DYNAMICS & CONTROL PDR 1 TEAM 4 Jared Hutter, Andrew Faust, Matt Bagg, Tony Bradford, Arun Padmanabhan, Gerald Lo, Kelvin Seah October 30, 2003

OVERVIEW Concept Review Recap of Class 1 Sizing Applicability of Class 2 Sizing Effects of Pod Location on Static Margin Stability Check Linearized Aircraft Response Follow-Up Actions

CONCEPT REVIEW Empennage High Wing Twin Booms Avionics Pod Twin Engine Horizontal and Vertical Tails sized using modified Class 1 Approach (per QDR 1) High Wing S = 47.8 ft2 b = 15.5 ft, c = 3.1 ft AR = 5 Twin Booms 3 ft apart; 7.3 ft from Wing MAC to HT MAC Avionics Pod 20 lb; can be positioned front or aft depending on requirements Twin Engine 1.8 HP each

CLASS 1 TAIL SIZING RESULTS HORIZONTAL TAIL VERTICAL TAIL = 14.14 ft2 1.26 ft 2.10 ft ½ = 4.04 ft2 1.80 ft 1.86 ft 3.0 ft 1.86 ft 6.727 ft 2.10 ft Volume Coefficients: = 0.70 = 0.08 Chord-wise Span-wise Elevator 1.05 ft 2.67 ft Rudder 0.35 cVT 1.62 ft

APPLICABILITY OF CLASS 2 SIZING For Horizontal Tail, Find aircraft aerodynamic center and aft-most C.G. location as functions of horizontal tail area. Select the horizontal tail area based on a desired static margin. For Vertical Tail, Find weathercock stability derivative as a function of vertical tail area. Select vertical tail area based on a desired value. Desired S.M. Desired C n

APPLICABILITY OF CLASS 2 SIZING Vertical Tail has been sized to a reasonable value of 0.113 rad-1 through a non-graphical approach (modified Class 1 Sizing). The rudders were verified to meet FAR 23, 25 requirements for One-Engine-Inoperative (OEI) flight conditions. The avionics pod was designed to have a variable x-location. This affects the aircraft C.G. and its static margin. As such, the operator can position the pod based on the desired static margin. A study was conducted to find the correlation between the pod location and the static margin achieved.

C.G. LOCATION ESTIMATION from the last Structures & Weights PDR … Aircraft C.G. location: x Wing W = 12.0 lb x = 1.55 ft Avionics Pod W = 20 lb x variable Tail Gear W = 0.5 lb x = 8.00 ft Main Gear W = 3.0 lb x = 0 ft Tail Booms W = 5.9 lb x = 4.05 ft Engines, Fuel & Casings W = 10.7 lb x = -0.30 ft Tail Section W = 2.2 lb x = 8.23 ft

AIRCRAFT AERODYNAMIC CENTER The following equation was used: ref. “Airplane Flight Dynamics and Automatic Flight Controls” (Roskam) Equation 3.38 where = 0.25 = 2.62 = 6.19 rad-1 = 6.01 rad-1 = 0.45 = 14.14 ft2 = 47.81 ft2 = 0.90 ref. “Airplane Design, Volume VI” (Roskam) Equation 8.45 ref. Raymer, p. 486 Typical Range given as 0.85 ~ 0.95. Typical Value is 0.90.

AIRCRAFT AERODYNAMIC CENTER was calculated to be 0.56 (non-dimensional). Static margin: Non-dimensional, expressed as a percentage of M.A.C. Static Margin is a function of payload C.G. location. Sensitivity study was conducted to examine the effect of the payload C.G. location on static margin.

SENSITIVITY STUDY Nominal Design Point where SM = 15% MAC Payload of 20 lb, with its C.G. @ x = +0.33 ft

STABILITY CHECK Symbol Computed Typical 1. -0.537  15% SM Description of Symbols Used: Variation of aircraft pitching moment coefficient with angle of attack. Variation of aircraft yawing moment coefficient with sideslip angle (Weathercock Stability). Variation of aircraft rolling moment coefficient with sideslip angle (Dihedral Effect). Variation of aircraft pitching moment coefficient with elevator deflection angle. Variation of aircraft yawing moment with rudder deflection angle. Variation of aircraft rolling moment with aileron deflection angle. All values are in units of rad-1. Symbol Computed Typical 1. -0.537  15% SM 2. 0.113 0.06 ~ 0.20 3. -0.111 -0.09 ~ -0.30 4. -1.56 -1 ~ -2 5. -0.144 -0.06 ~ -0.12 6. 0.184 0.05 ~ 0.20

LINEARIZED SYSTEM RESPONSE The 6-DOF motion was simulated in MATLAB. Aircraft was trimmed at loiter conditions, and linearized the system about trim. Obtained a state-space realization. Obtained the poles for the Lateral-Directional and Longitudinal subsystems. Compared the time responses of the linear and non-linear models to small perturbations to verify the accuracy of linearization.

LINEARIZED SYSTEM RESPONSE Lateral-Directional Subsystem Longitudinal Subsystem Mode Poles Natural Frequency (rad/sec) Damping Ratio Dutch Roll -0.207 ± 0.538i 0.359 0.576 Roll -1.077 1 Spiral 0.0272 Side Velocity Mode Poles Natural Frequency (rad/sec) Damping Ratio Phugoid -0.155 ± 0.426i 0.454 0.341 Short Period -2.910 ± 0.618i 2.970 0.978 Altitude ≈ 0 1

FOLLOW-UP ACTIONS Verification of desired static margin with Mark Peters’ thesis. Attempt to improve on the linearization. Comparison of modal parameters (damping ratios and natural frequencies) to FAR 23, 25 or MIL-F-8785C requirements. Further analysis on dihedral effect. Further analysis on aircraft performance.

QUESTIONS?

APPENDIX

RUDDER DEFLECTION IN OEI CONDITIONS Roskam (AAE 421 Textbook) Required rudder deflection: DRnO: = 28 ft/s Deflection Limit: = 25° FAR 23, 25 requires that In this case, = 32.86 ft/s for  = 0° 1.2

RUDDER DEFLECTION IN OEI CONDITIONS ref. “Airplane Flight Dynamics and Automatic Flight Controls” (Roskam) Section 4.2.6 [rad] where @ 2,000 ft  [slug/ft3] V [ft/sec] P [hp] yT [ft] for fixed pitch

IDENTIFICATION OF POLES Lateral-Directional Subsystem Dutch Roll Mode Only pole in this subsystem with both e and m parts. Roll Mode Pole is negative; relatively large magnitude. Spiral Mode Positive, thus instable. Small magnitude, so not a problem. Side Velocity Mode Only pole in this subsystem that has zero magnitude.

IDENTIFICATION OF POLES Longitudinal Subsystem Phugoid Mode Conjugate pair with both e and m parts. The m parts have a smaller magnitude than that of the other conjugate pair, indicating longer period (lower frequency). e parts are of small value, light damping. Short Period Mode Conjugate pair with both e and m parts. The m parts have a larger magnitude than that of the other conjugate pair, indicating shorter period (higher frequency). e parts are of larger value, heavier damping. Altitude Pole Only pole in this subsystem that is purely real.