The wing-tail configuration has a wing aspect ratio AR=10, wing area Sref=0.3m2, wing chord cref=0.1732m. Both the wing and the tail have a taper ratio of 1. The distance between the quarter-chord points of the wing and the tail is l=0.56m. A body-fixed coordinate system is placed at the wing quarter-chord point. The reference speed is ∣Vref∣=25m/s and the reference density, pressure, and viscosity are evaluated at an altitude of h=0m (ISA). An unstructured mesh with 6354 panels is used. The image below shows the pressure coefficient of the wing-tail configuration at an angle of attack α=5deg.
where qˉ=2Vrefqcref is the normalised pitch rate. The CX, CZ, and Cm are functions of the the angle of attack α, the normalised pitch rate qˉ and the elevator deflection angle δe. The aim is to obtain the CX0, CZ0, and Cm0 coefficients and the CX,α, CX,qˉ , CX,δe, CZ,α, CZ,qˉ, CZ,δe, Cm,α, Cm,qˉ, and Cm,δe derivatives using APM.
A simulation of the wing-tail configuration is performed at α=0 rad, p=0 rad/s, δe=0 rad. The values for the aerodynamic coefficients are recorded. They are the values of the 0- coefficients.
As already discussed, a simulation of the wing-tail configuration is performed at α=0rad, p=0 rad/s, δe=0rad. The values for the aerodynamic coefficients are recorded. A second simulation of the wing-tail configuration is performed at α=0.1 rad, p=0rad/s, δe=0 rad. The choice of value for α is arbitrary. Again the aerodynamic coefficients are recorded. To obtain the α- derivatives a simple difference is used:
CZ,α=ΔαCZ∣α=0.1−CZ∣α=0,
where Δα=0.1rad. The same method is used to obtain the δe- derivatives.
The coordinate system used for the analysis above is fixed to the wing quarter-chord point however, this point does not coincide with the centre of gravity. If one determines the qˉ- derivatives with respect to this coordinate system they will not be representative of the qˉ- derivatives with respect to the centre of gravity. To determine the position of the centre of gravity (often driven by a required static margin), the neutral point position must be found. By definition, the stick-fixed neutral point position is where:
Cm,α=0.
Using the difference method described above, the value of Cm,α is obtained for two different locations of the centre of gravity. The first location is when the centre of gravity coincides with the quarter-chord point of the wing xcg=0m and the second location is selected forward of the quarter-chord point of the wing xcg=0.05m. The choice of value for xcg is arbitrary. The equation for Cm,α as a function of xcg is then:
The solution of the equation gives xcg=−0.0645, for Cm,α=0. This is the location of the stick-fixed neutral point, xnp, (xnp=xcg for Cm,α=0). The centre of gravity location for a 15% static margin is then given by:
with xcg=−0.03852m. The qˉ- derivatives can now be obtained with respect to this point. A forced sinusoidal motion is imposed on the wing-tail combination:
α=Asin(ωt)ω=2πf,
where A=5deg is the amplitude of the motion and f=5Hz is the frequency of the motion. The video below shows the development of the wing-tail configuration wake during the forced sinusoidal motion.
The values of Cm with respect to α make a quasi-steady elliptical hysteresis. Two points exist at which the angular velocity is maximum or minimum. Cm,qˉ derivative is estimated from the values of the Cm at these points
where k is the reduced frequency. The same method is used to obtain the other qˉ- derivatives. Below are the wing-tail configuration derivatives estimated by the method.