# Wing-tail configuration The wing-tail configuration has a wing aspect ratio $$AR=10$$, wing area $$S\_{ref}=0.3$$m$$^2$$, wing chord $$c\_{ref}=0.1732$$m. 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 $$l=0.56$$m. A body-fixed coordinate system is placed at the wing quarter-chord point. The reference speed is $$|\mathbf{V}\_{ref}|=25$$m/s and the reference density, pressure, and viscosity are evaluated at an altitude of $$h=0$$m (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 $$\alpha=5$$deg. ![Wing-tail configuration pressure coefficient ](/files/-MVfbklO0WszKasUf74a) The lift, drag and pitching moment coefficients of the wing-tail configuration are assumed to be given by: $$ C\_{X}=C\_{X0}+C\_{X,\alpha}\alpha+C\_{X,\bar{q}}\bar{q}+C\_{X,\delta\_{e}}\delta\_{e}\ C\_{Z}=C\_{Z0}+C\_{Z,\alpha}\alpha+C\_{Z,\bar{q}}\bar{q}+C\_{Z,\delta\_{e}}\delta\_{e}\C\_{m}=C\_{m0}+C\_{m,\alpha}\alpha+C\_{m,\bar{q}}\bar{q}+C\_{m,\delta\_{e}}\delta\_{e} \text{,} $$ where $$\bar{q}=\frac{qc\_{ref}}{2V\_{ref}}$$ is the normalised pitch rate. The $$C\_{X}$$, $$C\_{Z}$$, and $$C\_{m}$$ are functions of the the angle of attack $$\alpha$$, the normalised pitch rate $$\bar{q}$$ and the elevator deflection angle $$\delta\_{e}$$. The aim is to obtain the $$C\_{X0}$$, $$C\_{Z0}$$, and $$C\_{m0}$$ coefficients and the $$C\_{X,\alpha}$$, $$C\_{X,\bar{q}}$$ , $$C\_{X,\delta\_{e}}$$, $$C\_{Z,\alpha}$$, $$C\_{Z,\bar{q}}$$, $$C\_{Z,\delta{e}}$$, $$C\_{m,\alpha}$$, $$C\_{m,\bar{q}}$$, and $$C\_{m,\delta\_{e}}$$ derivatives using APM. ## Obtaining the 0-coefficients A simulation of the wing-tail configuration is performed at $$\alpha=0$$ rad, $$p=0$$ rad/s, $$\delta\_{e}=0$$ rad. The values for the aerodynamic coefficients are recorded. They are the values of the $$0$$- coefficients. ## Obtaining the angle of attack and elevator deflection derivatives As already discussed, a simulation of the wing-tail configuration is performed at $$\alpha=0$$rad, $$p=0$$ rad/s, $$\delta\_{e}=0$$rad. The values for the aerodynamic coefficients are recorded. A second simulation of the wing-tail configuration is performed at $$\alpha=0.1$$ rad, $$p=0$$rad/s, $$\delta\_{e}=0$$ rad. The choice of value for $$\alpha$$ is arbitrary. Again the aerodynamic coefficients are recorded. To obtain the $$\alpha$$- derivatives a simple difference is used: $$ C\_{Z,\alpha}=\frac{C\_{Z}\rvert\_{\alpha=0.1}-C\_{Z}\rvert\_{\alpha=0}}{\Delta\alpha}\text{,} $$ where $$\Delta\alpha=0.1$$rad. The same method is used to obtain the $$\delta\_{e}$$- derivatives. ## Obtaining the pitch rate 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 $$\bar{q}$$- derivatives with respect to this coordinate system they will not be representative of the $$\bar{q}$$- 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: $$ C\_{m,\alpha}=0\text{.} $$ Using the difference method described above, the value of $$C\_{m,\alpha}$$ 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 $$x\_{cg}=0$$m and the second location is selected forward of the quarter-chord point of the wing $$x\_{cg}=0.05$$m. The choice of value for $$x\_{cg}$$ is arbitrary. The equation for $$C\_{m,\alpha}$$ as a function of $$x\_{cg}$$ is then: $$ C\_{m,\alpha}=ax\_{cg}+b\\ -2.0012=a0+b \\ -3.5526=a0.05+b \\ a=-31.028 \\ b=-2.0012 \text{.} $$ The solution of the equation gives $$x\_{cg}=-0.0645$$, for $$C\_{m,\alpha}=0$$. This is the location of the stick-fixed neutral point, $$x\_{np}$$, ($$x\_{np}$$=$$x\_{cg}$$ for $$C\_{m,\alpha}=0$$). The centre of gravity location for a 15% static margin is then given by: $$ SM=0.15=\frac{x\_{cg}-x\_{np}}{c\_{ref}}\\ x\_{cg}\rvert\_{SM=0.1}=x\_{np}+0.15c\_{ref} \text{,} $$ with $$x\_{cg}=-0.03852$$m. The $$\bar{q}$$- derivatives can now be obtained with respect to this point. A forced sinusoidal motion is imposed on the wing-tail combination: $$ \alpha=Asin(\omega t) \\ \omega=2\pi f\text{,} $$ where $$A=5$$deg is the amplitude of the motion and $$f=5$$Hz is the frequency of the motion. The video below shows the development of the wing-tail configuration wake during the forced sinusoidal motion. {% embed url="" %} Wing-tail configuration wake development {% endembed %} The values of $$C\_{m}$$ with respect to $$\alpha$$ make a quasi-steady elliptical hysteresis. Two points exist at which the angular velocity is maximum or minimum. $$C\_{m,\bar{q}}$$ derivative is estimated from the values of the $$C\_{m}$$ at these points $$ C\_{m,\bar{q}}=\frac{C\_{m}\rvert\_{q\_{max}}-C\_{m}\rvert\_{q\_{min}}}{2kA},\\ k=\frac{\omega c\_{ref}}{2V\_{ref}}\text{,} $$ where $$k$$ is the reduced frequency. The same method is used to obtain the other $$\bar{q}$$- derivatives. Below are the wing-tail configuration derivatives estimated by the method. | | $$\_{0}$$ | $$\frac{\partial}{\partial \alpha}$$ | $$\frac{\partial}{\partial \bar{q}}$$ | $$\frac{\partial}{\partial \delta\_{e}}$$ | | ---------- | --------- | ------------------------------------ | ------------------------------------- | ----------------------------------------- | | $$C\_{X}$$ | -0.0114 | 0.2555 | -0.2491 | -0.0195 | | $$C\_{Z}$$ | -0.3219 | -5.1439 | 2.7170 | -0.8377 | | $$C\_{m}$$ | 0.0556 | -0.9575 | -20.0581 | -2.3205 | ## Files {% file src="/files/yJfa7QnBRkawvL7jZX84" %} --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://docs.aviumtechnologies.com/wing-tail-configuration.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.