1. Stability Spectral Analysis Based on the Damping Spectral Analysis and the Data from Dryden flight tests, ATW_f5_m83h10-1

2. Location of the Test Wing

3. Details of the test wing

4. Test Video

5. The Full Data : atw_f5_m83h10_1 Details

6. IMF Data x83 : atw_f5_m83h10_1

8. Fourier Spectra of Various Sum of IMFs

9. Hilbert Filtered Data x83 : atw_f5_m83h10_1

10. Hilbert Spectrum : x83

11. Hilbert Spectrum : x83 Details

12. Spectrogram (512) : x83

13. Spectrogram (512) : x83 Details

14. Spectrogram (1024) : y83

15. Spectrogram (1024) : y83 Details

16. Re-sampled Hilbert Filtered Data : y(I)

17. Mean Hilbert Spectrum : y(i)

18. Hilbert Spectrum : x83 Details

19. Marginal Hilbert and Fourier Spectra : y83

20. 3D Mean Hilbert Spectrum : y(i)

21. 3D Spectrogram : y83

22. Instantaneous frequency : y(i)

23. Instantaneous frequency : y(i) Details

26. Mean Hilbert and Spectrogram : y83

27. Mean Hilbert and Spectrogram : y83 Details

28. Envelopes of Data x83 and Filtered Data y83

29. Instantaneous frequency and data Envelope

30. Stability Spectrum Problems of the previous approach: –1. Hilbert Envelope contains modulation in the amplitude –2. Define both positive and negative damping –3. How to define the instantaneous frequency

31. Time-Frequency Dependent Damping Analytic function of k th mode : (subscripts omitted for simplicity) Model time-dependent decay factor: Loss factor: where  (t) is critical damping ratio and  0 (t) is natural frequency is the (damped) system frequency If  = const.,  = 1/2  t --- Under damped harmonic oscillator

32. Hilbert Damping Spectrum Time and Frequency Dependent Damping -  (t) contoured on the time- frequency plane, i.e. [  (t),  (t), t]   ( , t), where  (t)=  (  (t), t), Time-averaged loss factor using root-mean-square: A frequency dependent damping loss factor can be calculated if the system is essentially linear: If  n is a resonant frequency of a structure,  (  n ) = loss factor obtained using conventional modal method

33. Difference in Envelopes Hilbert Transform vs Spline

34. Different Envelopes

35. Different Derivatives from Envelopes

36. Different Derivatives from Envelopes : S5

37. Different Derivatives from Envelopes : S21

38. Stability Spectrum [n, t, f]=isspec(imf(:, 1:7), 600, 0, 30, 0, tt(11750), 20, 0.01,[],'no','no'); Data from C1; Frequency resolution : 600 Frequency range : 0 to 30 Hz Smoothed temporally with 20 point = 0.2 Second In this study NT= 3,5,10,15,20 were used Cut-off magnitude set 0.01 In this study PER=0.1, 0.01, 0.005, 0.0001 were used

39. Effect of Magnitude Cut-off Varying percentage cut-off values

40. Hilbert Stability Spectrum : Per=0.001, NT=10

41. Hilbert Stability Spectrum : Per=0.005, NT=10

42. Hilbert Stability Spectrum : Per=0.01, NT=10

43. Hilbert Stability Spectrum : Per=0.1, NT=10

44. Stability Index as a Function of Frequency Per=0.1, 0.01, 0.005, 0.001

45. Stability Index as a Function of Time Per=0.1, 0.01, 0.005, 0.001

46. Effect of Smoothing Varying NT values

47. Hilbert Stability Spectrum : Per=0.01, NT=3

48. Hilbert Stability Spectrum : Per=0.01, NT=5

49. Hilbert Stability Spectrum : Per=0.01, NT=10

50. Hilbert Stability Spectrum : Per=0.01, NT=15

51. Hilbert Stability Spectrum : Per=0.01, NT=20

52. Hilbert Stability Spectrum : Per=0.01, NT=30

53. Stability Index as a Function of Frequency NT = 3, 5, 10, 15, 20

54. Stability Index as a Function of Time NT = 3, 5, 10, 15, 20

55. Nonlinearity Determined from various methods: HHT Teager’s Energy Operator Generalized Zero-crossing

56. IF from Various Methods

57. IF from Various Methods, More Details

58. Preliminary Conclusions The flutter is quite nonlinear. The flutter frequency increases with increasing Mach number. Even from Fourier point view, there is a faint sub-harmonics vibration for the flutter, which usually suggests nonlinearity. Nonlinearity becomes obvious toward the end of the test, after the flutter amplitude increases almost exponentially and starts to level off. The nonlinear vibration is confirmed by Fourier based spectrogram, which clearly shows second harmonics. Just before the shattering of the wing, the flutter frequency starts to decrease suggesting yielding of the wing. The frequency change at the end cannot be detected quantitatively by any method other than Hilbert Spectral Analysis. Stability spectra with different magnitude cut-off and smoothing: tentative guide: PER>0.01; NT<20. Over most of the range, the wing is unstable with negative stability index (i.e. negative damping).