The wing GP model is mostly based off of Mark Drela's wing bending notes from MIT's OpenCourseWare site. Basic assumptions are a constant tapered wing, no sweep, bending loads are taken by the spar, torsional loads are taken by the skin.

The usage is as follows: Wing in wing.py is created either by itself or in an aircraft model. It has two functions that return models to be created separately from Wing, WingAero and WingLoading (also in wing.py). WingAero is an erodynamic model of the JHO airfoil. $C_d$ is to be used in an overall aircraft aerodyanmic model. WingLoading creates loading models that govern the bending loads of the spar and the torsional loads of the skin. Inside Wing 2 objects are created CapSpar or TubeSpar and WingSkin with the option to create WingInterior if hollow=True. Their weights are summed to give an overall weight of the wing. Each submodel is described in detail.

Constrains basic shape of wing using

$$ b^2 = SAR$$ $$\bar{c}(y) \equiv \frac{c(y)}{S/b} = \frac{2}{1+\lambda} \left( 1 + (\lambda - 1) \frac{2y}{b} \right) $$

Calculates the wing drag from induced drag and wing profile drag. Wing profile drag assumes the JHO airfoil and is calculated using a posynomial fit to XFOIL data at various reynolds numbers.

Set up constraints for size of spar. Spar thickness is assumed to be no greater than the max thickness of the airfoil. Width is no greater than %10 of the chord. Moment of inerita is calculated using first order approximation.

Basic assumption is that the distributed load of lift and weight is proportional to the local chord.

$$ q(y) \approx K_q c(y) $$ $$ K_q = \frac{N_{\text{max}}W_{\text{cent}}}{S} $$

This model takes as an input CapSpar and Wcent, or the center weight of the aircaft. Uses discretized beam theory to predict moment and deflection. Stress and overall wing deflection are constrainted.

The main assumption is the the loading is the same as the ChordSparL case but there is an additional gust term that varies as $1-\cos$ along the span. So the distributed load that is used as an input to the discretized beam model is

$$\bar{q}(y) = \frac{q(y)b}{W_{\text{cent}}N_{\text{max}}} &\geq \bar{c}(y) \left[1 + \frac{c_{l_{\alpha}}}{C_L} \alpha_{\text{gust}} (y) \left(1 + \frac{W_{\text{wing}}}{W_{\text{cent}}} \right) \right] $$

where $\alpha_{\text{gust}}$ is

$$ \alpha_{\text{gust}}(y) = \tan^{-1}\left(\frac{V_{\text{gust}}(y)}{V} \right). $$

and $V_{\text{gust}}$ is:

$$V_{\text{gust}}(y) = V_{\text{ref}} \left(1-\cos\left(\frac{2y}{b} \frac{\pi}{2} \right) \right) $$

$\alpha_{\text{gust}}$ is approximated by a monomial fit on the range of gust velocities $V/V_{\text{gust}} \in [0, 0.7]$.

The wing skin calculates the weight of the wing based off of the surface aera of the wing

$$ W_{\text{skin}} \geq 2 \rho_{\text{cfrp}}} t S g. $$

The main assumptions here are that the wing has a constant thickness skin, the torsional loads are taken by the skin, and the highest torison occurs at the root. This model only has one constraint

$$ \tau_{CFRP} \geq \frac{C_{m_w}S\rhoV_{NE}^2}{\bar{J/t} c_{root}^2 t} $$

The $\bar{J/t}$ is taken from the JHO airfoil.

If this model is enabled, the wing is assummed to be filled with foam. The cross sectional aera of the wing is assumed to be the JHO airfoil.