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

2. 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

3. 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

4. CLASS 1 TAIL SIZING RESULTS
HORIZONTAL TAIL
VERTICAL TAIL
= 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 ft ft
Rudder cVT ft

5. 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

6. APPLICABILITY OF CLASS 2 SIZING
Vertical Tail has been sized to a reasonable value of 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.

7. 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 = ft
Tail Section
W = 2.2 lb
x = 8.23 ft

8. 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
= ft2
= 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.

9. 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.

10. SENSITIVITY STUDY Nominal Design Point where SM = 15% MAC
Payload of 20 lb,
with its x = ft

11. 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
 15% SM
~ 0.20 ~
~ -2 ~
~ 0.20

12. 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.

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

17. 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.

18. QUESTIONS?

19. APPENDIX

20. 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, = ft/s
for  = 0°
1.2

21. 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

22. 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.

23. 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.