AE4600-2022-Final-Exam (2)

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Western Michigan University *

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4600

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Mechanical Engineering

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Apr 3, 2024

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AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 1/10 AE 4600 Final Exam: Closed Book & Closed Notes (31/100) 1. (4pts) Read the question and answer (circle) True (T) or False (F). (1) (T, F) If a body disturbed from equilibrium remains in the disturbed position, it is statically unstable. (2) (T, F) With CG aft, an aircraft is less maneuverable and more stable. (3) (T, F) Aircraft CG position is only of a secondary importance when discussing longitudinal statistic stability. (4) (T, F) Elevator effectiveness and elevator power are the same thing. (5) (T, F) The neutral point of the entire aircraft is analogous to the aerodynamic center of the wing by itself. (6) (T, F) A positive E d is elevator deflection causing a nose-up pitching moment. (7) (T, F) In a “second order differential equation (spring-mass-damper) terms,” M C a is analogous to damping. (8) (T, F) L C  is increased if the area of conventional vertical stabilizer ( F S ) is decreased. 2. (4pts) Identify (a single check mark) the stability derivatives that best described by the following brief statements: (1) Wing Dihedral ______ L C  ______ N C  ______ L C  ______ r N C (2) Weathercock Stability ______ L C  ______ N C  ______ L C  ______ r N C (3) Pitch Stiffness ______ L C  ______ M C  ______ q L C ______ q M C (4) Adverse Yaw ______ R N C  ______ A N C  ______ L A C  ______ L R C  (5) Pitch Damping ______ L C  ______ M C  ______ q L C ______ q M C (6) Cross-coupling derivative ______ L p C ______ L r C ______ L A C  _____ R N C  (7) Roll Stiffness ______ L p C ______ L r C ______ C  L _____ R N C  (8) Yaw Damping ______ L p C ______ L r C ______ r N C _____ R N C  3. (1pt) How many degrees of freedom (DOF) does the aircraft longitudinal motion have? a. 1-DOF b. 3-DOF c. 4-DOF d. 6-DOF

AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 2/10 4. (1pt) Which of the following term is NOT a part of the horizontal tail volume coefficient? a. main wing area b. horizontal tail area c. horizontal tail moment arm d. horizontal tail lift curve slope 5. (1pt) The horizontal tail is subject to the same angle of attack as the wing-body except for a. tail incidence angle b. up-wash and tail incidence angle c. down-wash and tail incidence angle d. nothing – it is subject to the same angle of attack 6. (1pt) Increasing the tail volume would move the neutral point a. farther forward b. farther aft c. need more information to tell d. would not change location 7. (1pt) Increasing elevator control power a. decreases the amount of elevator required to trim b. increase the amount of elevator required to trim c. has no effect on elevator required to trim d. would only affect the stability level of the aircraft 8. (1pt) Given the graph on the left for a conventional aircraft, which of the following describes the relationship between Line 1 and Line 2 ? a. line 1 is for a more positive elevator deflection than line 2 b. line 2 is for more positive elevator deflection than line 1 c. the center of gravity is farther aft for line 2 than line 1 d. the center of gravity is farther aft for line 1 than line 2 9. (1pt) If an aircraft has positive longitudinal stability at a given center of gravity location, which of the following is true? a. 0 n K < b. 0 E L d dC d = c. 0 M C a < d. 0 E M C d < 10. (1pt) Which of the following represent a non-dimensional form of pitch damping stability derivative ? a. M Mq dC C dq = b. c. d. (a, b & c) a M C line 1 line 2

AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 3/10 11. (1pt) To obtain aircraft lateral-directional stability and control derivatives, the characteristic length used to nondimensionalize the aerodynamic forces and moments is a. the mean aerodynamic chord b. the wingspan c. the body length d. the vertical fin area 12. (1pt) The conditions for Lateral/Directional Stability are: a. 0 0  M C & 0   M C b. 0 C   L & 0 N C   c. 0 0  M C & 0   M C d. 0 C   L & 0 N C   13. (1pt) The degrees of freedom (DOF) of a mechanical system refers to a. the number of control variables to control the system motion b. the number of output variables to control the system motion c. the number of independent variables to describe the system motion d. the number of transfer functions to describe the system motion 14. (1pt) Which of the following provides the largest contribution to N C  ? a. wing b. horizontal tail c. vertical tail d. fuselage 15. (1pt) Consider a pure pitching motion of aircraft (imagine an airplane in the wind tunnel restrained to rotate in y-axis which passes through the CG). What determines the natural frequency of the motion? a. the CG location b. the value of moment coefficient at the zero angle of attack c. the aircraft trim lift coefficient d. all of above 16. (1pt) The most critical condition for pitch control is; the elevators must be able to a. rotate the airplane for takeoff b. flare for airplane for landing c. overcome a forward CG location d. all of above

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AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 4/10 17. (3pts) Looking at the location of 9 poles in s-plane shown, match associated time function shown below. Pole No. ( ) Pole No. ( ) Pole No. ( ) Pole No. ( ) Pole No. ( ) Pole No. ( ) Pole No. ( ) Pole No. ( ) Pole No. ( )

AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 5/10 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 0.22 0.42 0.6 0.74 0.84 0.91 0.96 0.99 0.22 0.42 0.6 0.74 0.84 0.91 0.96 0.99 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Pitch Angle to Elevator Impulse Real Axis Imaginary Axis S1 S1 S2 S2 Z2 Z2 The following five figures show the Boeing 747 longitudinal dynamic transfer function and its responses to the elevator impulse input at a steady level flight condition of h CG = 0.25MAC, Mach = 0.9 and Altitude = 40,000FT. Answer to the following questions.

AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 6/10 18. (1pt) Which of the following polynomial represents the denominator of the above transfer function? a. 0.016148* (s+13.68) (s+0.9194) (s-0.0195) b. (s+1.066) (s+0.04596) (s^2 + 0.05273s + 0.4638) c. -1.2117 *(s+0.3666) (s+0.008038) d. (s^2 + 0.003304s + 0.001766) (s^2 + 0.9384s + 1.779) 19. (1pt) Which of the followings is TRUE for the short-period motion? a. It is represented by the complex conjugate poles identified as S2 b. Its period is about 150 seconds. c. Its frequency is about 1.336 radian/sec d. It is unstable 20. (1pt) Which of the followings is TRUE for the phugoid motion? a. It is represented by the zeros identified as Z1 and Z2 b. Its frequency is about 1.336 radian/sec c. It is over damped d. None of above 21. (1pt) What are the dominant aircraft motion variables in the short-period motion? a. flight path angle & speed b. pitch angle & angle of attack c. flight path angle & angle of attack d. speed and angle of attack 22. (1pt) What are the dominant aircraft motion variables in the phugoid motion? a. flight path angle & speed b. pitch angle & angle of attack c. flight path angle & angle of attack d. speed & angle of attack 23. (1pt) If the aircraft CG is moved AFT but still ahead of the stick-fixed neutral point, the change in S1 is such that a. its real part become positive b. its imaginary part increases c. it is not changing d. all of above

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AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 7/10 AE 4600 Final Exam: Open Book & Open Notes (69/100) 1. (20pts) Consider a canard configured light general aviation aircraft as shown in the figure below. Both the wing and the canard of this aircraft have a rectangular planform. The aircraft has a mass of 1600 kg and is designed to fly as slow as 30 m/s at sea level ( 1.225 ssl r = kg/m^3) in level flight. At this speed, its cambered main wing generates ‒ 900N-m of pitching moment about its aerodynamic center. If the maximum lift coefficient for its canard is 1.5, how large (area) must the canard be to trim the aircraft at its minimum speed?

AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 8/10 2. (20 pts) A small general aviation aircraft has the following characteristics: ======================================================================= Geometric Data  Wing Area : 128.0 ft^2  Tail Area : 26.0 ft^2  Wing MAC : 5.0 ft  Distance from Wing-Body AC to Tail AC : 18.0 ft  Distance from Stick Fixed Neutral Point to Tail AC : 15.5 ft  Wing-body Aerodynamic Center Location : 30% MAC ======================================================================= Aerodynamic Data (No Propulsion Effect)  Wing-body Moment Coefficient about Wing-body AC : 0.0000  Total Aircraft Lift Curve Slope : 7.4459/rad  Total Aircraft Lift Curve Slope with Free Elevator : 7.0745/rad  Wing-body Lift Curve Slope : 6.4100/rad  Tail Lift Curve Slope : 6.0000/rad  Downwash Constant : 0.0000 deg  Downwash Derivative : 0.1500  Tail Incidence Angle : 4.0 deg (0.0698 rad) ======================================================================= Control Surface Data  Elevator Chord : 1.0 ft  Elevator Area : 6.0 ft^2  Elevator Effectiveness : 3.2/rad  Change in Hinge Moment Coefficient with Tail AOA : -0.40/rad  Change in Hinge Moment Coefficient with Elevator Angle : -0.70/rad  Change in Hinge Moment Coefficient with Tab Angle : -0.50/rad  Elevator Gearing Ratio : 1 rad/ft ======================================================================= Steady Level Trimmed Flight Condition  Weight : 2000 lbs  Stick Fixed Static Margin : 0.3500  Stick Free Static Margin : 0.3219  Design Altitude : Sea Level (0.00238 slug/ft^3)  Trim Speed : 120 ft/sec ======================================================================= (a) (10 pts) Find the elevator angle at the trim condition. (b) (10 pts) Find the tab angle for freeing the elevator hinge moment at this trim condition.

AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 9/10 3. (20pts) Consider a spring-mass-damper system shown below, where the input ( ) U t is displacement input applied to the mass 1 m , and 1 2 ( ), ( ) X t X t are the displacement of each mass, respectively. (Note that the input is displacement, NOT force) (a) (10pts) Derive the governing equations of motion of the system. (b) (10pts) Given that the system parameter values of m 1 = m 2 = 1, c 1 = 1, k 1 =100, k 2 =1, find the natural frequencies and damping ratios of the system. (c) (Bonus 10pts) Find the transfer function ( ) 2 ( ) X s U s , and describe the time response ( ) 2 t x when an impulsive displacement input is applied to the input U.

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AE 4600 Final Exam, Tuesday, 12/13/2022 (8:00AM – 10:00AM) Name:__________________________ WMU-MAE pg. 10/10 4. (9pts) Given below is the stability and control S&C data for A-4D Skyhawk fighter aircraft. Find the followings: (a) (3pts) The natural frequency of the pure (1-DOF) yawing motion. (b) (3pts) The damping ratio of the pure (1-DOF) yawing motion. (c) (3pts) The time constant of the pure (1-DOF) rolling motion. Flight Condition : M=0.4 at Sea Level, W = 17,578 lbs, h cg = 25%MAC Reference Geometry : S ref = 260 ft 2 , b = 27.5 ft, MAC = 10.8 ft Inertia Data : I xx = 8,090 slug- ft 2 , I yy = 25,900 slug- ft 2 , I zz = 29,200 slug- ft 2 , I xz = 1,300 slug- ft 2 Longitudinal Derivatives Lateral/Directional Derivatives L C 0.28 L C   0.72 Y C   0.98 r N C  0.35 D C 0.03 M C    1.1 l C   0.12 a l C  0.08 L C  3.45 q L C 0.0 N C  0.25 a N C  0.06 D C  0.30 q M C  3.6 p l C  0.26 Y r C  0.17 M C   0.38 e L C  0.36 p N C 0.022 r N C  0.032 e M C   0.50 r l C 0.14 r l C   0.105