## 2 Materials and Methods
### Twist Effectiveness  
For a wing with a given \(c(\theta)\), \(\omega(\theta)\), and \(C_{L}\), an optimal \(\Omega\) can be found to minimize induced drag by taking the partial derivative of Equation (9) with respect to \(\Omega\), setting the right-hand side equal to zero, and solving for the optimal \(\Omega\). Using the optimal \(\Omega\) in Equation (9) gives  
\[\left(C_{D_{i}}\right)_{\text{opt}}=\frac{C_{L}^{2}}{\pi R_{A}}(1+\kappa_{Do}),\ \ \kappa_{Do}=\kappa_{P}-\frac{\kappa_{PL}^{2}}{4\kappa_{P\Omega}} \tag{18}\]  
The parameter \(\kappa_{Do}\) is the _optimum induced-drag factor_ and shows the minimum induced drag possible for a given planform and twist distribution function. It is similar to \(\kappa_{P}\) in that it is the percent increase in drag from the absolute minimum induced drag given in Equation (13) to that of a wing with a given planform, twist distribution function, and an optimum amount of twist. In other words,  
\[\kappa_{Do}=\frac{\left(C_{D_{i}}\right)_{\text{opt}}-\left(C_{D_{i}}\right) _{\text{elliptic}}}{\left(C_{D_{i}}\right)_{\text{elliptic}}} \tag{19}\]  
The term \(\kappa_{Do}\) is a function of both \(c(\theta)\) and \(\omega(\theta)\). If both of these are known, \(\kappa_{Do}\) can be calculated from Equations (10)-(12) and (18). Note that up to this point, no constraint has been applied to the twist distribution function and \(\omega(\theta)\) is completely arbitrary. In order to compare the induced-drag performance of spanwise discrete and continuous control surfaces, the best twist distribution function possible should be used for each case. The best twist distribution function a wing can have is given in Equation (15), but this would require infinitely fine spanwise control. Realistic designs are limited by constraints on the twist distribution function and twist amount such that the twist distribution is always within the capabilities of the control mechanism. Determining an optimal twist distribution function for a given control mechanism can be challenging. Towards this goal, \(\kappa_{Do}\) can be redefined into a more useful form by  
\[\kappa_{Do}\equiv\kappa_{P}(1-\varepsilon_{T}),\ \ \varepsilon_{T}=\frac{\kappa_{ PL}^{2}}{4\kappa_{P}\Omega\kappa_{P}} \tag{20}\]  
where \(\varepsilon_{T}\) is the _twist effectiveness_. The twist effectiveness, \(\varepsilon_{T}\), is a measure of how well the twist distribution corrects for \(\kappa_{P}\). The twist effectiveness can vary from 0.0 to 1.0, with 0.0 meaning that the twist cannot correct for any of the planform penalty, and 1.0 meaning that the twist can completely correct for the planform penalty.  
For example, if we had a wing with a taper ratio of 1 and an aspect ratio of 18, the induced-drag planform penalty would be roughly 0.15, see Figure 2. This means that the wing without any twist will produce roughly 15% more induced drag than the optimum. However, if the twist distribution given in Equations (15) and (16) is used, \(\varepsilon_{T}=1\), and the control mechanism is capable of correcting for 100% of the planform penalty. If instead, the best the control mechanism was able to achieve was a twist effectiveness of 0.5, then the control mechanism would only correct for 50% of the planform penalty and the wing would produce roughly 7.5% more than the optimum, \(\kappa_{Do}=0.075\). The twist effectiveness term by itself is not enough information to determine actual changes in induced drag since it is the measure of how well the twist distribution corrects for the induced-drag planform penalty. Using Equation (20) in Equation (18) allows the induced-drag coefficient to be rewritten as  
\[\big{(}C_{D_{i}}\big{)}_{\text{opt}}=\frac{C_{L}^{2}}{\pi R_{A}}[1+\kappa_{P} (1-\varepsilon_{T})] \tag{21}\]  
This formulation combines all the components of \(\kappa_{Do}\) dependent on twist into one parameter, \(\varepsilon_{T}\), and helps in understanding and comparing the effectiveness of different control mechanisms.