Frequency Based Fatigue Professor Darrell F. Socie Mechanical Engineering University of Illinois at Urbana-Champaign © 2001 Darrell Socie, All Rights Reserved 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 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 1 of 57 Time Domain Bracket.sif-Strain_c56 750 Strain (ustrain) -750 Time (Secs) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 2 of 57 Frequency Domain ap_000.sif-Strain_b43 102 101 100 Strain (ustrain) 10-1 0 20 40 60 80 100 120 140 Frequency (Hz) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 3 of 57 Statistics of Time Histories ! Mean or Expected Value ! Variance / Standard Deviation ! Coefficient of Variation ! Root Mean Square ! Kurtosis ! Skewness ! Crest Factor ! Irregularity Factor AAR Seminar © 2001 Darrell Socie, All Rights Reserved 4 of 57 Mean or Expected Value Central tendency of the data N ∑ x i = Mean = µ = x = E()X = i 1 x N AAR Seminar © 2001 Darrell Socie, All Rights Reserved 5 of 57 Variance / Standard Deviation Dispersion of the data N − 2 ∑( xi x ) Var()X = i =1 N Standard deviation σ = x Var( X ) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 6 of 57 Coefficient of Variation σ COV = x µ x Useful to compare different dispersions µ =10 µ = 100 σ =1 σ =10 COV = 0.1 COV = 0.1 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 7 of 57 Root Mean Square N 2 ∑ x i = RMS = i 1 N The rms is equal to the standard deviation when the mean is 0 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 8 of 57 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. N − 3 ∑( xi x ) = Skewness()X = i 1 Nσ3 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 9 of 57 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. N − 4 ∑( xi x ) = Kurtosis()X = i 1 Nσ4 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 10 of 57 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 1.414. AAR Seminar © 2001 Darrell Socie, All Rights Reserved 11 of 57 Irregularity Factor Positive zero crossing E(0+ ) 4 IF= = Peak E(P ) 7 IF → 1 is narrow band signal IF → 0 is wide band signal AAR Seminar © 2001 Darrell Socie, All Rights Reserved 12 of 57 Time Domain Nomenclature ! Random ! Stochastic ! Stationary ! Non-stationary ! Gaussian ! Narrow-band ! Wide-band AAR Seminar © 2001 Darrell Socie, All Rights Reserved 13 of 57 Random The instantaneous value can not be predicted at any future time. AAR Seminar © 2001 Darrell Socie, All Rights Reserved 14 of 57 Stochastic Stochastic processes provide suitable models for physical systems where the phenomena is governed by probabilities. AAR Seminar © 2001 Darrell Socie, All Rights Reserved 15 of 57 Stationary ∆t The properties computed over short time intervals, t + ∆t, do not significantly vary from each other AAR Seminar © 2001 Darrell Socie, All Rights Reserved 16 of 57 Non-stationary AAR Seminar © 2001 Darrell Socie, All Rights Reserved 17 of 57 Gaussian Normally distributed around the mean AAR Seminar © 2001 Darrell Socie, All Rights Reserved 18 of 57 Narrow-band AAR Seminar © 2001 Darrell Socie, All Rights Reserved 19 of 57 Wide-band AAR Seminar © 2001 Darrell Socie, All Rights Reserved 20 of 57 Frequency Domain Nomenclature Fourier Nyquist frequency Aliasing Anit-aliasing filter FFT Inverse FFT Autospectral density Transfer Function AAR Seminar © 2001 Darrell Socie, All Rights Reserved 21 of 57 Fourier 1768 - 1830 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. AAR Seminar © 2001 Darrell Socie, All Rights Reserved 22 of 57 Fourier Series ∞ = + ω + ω X(t) ao ∑ak sin(k ot) bk cos(k ot) k =1 ∞ = + ω + θ X(t) co ∑ck cos(k ot k ) k =1 ∞ ω = jk ot X(t) ∑ XS [k]e k = −∞ ak,bk,ck,XS[k] are Fourier coefficients frequency, magnitude, and phase are all described by the coefficients AAR Seminar © 2001 Darrell Socie, All Rights Reserved 23 of 57 Digital Sampling ∆t What ∆t should be used for sampling the data? AAR Seminar © 2001 Darrell Socie, All Rights Reserved 24 of 57 Aliasing 2 signals that differ in frequency by a factor of 2 f 2f Both signals have the same digital samples. AAR Seminar © 2001 Darrell Socie, All Rights Reserved 25 of 57 Nyquist frequency Real spectrum Aliased spectrum 1magnitude magnitude f = c 2∆t frequency frequency fc 1 f = c 2∆t Sample at 2 times the frequency of interest AAR Seminar © 2001 Darrell Socie, All Rights Reserved 26 of 57 Anti-aliasing Filter 180 0 Phase -180 0 50 100 150 0 frequency -20 -40 -60 -80 Magnitude, dB -100 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 27 of 57 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. AAR Seminar © 2001 Darrell Socie, All Rights Reserved 28 of 57 Fast Fourier Transform - FFT 16 14 12 10 Strain (ustrain) 8 6 FFT 4 2 0 0 20 40 60 80 100 120 140 Frequency (Hz) N/ 2 = + ω + θ X(t) co ∑ck cos(k ot k ) k =1 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 29 of 57 Fast Fourier Transform - FFT 150 100 50 0 -50 Phase (Degrees) -100 -150 0 20 40 60 80 100 120 140 Frequency (Hz) N/ 2 = + ω + θ X(t) co ∑ck cos(k ot k ) k =1 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 30 of 57 Inverse FFT N/ 2 = + ω + θ X(t) co ∑ck cos(k ot k ) k =1 θ ω ck, k, and k o are all known 750 Strain (ustrain) -750 Time (Secs) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 31 of 57 Autospectral Density Function Units Time History EU X(n) Linear Spectrum EU S(n) = DFT(X(n)) AutoPower EU^2 AP(n) = S(n) · S(n) PSD (EU^2)/Hz ∆ PSD(n) = AP(n) / ( Wf · f) · ∆ ESD (EU^2*sec)/Hz ESD(n) = AP(n) T/(Wf · f) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 32 of 57 Calculating Autospectral Density ND T2T3T nDT ND Block size n 1 D 2 S()f = X()f k ∆ ∑ k nD ND t i=1 Magnitude only no phase information AAR Seminar © 2001 Darrell Socie, All Rights Reserved 33 of 57 Power Spectral Density - PSD Log Magnitude-Power-Strain_c56 105 104 103 102 101 Strain (ustrain^2)/Hz 100 100 101 102 Frequency (Hz) Average power associated with a 1 Hz frequency window centered at each frequency, f . Phase information is lost. AAR Seminar © 2001 Darrell Socie, All Rights Reserved 34 of 57 Linear Spectrum Log Magnitude-Linear-Strain_c56 102 101 Strain (ustrain) 100 100 101 102 Frequency (Hz) Sometimes called Amplitude Spectral Density AAR Seminar © 2001 Darrell Socie, All Rights Reserved 35 of 57 Comparison 105 104 103 PSD 102 101 Strain (ustrain^2) 0 10 Linear Spectrum 10-1 100 101 102 Frequency (Hz) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 36 of 57 Frequency Domain Limitations Non-stationary signals EASE1.EDT-Strain_c56 800 600 400 200 0 Strain (ustrain) -200 -400 Time (Secs) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 37 of 57 Linear Spectrum 103 102 101 Strain (ustrain) 100 10-1 0 20 40 60 80 100 120 140 Frequency (Hz) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 38 of 57 Transfer Function output 104 m output 3 input 10 input 102 100 101 102 Frequency (Hz) AAR Seminar © 2001 Darrell Socie, All Rights Reserved 39 of 57 Frequency Based Fatigue ! Stationary Loading ! Wind ! Sea State ! Vibration AAR Seminar © 2001 Darrell Socie, All Rights Reserved 40 of 57 Assumptions Random Gaussian Stationary AAR Seminar © 2001 Darrell Socie, All Rights Reserved 41 of 57 Fatigue Analysis Load Stress Structural time time model history history Time Domain Miner Rainflow linear Durability damage Frequency Domain ? Acceleration Transfer Stress PSD function PSD AAR Seminar © 2001 Darrell Socie, All Rights Reserved 42 of 57 Dynamics Model F(t) y(t) d2y m = F(t) m dt2 F(t) z(t) = y(t) – x(t) y(t) d2z dz d2x m m + c + k z = F(t) − m dt2 dt dt2 x(t) c k d2z dz d2x + 2ζω + ω 2 z = − dt2 n dt n dt2 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 43 of 57 PSD Moments fk /Hz 2 Gk(f) Magnitude δf frequency N = n δ mn ∑ fk Gk ( fk ) f k =1 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 44 of 57 Expected Values m zero crossings E(0)= 2 m0 m peaks E(P )= 4 m2 E(0 ) m 2 Irregularity factor IF= = 2 E(P ) m0 m4 AAR Seminar © 2001 Darrell Socie, All Rights Reserved 45 of 57 Probability Density Function ∆ Si ∆ p( Si ) Probability density δS Stress range ∆ The probability P( Si ) of a stress range occurring between δS δS ∆S − and ∆S + is P( ∆S ) = p( ∆S )δS i 2 i 2 i i AAR Seminar © 2001 Darrell Socie, All Rights Reserved 46 of 57 Fatigue Damage ∆ δ Cycles at level i ni =p( Si ) SNT Total cycles NT =E(P)T Total time 1 ∆S b Fatigue life N ( ∆S )= i f i ' 2Sf N n 1 N p( ∆S ) Fatigue Damage D= i = E(P)T δS(2S' )b i ∑ f ∑ 1 = ∆ = i 1 Nf ( Si ) i 1 ()∆