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

Location of the Test Wing

Details of the test wing

Test Video

The Full Data : atw_f5_m83h10_1 Details

IMF Data x83 : atw_f5_m83h10_1

Fourier Spectra of Various Sum of IMFs

Hilbert Filtered Data x83 : atw_f5_m83h10_1

Hilbert Spectrum : x83

Hilbert Spectrum : x83 Details

Spectrogram (512) : x83

Spectrogram (512) : x83 Details

Spectrogram (1024) : y83

Spectrogram (1024) : y83 Details

Re-sampled Hilbert Filtered Data : y(I)

Mean Hilbert Spectrum : y(i)

Hilbert Spectrum : x83 Details

Marginal Hilbert and Fourier Spectra : y83

3D Mean Hilbert Spectrum : y(i)

3D Spectrogram : y83

Instantaneous frequency : y(i)

Instantaneous frequency : y(i) Details

Mean Hilbert and Spectrogram : y83

Mean Hilbert and Spectrogram : y83 Details

Envelopes of Data x83 and Filtered Data y83

Instantaneous frequency and data Envelope

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

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

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

Difference in Envelopes Hilbert Transform vs Spline

Different Envelopes

Different Derivatives from Envelopes

Different Derivatives from Envelopes : S5

Different Derivatives from Envelopes : S21

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, were used

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

Hilbert Stability Spectrum : Per=0.001, NT=10

Hilbert Stability Spectrum : Per=0.005, NT=10

Hilbert Stability Spectrum : Per=0.01, NT=10

Hilbert Stability Spectrum : Per=0.1, NT=10

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

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

Effect of Smoothing Varying NT values

Hilbert Stability Spectrum : Per=0.01, NT=3

Hilbert Stability Spectrum : Per=0.01, NT=5

Hilbert Stability Spectrum : Per=0.01, NT=10

Hilbert Stability Spectrum : Per=0.01, NT=15

Hilbert Stability Spectrum : Per=0.01, NT=20

Hilbert Stability Spectrum : Per=0.01, NT=30

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

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

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

IF from Various Methods

IF from Various Methods, More Details

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).