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  • Obtaining the 0-coefficients
  • Obtaining the angle of attack and elevator deflection derivatives
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Wing-tail configuration

Example simulation of a fixed-wing configuration

PreviousAGARD-AR-303 E6NextFixed-wing UAV

Last updated 1 month ago

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The wing-tail configuration has a wing aspect ratio AR=10AR=10AR=10, wing area Sref=0.3S_{ref}=0.3Sref​=0.3m2^22, wing chord cref=0.1732c_{ref}=0.1732cref​=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=0.56l=0.56l=0.56m. A body-fixed coordinate system is placed at the wing quarter-chord point. The reference speed is ∣Vref∣=25|\mathbf{V}_{ref}|=25∣Vref​∣=25m/s and the reference density, pressure, and viscosity are evaluated at an altitude of h=0h=0h=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=5\alpha=5α=5deg.

The lift, drag and pitching moment coefficients of the wing-tail configuration are assumed to be given by:

CX=CX0+CX,αα+CX,qˉqˉ+CX,δeδeCZ=CZ0+CZ,αα+CZ,qˉqˉ+CZ,δeδeCm=Cm0+Cm,αα+Cm,qˉqˉ+Cm,δeδe,C_{X}=C_{X0}+C_{X,\alpha}\alpha+C_{X,\bar{q}}\bar{q}+C_{X,\delta_{e}}\delta_{e}\\ C_{Z}=C_{Z0}+C_{Z,\alpha}\alpha+C_{Z,\bar{q}}\bar{q}+C_{Z,\delta_{e}}\delta_{e}\\C_{m}=C_{m0}+C_{m,\alpha}\alpha+C_{m,\bar{q}}\bar{q}+C_{m,\delta_{e}}\delta_{e} \text{,} CX​=CX0​+CX,α​α+CX,qˉ​​qˉ​+CX,δe​​δe​CZ​=CZ0​+CZ,α​α+CZ,qˉ​​qˉ​+CZ,δe​​δe​Cm​=Cm0​+Cm,α​α+Cm,qˉ​​qˉ​+Cm,δe​​δe​,

Obtaining the 0-coefficients

Obtaining the angle of attack and elevator deflection derivatives

Obtaining the pitch rate derivatives

-0.0114

0.2555

-0.2491

-0.0195

-0.3219

-5.1439

2.7170

-0.8377

0.0556

-0.9575

-20.0581

-2.3205

Files

where qˉ=qcref2Vref\bar{q}=\frac{qc_{ref}}{2V_{ref}}qˉ​=2Vref​qcref​​ is the normalised pitch rate. The CXC_{X}CX​, CZC_{Z}CZ​, and CmC_{m}Cm​ are functions of the the angle of attack α\alphaα, the normalised pitch rate qˉ\bar{q}qˉ​ and the elevator deflection angle δe\delta_{e}δe​. The aim is to obtain the CX0C_{X0}CX0​, CZ0C_{Z0}CZ0​, and Cm0C_{m0}Cm0​ coefficients and the CX,αC_{X,\alpha}CX,α​, CX,qˉC_{X,\bar{q}}CX,qˉ​​ , CX,δeC_{X,\delta_{e}}CX,δe​​, CZ,αC_{Z,\alpha}CZ,α​, CZ,qˉC_{Z,\bar{q}}CZ,qˉ​​, CZ,δeC_{Z,\delta{e}}CZ,δe​, Cm,αC_{m,\alpha}Cm,α​, Cm,qˉC_{m,\bar{q}}Cm,qˉ​​, and Cm,δeC_{m,\delta_{e}}Cm,δe​​ derivatives using APM.

A simulation of the wing-tail configuration is performed at α=0\alpha=0α=0 rad, p=0p=0p=0 rad/s, δe=0\delta_{e}=0δe​=0 rad. The values for the aerodynamic coefficients are recorded. They are the values of the 000- coefficients.

As already discussed, a simulation of the wing-tail configuration is performed at α=0\alpha=0α=0rad, p=0p=0p=0 rad/s, δe=0\delta_{e}=0δe​=0rad. The values for the aerodynamic coefficients are recorded. A second simulation of the wing-tail configuration is performed at α=0.1\alpha=0.1α=0.1 rad, p=0p=0p=0rad/s, δe=0\delta_{e}=0δe​=0 rad. The choice of value for α\alphaα is arbitrary. Again the aerodynamic coefficients are recorded. To obtain the α\alphaα- derivatives a simple difference is used:

CZ,α=CZ∣α=0.1−CZ∣α=0Δα,C_{Z,\alpha}=\frac{C_{Z}\rvert_{\alpha=0.1}-C_{Z}\rvert_{\alpha=0}}{\Delta\alpha}\text{,}CZ,α​=ΔαCZ​∣α=0.1​−CZ​∣α=0​​,

where Δα=0.1\Delta\alpha=0.1Δα=0.1rad. The same method is used to obtain the δe\delta_{e}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\bar{q}qˉ​- derivatives with respect to this coordinate system they will not be representative of the qˉ\bar{q}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.C_{m,\alpha}=0\text{.}Cm,α​=0.

Using the difference method described above, the value of Cm,αC_{m,\alpha}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=0x_{cg}=0xcg​=0m and the second location is selected forward of the quarter-chord point of the wing xcg=0.05x_{cg}=0.05xcg​=0.05m. The choice of value for xcgx_{cg}xcg​ is arbitrary. The equation for Cm,αC_{m,\alpha}Cm,α​ as a function of xcgx_{cg}xcg​ is then:

Cm,α=axcg+b−2.0012=a0+b−3.5526=a0.05+ba=−31.028b=−2.0012. C_{m,\alpha}=ax_{cg}+b\\ -2.0012=a0+b \\ -3.5526=a0.05+b \\ a=-31.028 \\ b=-2.0012 \text{.}Cm,α​=axcg​+b−2.0012=a0+b−3.5526=a0.05+ba=−31.028b=−2.0012.

The solution of the equation gives xcg=−0.0645x_{cg}=-0.0645xcg​=−0.0645, for Cm,α=0C_{m,\alpha}=0Cm,α​=0. This is the location of the stick-fixed neutral point, xnpx_{np}xnp​, (xnpx_{np}xnp​=xcgx_{cg}xcg​ for Cm,α=0C_{m,\alpha}=0Cm,α​=0). The centre of gravity location for a 15% static margin is then given by:

SM=0.15=xcg−xnpcrefxcg∣SM=0.1=xnp+0.15cref,SM=0.15=\frac{x_{cg}-x_{np}}{c_{ref}}\\ x_{cg}\rvert_{SM=0.1}=x_{np}+0.15c_{ref} \text{,}SM=0.15=cref​xcg​−xnp​​xcg​∣SM=0.1​=xnp​+0.15cref​,

with xcg=−0.03852x_{cg}=-0.03852xcg​=−0.03852m. The qˉ\bar{q}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,\alpha=Asin(\omega t) \\ \omega=2\pi f\text{,}α=Asin(ωt)ω=2πf,

where A=5A=5A=5deg is the amplitude of the motion and f=5f=5f=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 CmC_{m}Cm​ with respect to α\alphaα make a quasi-steady elliptical hysteresis. Two points exist at which the angular velocity is maximum or minimum. Cm,qˉC_{m,\bar{q}}Cm,qˉ​​ derivative is estimated from the values of the CmC_{m}Cm​ at these points

Cm,qˉ=Cm∣qmax−Cm∣qmin2kA,k=ωcref2Vref,C_{m,\bar{q}}=\frac{C_{m}\rvert_{q_{max}}-C_{m}\rvert_{q_{min}}}{2kA},\\ k=\frac{\omega c_{ref}}{2V_{ref}}\text{,} Cm,qˉ​​=2kACm​∣qmax​​−Cm​∣qmin​​​,k=2Vref​ωcref​​,

where kkk is the reduced frequency. The same method is used to obtain the other qˉ\bar{q}qˉ​- derivatives. Below are the wing-tail configuration derivatives estimated by the method.

0_{0}0​
∂∂α\frac{\partial}{\partial \alpha}∂α∂​
∂∂qˉ\frac{\partial}{\partial \bar{q}}∂qˉ​∂​
∂∂δe\frac{\partial}{\partial \delta_{e}}∂δe​∂​
CXC_{X}CX​
CZC_{Z}CZ​
CmC_{m}Cm​
Wing-tail configuration wake development
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Wing-tail configuration pressure coefficient