Trim Strategies and Dynamic Cross-Coupling During Aerial Refueling

Abstract

This paper presents a multi-modal analysis framework which is used to examine the aerodynamic interactions between a tanker aircraft and a receiver during AAR for the purposes of predicting receiver dynamic modes and associated flying qualities. The data produced under this framework can then be used to predict handling qualities during the AAR task given appropriate handling qualities data. This work is motivated by the need to rapidly clear tanker–receiver pairs for safe operation in the event of multilateral operation between groups with different refueling platforms. The framework uses a vortex panel method with an actuator disk propulsion model to predict air velocities in the tanker wake. A vortex panel representation of the receiver is then combined with a closed-form aerodynamic model to find the trim points at various locations in the tanker’s wake. The closed-form aerodynamic model is used for speed of analysis and for better results at higher aerodynamic angles. This analysis framework is applied to the case of a C-130 tanker with a F/A-18 receiver. Two different trim strategies are examined: one where the yaw angle of the fighter is zero degrees, and the other where the roll angle of the fighter is zero degrees. The positional stability of the receiver is examined, finding areas of positional stability within the bounds of the tanker’s wingspan. A simplified controller model is combined with 6-DOF, 9-state equations to predict the closed-loop natural modes at the trim points within the tanker wake. The closed-loop eigenvalues of the 9-state system are not predicted to change appreciably during refueling compared to steady-level flight, and the examination of the eigenvectors shows cross-coupling effects. It is theorized that a purely eigenvalue-based analysis will be insufficient to predict handling qualities during AAR using existing decoupled approaches due to this cross-coupling.

Appendix A: Aerodynamic Model

The appendix contains the details of the aerodynamic model used in this paper based on the model published by Chakraborty et al. [23]

$$\begin{aligned} \begin{aligned} C_m =&\big (C_{m\alpha 3}\alpha ^3 + C_{m\alpha 2}\alpha ^2 + C_{m\alpha 1}\alpha \big ) + \big (C_{m\delta _{e}2}\alpha ^2 + C_{m\delta _{e}1}\alpha + C_{m\delta _{e}0} \big )\delta _{\text {elev}} \\&+\frac{\bar{c}}{2V}\big (C_{mq3}\alpha ^3 + C_{mq2}\alpha ^2 + C_{mq1}\alpha + C_{mq0}\big )q. \end{aligned} \end{aligned}$$

(A1)

See Tables 4, 5, 6, 7 and 8.

Table 4 Pitching moment coefficient model data

$$\begin{aligned} \begin{aligned} C_l =&\big (C_{l\beta 4}\alpha ^4 + C_{l\beta 3}\alpha ^3 + C_{l\beta 2}\alpha ^2 + C_{l\beta 1}\alpha + C_{l\beta 0}\big )\beta \\&+ \big (C_{l\delta _{a}3}\alpha ^3 + C_{l\delta _{a}2}\alpha ^2 + C_{l\delta _{a}1}\alpha + C_{l\delta _{a}0} \big )\delta _{\text {ail}} \\&+ \big (C_{l\delta _{r}3}\alpha ^3 + C_{l\delta _{r}2}\alpha ^2 + C_{l\delta _{r}1}\alpha + C_{l\delta _{r}0} \big )\delta _{\text {rud}} \\&+\frac{b}{2V}\big (C_{lp1}\alpha + C_{lp0}\big )p +\frac{b}{2V}\big (C_{lr2}\alpha ^2 + C_{lr1}\alpha + C_{lr0}\big )r. \end{aligned} \end{aligned}$$

(A2)

Table 5 Rolling moment coefficient model data

$$\begin{aligned} \begin{aligned} C_n =&\big (C_{n\beta 2}\alpha ^2 + C_{n\beta 1}\alpha + C_{n\beta 0}\big )\beta \\&+ \big (C_{n\delta _{a}3}\alpha ^3 + C_{n\delta _{a}2}\alpha ^2 + C_{n\delta _{a}1}\alpha + C_{n\delta _{a}0} \big )\delta _{\text {ail}} \\&+ \big (C_{n\delta _{r}4}\alpha ^4 + C_{n\delta _{r}3}\alpha ^3 + C_{n\delta _{r}2}\alpha ^2 + C_{n\delta _{r}1}\alpha + C_{n\delta _{r}0} \big )\delta _{\text {rud}} \\&+\frac{b}{2V}\big (C_{np1}\alpha + C_{np0}\big )p +\frac{b}{2V}\big (C_{nr1}\alpha + C_{nr0}\big )r. \end{aligned} \end{aligned}$$

(A3)

Table 6 Yawing moment coefficient model data

$$\begin{aligned} \begin{aligned} C_Y =&\big (C_{Y\beta 2}\alpha ^2 + C_{Y\beta 1}\alpha + C_{Y\beta 0}\big )\beta \\&+ \big (C_{Y\delta _{a}3}\alpha ^3 + C_{Y\delta _{a}2}\alpha ^2 + C_{Y\delta _{a}1}\alpha + C_{Y\delta _{a}0} \big )\delta _{\text {ail}} \\&+ \big (C_{Y\delta _{r}3}\alpha ^3 + C_{Y\delta _{r}2}\alpha ^2 + C_{Y\delta _{r}1}\alpha + C_{Y\delta _{r}0} \big )\delta _{\text {rud}} \end{aligned} \end{aligned}$$

(A4)

Table 7 Side force coefficient model data

$$\begin{aligned} \begin{aligned} C_D =&\big (C_{D\alpha 4}\alpha ^4 + C_{D\alpha 3}\alpha ^3 + C_{D\alpha 2}\alpha ^2 + C_{D\alpha 1}\alpha + C_{D\alpha 0}\big )\cos \beta + C_{D_{0}}\\&+ \big (C_{D\delta _{e}3}\alpha ^3 + C_{D\delta _{e}2}\alpha ^2 + C_{D\delta _{e}1}\alpha + C_{D\delta _{e}0} \big )\delta _{\text {elev}} \end{aligned} \end{aligned}$$

(A5)

Table 8 Drag coefficient model data