1. Dynamic Modeling PDR 17 October, 2000 Keith R. Hout Patrick Dempsey Bridget Fitzpatrick Heather Garber J.S. Mok

2. Loop Closure Description  Rate feedback in the pitch axis  Objective: To vary the damping ratio of the short period mode  Short Period Mode  Describes airplane pitch response to elevator inputs  Characterized by near constant velocity and oscillations in angle of attack and pitch angle  Tends to be highly damped for for flight conditions such as ours  I.e. ~0.7 < ξ < ~1

3. Loop Closure Description  Block Diagram TX RX Servo Aircraft Pitch Rate Gyro Pilot +/ - ? + Pilot inputs elevator command Servo converts voltage to elevator deflection Sign of feedback gain is chosen to stabilize or destabilize the mode

4. Dynamic Models  Servo Model  2 nd Order Approximation  Frequency and damping ratio are still unknown and are to be obtained from testing

5. Dynamic Models  Pitch Rate Gyro  2 nd Order Approximation  Gyro frequency and damping ratio still needed, are to be obtained from testing

6. Dynamic Models  Aircraft Transfer Function  Short Period Approximation  2 nd Order - Flight speed

7. Dynamic Models  Longitudinal Dimensional Stability Derivatives  SM=0.18 Equation Pitch angular acceleration due to elevator deflection Vertical acceleration due to elevator deflection Pitch angular acceleration due to rate of change of α Pitch angular acceleration due to α Vertical acceleration due to α Pitch angular acceleration due to pitch rate DescriptionUnits [(rad/s 2 )/rad] [(rad/s 2 )/(rad/s)] [(ft/s 2 )/rad] [(rad/s 2 )/rad] [(ft/s 2 )/rad] [(rad/s 2 )/(rad/s)] Value -16.07 -13.61 -1.87 -14.12 -186.82 -5.14

8. Dynamic Models  Longitudinal Non-Dimensional Stability Derivatives  SM=0.18 Equation Δ Pitching moment coefficient due to elevator deflection Δ Lift coefficient due to elevator deflection Δ Pitching moment coefficient with α Δ Pitching moment coefficient due to rate of change of α Δ Pitching moment coefficient due to pitch rate Description Value[rad -1 ] SID5 (MPX5) -0. 81 (-1.15) 0.30 (0.42) -0.71 (-1.13) -3.11 (-3.59) -8.54 (-10.57)

9. Dynamic Models  Damping Ratio and Natural Frequency  SM=0.18  Mode is very highly damped

10. Root Locus

12. Dynamic Models  Tasks Remaining  Complete transfer function models for remaining elements of block diagram  Obtain parameters for the servo and gyro  Construct equivalent transfer function for entire loop and obtain a root locus  Determine gain magnitude and sign to achieve either a higher or lower damping ratio  Determine the exact method the pilot will use to implement the feedback gain  i.e. what channel will be used to switch the gyro gain from off to nominal