Frequency Based Fatigue

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2 Deterministic versus Random Deterministic from past measurements the future position of a satellite can be predicted with reasonable accuracy Random from past measurements the future position of a car can only be described in terms of probability and statistical averages Frequency Based Fatigue Darrell Socie, All Rights Reserved 1 of 57

7 Statistics of Time Histories Mean or Expected Value Variance / Standard Deviation Root Mean Square Kurtosis Skewness Crest Factor Irregularity Factor Frequency Based Fatigue Darrell Socie, All Rights Reserved 6 of 57

11 Skewness Skewness is a measure of the asymmetry of the data around the sample mean. If skewness is negative, the data are spread out more to the left of the mean than to the right. If skewness is positive, the data are spread out more to the right. The skewness of the normal distribution (or any perfectly symmetric distribution) is zero. Skewness N ( x i 1 X 3 i N x ) 3 Frequency Based Fatigue Darrell Socie, All Rights Reserved 10 of 57

12 Kurtosis Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 3; distributions that are less outlier-prone have kurtosis less than 3. Kurtosis N ( x i 1 X 4 i N x ) 4 Frequency Based Fatigue Darrell Socie, All Rights Reserved 11 of 57

13 Crest Factor The crest factor is the ration of the peak (maximum) value to the root-mean-square (RMS) value. A sine wave has a crest factor of Frequency Based Fatigue Darrell Socie, All Rights Reserved 1 of 57

Fourier 1768-1830 Fourier studied the mathematical theory of heat conduction.

24 Fourier Fourier studied the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions. Frequency Based Fatigue Darrell Socie, All Rights Reserved 3 of 57

25 Fourier Series X(t) a o ak sin(kot) bk cos(kot) k 1 X(t) c o ck cos(kot k ) k 1 X(t) k X S [k]e jk t o a k, b k, c k, X S [k] are Fourier coefficients frequency, magnitude, and phase are all described by the coefficients Frequency Based Fatigue Darrell Socie, All Rights Reserved 4 of 57

26 Fourier Transform Magnitude FFT Magnitude f Time Frequency The area under each spike represents the magnitude of the sine wave at that frequency. The magnitude of the FFT depends on the frequency window f. Frequency Based Fatigue Darrell Socie, All Rights Reserved 5 of 57

30 Autospectral Density Function Units Time History EU X(n) Linear Spectrum EU S(n) = DFT(X(n)) AutoPower EU^ AP(n) = S(n) S(n) PSD (EU^)/Hz PSD(n) = AP(n) / ( W f f ) ESD (EU^*sec)/Hz ESD(n) = AP(n) T / ( W f f ) Frequency Based Fatigue Darrell Socie, All Rights Reserved 9 of 57

31 Calculating Autospectral Density N D N D Block size T T 3T n D T S f k n D 1 N D t n D i 1 X f k Magnitude only no phase information Frequency Based Fatigue Darrell Socie, All Rights Reserved 30 of 57

32 Power Spectral Density - PSD Log Magnitude-Power-Strain_c56 Strain (ustrain^)/hz Frequency (Hz) Average power associated with a 1 Hz frequency window centered at each frequency, f. Phase information is lost. Frequency Based Fatigue Darrell Socie, All Rights Reserved 31 of 57

Frequency Based Fatigue Stationary Loading Wind Sea State Vibration

40 Fatigue Analysis Load time history Structural model Time Domain Frequency Domain Stress time history Rainflow? Miner linear damage Durability Acceleration PSD Transfer function Stress PSD Frequency Based Fatigue Darrell Socie, All Rights Reserved 39 of 57

41 Dynamics Model F(t) m y(t) d y dt m F(t) k m F(t) c y(t) x(t) d z m dt d z dt c z(t) = y(t) x(t) dz dt n dz dt k z n F(t) z d x m dt d x dt Frequency Based Fatigue Darrell Socie, All Rights Reserved 40 of 57

45 Probability Density Function Probability density S i S p( S i ) Stress range The probability P( S i ) of a stress range occurring between S i S and S i S is P( S i ) p( S i ) S Frequency Based Fatigue Darrell Socie, All Rights Reserved 44 of 57

46 Fatigue Damage Cycles at level i n i = p( S i ) S N T Total cycles N T = E( P ) T Total time Fatigue Damage Fatigue life D N i 1 N ni N ( S f f i ( S ) i S ) S i ' f 1 b ' E(P )T S(S ) Fatigue damage is determined by p( S i ) f 1 b N i 1 p( S S i i 1 b ) Frequency Based Fatigue Darrell Socie, All Rights Reserved 45 of 57

47 Narrow Band Solution Rayleigh distribution S p( S ) 4m exp S 8 0 m0 IF 1 is narrow band signal IF 0 is wide band signal This wide band signal has the same peak distribution as this narrow band signal Frequency Based Fatigue Darrell Socie, All Rights Reserved 46 of 57

48 Frequency Based Fatigue Darrell Socie, All Rights Reserved 47 of 57 Dirlik Solution m Z Zexp D R Z exp R D Q Z exp Q D S ) p( m 0 S Z 4 0 m m m IF m m m m m X m 1 IF 1 IF X D R 1 D D IF 1 D m D D IF 1 D X IF R 1 1 4D R D D IF 5 Q 1 3 D D 1 D p( S ) = f ( m 0, m 1, m, m 4 )

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Optimized PSD Envelope for Nonstationary Vibration Revision A

ACCEL (G) Optimized PSD Envelope for Nonstationary Vibration Revision A By Tom Irvine Email: tom@vibrationdata.com July, 014 10 FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE 5 0-5 -10-5 0 5 10