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  • / NASA Technical Memoi-an_lum 100464

    Random Process Simulation For

    Stochastic Fatigue Analysis

    |NAS&-TM-100_64) RANDOM PROCESS SIMULATION

    FOE STOCHASTIC FATIGUE ANALYSIS Ph°D° :Thesis

    -Rice Univ., Houston, Tex° , [NASA) 150 p CSCL 12A

    G3/65

    N88-22654

    Unclas

    0131795

    Curtis E. Larsen

    March 1988

    Nt A

  • NASA Technical Memorandum 100464

    Random Process Simulation For

    Stochastic Fatigue Analysis

    Curtis E. Larsen

    Lyndon B. Johnson Space Center

    Houston, Texas

  • TABLE OF CONTENTS

    Chapter

    I

    Page

    INTRODUCTION ....................... i

    2 BACKGROUND ........................ 3

    2.1 Stochastic Processes ................. 3

    2.2 Concepts in Fatigue ................. 10 2.3 Gaussian Process Simulation ............. 15

    2.4 Autoregressive (AR) Processes and Simulation .... 17

    3 GAUSSIAN SIMULATION RESULTS ............... 22

    3.1PSD Component Contributions to Damage ........ 25 3.2 High Frequency Truncation Effect .......... 32 3.3 Extrema Correlations ................ 34

    4 AUTOREGRESSIVE SIMULATION: UNIMODAL PSD'S ........ 57

    4.1AR(1) Processes ................... 57

    4.2 AR(1) Application to Extrema Processes ....... 58

    4.3 Simulation Results ................. 63

    5 AUTOREGRESSIVE SIMULATION FOR BIMODAL PSD'S ....... 69

    5.1Bimodal PSD Process Modeling Difficulties ...... 69 5.2 Adaptation of the AR(1) Model ............ 70

    5.3 Simulation Results ................. 75

    CONCLUSION ....................... 93

    APPENDIX A PSD CASES STUDIED USING GAUSSIAN TECHNIQUE ....... 96

    APPENDIX B A NOTE ON THE CORRECTION OF RANGE MOMENTS FOR

    SIMULATED STRESS TIME HISTORIES ............. 102

    APPENDIX C AR(1) TECHNIQUE SIMULATION CASES ............ 104

    APPENDIX D AR(1) SUPERPOSITION TECHNIQUE SIMULATION CASES ..... 105

    APPENDIX E RELATING EXTREMA CORRELATIONS TO THE AUTOCORRELATION

    FUNCTION ........................ 107

    E.I The Step-Type Model ................. 107 E.2 A Normal Approximation for the Distribution of N*..123

    E.3 A Phase-Angle Approximation ............. 125

    E.4 A Second Simplification to the Step-Type

    Approximation .................... 132

    REFERENCES .............................. 136

    PRECEDING PAGE BLANK NOT FILMED

    £J.J.

  • 4-I

    E-I

    E-2

    E-3

    E-4

    E-5

    LIST OF TABLES

    Page

    Effect of Mapping Procedure on Correlation

    Pl .......................... 65

    Comparison of RXX(T ) Extrema Values to Values

    Predicted by Step-Type Model ............. 115

    Comparison of VN* to VT for 5 Unimodal Psd's ..... 129

    VN Compared to VT for E[N*] = 1,2,3,4 ........ 129

    Comparison of VN* = (i - B)I/2 to VN* using Equation E-46 .................. 134

    Comparison of B Values ................ 135

    iv

  • LIST OF FIGURES

    2.1

    2.2

    3.1

    3.2

    3.3

    3.4

    3.5

    3.8

    3.9

    3.10

    3.11

    3.12

    3.13

    3.14

    3.15

    Typical S-N Curve ...................... 11

    Rainflow Cycle-ldentification ................ 14

    PSD Types Studied ...................... 23

    Normalized Damage Rate vs Normalized Area Ratio: m = 3.... 26

    Normalized Damage Rate vs Normalized Area Ratio: m=5. . .27

    Normalized Damage Rate vs Normalized Area Ratio: m =7 .... 28

    Vanmarcke's Parameter vs Area and Frequency Ratios:

    Bimodal Region ....................... 31

    Normalized Damage Rate vs Normalized Cut-Off Frequency . . .33

    Normalized Damage Rate vs Normalized Vanmarcke's Parameter .......................... 35

    Correlation Pl vs Vanmarcke's Parameter: Scattergraph ........................ 37

    Correlation Pl vs Vanmarcke's Parameter: Frequency Ratlo Effect ................... 38

    Correlation Pl vs Vanmarcke's Parameter: Area Ratio Effect, r = 15 .................. 39

    Correlation Pl vs Vanmarcke's Parameter: Linear RegresSion ...................... 41

    Typical Extrema Correlations: Unimodal PSD ......... 42

    Extrema Correlations: r=7 ................. 43

    a) b = 0.001

    b) b : 0.01

    Extrema Correlations: r =7 ................. 44

    a) b = 0.1

    b) b : 1.0

    Extrema Correlations: r =7 ................. 45

    a) b = 5.2

    b) b : 115

    v

  • 3.16

    3.17

    3.18

    3.19

    3.20

    3.21

    3.22

    3.23

    3.24

    4.1

    4.2

    4.3

    4.4

    4.5

    4.6

    5.1

    5.2

    5.3

    5.4

    Page

    Typical Time History: Bimodal PSD, r =3, b=0.0016 ..... 47

    Typical Time History: Bimodal PSD, r:3, b =0.63 ...... 48

    Typical Time History: Bimodal PSD, r:3, b:70 ....... 49

    Autocorrelation Function: r =7, b =0.001 ............ 50

    Autocorrelation Function: r =7, b =0.01 .......... 51

    Autocorrelation Function: r =7, b =0.I ........... 52

    Autocorrelation Function: r =7, b:loO ........... 53

    Autocorrelation Function: r =7, b:5.2 ........... 54

    Autocorrelation Function: r =7, b:115 ........... 55

    Mapping from Normal to S.O. Rice Distribution ....... 60

    Autoregressive Simulation Technique ............ 61

    AR(1) Simulation Algorithm ................. 62

    Range Moment Comparison, AR(1) and Gaussian Techniques: m = 3 ............................ 66

    Range Moment Comparison, AR(1) and Gaussian Techniques: m = 5............................ 67

    Range Moment Comparison, AR(1) and Gaussian Techniques: m = 7 ............................ 68

    Superposition of Time History Components .......... 72

    a) r=3 b) r=2 c) r=6

    General Component Superposition .............. 74

    AR(1) Superposition Algorithm ............... 76

    Range Moment Comparison, AR(1) Superposition and Gaussian Techniques: m =3 ....................... 78

    a) r = 1.5 ......................... 78 b) r = 2.0 ......................... 78 c) r = 2.5 ......................... 79 d) r : 3.0 ......................... 79 e) r = 3.5 ......................... 80

    vi

  • 5.4

    5.5

    5.6

    E-1

    E-2

    E-3

    E-4

    E-5

    E-6

    E-7

    E-8

    E-9

    Page Continued

    f) r = 4.0 .......................... 80

    g) r = 4.5 .......................... 81

    h) r = 5.0 .......................... 81

    i) r=7.5 .......................... 82

    j) r : 10.0 ......................... 82

    Range Moment Comparison, AR(1) Superposition and Gaussian

    Techniques: m =5 ...................... 83

    a) r-- 1.5 .......................... 83

    b) r = 2.0 .......................... 83

    c) r = 2.5 .......................... 84

    d) r = 3.0 .......................... 84

    e) r = 3.5 .......................... 85

    f) r = 4.0 .......................... 85

    g) r = 4.5 .......................... 86 h) r = 5.0 .......................... 86

    i) r=7.5 .......................... 87

    j) r = 10.0 ......................... 87

    Range Moment Comparison, AR(1) Superposition and Gaussian

    Techniques: m = 7 ...................... 88

    a) r = 1.5 .......................... 88

    b) r = 2.0 ............ .............. 88

    c) r = 2.5 .......................... 89

    d) r = 3.0 .......................... 89

    e) r-- 3.5 .......................... 90

    f) r = 4.0 .......................... 90

    g) r = 4.5 .......................... 91 h) r = 5.0 .......................... 91

    i) r=7.5 .......................... 92

    j) r = 10.0 ......................... 92

    Comparlson of RXX(T) to Approximation ........... 116

    Comparlson of RXX(T) to Approximation ........... 117

    Comparlson of RXX(T) to Approximation ........... 118

    Comparlson of RXX(T) to Approximation ........... 119

    Comparlson of RXX(T) to Approximation ........... 120

    Comparlson of Rxx(T) to Approximation ........... 121

    Comparlson of RXX(T) to Approximation ........... 122

    Correction Factor, K vs. Vanmarcke's Parameter ...... 130

    2 a0 vs. Vanmarcke's Parameter ............... 131

    vii

  • AR

    ARMA

    at

    e2

    b

    C

    C 0

    d/d o

    d. J

    d/t

    D(t)

    E(.)

    E[Rm]

    E[Sm]

    erf(x)

    erfc(x)

    fx (.), px(.)

    FX(-)

    G

    Gx(m)

    r(x)

    rxx(_)

    INT(r)

    LIST OF SYMBOLS

    Autoregressive

    Autoregressive Moving Average

    Random variate in ARMA model

    Bandwidth parameter, irregularly factor for

    processes

    2-block psd area ratio

    Material constant

    Estimate for o_

    Normalized damage rate

    Fractional damage due to nj cycles

    Expected damage rate

    Total damage at time t

    Mathematical expectation

    Rainflow stress range moment

    Theoretical stress range moment

    Error function

    Complementary error function

    Probability density function for X

    Cumulative distribution function for X

    Constant magnitude of psd

    1-sided psd, defined on (0,_)

    Gamma function

    Autocovariance function, stationary processes