Propeller Modeling

Many thrust and power calculations use nondimensional parameters; these are defined by Equations 68-70.

   

$$\equiv \frac{T}{\rho N^2 D^4}$$ C_T (68)

$$\equiv \frac{P}{\rho N^3 D^5}$$ C_P (69)

$$\equiv \frac{V}{N D}$$ J (70)

Propeller efficiency charts, which are useful in performance analysis of steady, level flight, are not sufficient for flight simulation. The reason is that, in performance calculations, thrust is considered equal to drag. This is not true in general for a flight simulator.

Therefore, to model the propeller, one needs to determine the thrust produced. The thrust produced by the propeller is a function of the forward airspeed, the air density, the blade geometry (including the variable pitch setting), and the propeller's angular speed. The thrust can be calculated with a theory such as blade-element theory or vortex theory. One can also use empirical data (such as the chart on Page 306 of Reference 7) to determine the thrust.

Once the thrust is determined, the power consumed by the propeller can be found using the propeller efficiency charts in tabular form, by solving the following relation for C_P:

 

$$\eta_p = C_TJ/C_P$$ (71)

Or, the power consumed can be calculated directly, much as the thrust was, as a function of airspeed, density, blade geometry, engine speed, and thrust.

Note that the power consumed by the propeller does not necessarily equal the power delivered to it by the engine. Any excess power delivered through the shaft increases the propeller's angular speed. Similarly, if the propeller does not receive enough power from the engine, its angular speed decreases.

Newton's Second Law for angular momentum, applied to the propeller, is $Q_p = I_p \dot N$. The excess power is related to the propeller torque by the relation P_e = Q_p N. Combining the two equations yields the differential equation for engine speed:

$$\dot N = \frac{P_e}{I_p N}$$ (72)

Using this equation, the simulator calculates N each time step by numerical integration.

Assuming the shaft axis is parallel to the body x-axis, and lies in the x-z-plane, the propulsive force and moment components are:

 

X_P = T (73)

$$\pm Q_p$$ L_P = (74)

M_P = T(z^B_c - z_p) (75)

The sign in front of Q_p in Equation 74 depends on which way the propeller spins.