/ 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