airflow around their blades. The blades are essentially wings. Unlike airplanes this is the reason why helicopters can hover and do not need to move forward to produce lift and stay airborne.

Propeller or rotor thrust is calculated using a equation which could be derived from the lift coefficient equation, see Ref [1]. The velocity, V_Tip in this case is based on propeller or rotor tip speed:

V_Tip=(R ω)²

ω is the rotation speed of the propeller or rotor. R is the radius of the propeller or rotor. The area, A_Rotor in the case of thrust is based on the area swept out by the rotor:

A_Rotor=(π R²)

Therefore thrust can calculated using the equation below:

T=ρ (π R²) (R ω)² C_T

Hopefully you can see how this equation is related to the lift coefficient equation. A future article will explain in more detail circle geometry and rotation speeds, etc. ρ, π are density and the mathematic constant number, 3.14.... as defined in previous articles see Ref [1] and [2]. C_T is thrust coefficient it can be estimated using the following equation:

\[C_T=C_L \frac{\sigma}{6}\]

The purpose of calculating thrust is to estimate your required rotor or propeller radius, diameter. This can be done by rearranging the thrust coefficient equation in the following form:

\[ R= \left( \frac{T}{\rho \pi \omega^2 C_T} \right) ^\frac14 \]

The above equation requires a quartic root to be determined which will require a scientific calculator or maybe a computer program like microsoft excel. A quartic root is similar to a square root, see below for an example of each and the difference between them.

√49=49^1/2=7, therefore 7x7=49

∜2401=2401^1/4=7, therefore 7x7x7x7=2401

Diameter is related to radius by D=2R. T is the thrust which for VTOL (Vertical Take Off Landing) aircraft or helicopter would be slightly more that the aircraft's weight depending on vertical climb rate you want to achieve. For cruise the thrust would be equal to the drag in cruise as defined in a previous article Ref [2].

C_L is lift Coefficient as defined previously in Ref [1] with the exception it is not the main aircraft wing this time. As explained earlier propeller or rotor is a essentially a rotating wing so a related equation can be used as can a similar value of 0.2. Sigma is the solidary ratio which is the area of the blades divided by the total area the propeller or rotors sweep out when it's spinning. Sigma is defined using the following equation:

\[\sigma =\frac{N c}{\pi R} \]

N is the number of blades, c is the average chord or width of the blades, π and R are the mathematical constant 3.14... and propeller or rotor as explained earlier.

The following table gives an example of how to calculate the propeller radius including using some assumed values.

Symbol Property Example Value Units

π Mathematical constant 3.14159265 None

C_L Lift coefficent 0.2 None

σ Solidary ratio 0.06 None

C_T Thrust coefficent 0.2x(0.06/6)=0.002 None

ρ Air density 1.2256 kgm^-1

ω Angular velocity 2×π×(10000/60)=1047.2 rads^-1

D Drag 6.292 N

R Propeller radius ∜[6.292/(1.2256×π×1047.2²×0.002)]=0.165 m

Symbol	Property	Example Value	Units
π	Mathematical constant	3.14159265	None
C_L	Lift coefficent	0.2	None
σ	Solidary ratio	0.06	None
C_T	Thrust coefficent	0.2x(0.06/6)=0.002	None
ρ	Air density	1.2256	kgm^-1
ω	Angular velocity	2×π×(10000/60)=1047.2	rads^-1
D	Drag	6.292	N
R	Propeller radius	∜[6.292/(1.2256×π×1047.2²×0.002)]=0.165	m

We have only touched on propeller and rotor thrust, future articles will go into detail considering things such as inflow. Ref [3] gives a very comprehensive and extensive overview of calculating propeller and helicopter rotor thrust.

Please leave a comment on my facebook page or via email and let me know if you understand the principle of rotor or propeller thrust and how it's related to lift.

References:

[1] http://www.eiteog.com/No1EiteogBlogLiftCL.html

[2] http://www.eiteog.com/No2EiteogBlogDragCD.html

[3] Helicopter Theory, Wayne Johnson, 1980, Dover Publications

Disclaimer: Eiteog makes every effort to provide information which is as accurate as possible. Eiteog will not be responsible for any liability, loss or risk incurred as a result of the use and application of information on its website or in its products. None of the information on Eiteog's website or in its products supersedes any information contained in documents or procedures issued by relevant aviation authorities, manufacturers, flight schools or the operators of aircraft, UAVs.