The force, or static thrust, generated by a propeller is described by the following equation where \(k_F\) is a constant and \(\omega\) is the angular velocity of the motor.
$$F_i=k_F\omega^2_i$$
The moment equation is of a similar form:
$$M_i=k_M\omega_i^2$$
The constants depend on the number of blades the propeller has, the propeller diameter, the propeller pitch, the material, and air viscosity. For the dynamic thrust, an additional term is added that includes the axial inflow speed \(V\). This new equation adds another constant and becomes
$$F=k_1\omega^2+k_2V\omega$$
Which can be represented by the differential equation
$$\frac{\partial F}{\partial\omega}=2k_1\omega+k_2V$$
With the values of the correction, constants interpolated from experimental data. If experimental data is not available, then the constant \(k_2\) can be determined using an analytical approach such as blade-element theory or the following equation
$$k_2=p_{11}D^2+p_{10}D+p_{20}S+p_0$$
Where the values \(p_n\) are available in the APC or UIUC propeller databases, \(D\) is the propeller diameter, and $S$ is the pitch of the propeller. \(k_1\) can be estimated from the static motor model.
[[Angular Velocity]]
[[Slip-Stream Velocity]] – used to calculate the velocity in the wake of a propeller.
[[Blade-Element Method]] – also used to calculate the force on a propeller
[[Glauert’s R&M 111] – older inflow model
- prgumdteachingClassQuadrotorDynamics2019
- binzActuatorModellingAttitude2020
Backlinks:
[[Electric Aircraft Powertrain Model]]
[[Propeller]]
[[Quadcopter Newton-Euler Equations]]
[[Streamtube Analysis of the Actuator Disk Model]]