## 4 Discussion  
A theoretical development based on lifting-line theory is presented which gives three methods for comparing the ability of continuous and discrete control surfaces to minimize induced drag. These include the twist effectiveness \(\varepsilon_{T}\), the optimal induced-drag factor \(\kappa_{D\omega}\), and a percent change in induced drag \(\Delta\big{(}C_{D_{j}}\big{)}_{\text{opt}}\). A numerical process is also presented which is capable of calculating these parameters for any given planform shape and control mechanism. This methodology was used to study discrete and continuous twist distributions on wings with evenly spaced actuators. Wings without twist experience an induced-drag planform penalty when the planform is not elliptic. This induced-drag planform penalty is shown in Figure 2. The twist effectiveness \(\varepsilon_{T}\) is a measure of how well the wing's control mechanism is able to negate the induced-drag planform penalty for any given flight condition and is shown in Figure 3 for discrete and continuous control surfaces with varying number of actuators per semispan. These plots show that the continuous control surfaces generally perform better than discrete control surfaces. These results also show that, in general, control mechanisms are less effective at minimizing induced drag in the taper ratio range of 0.3-0.4. This is insightful, since this is near the taper ratio range that gives the smallest induced-drag planform penalty. Therefore, both the wing planform and control-mechanism design should be taken into  
Figure 5: Percent change in induced drag of using a continuous instead of a discrete control mechanism with 2 (**a**), 3 (**b**), 4 (**c**), or 5 (**d**) actuators, shown as a function of taper ratio for aspect ratios ranging from 4 to 20 in increments of 2.  
account when designing a wing that uses the control mechanism to actively reduce induced drag. Figure 3 also shows that as the number of actuators within the control mechanism increase, twist effectiveness improves. This is because higher degrees of freedom within the control mechanism, allow it to better match the optimal twist distribution given in Equations (15) and (16). The optimal induced-drag factor \(\kappa_{Do}\) combines the effects of planform and control-mechanism design and is given in Equation (20). This allows for both the induced-drag planform penalty and twist effectiveness to be considered when designing a wing. Figure 4 shows the optimal induced-drag factor which is the optimal induced drag possible for a given wing planform and control-mechanism design relative to the absolute minimum induced drag for the same aspect ratio. When viewed from this perspective, the continuous control surfaces again noticeably outperform discrete control surfaces. These results also show that the desired taper ratio is highly dependent on the control mechanism design, and that discrete control surfaces are most effective with taper ratios in the range of 0.1-0.2, while continuous control surfaces are most effective at either relatively high or low taper ratios. The percent difference in induced drag between using a continuous and discrete control surface \(\Delta\big{(}C_{D_{i}}\big{)}_{\text{opt}}\) can be helpful to determine whether there is sufficient aerodynamic benefit to offset the likely increase in manufacturing complexity and weight for using the continuous control surface. This percent difference is given in Equation (22), and results are shown in Figure 5. From the best case of the results presented here, wings with a continuous control surface can have an induced-drag reduction of just over 5% compared to an equivalent wing with discrete control surfaces. The difference in drag between the two control surfaces diminishes with increasing number of actuators with less than 3% benefit for the three-actuator cases, less than 2% benefit for the four-actuator cases, and less than about 1% for the five-actuator cases.