Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test 9811636931, 9789811636936

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Yu Jiang Junyong Tao Xun Chen

Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test

Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test

Yu Jiang · Junyong Tao · Xun Chen

Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test

Yu Jiang Laboratory of Science and Technology on Integrated Logistics Support College of Intelligence Science and Technology National University of Defense Technology Changsha, China

Junyong Tao Laboratory of Science and Technology on Integrated Logistics Support College of Intelligence Science and Technology National University of Defense Technology Changsha, China

Xun Chen Laboratory of Science and Technology on Integrated Logistics Support College of Intelligence Science and Technology National University of Defense Technology Changsha, China

Translated by Yu Jiang Laboratory of Science and Technology on Integrated Logistics Support College of Intelligence Science and Technology National University of Defense Technology Changsha, China

Zhengwei Fan Laboratory of Science and Technology on Integrated Logistics Support College of Intelligence Science and Technology National University of Defense Technology Changsha, China

Wuyang Lei Laboratory of Science and Technology on Integrated Logistics Support College of Intelligence Science and Technology National University of Defense Technology Changsha, China

ISBN 978-981-16-3693-6 ISBN 978-981-16-3694-3 (eBook) https://doi.org/10.1007/978-981-16-3694-3

Jointly published with National Defense Industry Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: National Defense Industry Press. Translation from the Chinese Simplified language edition: Fei Gao Si Sui Ji Zhen Dong Pi Lao Fen Xi Yu Shi Yan Ji Shu by Yu Jiang, et al., © National Defense Industry Press 2019. Published by National Defense Industry Press. All Rights Reserved. © National Defense Industry Press 2022 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Statistic Analysis of Non-Gaussian Random Load . . . . . . . . . . . . . . . . . 2.1 Common Non-Gaussian Statistical Parameters . . . . . . . . . . . . . . . . . . 2.2 Symmetrical Non-Gaussian Probability Density Function . . . . . . . . 2.2.1 Gaussian Mixed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Estimation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Asymmetrical Non-Gaussian Probability Density Function . . . . . . . 2.3.1 Gaussian Mixed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Estimation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Case Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Case of Symmetrical Non-Gaussian Random Process . . . . . 2.4.2 Case of Asymmetrical Non-Gaussian Random Process . . . . 2.5 Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 6 6 8 8 8 11 11 16 21 22

3 Simulation of Non-Gaussian/Non-stationary Random Vibration . . . . 3.1 Simulation of Non-Gaussian Stochastic Processes by Amplitude Modulation and Phase Reconstruction . . . . . . . . . . . . 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Experiments of Coupling Effect of Fourier Coefficients on the Non-Gaussian Features . . . . . . . . . . . . . . 3.1.4 Simulation Algorithm of Non-Gaussian Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Cases Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modeling and Simulation of Non-stationary Random Vibration Signals Based on Hilbert Spectrum . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Modeling of Non-stationary Random Data . . . . . . . . . . . . . . .

23 23 24 26 30 36 37 41 41 42 43 45 v

vi

Contents

3.2.4 Simulation of Non-stationary Random Vibration Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 59 59

4 Dynamic Stress Response and Fatigue Life of Cantilever Beam Under Non-Gaussian Base Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Non-Gaussian Random Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dynamic Stress Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Displacement Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Stress Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Input Base Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Fatigue Life Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 64 65 65 67 70 71 71 72 73 74 77 78

5 Fatigue Life Analysis Under Non-Gaussian Random Vibration Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 On the Sampling Frequency of Random Loadings for Fatigue Damage Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Random Vibration Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Time-Domain Method for Fatigue Damage Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Sampling Frequency and Calculation Accuracy . . . . . . . . . . . 5.1.5 Verification Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Method for Estimating Rain-Flow Fatigue Damage of Narrowband Non-Gaussian Random Loadings . . . . . . . . . . . . . . . 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Non-Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Nonlinear Transformation Models . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Fatigue Damage Estimation Based on Nonlinear Transformation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Direct Method for Fatigue Damage Estimation . . . . . . . . . . . 5.2.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A Spectral Method to Estimate Fatigue Life Under Broadband Non-Gaussian Random Vibration Loading . . . . . . . . . . . . . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Non-Gaussian Random Vibration Loadings . . . . . . . . . . . . . .

81 81 82 83 83 85 89 90 90 90 92 93 95 96 99 106 106 107 108

Contents

vii

5.3.3 Gaussian Mixture Model (GMM) . . . . . . . . . . . . . . . . . . . . . . 5.3.4 PSD Decomposition of Non-Gaussian Vibration Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Modified Dirlik’s Formula and Fatigue Damage Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112 114 119 120

6 Fatigue Reliability Evaluation of Structural Components Under Random Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Uncertainty Quantification from External Loading . . . . . . . . . . . . . . 6.2.1 The Number of Rainflow Cycles . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The Rainflow Cycle Distribution . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Uncertainty Quantification of the Fatigue Property . . . . . . . . . . 6.4 Fatigue Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Fatigue Reliability Expectation . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Fatigue Reliability Confidence Interval . . . . . . . . . . . . . . . . . . 6.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Monte-Carlo Simulation and Bootstrap Method . . . . . . . . . . 6.5.2 Theoretical Results from the Proposed Method . . . . . . . . . . . 6.5.3 Results Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 124 125 127 129 129 130 131 132 133 134 136 136

7 Non-Gaussian Random Vibration Accelerated Test . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Model of Random Vibration Accelerated Test . . . . . . . . . . . . . . . . . . 7.2.1 Model of Gaussian Random Vibration Accelerated Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Model of Non-Gaussian Random Vibration Accelerated Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Solving Unknown Parameters for Model of Non-Gaussian Random Vibration Accelerated Test . . . . . 7.3 Non-Gaussian Random Vibration Accelerated Test System . . . . . . . 7.3.1 Design of Random Vibration Fatigue Test System . . . . . . . . 7.3.2 Design of Test Specimen and Fixture . . . . . . . . . . . . . . . . . . . 7.4 Experimental Design of Non-Gaussian Random Vibration Accelerated Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Design Considerations for Non-Gaussian Random Vibration Fatigue Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Hybrid Test Strategy for Accelerated Random Vibration Fatigue Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Experimental Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . .

109 111

139 139 142 142 145 148 149 149 150 152 152 154 160

viii

Contents

7.6 Application Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 164 165 166

Chapter 1

Introduction

Abstract A general introduction about the book ‘Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test’ is given, and the contents of chapters are summarized.

As a common problem in the engineering field, fatigue failure caused by vibration seriously endangers the safe and reliable operation of important equipment and structures. Vibration test is an important method for environmental adaptability, safety, reliability and life assessment of large equipment and structures in the fields of aviation, aerospace, shipbuilding, and mechanical engineering. How to ensure that the vibration test performed in the laboratory conforms to the actual service or transportation vibration environment of the equipment, and avoid under-testing and over-testing, has become an urgent problem to be solved. In order to solve the above problem, we first need to understand the description of random vibration. The most common parameter used to describe random vibration is Power Spectral Density (PSD). However, PSD is not able to adequately portray all the characteristics of random vibration. For example, with the same PSD (also the same Root Mean Square), random vibration signals can have completely different probability density distribution, as shown in Fig. 1.1. Because the higher-order statistics over the second order are constantly zero for a Gaussian random process, only using PSD function or self-correlation function is enough to fully describe the characteristics of Gaussian random vibration. But for non-Gaussian (including super-Gaussian and sub-Gaussian) random vibration, in addition to PSD, higher-order statistics (above the second order) are necessary to describe its comprehensive characteristics, which will be discussed in detail in Chap. 2. After understanding how to fully describe the characteristics of non-Gaussian random vibration, the next step is to simulate non-Gaussian random vibration. This is the key to the reproduction of the actual vibration environment experienced by the equipment, and it can also be used for structural dynamics and vibration fatigue simulation analysis. Because non-Gaussian random vibration also often appears in the form of non-stationary random vibration, in Chap. 3, besides discussing the simulation of non-Gaussian random vibration, the simulation of non-stationary random vibration is also discussed. © National Defense Industry Press 2022 Y. Jiang et al., Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test, https://doi.org/10.1007/978-981-16-3694-3_1

1

2

1 Introduction g2/Hz

0.6

Amplitude Probability Density Function

PSD

0.02

super-Gaussian

sub-Gaussian

Gaussian

0.3

20

2000

(Hz)

0

-4 -3 -2 -1 0 1 2 3 4 Multiple of Sigma

(b) Amplitude Probability Density Function

80

80

60

60

40

40

Acceleration (g)

Acceleration (g)

(a) Power Spectral Density

20 0 -20 -40 -60 -80

20 0 -20 -40 -60

0

2

4

Time (s)

6

(c) Gaussian random vibration signal

8

-80 0

2

4

Time (s)

6

8

(d) Super-Gaussian random vibration signal

Fig. 1.1 Gaussian and non-Gaussian vibration signals with the same PSD

After realizing the simulation of non-Gaussian random vibration, using it for input excitation to act on the structure will produce a dynamic response, which is the root cause of the structure’s vibration fatigue failure. Therefore, in order to study the influence of non-Gaussian random vibration on the reliability of the structure, we must first study the dynamic response of the structure under non-Gaussian random vibration. This is the main content of Chap. 4. Chapter 5 mainly studies how to obtain accurate and reliable calculation results of fatigue life after obtaining the non-Gaussian random stress response. First, the influence of sampling frequency on the calculation accuracy of fatigue damage is studied, and a stress signal reconstruction method based on Shannon’s formula is proposed. Then, based on the rain-flow counting method, the calculation methods of the fatigue life of the structure under the action of narrowband and wideband nonGaussian random stress are, respectively, proposed. Finally, the applicability and accuracy of the calculation method of non-Gaussian fatigue life are verified through specific example analysis. Chapter 6 refers to fatigue reliability evaluation of structural components under random loadings. The randomness of fatigue damage is treated in two aspects. The first one is the uncertainty quantification from the external random loading. The

1 Introduction

3

second one is the uncertainty quantification of the fatigue property of the structural component. The former is characterized by Gaussian distribution derived from the rainflow cycle distribution, medium stress-life (S–N) curve, and the linear damage accumulation rule. The latter is described with the probabilistic stress-life (P–S–N) curve based on log-normal distribution. The proposed method has colligated these two aspects together to evaluate the expectation and confidence interval of fatigue reliability. In Chap. 7, a novel non-Gaussian random vibration accelerated test methodology was proposed, which can significantly reduce the test time and the sample size. First, fatigue life prediction models of Gaussian and non-Gaussian random vibration were proposed based on random vibration and fatigue theory. Meanwhile a detailed solving method was also presented for determining the unknown parameters in the models. Second, a non-Gaussian random vibration accelerated test system was designed. Third, several groups of random vibration fatigue tests were designed and conducted with the aim of investigating effects of both Gaussian and non-Gaussian random excitation on the vibration fatigue. Finally, an application case for the fatigue life prediction of electronic product structures verified the effectiveness of the above non-Gaussian random vibration accelerated test method.

Chapter 2

Statistic Analysis of Non-Gaussian Random Load

Abstract Statistic analysis of non-Gaussian random load underpins the calculation and reliability analysis of the fatigue life for products under non-Gaussian loads. Skewness and kurtosis only are insufficient to express the non-Gaussian statistical properties. In view of this, chapter sets up the mathematical model of probability density function (PDF) for the symmetrical and oblique non-Gaussian random load amplitude based on the Gaussian Mixture Model (GMM). Higher-order statistics are sub-stituted into the math model to determine the parameters and obtain the analytical expression of non-Gaussian PDF. Validity of this method is verified by simulation and measured signals. The analytical expression of PDF contributes to a precise definition of the non-Gaussian random load, and underpins the calculation and reliability analysis of the fatigue life as well as the testing program.

2.1 Common Non-Gaussian Statistical Parameters In theory, statistics fully describing the non-Gaussian properties in the random process include higher moment m n (τ1 , . . . , τn−1 ) or high-order cumulants cn (τ1 , . . . , τn−1 ) [4] (n > 2). m n (τ1 , . . . , τn−1 ) and cn (τ1 , . . . , τn−1 ) are multivariate functions of the time interval {τi }, which has complicated computation with calculation errors. Higher-order statistics is rarely applied in the quantitative analysis of nonGaussian random load due to its complexity. For the purpose of convenience, static higher-order statistics, as a result of setting the time interval τi of m n (τ1 , . . . , τn−1 ) or cn (τ1 , . . . , τn−1 ) as 0, is used to express the non-Gaussian properties. The static higher-order statistics of the random process, in essence, amounts to the higher-order statistics of the random variable. Normalized 3-order and 4-order static matrix are called skewness γ 3 and kurtosis γ 4 , respectively. Skewness of the Gaussian random process γ 3 = 0, and kurtosis γ 4 = 3; skewness of symmetrical non-Gaussian random process γ 3 = 0, and kurtosis γ 4 > 3 (Strictly, γ 4 = 3, but the non-Gaussian random load in practice is usually γ 4 > 3 [1]). Skewness and kurtosis of the gaussian random process X(t) can be estimated by the time series x(t) of the sample:

© National Defense Industry Press 2022 Y. Jiang et al., Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test, https://doi.org/10.1007/978-981-16-3694-3_2

5

6

2 Statistic Analysis of Non-Gaussian Random Load

γˆ3 =

1 T

 T 0

x 3 (t)dt

 mˆ 3 (0, 0) = ; σˆ X3

σˆ X3

γˆ4 =

1 T

 T

x 4 (t)dt

0



σˆ X4

=

mˆ 4 (0, 0, 0) (2.1) σˆ X4

Of which, T is time span of the sample. Skewness and kurtosis only are insufficient to characterize the probability statistics of the non-Gaussian random process, therefore, higher-order statistics is required for the analytical expression of PDF of the non-Gaussian amplitude.

2.2 Symmetrical Non-Gaussian Probability Density Function 2.2.1 Gaussian Mixed Model Middleton [5] proposed GMM for the probability distribution of the amplitude of the noise signal with multi-source superimposition in the communication system, expressed as: f NG (x) =

N 

αi f i (x)

(2.2)

i=0

Of which, f NG (x) is the non-Gaussian PDF, f i (x) is the PDF of the ith Gaussian component, N is dimension, α i is weight of the ith Gaussian component, 0 ≤ α i ≤ 1,  α i = 1. Generally, 2D or 3D GMM can obtain precise non-Gaussian PDF. Here, we adopt the 2D GMM. f NG (x) = α f 1 (x|σ1 ) + (1 − α) f 2 (x|σ2 )

(2.3)

2.2.2 Estimation of Parameters Since the PDF of the Gaussian component is determined by the standard deviation σ , the 2D GMM of the zero-mean non-Gaussian process X(t) can be expressed as:     1 x2 x2 exp − 2 + (1 − α) √ exp − 2 f NG (x) = α √ 2σ1 2σ2 2π σ1 2π σ2 1

(2.4)

Of which, σ 1 and σ 2 are standard deviation of the Gaussian component 1 and 2, respectively; α and 1–α are weight of the Gaussian component 1 and 2, respectively. Three unknowns in Eq. (2.4) are σ 1 , σ 2, and α. For the zero-mean process X(t), the

2.2 Symmetrical Non-Gaussian Probability Density Function

7

2nd, 4th, and 6th central moment can be estimated via the time series x(t) of the sample: ⎧

1 T 2 ⎪ ⎪ mˆ 2 = x (t)dt ⎪ ⎪ T 0 ⎪ ⎪ ⎪

⎨ 1 T 4 x (t)dt mˆ 4 = ⎪ T 0 ⎪ ⎪ ⎪

⎪ ⎪ 1 T 6 ⎪ ⎩ mˆ 6 = x (t)dt T 0

(2.5)

With sufficient time span T, estimated value mˆ n can well approach the truth-value mn [2]. With 2D GMM (Eq. 2.4), we obtain the the following: ⎧ (2) ⎪ m = αm (1) ⎪ 2 + (1 − α)m 2 ⎨ 2 (2) m 4 = αm (1) 4 + (1 − α)m 4 ⎪ ⎪ ⎩ (2) m 6 = αm (1) 6 + (1 − α)m 6

(2.6)

(1) (1) Of which, m (1) 2 , m 4 , and m 6 are the 2nd, 4th, and 6th moment of the Gaussian (2) (2) component 1, respectively; m 2 , m (2) 4 , and m 6 are the 2nd, 4th, and 6th moment of the Gaussian component 2, respectively. 2-order matrix is quadratic mean. For the zero-mean gaussian random process, there is the relationship among the moments:

⎧ (1) 2 ⎪ m = σ12 ; m (2) ⎪ 2 = σ2 ⎨ 2 (2) 4 4 m (1) 4 = 3σ1 ; m 4 = 3σ2 ⎪ ⎪ ⎩ (1) 6 m 6 = 15σ16 ; m (2) 6 = 15σ2

(2.7)

Equation (2.7) is substituted into (2.6), then: ⎧ 2 2 ⎪ ⎨ m 2 = ασ1 + (1 − α)σ2 m 4 = 3ασ14 + 3(1 − α)σ24 ⎪ ⎩ m 6 = 15ασ16 + 15(1 − α)σ26

(2.8)

The truth-value is replaced by the estimated results of high-order moment of the non-Gaussian random process given by Eq. (2.5), then: ⎧ 2 2 ⎪ ⎨ mˆ 2 = ασ1 + (1 − α)σ2 mˆ 4 = 3ασ14 + 3(1 − α)σ24 ⎪ ⎩ mˆ 6 = 15ασ16 + 15(1 − α)σ26

(2.9)

8

2 Statistic Analysis of Non-Gaussian Random Load

Ternary system of nonlinear equations defined by Eq. (2.9) can be determined through the calculation software (Matlab symbolic operation). The determined parameters α, σ1 , and σ2 are substituted into Eq. (2.4) to obtain PDF of the symmetrical non-Gaussian amplitude of GMM. It can be seen that the PDF of the symmetrical non-Gaussian amplitude set by Eqs. (2.4) and (2.9) considers the 6th moment of the non-Gaussian random process, unlike the single application of kurtosis, which improves the description accuracy of non-Gaussian statistical property.

2.3 Asymmetrical Non-Gaussian Probability Density Function 2.3.1 Gaussian Mixed Model Uniform expression of GMM is shown by Eq. (2.2), but the math model of asymmetrical non-Gaussian PDF differs from the symmetrical non-Gaussian PDF. Take the 2D GMM as example, the Gaussian component f 1 (x) and f 2 (x) of the symmetrical non-Gaussian random load are zero-mean Gaussian PDF while the Gaussian component f 1 (x) and f 2 (x) of the asymmetrical non-Gaussian random process have different means μ and standard deviations σ . f NG (x) = α f 1 (x|μ1 , σ1 ) + (1 − α) f 2 (x|μ2 , σ2 )

(2.10)

In this way, the odd higher moment of the asymmetrical non-Gaussian random process cannot be 0.

2.3.2 Estimation of Parameters We first presume the analysis based on zero-mean, because the effects of means on the probability distribution can be solved by translation. By Eq. (2.10), the amplitude PDF of the asymmetrical non-Gaussian random process can be expanded as:    1 (x − μ1 )2 (x − μ2 )2 + (1 − α) √ exp − exp − f NG (x) = α √ 2σ12 2σ22 2π σ2 2π σ1 (2.11) 1



Of which, α, μ1 andσ 1 are weight, mean and standard deviation of the Gaussian component 1, respectively; 1−α, μ2, and σ 2 are weight, mean, and standard deviation of the Gaussian component 2. The means μ1 and μ2 are introduced for GMM to fit the asymmetrical non-Gaussian PDF curve. Equation (2.11) has five unknowns: μ1 , μ2, σ 1, σ 2, and α. As for the zero-mean, the central moment of the non-Gaussian

2.3 Asymmetrical Non-Gaussian Probability Density Function

9

process amounts to the origin moment, hereinafter uniformly referred to as moment; 1st moment is 0.

∞

∞

∞ = x f x f x f 2 (x)dx = α + − α) m (NG) (x)dx (x)dx (1 NG 1 1 (2.12) −∞ −∞ −∞ = αμ1 + (1 − α)μ2 = 0 The 2nd moment of the non-Gaussian process equals to quadratic mean. m (NG) 2

=

∞

−∞

x f NG (x)dx = α 2

∞

−∞

x f 1 (x)dx + (1 − α)

∞

2

−∞

x 2 f 2 (x)dx

= α2(1) (x) + (1 − α)2(2) (x) (2.13) Of which, 2(1) (x) and 2(2) (x) are quadratic mean of the Gaussian component 1 and 2, respectively, also the function of quadratic mean and variance: 

2(1) (x) = μ21 + σ12

(2.14)

2(2) (x) = μ22 + σ22

Equation (2.14) is substituted into Eq. (2.13), then the 2nd moment of the nonGaussian random process is expanded as:     = α μ21 + σ12 + (1 − α) μ22 + σ22 m (NG) 2

(2.15)

Similarly, the 3rd moment of the non-Gaussian process is: m (NG) 3

=

∞

−∞

x f NG (x)dx = α 3

= α3(1) (x) + (1 −

∞

x f 1 (x)dx + (1 − α) 3

−∞ α)3(2) (x)

∞

−∞

x 3 f 2 (x)dx

(2.16) Of which, 3(1) and 3(2) are the 3rd-moment origin moment of the Gaussian component 1 and 2, respectively: 

3(1) (X ) = 3μ1 2(1) (X ) − 2μ31 = μ31 + 3μ1 σ12 3(2) (X ) = 3μ2 2(2) (X ) − 2μ32 = μ32 + 3μ2 σ22

(2.17)

Equation (2.17) is substituted into Eq. (2.16), then the 3rd moment of the nonGaussian random process is expanded as:     = α μ31 + 3μ1 σ12 + (1 − α) μ32 + 3μ2 σ22 m (NG) 3

(2.18)

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2 Statistic Analysis of Non-Gaussian Random Load

Similarly, the 4th and 5th moment of the non-Gaussian random process are shown in Eqs. (2.19) and (2.20): = α4(1) + (1 − α)4(2) m (NG) 4

(2.19)

m (NG) = α5(1) + (1 − α)5(2) 5

(2.20)

For Eq. (2.19): 

4(1) (X ) = 4μ1 3(1) (X ) − 6μ21 2(1) (X ) + 3μ41 + 3σ14 = μ41 + 6μ21 σ12 + 3σ14 4(2) (X ) = 4μ2 3(2) (X ) − 6μ22 2(2) (X ) + 3μ42 + 3σ24 = μ42 + 6μ22 σ22 + 3σ24 (2.21)

Equation (2.21) is substituted into Eq. (2.19), then the 4th moment of the nonGaussian random process is:     = α μ41 + 6μ21 σ12 + 3σ14 + (1 − α) μ42 + 6μ22 σ22 + 3σ24 m (NG) 4

(2.22)

For Eq. (2.20) : 

5(1) = 5μ1 4(1) − 10μ21 3(1) + 10μ31 2(1) − 4μ51 = μ51 + 10μ31 σ12 + 15μ1 σ14 5(2) = 5μ2 4(2) − 10μ22 3(2) + 10μ32 2(2) − 4μ52 = μ52 + 10μ32 σ22 + 15μ2 σ24 (2.23)

Equation (2.23) is substituted into Eq. (2.20), then the 5th moment of the nonGaussian random process is:     = α μ51 + 10μ31 σ12 + 15μ1 σ14 + (1 − α) μ52 + 10μ32 σ22 + 15μ2 σ24 m (NG) 5 (2.24) In practice, every moment of the non-Gaussian random load is unknown and usually estimated as per the sample records. Presuming the time series of the sample for the zero-mean non-Gaussian process is x(t), then estimation of the ith moment is:   mˆ i(NG) = E x i (t) , i = 1, 2, . . .

(2.25)

The estimation is replaced by the theoretical value, then we can obtain a system of quintic Equations for α, μ1 , μ2 , σ1 , σ2 , based on (2.12), (2.15), (2.18), (2.22), and (2.24):

2.3 Asymmetrical Non-Gaussian Probability Density Function

11

⎧ 0 = αμ1 + (1 − α)μ2 ⎪ ⎪ ⎪     ⎪ (NG) ⎪ ⎪ mˆ 2 = α μ21 + σ12 + (1 − α) μ22 + σ22 ⎪ ⎨     = α μ31 + 3μ1 σ12 + (1 − α) μ32 + 3μ2 σ22 mˆ (NG) 3 ⎪ ⎪ ⎪ mˆ (NG) = α μ4 + 6μ2 σ 2 + 3σ 4  + (1 − α)μ4 + 6μ2 σ 2 + 3σ 4  ⎪ ⎪ 1 1 1 1 2 2 2 2 ⎪ ⎪ 4   5   5 ⎩ (NG) 3 2 4 3 2 = α μ1 + 10μ1 σ1 + 15μ1 σ1 + (1 − α) μ2 + 10μ2 σ2 + 15μ2 σ24 mˆ 5 (2.26) Solving the above quintic nonlinear equation s can give estimation of the parameˆ Substituting the estimation into the Eq. (2.11) can get the ters: μˆ 1 , μˆ 2 , σˆ 1 , σˆ 2 , and α. expression of the PDF of non-Gaussian random load. The above nonlinear equation s can be solved by means of the numerical method or symbolic operation software. It can be seen that, compared with skewness and kurtosis only, the skewness nonGaussian PDF considers the 5th moment of the non-Gaussian random process, thus improving the description accuracy for the non-Gaussian statistical property.

2.4 Case Analysis 2.4.1 Case of Symmetrical Non-Gaussian Random Process Two cases are given in the study to verify the validity of the method proposed: 1. 2.

Simulated non-Gaussian random signal with small kurtosis; Measured non-Gaussian random vibration signal of vehicle with big kurtosis;

The PDF of the four non-Gaussian random signals (two are simulation signals, and other other two are measured signals) are estimated by GMM. The method proposed is compared with other methods for a quantitative verification of the validity and accuracy.

2.4.1.1

Simulated Signal

The non-Gaussian random signals with symmetrical distribution are obtained by simulation. Figure 2.1 shows that mean μ = 0, variance σ 2 = 1.1976 × 103 , skewness γ3 = 0, kurtosis γ4 = = 8.1394. Figure 2.1a is the time series of the sample; Fig. 2.1b is PSD, a typical broadband non-Gaussian random process. The sample series shown in Fig. 2.1a is substituted into (2.5), then we can obtain: ⎧ 3 ⎪ ⎨ mˆ 2 = 1.1976 × 10 mˆ 4 = 1.1673 × 107 ⎪ ⎩ mˆ 6 = 2.5042 × 1011

(2.27)

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2 Statistic Analysis of Non-Gaussian Random Load

Fig. 2.1 Simulated symmetrical non-Gaussian random signal (kurtosis = 8.14)

The above results are substituted into (2.9), then we can obtain: ⎧ ⎪ ⎨ αˆ = 0.8120 σˆ 12 = 443.4025 ⎪ ⎩ 2 σˆ 2 = 4.4552 × 103

(2.28)

Results of Eq. (2.28) are substituted into Eq. (2.4), then the PDF of the nonGaussian amplitude is:     0.1880 0.8120 x2 x2 + f NG (x) = √ exp − √ exp − 886.8050 8.9104 × 103 21.0571 2π 66.7472 2π (2.29) Figure 2.2 show the non-Gaussian PDF curve based on four methods: (1) empirical distribution of sample series; (2) GMM; (3) Gaussian distribution; (4) 4th moment Edgeworth series expansion method [3]. Since the time-domain sample series have sufficient length, the empirical distribution is considered reliable and can serve as the benchmark of other results. Figure 2.2a gives the PDF curve at the linear coordinate, explicitly showing the differences of the middle peak; Fig. 2.2b gives the PDF curve at the semi-logarithmic coordinate, explicitly showing the differences of the tail of the curve. It can be seen that the PDF curve based on the Gaussian hypothesis has a big difference with the curve of empirical distribution, therefore, processing the non-Gaussian signals based on the Gaussian hypothesis would bring about errors. The PDF curve of non-Gaussian amplitude, which is expanded based on the 4th moment Edgeworth series expansion, has negative values and multiple kurtosis [3]. The Edgeworth series expansion method is available only for the nonGaussian random signal with small kurtosis. Methods based on GMM can always approach the curve of empirical distribution no matter near the peak or the tail.

2.4 Case Analysis

13

Fig. 2.2 Amplitude PDF curve of simulated symmetrical non-Gaussian random signal

2.4.1.2

Measured Signal

Figure 2.3 shows that measured non-Gaussian random vibration signals at the loading platform of a military vehicle driving at 25 km/h on the asphalt road. Mean of the vibration signal is μ = 0, variance σ 2 = 38.6462, skewness γ 3 = 0, and kurtosis γ 4 = 22.9716. Figure 2.3a is the time series of the sample; Fig. 2.3b is PSD, the narrowband non-Gaussian random process. The sample series shown in Fig. 2.3a is substituted into (2.5), then we can obtain: ⎧ ⎪ ⎨ mˆ 2 = 38.6462 mˆ 4 = 3.4308 × 104 ⎪ ⎩ mˆ 6 = 1.1175 × 108

(2.30)

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2 Statistic Analysis of Non-Gaussian Random Load

Fig. 2.3 Measured symmetrical non-Gaussian random signal (kurtosis = 22.9716)

The above results are substituted into (2.9) to solve the system of equations, then we can obtain the estimation of parameters: ⎧ ⎪ ⎨ αˆ = 0.9765 σˆ 12 = 23.1847 ⎪ ⎩ 2 σˆ 2 = 681.6913

(2.31)

Results of Eq. (2.31) in substituted into Eq. (2.4), then the PDF of the non-Gaussian amplitude is:     0.0235 0.9765 x2 x2 + f NG (x) = √ exp − √ exp − 46.3695 1.3634 × 103 4.8151 2π 26.1092 2π (2.32) Figure 2.4 gives the PDF curve based on four methods: (1) empirical distribution of sample series; (2) GMM; (3) Gaussian distribution; (4) 4th moment Edgeworth series expansion method [3]. The empirical distribution of the sample serves as the benchmark for other results. Compared with results in Sect. 2.4.1.1, errors of the PDF based on the Gaussian hypothesis and the Edgeworth series expansion method increase as kurtosis increases while GMM can still well fit the probability distribution of the amplitude of signals.

2.4.1.3

Result Analysis

To further verify the accuracy of our method, relative mean-square error is used to analyze the degree of deviation of the amplitude PDF curve to the empirical distribution curve, of which, the relative mean-square error is defined as:

2.4 Case Analysis

15

Fig. 2.4 Amplitude PDF curve of measured symmetrical non-Gaussian random signal

  E ( f − f EM )2  2  r= E f EM

(2.33)

Of which, f is the non-Gaussian PDF based on a specific method, and f EM is the results of empirical distribution. To fully verify the validity of our method, a simulated signal and a measured vibration signal of one plane are added to the cases in Sect. 2.4.1.1 and 2.4.1.2. The PDF based on GMM and the Gaussian hypothesis are calculated, respectively; and the relative mean-square error between the PDF based on the Edgeworth expansion method and the sample empirical distribution is computed; results are seen in Table 2.1. The comparison has shown that calculation of GMM has the minimum relative error, but the Edgeworth expansion method is available only for the situations with small kurtosis and the computation error increases quickly as kurtosis increases (Fig. 2.4).

16

2 Statistic Analysis of Non-Gaussian Random Load

Table 2.1 Relative errors of the amplitude PDF of the symmetrical non-Gaussian signal Method

Simulation r (%)

Measured r (%)

γ4 = 5.0191

γ4 = 8.1394

Vibration of the plane γ4 = 4.3044

GMM

0.0242

0.0442

0.72

0.19

Gaussian hypothesis

9.18

7.66

0.84

3.78

Edgeworth

2.67

4.19

1.25

297.95

Vibration of the vehicle γ4 = 22.9716

2.4.2 Case of Asymmetrical Non-Gaussian Random Process Two cases are given to verify the validity of the calculation methods for the PDF of the asymmetrical non-Gaussian amplitude: (1) Simulated signals based on nonlinear transformation; (2) Response signal of non-Gaussian stress obtained from the vibration test on cantilever.

2.4.2.1

Simulated Signal

Gaussian signal of zero-mean x(t), standard deviation of signal σx = 74.85, the time series, and PSD are first generated, as shown in Fig. 2.5a and b. Nonlinear transformation of the Gaussian signal is performed as follows, and means are deleted to obtain the zero-mean non-Gaussian signal z 0 (t): z(t) = x(t) + 0.002x 2 (t); Fig. 2.5 Simulated Gaussian random signal

z 0 (t) = z(t) − mean(z)

(2.34)

2.4 Case Analysis

17

Fig. 2.6 Simulated asymmetrical non-Gaussian random signal

The time series and PSD of z0 (t) are shown in Fig. 2.6 a and b. Standard deviation σ z0 , kurtosis γ 3 , and skewness γ 4 of z0 (t) are 76.78, 0.9150, and 4.1969, respectively. The non-Gaussian time series z0 (t) is substituted into Eq. (2.25) to obtained the estimation of every moment: 

= 5.8952 × 103 ; mˆ (NG) = 4.1415 × 105 mˆ (NG) 2 3 mˆ (NG) = 1.4586 × 108 ; mˆ (NG) = 2.9389 × 1010 4 5

(2.35)

The above results are substituted into (2.26) to obtain the estimation of GMM: 

μˆ 1 = −25.2518;μˆ 2 = 52.3119 σˆ 1 = 58.0970; σˆ 2 = 101.6851;

(2.36)

The above results are substituted into (2.11) to obtain the amplitude PDF of the non-Gaussian random process Z(t), as shown in Fig. 2.6a. f NG (z) =

  0.7729 (z + 25.2518)2 √ exp − 6.7505 × 103 58.0970 2π   0.2271 (z − 52.3119)2 + √ exp − 2.0680 × 104 101.6851 2π

(2.37)

Figure 2.7 shows the non-Gaussian PDF curve defined by Eq. (2.37), together with the PDF curves based on the Edgeworth expansion method [3] and the W–H (Winterstein–Hermite) model [6] as well as the empirical distribution curve based on the sample series. Figure 2.7a is linear coordinate, showing the differences at the middle peak of the curve; Fig. 2.7b is semi-logarithmic coordinate, showing the differences at the tail of the curve. Comparison has shown that GMM and W–H model

18

2 Statistic Analysis of Non-Gaussian Random Load

Fig. 2.7 Amplitude PDF curve of simulated asymmetrical non-Gaussian random signal

generate better results with a regionally fluctuant curve; for the non-Gaussian signals with bigger kurtosis and skewness, these fluctuations shall proceed, thus generating the multiple peaks and even negative values. To further verify the accuracy of those methods, the relative mean-square error (Eq. 2.3) is used to analyze the degree of deviation of the amplitude PDF curve to the empirical distribution results, as shown in Table 2.2. With the calculation in Fig. 2.7 and Table 2.2, we can see that GMM proposed in this chapter can give the Table 2.2 Relative errors r of the amplitude PDF of the asymmetrical non-Gaussian signal

Method

Simulation r (%)

Measured r (%)

γ 3 = 0.9150, γ 4 = 4.1969

γ 3 = −0.7102, γ 4 = 4.7977

GMM

1.03

0.30

W–H model

1.43

0.91

Edgeworth

2.13

1.39

2.4 Case Analysis

19

most precise analytical expression of the PDF of the non-Gaussian signal shown in Fig. 2.6.

2.4.2.2

Measured Signal

Figure 2.8 shows the structure of the cantilever made of Al2024-T3. The bottom of the cantilever is clamped on the vibrostand for the primary excitation vibrating system shown in the picture. The acceleration signal of the primary excitation is shown in Fig. 2.9, and it is symmetrical non-Gaussian signal; skewness γ 3 = 0, kurtosis γ 4 = 6, and standard deviation σ = 10 g. Considering the effects of gravity and nonlinear factors, stress response at the bottom of the cantilever should be asymmetrical nonGaussian process, and the stress signal after deleting the means is shown in Fig. 2.10. The standard deviation σ = 41 MPa, skewness γ3 = −0.7102, and kurtosis γ4 = 4.7977. The non-Gaussian stress response series in Fig. 2.10 is substituted into Eq. (2.25) to obtain the estimation of each moment:

Fig. 2.8 Primary excitation vibration test on cantilever structure (unit: mm)

Fig. 2.9 Non-Gaussian random excitation signal of vibration

20

2 Statistic Analysis of Non-Gaussian Random Load

Fig. 2.10 Non-Gaussian random stress response signal of vibration



mˆ (NG) = 1.6783 × 103 ; mˆ (NG) = − 4.8831 × 104 2 3 mˆ (NG) = 1.3514 × 107 ; mˆ (NG) = − 1.2172 × 109 4 5

(2.38)

The above results are substituted into (2.26) to obtain the estimation of GMM: 

μˆ 1 = 6.6934; μˆ 2 = −33.8174 σˆ 1 = 32.1114; σˆ 2 = 59.8164; αˆ = 0.8348

(2.39)

Equation (2.39) is substituted into (2.11) to obtain the amplitude PDF of the non-Gaussian random load X(t), as shown in Fig. 2.10a. f NG (x) =

  0.8348 (x − 6.6934)2 √ exp − 2.0623 × 103 32.1114 2π   0.1652 (x + 33.8174)2 + √ exp − 7.1560 × 103 59.8164 2π

(2.40)

Figure 2.11 shows the empirical distribution curves of GMM, Edgeworth series expansion method, W–H model, and sample series, respectively. As per Eq. (2.33), the relative mean-square error of every method compared with the empirical distribution is shown Table 2.2. Based on Fig. 2.11 and Table 2.2, the method based on GMM can generate the optimal analytical expression of the PDF curve for the non-Gaussian stress signal shown in Fig. 2.10. The general skewness of the asymmetrical non-Gaussian random signal of vibration in practical engineering is −1.2 < γ3 < 1.2. The skewness of simulated signal in case 2.4.2.1 is 0.9150, and the skewness in this case is −0.7102. Both are close to

2.4 Case Analysis

21

Fig. 2.11 Amplitude PDF curve of measured asymmetrical non-Gaussian random stress signal of vibration

the upper limit and the lower limit of the skewness of common non-Gaussian signals. By analyzing these cases we can conclude that GMM proposed in this chapter can precisely express the amplitude PDF of the asymmetrical non-Gaussian signal.

2.5 Brief Summary This chapter studies the statistic analysis methods of non-Gaussian random process which underpins the calculation of fatigue life of the non-Gaussian random load, reliability analysis, and acceleration test of random vibration. First, a review on the

22

2 Statistic Analysis of Non-Gaussian Random Load

definition and simplified hypotheses of common non-Gaussian higher-order statistics was made, the definition of skewness and kurtosis with their estimation methods were analyzed, and the existing problems in practical engineering were discussed. Then, the PDF model for the symmetrical non-Gaussian process and the asymmetrical non-Gaussian random process were built based on GMM. A system of equations was set to solve the unknown parameters of the model by means of the quantitative relation between moments of the non-Gaussian random process and the higher moment of the non-Gaussian random process. Then the analytical expression of PDF for the symmetrical and asymmetrical non-Gaussian random process were obtained. Last, our method was verified by different simulated signals and measures signals, and compared with common approaches qualitatively and quantitatively. Results have shown that the method based on GMM proposed by this chapter can generate the optimal analytical expression of non-Gaussian PDF.

References 1. Benasciutti D (2004) Fatigue analysis of random loadings. University of Ferrara, Ferrara 2. Bendat JS, Piersol AG (1971) Random data: analysis and measurement procedures. Wiley 3. Harremoes P (2005) Maximum entropy and the edgeworth expansion. In: Proceedings of the Information Theory Workshop. p 4 4. Mendel JM (1991) Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications. 49:278–305 5. Middleton D (1999) Non-Gaussian noise models in signal processing for telecommunications: new methods an results for class A and class B noise models. Inform Theory IEEE Trans 45:1129–1149 6. Winterstein SR (1985) Non-normal responses and fatigue damage. J Eng Mech 111:1291–1295

Chapter 3

Simulation of Non-Gaussian/Non-stationary Random Vibration

Abstract Stochastic processes are used to represent phenomena in many fields. Numerical simulation method is widely applied for the solution to stochastic problems of complex structures when alternative analytical methods are not applicable. In some practical applications the stochastic processes show non-Gaussian properties. When the stochastic processes deviate significantly from Gaussian, techniques for their accurate simulation must be available. In this chapter, a novel approach for the simulation of non-Gaussian stochastic processes with the prescribed amplitude probability density function (PDF) and power spectral density (PSD) by amplitude modulation and phase reconstruction was presented. Because non-Gaussian random vibration also often appears in the form of non-stationary random vibration, a novel method to model and simulate non-stationary vibration signals based on Hilbert spectrum was proposed. The above methods of simulating non-Gaussian and nonstationary random vibration have been verified the effectiveness and feasibility by numerical simulations and practical experiments, which further could be applied in many fields such as structural dynamics, and vibration testing.

3.1 Simulation of Non-Gaussian Stochastic Processes by Amplitude Modulation and Phase Reconstruction Stochastic processes are used to represent phenomena in many diverse fields. Numerical simulation method is widely applied for the solution to stochastic problems of complex structures when alternative analytical methods are not applicable. In some practical applications the stochastic processes show non-Gaussian properties. When the stochastic processes deviate significantly from Gaussian, techniques for their accurate simulation must be available. The various existing simulation methods of non-Gaussian stochastic processes generally can only simulate super-Gaussian stochastic processes with the high-peak characteristics. And these methodologies are usually complicated and time consuming, not sufficiently intuitive. By revealing the inherent coupling effect of the phase and amplitude part of discrete Fourier representation of random time series on the non-Gaussian features (such as skewness and kurtosis) through theoretical analysis and simulation experiments, this book © National Defense Industry Press 2022 Y. Jiang et al., Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test, https://doi.org/10.1007/978-981-16-3694-3_3

23

24

3 Simulation of Non-Gaussian/Non-stationary Random Vibration

presents a novel approach for the simulation of non-Gaussian stochastic processes with the prescribed amplitude probability density function (PDF) and power spectral density (PSD) by amplitude modulation and phase reconstruction. As compared to previous spectral representation method using phase modulation to obtain a nonGaussian amplitude distribution, this non-Gaussian phase reconstruction strategy is more straightforward and efficient, capable of simulating both super-Gaussian and sub-Gaussian stochastic processes. Another attractive feature of the method is that the whole process can be implemented efficiently using the Fast Fourier Transform. Cases studies demonstrate the efficiency and accuracy of the proposed algorithm.

3.1.1 Introduction Engineering structures are often subjected to stochastic loadings such as earthquakes, winds or ocean waves. Most loadings in nature can be modeled as Gaussian processes in light of the central limit theorem. However, the stochastic processes show nonGaussian properties in some practical applications, such as the random vibrations generated by wheeled vehicles travelling over irregular terrain, and wind pressure fluctuations on building envelopes. Numerical simulation method is widely applied for the solution to stochastic problems of complex structures when alternative analytical methods are not applicable. When the stochastic processes deviate significantly from Gaussian, techniques for their accurate simulation must be available. The simulation of Gaussian processes has been explored for several decades, while non-Gaussian simulation has not been as widely addressed. Yamazaki and Shinozuka [1] proposed a method of simulating non-Gaussian time series by a nonlinear static transformation from Gaussian to non-Gaussian with the aid of an iterative procedure. Spectral Correction method based on modified Hermite polynomial transformation for the simulation of a class of non-normal random processes has been presented by Gurley [2]. Seong and Peterka [3] presented a new method based on the Fourier representation of random time series for generating artificial surface-pressure time series focused on the simulation of non-Gaussian spiky fluctuation features. Kumar and Stathopoulos [4] presented an efficient and practical method based on the Fast Fourier Transform (FFT) for the digital generation of univariate non-Gaussian wind pressure time series on low building roofs. Gioffrè et al. [5] proposed a simulation algorithm using the correlation-distortion method based on translation vector processes to generate non-Gaussian wind pressure fields. Kanda and Hang [6] reviewed several simulation methods based on the translation method using logarithmic and polynomial functions for simulating the non-Gaussian stationary process. Phoon and Huang [7] proposed a simulation algorithm using Karhunen–Loeve expansion to generate strongly non-Gaussian processes. Bocchini and Deodatis [8] reviewed and introduced the latest developments of a class of simulation algorithms for strongly non-Gaussian random fields. Poirion and Puig [9] presented a simulation technique of multivariate non-Gaussian random processes and fields based on Rosenblatt’s transformation of

3.1 Simulation of Non-Gaussian Stochastic Processes by Amplitude …

25

Gaussian processes. Shields and Deodatis [10] presented a technique for simulation of strongly non-Gaussian stochastic vector processes using translation process theory. Zentner et al. [11] presented a new method for generating synthetic ground motion based on Karhunen–Loeve decomposition and a non-Gaussian stochastic model. Aung et al. [12] presented a stochastic non-Gaussian simulation algorithm using a cumulative distribution function (CDF) mapping technique that converges to a desired target power spectral density. Yura and Hanson [13] proposed a simulation method of two-dimensional random fields with arbitrary power spectra and non-Gaussian probability distribution functions. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. Li and Li [14] presented a direct simulation algorithm by expanding the autoregressive (AR) model and the autoregressive moving average (ARMA) model for the generation of a class of non-Gaussian stochastic processes according to target lower-order moments and prescribed power spectral density (PSD) function. Jihong et al. [15] presented a simplified simulation method of non-Gaussian wind load based on the inverse fast Fourier transform (IFFT), in which the amplitude spectrum is established via a target power spectrum and the phase spectrum is constructed by introducing the exponential peak generation (EPG) model. Luo et al. [16] presented a simulation methodology of the stationary nonGaussian stochastic wind pressure field based on the zero memory nonlinearity translation method and the spectral representation method. Jing and Xin [17] presented an exponential model for fast simulation of multivariate non-Gaussian processes with application to structural wind engineering. Vargas-Guzmán [18] presented new parametric heavy-tailed distributions for non-Gaussian simulations with higher-order cumulant parameters predicted from sample data. Aung et al. [19] developed a new wavelet-based CDF mapping technique for simulation of multivariate non- Gaussian wind pressure process. Shields and Deodatis [20] presented a simple and efficient methodology to approximate a general non-Gaussian stationary stochastic vector process by a translation process. As mentioned above, there is a wide range of different methodologies in the literature for simulation of non-Gaussian stochastic processes. In general, these previous methods can be broadly classified into three categories as follows: (1) ARMA class model with non-Gaussian white noise, (2) translation process-based, or nonlinear static transformation (NST), from the Gaussian random processes to non-Gaussian processes and (3) the spectral representation method. For category 1, the ARMA approach is based on the simple and well-known theory of linear difference equations and is computationally efficient. However, ARMA models cannot represent data exhibiting sudden spikes of very large amplitude at irregular intervals and having negligible probability of very high level crossings; therefore, these are not suitable for representing strong non-Gaussian time series. For category 2, the basic feature of translation process-based method is to generate a non-Gaussian translation field by mapping an underlying Gaussian process to the desired non-Gaussian marginal probability distribution function (PDF) according to a prescribed power

26

3 Simulation of Non-Gaussian/Non-stationary Random Vibration

spectral density (PSD). The translation process-based method usually employs iterative schemes to match both the prescribed non-Gaussian marginal PDF and PSD function. But because the transformation of Gaussian process to non-Gaussian process will change both the PDF and PSD in the same time, it is unable to ensure the compatibility between prescribed non-Gaussian PDF and PSD. The spectral representation method, of category 3, is based on discrete Fourier representation of random time series, which consists of the superposition of sinusoids at discrete frequencies that possess deterministic amplitude and random phase. The spectral representation method calculates the amplitude part of the Fourier coefficient (modulus of sinusoids) by the specified PSD, and approximates the non-Gaussian parameters (such as skewness and kurtosis) of a given amplitude PDF by changing the phase part of the Fourier coefficient (phase angles of sinusoids). Since the phase part is free from the second-order properties of random time series (such as PSD), the spectral representation simulation methodology can ensure the compatibility between prescribed non-Gaussian PDF and PSD. However, previous spectral representation methods only consider and utilize the impact of the Fourier phase coefficients for nonGaussian characteristics. Therefore, according to the central limit theorem, changing the randomness of the uniformly distributed random phases at different frequencies will strengthen the fluctuating features such as the sharp spike events of the generated random signal. This means that will increase the kurtosis value of the original Gaussian signal, resulting in super-Gaussian random signal. So the various existing methods of changing the phase part of the Fourier coefficient according to the non-Gaussian parameters (such as skewness and kurtosis) can only simulate non-Gaussian stochastic processes with the high-peak characteristics, that is superGaussian. This can be confirmed by the simulation examples of the papers listed in the references. In some cases, sub-Gaussian stochastic processes with less high peaks will be encountered or be useful for important application. For example, shaker power will be increased by the sub-Gaussian random control with decreased kurtosis for the purpose of less risk of damage to the test item and shaker in modal testing Steinwolf [21, 22]. Hence in order to simulate sub-Gaussian random processes, it is necessary to study and analysis the coupling effect of both the Fourier phase and amplitude coefficients on the non-Gaussian features of simulated random process. We will discuss in detail on this issue through theoretical analysis and simulation experiments in the following sections.

3.1.2 Theoretical Background The well known spectral representation is based on a discretized model of the target power spectral density (PSD) function G T ( f ) for the desired process. The simulation consists of the superposition of harmonics at discrete frequencies that possess deterministic amplitude and random phase (DARP). A zero mean stationary Gaussian realization can be simulated by DARP as N → ∞:

3.1 Simulation of Non-Gaussian Stochastic Processes by Amplitude …

x(t) =

N −1 

Ak cos(2π f k t + φk )

27

(3.1)

k=0

where Ak are determined by the target one sided PSD values G T ( f ) at the corresponding frequencies f k Ak =



2G T ( f k ) f

(3.2)

 f = f u /N

(3.3)

f k = k f, k = 0, 1, . . . , N − 1

(3.4)

f u is the upper cutoff frequency beyond which G T ( f ) can be considered to be zero. φk is the kth realizations of a uniformly distributed random phase angles from 0 to 2π or from −π to π . In order to improve simulation efficiency by employing the Fast Fourier Transform (FFT), Eq. (3.1) can be rewritten as x(nt) = Re

 M−1 

 Ak e

iφk i2π f k nt

e

, n = 0, 1, . . . , M − 1

(3.5)

k=0

where Re{.} represents the real part of the expression enclosed in brace, and M is the number of time intervals of length t, which is defined by sampling frequency f s t =

1 fs

(3.6)

According to sampling theorem, f s must satisfy fs ≥ 2 fu

(3.7)

It is obvious that f =

fs fu = N M

(3.8)

Thus, M and N must satisfy M ≥ 2N . Inserting Eqs. (3.4), (3.6) and (3.8) into (3.5) yields a discrete fourier representation of time series

28

3 Simulation of Non-Gaussian/Non-stationary Random Vibration

x(nt) = Re

 M−1 

 Ak e

iφk i2πnk/M

e

= Re{IDFT(Ck )}

(3.9)

k=0

where IDFT means Inverse Discrete Fourier Transform, and Ck is Ck = Ak eiφk = Ak (cosφk + isinφk ), k = 0, 1, . . . , M − 1

(3.10)

Note that M is selected to be a power of 2 to use the IFFT algorithm. The realizations of x(t) obtained using either Eq. (3.1) or (3.5) follow a Gaussian distribution in the limit as N → ∞ due to the Central Limit Theorem and can be considered approximately Gaussian for most practical applications if N is greater than approximately 100. Zero mean Gaussian stochastic processes can be adequately described by power spectral density, while for non-Gaussian stochastic processes, higher marginal moments, more precisely the marginal skewness and kurtosis, are additionally used to describe the non-Gaussian features. The kurtosis (K) and skewness(S) value of a stochastic processes X is defined by the expression K =

E[X − E(X )]4 2 − 3 E[X − E(X )]2

(3.11)

E[X − E(X )]3 3/2 E[X − E(X )]2

(3.12)

S=

It should be noted that the kurtosis defined in the Eq. (3.11) is a normalized value. Under this definition,the kurtosis value of a Gaussian random signal is 0, which is equivalent to a kurtosis value 3 defined in some other literatures. Kurtosis (K) is used to describe the tail distribution feature of amplitude probability density function, and skewness (S) is used to describe the asymmetry of amplitude probability density function. In case of symmetrical distributions, kurtosis is the most important coefficient used to find out how much the distribution differs from the normal distribution. It is well known that both skewness and kurtosis of Gaussian stochastic processes are equal to zero, and the stochastic processes with K > 0 are said over-Gaussian or super-Gaussian stochastic processes, while stochastic processes with K < 0 are said sub-Gaussian stochastic processes. To simulate non-Gaussian processes with given PSD, skewness and kurtosis values by suitable amplitude modulation and phase reconstruction, the relationships between the target characteristics(PSD, skewness, and kurtosis) and the variables in the fourier series model in Eq. (3.10) (amplitude and phase part) should be considered. Since the PSD does not depend on the phase angles (see Eq. 3.2), the variation of phase part does not affect the second order characteristics (variance, PSD) of the time series. On the other hand, an earlier study by shows that the spikes in the time domain, responsible for non-Gaussian nature, are strongly dependent on the phase part of the

3.1 Simulation of Non-Gaussian Stochastic Processes by Amplitude …

29

fourier transform [23]. In this book, further analysis showed that the skewness and kurtosis of the simulated random process depend not only on the phase part of the fourier transform, but also on the amplitude part. This book presents a theoretical background of the method and a series of simulation experiments in Sect. 3.1.3 to understand the mechanism of phase reconstruction by amplitude modulation, which provides a fundamental framework for the simulation method. The kurtosis of the random process x(t) can be expressed as the following form [21, 22] ⎧ N −1  ⎪ 3 4 ⎪ ⎪ − Ak + 2 A j A3k cos(φ j − 3φk )+  N −1 −2 ⎪ ⎪ ⎨ 2  k=0 j=3k A2k K =  ⎪ ⎪ 6 A j Ak A2n cos(φ j − φk − 2φn )+ k=0 ⎪ ⎪ ⎪ ⎩ j=k+2n k=n



6

A j Ak A2n cos(φ j + φk − 2φn )+

j+k=2n j 2. f M-C (·) indicates the functional relations. The first four concrete expressions of the M-C formula are

5.2 A Method for Estimating Rain-Flow Fatigue Damage …

⎧ c1 ⎪ ⎪ ⎨ c2 ⎪ c ⎪ ⎩ 3 c4

= m1 = m 2 − m 21 = m 3 − 3m 1 m 2 + 2m 31 = m 4 − 3m 22 − 4m 1 m 3 + 12m 21 m 2 − 6m 41

93

(5.12)

Hence, for a non-Gaussian process, only taking kurtosis into consideration means accepting the assumption that ci = 0, if i > 4, then all moments above fourth-order can be defined by m1 , m2 , m3 and m4 : m 1 = g(m 1 , m 2 , m 3 , m 4 ), for i > 4

(5.13)

where g(·) signifies the functional relations. Based on this assumption, the nonnormality is determined entirely by {m1 , m2 , m3 , m4 }. In practice, non-Gaussian random processes vary, and the assumption described above does not always hold. As previously indicated, the fatigue damage is most sensitive to the bth-order moment of the loading process instead of the kurtosis for a specified S–N curve NS b = A. If the value of b is approximately three or four, the kurtosis-based models will provide rational estimations. However, the kurtosis-based model will produce substantial inaccuracies when b > 4, and the assumptions listed in Eq. (5.13) do not hold true.

5.2.3 Nonlinear Transformation Models There are several transformation models for non-Gaussian fatigue damage estimations [16, 20, 30, 42]. The two most common are the Kihl et al. [20] and Winterstein-Hermite (W–H) models [42].

5.2.3.1

Kihl Model

The expression of the Kihl Model is X + β(sgn(X ))(|X |n ) , Z = G Kihl (X ) = C ⎧    ⎫ 2(n+1)/ 2 n n 2 σ X(n−1) ⎪ ⎪ ⎪ ⎪ β⎪ √ ⎨1 + ⎬ ⎪ π C =    2(n−1) ⎪

⎪ ⎪ 2n n + 1 2 σ X ⎪ ⎪ ⎩+ ⎭ β2 ⎪ √ π

(5.14)

94

5 Fatigue Life Analysis Under Non-Gaussian Random…

where sgn(·) is the signum function (sgn(x) = 1 for x > 0; 0 for x = 0 and −1 for x < 0). The variables β and n are the parameters used to control the intensity of the non-normality of Z(t), X(t) is the underlying Gaussian process, C is a normalized parameter used to guarantee that the transformed process has the same variance as the underlying Gaussian process, (·) denotes the gamma function and GKihl (·) is a monotonically non-decreasing function. The transformed process Z(t) retains the same peak occurrence rate vp as the underlying Gaussian process X(t). The kurtosis of Z(t) is

Kz =

  E Z4 σ Z4

=

  E (X + βsgn(X )|X |n )4 C 4 σ X4

(5.15)

In the Kihl model, n defines the intensity of non-normality and β defines the size of the nonlinear portion. Different sets of {β, n} values may result in different transformed processes but the same kurtosis value (see Fig. 5.8 in Ref. [20]).

5.2.3.2

W–H Model

The W–H model is expressed as   I  Z − μZ = Z 0 = G WH (X ) = k X + h˜ i H ei−1 (X ) σZ i=3       = k X + h˜ 3 X 2 − 1 + h˜ 4 X 3 − 3X + · · ·

(5.16)

where μZ and σ Z are the mean and standard deviation of the non-Gaussian process, h˜ i controls the shape of the distribution curves, and k is a scaling factor used to guarantee that the variance of Z 0 (t) is unity. In practical applications, the W–H model is always truncated by I = 4 and h˜ 4 =

√

1 + 1.5(γ4 − 3) − 1 ˜ γ3  , h3 =  18 6 1 + 6h˜ 4

(5.17)

where γ 3 and γ 4 are the skewness and kurtosis of the standardized non-Gaussian process Z 0 (t), and 1 k= 1 + 2h˜ 23 + 6h˜ 24

(5.18)

The applicable range of the W–H model can be found in Fig. 5.11 in Ref. [11].

5.2 A Method for Estimating Rain-Flow Fatigue Damage …

95

The inverse transformation of Eq. (5.16) for leptokurtic non-Gaussian processes (γ 4 > 3) G −1 WH (·) defines the standardized Gaussian process X 0 : 1/ 3  ξ 2 (Z ) + c + ξ (Z ) X 0 = G −1 WH (Z ) =  1/ 3 − ξ 2 (Z ) + c − ξ (Z ) −a  where ξ (Z ) = 1.5d a +

Z −μ Z kσ Z



(5.19)

3  − a 3 , a = h˜ 3 /3h˜ 4 , d = 1/3h˜ 4 , c = d − 1 − a 2 .

5.2.4 Fatigue Damage Estimation Based on Nonlinear Transformation Models For a narrowband Gaussian process, the rain-flow cycle distribution is modeled by a Rayleigh distribution: f Gau (S X ) =

  S X2 SX exp − σ X2 2σ X2

(5.20)

where S X is the amplitude of the rain-flow cycle, and σ X2 is the variance of a Gaussian loading process. Based on the monotonic transformation G(·) (GKihl (·) or GWH (·)), the rain-flow cycle distribution for a narrowband non-Gaussian process can be derived as   2    −1 G −1 (S Z ) G −1 (S Z ) exp − (5.21) f NG (S Z ) = G (S Z ) 2 2 σX 2σ X where S Z is the amplitude of rain-flow cycle of the non-Gaussian process. Using the rain-flow cycle distribution and Miner’s rule for a prescribed S–N curve (NS b = A), the expected fatigue damage from a single loading cycle is ∞ E[D N G ] =

S Zb f N G (S Z )dS Z A

(5.22)

0

For a time period T, the expected accumulation of fatigue damage is vpT E[DNG ] = A

∞ S Zb f NG (S Z )dS Z = 0

vpT b M S (Z ) A

(5.23)

96

5 Fatigue Life Analysis Under Non-Gaussian Random…

where M Sb (Z ) is the bth-order original moment of the amplitude of the rain-flow cycles, and vp is the expected rate of peak occurrence derived from the PSD [12]. For a narrowband Gaussian random process, there is an exact quantitative relationship between M Sb (X ) and m b|X | , which is defined as ∞ |x|b f (x)dx

m b|X | =

(5.24)

−∞

where f (·) is the probability density function of the Gaussian process X(t). This relationship is usually derived through theoretical analysis or numerical simulations. Furthermore, it is not difficult to imagine that the quantitative relationship will also hold for a non-Gaussian process. We therefore assume that the relationship between m b|Z | and E[D N G ] is E[D N G ] =

vpT b M S (Z ) ∝ m b|Z | A

(5.25)

where ∝ indicates the proportional relationship. However, kurtosis corresponds to m 4|Z | , and this is why kurtosis-based methods are accurate only when b ≤ 4. The bthorder moment must be used instead of the kurtosis to ensure that an accurate solution is obtained for b > 4. Unfortunately, these two models are unable to inherently treat the bth-order moment.

5.2.5 Direct Method for Fatigue Damage Estimation 5.2.5.1

Cycle Characteristics of Narrowband Processes

A narrowband process is relatively regular despite the fact that it is random (see Fig. 5.7a). It is accepted that the rain-flow cycle distribution is identical to the peak distribution for a narrowband process. The essence of this hypothesis is that the continuous cycles (the portion between two adjacent broken lines in the narrowband process in Fig. 5.7b) approximate the rain-flow cycles. If the time history is examined in detail, we see that every continuous cycle in the process is very regular, as shown in Fig. 5.7b. Intuitively, every continuous cycle in a narrowband process is similar, containing a cosine curve of amplitude equal to the magnitude of the peak. As seen in Fig. 5.7b, the continuous cycles on the left side are smaller than the corresponding cosine curves, whereas the cycles are larger than the cosine curves on the right side. If cosine curves are used to represent the shapes of the loading cycle curves in narrowband random processes, the errors can be empirically counteracted.

5.2 A Method for Estimating Rain-Flow Fatigue Damage …

97

Fig. 5.7 A representative narrowband non-Gaussian process: a a sample time history, b a close up of the bracketed portion shown in a

Based on this analysis, we assume that a symmetrical narrowband process is equivalent to a sequence of cosine curves of variable amplitude, shown as a discontinuous dashed line in Fig. 5.7b. Without loss of generality, we assume that the loading process starts from a maximum at t = 0. The narrowband process may be expressed as Z (t) ∼ =

∞ 

P(Ti )Q i (t|Ti , Ti )

(5.26)

i=0

⎧   ⎪ ⎨ cos 2π (t − Ti ) , Ti ≤ t < Ti+1 Ti Q i (t|Ti , Ti ) = ⎪ ⎩ 0, otherwise

(5.27)

where P(T i ) is the magnitude of the peak at T i , T i is random for i > 0, and T 0 = 0. The expression T i = T i+1 − T i signifies the random time interval between two adjacent peaks.

98

5 Fatigue Life Analysis Under Non-Gaussian Random…

∞   The bth-order central moment is expressed as m bZ = E Z b = −∞ z b f Z (z)dx for a zero mean process, where f Z (·) is the amplitude probability density function of the process described by Z(t). Based on Eqs. (5.26) and (5.27), the bth-order moment of Z(t) can also be expressed as ∞ m bZ

=

p b E[Q i (t|Ti , Ti )]b f P ( p)d p 0

= M Sb (Z )E[Q i (t|Ti , Ti )]b

(5.28)

where f P (p) indicates the probability density function of P(T i ) in Eq. (5.26). From Eq. (5.28), the relationship between m bZ , and the bth-order moment of the rain-flow amplitude, M Sb (Z ) is obtained from M Sb (Z ) =

mb m bZ = Z b Cb E[Q i (t|Ti , Ti )]

(5.29)

The parameter C b is a constant for a specified value of b. Equation (5.29) is applicable only when the value of b is an even number. However, in practice b may be an odd number and/or a fraction. In this case, we introduce the absolute moment, which is defined by Eq. (5.24). The bth-order absolute moment for a narrowband process Z(t) can be expressed as   m b|Z | = M Sb (Z )E |Q i (t|Ti , Ti )|b M Sb (Z ) =

m b|Z | C˜ b

(5.30)

(5.31)

  where C˜ b = E |Q i (t|Ti , Ti )|b is constant for a specified parameter b. From this, the parameter b can be taken arbitrarily from the interval [0, ∞] in Eq. (5.31).

5.2.5.2

Fatigue Damage Calculation

By combining Eqs. (5.23) and (5.31), the fatigue damage occurring during a time period T may be expressed as E[DNG ] =

v p T m b|Z | vpT b M S (Z ) = A AC˜ b

(5.32)

where b is the material fatigue constant and is defined as 2 ≤ b ≤ 6 for common engineering materials. However, according to Kihl [20], b may be larger for some

5.2 A Method for Estimating Rain-Flow Fatigue Damage …

99

deteriorating materials. Therefore, b will be examined from 2 to 10 to validate the capability and flexibility of the method proposed herein. Equation (5.32) shows how the direct relationship between the absolute moments of Z(t) and the expected fatigue damage. This new method neglects complicated non-linear transformation and the rain-flow counting procedure.

5.2.6 Numerical Examples Two examples are analyzed. In the first example, the experimental data are from Ref. [20]. For Gaussian random loadings, the proposed method (Eq. 5.32) and the method based on Rayleigh distribution were used to estimate the number of cycles to failure. For non-Gaussian random loadings, the proposed method together with Kihl model and W–H model were used. Comparisons were carried out extensively. In the second example, we simulated a narrowband non-Gaussian process (γ 4 > 3) with one thousand sample time histories. The material parameters in the expression of S–N curve, b = 2, 4… 10 and A = 2.23 × 1015 are used. The observed fatigue damage is calculated using WAFO [8] for each time history. The mean value from the one thousand observed results is used to provide a comparison to the theoretical estimations from different methods.

5.2.6.1

Example 1

The experimental results are from Ref. [20]. The cruciform joint shown in Fig. 5.8 was extensively tested under narrowband Gaussian and non-Gaussian random loadings. The welds in the specimens are common locations for initiation and propagation of fatigue cracks in actual structures. The configuration and dimensions of the test specimens are shown in Fig. 5.8. The yield and ultimate stress of the steel plate are 638 and 683 MPa, respectively. During the fatigue tests, the cruciforms were loaded axially in the vertical direction with loads applied to the ends of the vertical legs by means of hydraulic grips. Owing to the presence of stress concentrations and residual stresses at the weld toe, the fatigue cracks normally began at the toe of the welds which are shown in Fig. 5.8. The S–N curve of the structural detail is N S 3.210 = 1.7811 × 1012

(5.33)

The expression in Eq. (5.33) is fitted based on the results of constant-amplitude tests in four different levels where the lowest and highest levels are 83 and 310 MPa, respectively. The narrowband Gaussian random process is generated using the following autoregressive model, xt = −0.95xt−1 + 0.05wt

(5.34)

100

5 Fatigue Life Analysis Under Non-Gaussian Random…

Fig. 5.8 Fatigue test specimen (Dimensions are in millimeters)

Fig. 5.9 The standardized Gaussian and non-Gaussian processes

where wt is white Gaussian noise with σw2 = σx2 = 1. The non-Gaussian loadings with the same RMS values were obtained by transforming the Gaussian ones through Kihl model. The targeted kurtosis value is five, then the transformation parameters in Eq. (5.14), n = 2, β = 0.342, C = 1.563. The standardized Gaussian and nonGaussian processes are shown in Fig. 5.9. Then a scaling factor can be used to control the RMS values of the Gaussian and non-Gaussian processes. The RMS stress levels used in the experiments is 52, 69 and 103 MPa. Each simulated load was used as

5.2 A Method for Estimating Rain-Flow Fatigue Damage …

101

Table 5.2 Narrowband Gaussian fatigue test results RMS stress level (MPa)

Cycles to failure, N f,exp Exp. 1

Exp. 2

Exp. 3

Exp. 4

52

1504 200

1111 300

1178 100

1216 300

69

488 000

686 700

901 700

463 000

103

93 600

112 600

128 000

141 200

input in the fatigue test and, treated as a loading block, was repeated many times until failure [20]. Four specimens are tested for each loading process. The Gaussian fatigue test results are shown in Table 5.2. For all the fatigue tests, vp = 11,603 day−1 . The critical fatigue damage is assumed to Dcr = 1. Then the predicted number of cycles to failure based on the proposed method (Eq. 5.32) is NNew = T vp = Dcr

AC˜ b AC˜ b = b b m |X | m |X |

(5.35)

where A = 1.7811 × 1012 , b = 3.210. The mean values of N f,exp in Table 5.1 together with the predicted values based on Eq. (5.35) and Rayleigh distributions [20] are show in Table 5.2, where N¯ f,exp indicates the mean values of N f,exp in Table 5.1, N New indicates the results from the proposed method, N Ray indicates the results based on Rayleigh distributions. The predicted results of the proposed method seem to agree very well with the mean values of the experimental results for narrowband Gaussian random loadings. Compared to the mean values of the experimental results in second column in Table 5.2, the relative deviations of the results of the proposed methods are 1.27, −19.10 and 15.63% for the RMS values 52, 69 and 103 MPa, respectively. While the relative deviations of the Rayleigh distributions based results are 2.87, −19.40 and 17.12%. The magnitude of the relative deviations from the two methods on different RMS stress levels are shown in Fig. 5.10. The results of the proposed model are substantially comparable with the results based on Rayleigh distributions. But the proposed method is more computationally concise. The non-Gaussian fatigue test results are shown in Table 5.3, where the kurtosis of the random loadings is five. The mean values of N f,exp in Table 5.3 together with the predicted values obtained from Eq. (5.26), Rayleigh distributions, Kihl model [20] and W–H model are show in Table 5.4, where

102

5 Fatigue Life Analysis Under Non-Gaussian Random…

Fig. 5.10 The relative deviations of predictions for Gaussian loadings

Table 5.3 Comparison of the fatigue lives for Gaussian loadings RMS stress level (MPa)

Cycles to failure N¯ f,exp

N New

N Ray

52

1252 475

1268 441

1288 400

69

634 850

513 613

511 700

103

118 850

137 425

139 200

Table 5.4 Narrowband non-Gaussian fatigue test results RMS stress level (MPa)

Cycles to failure, N f,exp Exp. 1

Exp. 2

Exp. 3

Exp. 4

52

693 000

903 100

1013 100

922 500

69

256 700

337 500

229 300

424 300

103

30 700

27 100

31 200

28 400

N¯ f,exp indicates the mean values of N f,exp in Table 5.3, N New indicates the results from the proposed method, N Ray indicates the results based on Rayleigh distributions, N Kihl indicates the results from Kihl model, N WH indicates the results from W–H model. Compared to the mean values of the experimental results in second column in Table 5.4, the relative deviations of the results based on the proposed methods are 1.15, 3.30 and 207.99% for the RMS values, 52, 69 and 103 MPa, respectively. The relative deviations are 45.92, 64.30 and 374.28% for the results based on Rayleigh distributions. The relative deviations are −2.34, 9.79% and 217.55 for Kihl model.

5.2 A Method for Estimating Rain-Flow Fatigue Damage …

103

The relative deviations are 2.44, 13.20 and 233.00% for W–H model. The magnitude of the relative deviations of all the models in different RMS stress levels are shown in Fig. 5.11. We can see that the predictions of the proposed method agree fairly well with the experimental results for the RMS stress levels are 52 and 69 MPa. The results of the Kihl model and the W–H model are acceptable. While the results based on Rayleigh distributions are unacceptable for the reason of ignoring the non-normality of the random loadings. Large deviations between the experimental results and the predicted results present for all the methods when RMS stress level is 103 MPa. There are two reasons for this phenomenon. Firstly, the S–N curve of the structure is fitted based on constantamplitude tests where the lowest and highest stress levels are 83 and 310 MPa, respectively. But in the narrowband non-Gaussian random loading, the extrema are much greater than 310 MPa, as shown in Fig. 5.12. Secondly, some extrema in the loading process have exceeded the yield stress (638 MPa) of the material as shown in Fig. 5.12. These larger extrema will produce dominant fatigue damage in the structures. And the fatigue mechanism was changed where the linear damage accumulation rule is not available. Maybe one can refer to Manson’s double linear damage rule [31] and the strain-life methodology [17] for this condition. Fig. 5.11 The relative deviations of predictions for non-Gaussian loadings

Fig. 5.12 Sample time history of the narrowband non-Gaussian random loading in the fatigue test (γ 4 = 5)

104

5 Fatigue Life Analysis Under Non-Gaussian Random…

Analyzing the results in Tables 5.2 and 5.4, one can see that the proposed method is available for both Gaussian and non-Gaussian loadings. In this example the parameter b = 3.210, less than four. For this condition, the Kihl and W–H models can give reasonable predictions. In the next subsection, we will analysis the accuracy of all the methods for narrowband non-Gaussian random loadings when the parameter b is greater than four.

5.2.6.2

Example 2

A narrowband non-Gaussian zero mean process Z(t) with values σ 2 Z = 54.7158 MPa2 and γ 4 = 7.3 is analyzed in this example, as shown in Fig. 5.13. The mean values of the one thousand observed results are listed in the second column of Table 5.5 for the parameter b = 2, 4… 10. The results of the proposed method are in the third column. The estimated fatigue damages of the Kihl model with different parameters are listed in the fourth to sixth columns. The results of the W–H model are in the last column. The predicted values from the proposed method agree well with the mean values of the observed results. In addition, we define a parameter ζ to be the ratio of the fatigue estimations of the theoretical methods to the mean value of the observed results. For example, using the Kihl model (n = 2, β = 1.15), the fatigue damage ratio is defined as Fig. 5.13 The narrowband non-Gaussian random loading: a sample time history, b PSD

5.2 A Method for Estimating Rain-Flow Fatigue Damage …

105

Table 5.5 Comparison of the fatigue lives for non-Gaussian loadings (γ 4 = 5) RMS stress level (MPa)

Cycles to failure N¯ f,exp

N New

N Ray

N Kihl

N WH

52

882 925

893 069

1288 400

862 300

904 502

69

311 950

322 251

511 700

342 500

353 117

103

29 350

90 394

139 200

93 200

97 736

ζKihl (b|2, 1.15) =

DKihl (b|2, 1.15) Dobs (b)

(5.36)

where DKihl (b|2, 1.15) is the fatigue estimation of the Kihl model. Fatigue damage is stochastic, and using the bootstrap method, we obtain a 90% confidence interval, [DL obs(b), DU obs(b)], from the one thousand observed fatigue damage results with the upper and lower limits of the fatigue damage ratio defined as L ζobs (b) =

L Dobs D U (b) (b) U , ζobs (b) = obs Dobs (b) Dobs (b)

(5.37)

Figure 5.14 shows a comparison of the fatigue damage ratio curves for the different U L models together with the fatigue damage ratios ζobs (b) and ζobs (b). The fatigue damage ratio for the proposed method is somewhat greater than one. This suggests that the result is very accurate and slightly conservative. For the Kihl model, the fatigue damage ratio curves are very different when different parameters {n, β} are chosen. For example, the fatigue damage ratio curve corresponding to a

Fig. 5.14 Comparison of fatigue damage ratio curves—ζ new : the proposed new method, ζ k2 : the Kihl model with parameters {2, 1.15}, ζ k3 : the Kihl model with parameters {3, 0.135}, ζ k4 : the Kihl model with parameters {4, 0.033}, ζ WH : the W–H model, ζ up and ζ low : the upper and lower limits of the 90% confidence interval, respectively

106

5 Fatigue Life Analysis Under Non-Gaussian Random…

U parameter set of {4, 0.033} will intersect ζobs (b) as the value of b increases, whereas the curve corresponding to {2, 1.15} decreases along the b axis. The result from the W–H model is conservative. The Kihl model is unstable depending on the parameter set {n, β} chosen during the nonlinear transformation.

5.2.7 Conclusions This work focused on the fatigue problem induced by narrowband random loadings. We have examined the reasons kurtosis-based transformed models introduce errors in analyses of materials or structures where parameter b is greater than four. This is mainly due to a mismatch in the moments greater than fourth-order between the real and the transformed processes. The fatigue damage is most sensitive to the bth-order moment and not the kurtosis. A new method is proposed to estimate the fatigue damage of narrowband loadings. For non-Gaussian random loadings, the new method neglects complicated non-linear transformations and the rain-flow counting procedure to permit fatigue damage to be directly estimated from the bth-order moment of the loading process itself. The proposed method has been validated available for narrowband Gaussian and non-Gaussian random loadings by comparing experimental results with the estimations on different RMS stress levels. Thousands of simulated stress time histories are used to validate the availability and advantage of the presented method when the parameter b, in the S–N expression, is greater than four. The predictions from the proposed method show a pronounced advantage over other commonly used methods for the condition that the kurtosis is greater than three and the parameter b is greater than four.

5.3 A Spectral Method to Estimate Fatigue Life Under Broadband Non-Gaussian Random Vibration Loading The aim of this Section is to propose a spectral method for assessing the fatigue lives of mechanical components under non-Gaussian random vibration load-ings. Efforts are made to extend the Dirlik’s method to non-Gaussian vibra-tion field by introducing the Gaussian mixture model. A symmetric non-Gaussian random vibration can be decomposed into a series of Gaussian com-ponents through Gaussian mixture model. Then the rain-flow cycle distribu-tions of the Gaussian components can be obtained using Dirlik’s method. The cycle distribution of the underlying non-Gaussian process is derived by com-pounding the distributions of Gaussian components together. The non-Gaussian cycle distribution, combined with Palmgren–Miner rule is used to predict the fatigue lives of specimens. Comparisons among the proposed method, Dirlik’s solution, nonlinear model in literature, and

5.3 A Spectral Method to Estimate Fatigue Life …

107

the experimental data, are carried out extensively. The results have confirmed good accuracy of the proposed method.

5.3.1 Introduction For some mechanical components, the service loadings are induced by random vibrations. The randomness of stress-time histories makes the assessment of fatigue damage quite difficult. Among all the cycle counting methods, the rain-flow method is regarded as the best one [24]. In the time domain, the rain-flow method is applicable for any kind of random process, but it usually requires a large amount of loading records from lengthy and expensive experimental data acquisition programs [4]. Furthermore, we cannot get a stable distribution of rain-flow cycles from the time domain data [6]. The frequency-domain representations of random processes, normally power spectral densities (PSDs), are easier to apply and more flexible in engineering applications. For spectral methods, the rain-flow cycle distributions are usually estimated based on the PSDs. Based on Gaussian assumption some spectral methods have been proposed in the literature, such as narrow-band approximation method [34], Dirlik’s solution [12] and the methods presented in [28, 37]. A comparison of several spectral methods was presented in [10], where the precision of Dirlik’s method was validated. In many cases, however, the dynamic vibration loadings of mechanical components do not follow Gaussian distributions [16, 38, 46]. The non-Gaussian nature of the stress response results from non-Gaussian external excitation, nonlinearity, or both [4]. The non-Gaussian random vibration loadings can accelerate the fatiguedamage accumulation because of the presence of high-excursion loading cycles. Hence, the spectral methods applicable to Gaussian loadings are not useful in nonGaussian case. The methods based on Gaussian assumption will overestimate the fatigue lives of mechanical components subjected to non-Gaussian random loadings, possibly leading to serious accidents. Hence, new method which is effective for non-Gaussian stress-time histories is required. Some spectral methods for non-Gaussian random loadings have been presented in literature. A narrow-band approximation method modified by non-normality and bandwidth correction coefficients was presented in [45]. Some methods based on nonlinear transformations of Gaussian processes were proposed in [4, 16, 42]. There are many damage accumulation rules proposed in the past. Generally speaking, the Palmgren–Miner rule [27, 43] could provide reliable fatigue damage estimation for stationary random loadings [4, 37]. Although some spectral methods for stationary non-Gaussian loadings have been proposed, simpler and more efficient methods are still needed in engineering practice. The method proposed in this book is based on Dirlik’s formula. On the basis of the mathematical treatment of Gaussian mixture models (GMMs) for non-Gaussian noise in telecommunications applications [7, 26], a Gaussian mixture model is proposed here which is available for symmetric non-Gaussian loadings whose skewness values

108

5 Fatigue Life Analysis Under Non-Gaussian Random…

are zero and kurtosis values are three. Using the proposed Gaussian mixture model, a non-Gaussian loading can be decomposed into a series of Gaussian components with different probability weighting factors. Then Dirlik’s formula is used to obtain the cycle distributions of the Gaussian components. The cycle distribution of the nonGaussian loading is obtained by compounding the distributions of the components with the proposed Gaussian mixture model. The non-Gaussian cycle distribution, combined with Palmgren–Miner rule is used to predict the fatigue lives of test specimens. Comparisons among the proposed method, nonlinear model in [20], Dirlik’s solution, and the experimental data, are carried out extensively. The results have confirmed good accuracy of the proposed method.

5.3.2 Non-Gaussian Random Vibration Loadings This study focuses on symmetrical non-Gaussian random loadings. Non-Gaussian vibrations are widely present in real-world environments. Theoretically, the statistical parameters that can thoroughly represent a non-Gaussian process are higher-order statistics: higher-order moments or higher-order cumulants [25]. The higher-order statistics of a random process are functions of the sequence of time lags {τ i }, i = 1, 2, …, n. The estimation of higher-order statistics is a highly complex problem. In vibration engineering, the higher-order statistics by setting the time lags {τ i } to be zero are always used as substitutions. For this reason, certain statistical properties of the non-Gaussian random processes are ignored. This means that most spectral methods for non-Gaussian loadings are empirical or semi-empirical solutions. The most used statistics are the normalized third- and fourth-order central moments: skewness (γ 3 ) and kurtosis (γ 4 ). Denoting by X(t) a non-Gaussian random loading, skewness and kurtosis are defined as follows:     E (X − μ X )3 E (X − μ X )4 m3 m4 = 3 , γ4 = = 4, (5.38) γ3 = 3 4 σX σX σX σX where E[·] denotes mathematical expectation, μX and σ X are the mean value and the standard deviation of X(t) and m3 and m4 are the third- and fourth-order central moments. In fact, skewness and kurtosis cannot represent the non-normality of a non-Gaussian process completely because the statistics higher than fourth-order are ignored and the properties of temporal correlation are neglected. It is not difficult to imagine a case that two different stationary non-Gaussian processes having identical variance, skewness, and kurtosis. In engineering filed, however, some simplifications are unavoidable. For Gaussian processes, the skewness and kurtosis values are zero and three respectively. The non-Gaussian properties of vehicular vibrations are investigated in [32, 38], where it has been pointed out that most of the non-Gaussian loadings encountered in engineering practice are non-stationary from a short-duration viewpoint,

5.3 A Spectral Method to Estimate Fatigue Life …

109

Fig. 5.15 The standardized Gaussian and non-Gaussian processes: a Gaussian, b non-Gaussian (γ 4 = 8)

but stationary from a longer-duration viewpoint. In engineering practice, these kinds of loadings are always treated as stationary processes for simplicity. This study is also partially based on this assumption. A comparison between standardized Gaussian and non-Gaussian processes is shown in Fig. 5.15. The discrepancy is prominent, and the non-Gaussian process has many higher-excursion peaks.

5.3.3 Gaussian Mixture Model (GMM) In the study of non-Gaussian noise models in signal processing, Middleton proposed the Gaussian mixture model [26]. From the viewpoint of this model, the underlying non-Gaussian process consists of a series of Gaussian components, with different probability weight factors and other parameters. The original GMM was proposed mainly for estimating the non-Gaussian noise probability density function (PDF) in telecommunications applications. The general form of the GMM is: f NG (x) =

N 

αi f i (x),

(5.39)

i=1

where f NG (x) is the PDF of the non-Gaussian process; f i (x) is the Gaussian term, namely the PDF of the ith Gaussian component; α i is the probability weighting factor, 0 ≤ α i ≤ 1, αi = 1; and N is the dimension of GMM. For the original GMM in telecommunications applications, the weighting factor α i is quantified by a Poisson distribution based on thorough understanding of each Gaussian noise source. This is impossible for non-Gaussian random loadings in mechanical engineering. Hence, a modified GMM is needed which is available for non-Gaussian random loadings. Normally, a rather small value of N in Eq. (5.39), is sufficient to provide an excellent approximation of the real distribution function [7]. The two-term GMM will be used here,

110

5 Fatigue Life Analysis Under Non-Gaussian Random…

f NG (x) = α f 1 (x) + (1 − α) f 2 (x).

(5.40)

For a zero-mean stationary non-Gaussian process X(t), the GMM can be expressed as:     1 x2 x2 1 exp − 2 + (1 − α) √ exp − 2 , f NG (x) = α √ 2σ1 2σ2 2π σ1 2π σ2

(5.41)

where σ 1 and σ 2 are the standard deviations of the two Gaussian components and α and (1−α) are the probability weighting factors of the two terms. There are three unknown quantities, σ 1 , σ 2 , and α, in Eq. (5.41). Therefore, a three-variable set of equations is needed to derive the unknown parameters. For real non-Gaussian random loadings in engineering practice, the true values of the higher-order moments cannot be known. The estimated values are always used as substitutes. For a zero-mean process, the second-, fourth-, and sixth-order moments can be calculated as follows, ⎧ ∞ ⎪ ⎪  2 ⎪ ⎪ m =E x = x 2 f NG (x)dx ⎪ ⎪ ⎪ 2 ⎪ ⎪ −∞ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ⎨  4 m4 = E x = x 4 f NG (x)dx ⎪ ⎪ ⎪ −∞ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪  6 ⎪ ⎪ m6 = E x = x 6 f NG (x)dx ⎪ ⎪ ⎩ −∞

1 ∼ = mˆ 2 = T

T x 2 (t)dt 0

1 ∼ = mˆ 4 = T

T x 4 (t)dt ,

(5.42)

0

1 ∼ = mˆ 4 = T

T x 6 (t)dt 0

where T is the duration of the sample time history. When T is long enough, these estimates will converge to the true values with sufficient precision [21]. By substituting Eq. (5.41) into Eq. (5.42), the following equations can be obtained: ⎧ (2) ⎪ m = αm (1) ⎪ 2 + (1 − α)m 2 ⎨ 2 ⎪ ⎪ ⎩

(2) m 4 = αm (1) 4 + (1 − α)m 4 ,

m6 =

αm (1) 6

+ (1 −

(5.43)

α)m (2) 6

(2) where m (1) 2 and m 2 are the second-order moments of the two Gaussian components, (1) (2) (2) m 4 and m 4 are the fourth-order moments, and m (1) 6 and m 6 are the sixth-order moments. The second-order moments are equal to the variances, σ12 and σ22 . For a zero-mean stationary Gaussian process, the following relationship exists between the various ordered moments:

5.3 A Spectral Method to Estimate Fatigue Life …

! mk =

[1 × 3 × 5 · · · (k − 1)]σ k , k is even , 0, k is odd

111

(5.44)

where σ is the standard deviation or root mean square (RMS) and k is a positive integer, 1 ≤ k < ∞. Then for the two Gaussian components: ⎧ (1) (2) ⎨ m 2 = σ12 ; m 2 = σ22 (1) 4 4 . m 4 = 3σ1 ; m (2) 4 = 3σ2 ⎩ (1) (2) 6 m 6 = 15σ1 ; m 6 = 15σ26

(5.45)

Substituting Eq. (5.45) into Eq. (5.43) results in, ⎧ ⎨ m 2 = ασ12 + (1 − α)σ22 m = 3ασ14 + 3(1 − α)σ24 . ⎩ 4 m 6 = 15ασ16 + 15(1 − α)σ26

(5.46)

The unknown parameters σ 1 , σ 2 , and α can be derived through Eq. (5.46) by substituting the theoretical values of m1 , m2 , and m3 by the estimated ones in Eq. (5.42). Then the two-term mixture PDF of a non-Gaussian process is obtained. This is a new method for estimating the PDFs of symmetric non-Gaussian loadings. However, to assess the fatigue cycle distribution of non-Gaussian random loadings based on spectral data, a further step must be taken. Hence, the GMM will be introduced into the frequency domain.

5.3.4 PSD Decomposition of Non-Gaussian Vibration Loadings It is clear that a PSD cannot define a non-Gaussian process, unlike the Gaussian case. Based on the GMM, a probabilistic explanation of a non-Gaussian process has been proposed. In Eq. (5.41), α and (1−α) represent the probabilities of existence of the two Gaussian components in the time domain. Furthermore, in the frequency domain, the underlying PSD is decomposed into two different-valued PSDs to account for non-normality. For a non-Gaussian zero-mean stationary process, X(t), the variance can be expressed as: ∞ σ X2

=

S X ( f )d f, 0

(5.47)

112

5 Fatigue Life Analysis Under Non-Gaussian Random…

where S X (f ) is the one-sided PSD, and f is the frequency. For the two Gaussian components: ∞ σ12

=

∞ S1 ( f )d f,

0

σ22

=

S2 ( f )d f,

(5.48)

0

where S 1 (f ) and S 2 (f ) are the PSDs of the two components. According to Eq. (5.46), σ X2 = ασ12 + (1 − α)σ22 .

(5.49)

Substituting Eqs. (5.47) and (5.48) into Eq. (5.49), results in, S X ( f ) = αS1 ( f ) + (1 − α)S2 ( f ).

(5.50)

To derive the PSD-based rain-flow cycle distribution, the magnitudes of S 1 (f ) and S 2 (f ) must be determined. Here we assume that S 1 (f ) and S 2 (f ) are proportional to S X (f ) along the frequency axis: S1 ( f ) = η1 S X ( f ), S2 ( f ) = η2 S X ( f ),

(5.51)

where η1 and η2 are the constants of proportionality, which can be derived by combining Eq. (5.51) with Eqs. (5.47) and Eq. (5.48): η1 =

σ12 σ2 , η2 = 22 . 2 σX σX

(5.52)

Then, substituting Eqs. (5.51) and (5.52) into Eq. (5.50), the PSD decomposition of symmetric non-Gaussian random loadings is obtained. The expression in Eq. (5.50) is defined as the probabilistic PSD (p-PSD).

5.3.5 Modified Dirlik’s Formula and Fatigue Damage Estimation 5.3.5.1

Dirlik’s Formula

Dirlik’s formula is an approximate closed-form expression of the PDF of the normalized amplitude of rain-flow cycles. This method has been developed based on extensive numerical simulations with computers [12]. First, let us introduce the definition of spectral moment. For the PSD of a given Gaussian process, X(t), the spectral moment is defined as:

5.3 A Spectral Method to Estimate Fatigue Life …

113

∞ λn =

f n S X ( f )d f .

(5.53)

0

From spectral moments, it is possible to derive some important characteristics √ of the random process itself. For example, the√standard deviation is σ X = λ0 , the expected √ rate of zero-upcrossing is v0 = λ2 /λ0 , the expected rate of peaks /λ2 , the bandwidth factor is B = v0 /vp , and the average frequency is is vp = λ4√ f m = λ1 /λ0 λ2 /λ4 . The normalized amplitude of the loading cycle is defined as  z = s σX ,

(5.54)

where s is the amplitude of the rain-flow cycle. Then the distribution of the normalized rain-flow cycle, based on Dirlik’s solutions, is [12]:    2  z  1 z z2 z + c2 2 exp − 2 + c3 z exp − , p(z) = c1 exp −   ξ 2ξ 2 B− f −c2

(5.55)

1−B−c +c2

f m −B ) 1 1 where c1 = 2( 1+B , ξ = 1−B−cm +c1 2 , c2 = , c3 = 1 − c1 − c2 , 2 1−ξ 1 1  = 1.25(B − c3 − c2 ξ )/c1 . Previous studies have proved that Dirlik’s empirical formula can precisely approximate the rain-flow cycle distributions of Gaussian random loadings [10]. 2

5.3.5.2

Dirlik’s Formula for Non-Gaussian Loadings

Equation (5.50) gives the p-PSD of a non-Gaussian random loading. Then the rainflow cycle distribution of each component can be calculated based on Dirlik’s formula through a simple variable change: p1 (s) =

" p1 (z) "" , σ1 " z = s/σ1

p2 (s) =

" p2 (z) "" , σ2 " z = s/σ2

(5.56)

where p1 (z) and p2 (z) are the normalized cycle distributions of the Gaussian components, s is the amplitude of the rain-flow cycle, and σ 1 and σ 2 are the standard deviations of the two Gaussian components. The cycle distribution of the symmetric non-Gaussian random loading is defined as follows, f GMM (s) = αp1 (s) + (1 − α) p2 (s).

(5.57)

114

5.3.5.3

5 Fatigue Life Analysis Under Non-Gaussian Random…

Fatigue Damage Estimation

For a random loading, fatigue damage is caused by amplitudes and mean values of loading cycles. The counted cycles are random events. For zero mean non-Gaussian random loadings, the rain-flow cycles follow the distribution expressed by Eq. (5.57). For nonzero mean loadings, the rain-flow cycle distribution should be modified based on the correction models, such as Goodman model, Gerber model, and Soderberg model [21]. The expected rate of occurrence of rain-flow cycles is denoted with vc which is equal to the expected rate of occurrence of peak, vp . It is can be derived from the spectral moments of PSD, as shown in Sect. 5.3.5.1. Furthermore, a damage accumulation rule must be selected to collect the fatigue damage caused by each cycle together in a specified manner. There are many damage accumulation rules reviewed in [14]. The linear damage accumulation rule, namely Palmgren–Miner rule can give reasonable results for stationary random loadings according to [19]. We will adopt Palmgren–Miner rule in this study. Normally, the stress life curve, namely S–N curve is defined in the form of power-low model, N S b = A,

(5.58)

where, N is the number of cycles to failure at amplitude S, b and A are the fatigue parameters of material or structure. Then the expected fatigue damage can be expressed as follows, vc T E[D N G ] = A

∞ s b f GMM (s)ds,

(5.59)

0

where T is the time duration of the non-Gaussian random loading, f GMM (s) is the non-Gaussian rain-flow cycle distribution defined in Eq. (5.57).

5.3.6 Examples To validate the capability of the proposed method, non-Gaussian random loadings and the corresponding fatigue data in [20] are analyzed. The proposed method is used to predict the fatigue lives of the fatigue test specimens (Fig. 5.16). Comparisons among experimental data, results from the proposed method, nonlinear transformation model, and Gaussian assumption are carried out extensively. In addition, a rain-flow counting procedure based on the time history is carried out to evaluate the empirical distribution of rain-flow cycles. The rain-flow counting procedure is based mainly on the WAFO toolbox [8].

5.3 A Spectral Method to Estimate Fatigue Life …

115

Fig. 5.16 Fatigue test specimen [18] (Dimensions are in millimeters)

The cruciform joint shown in Fig. 5.16 was extensively tested under non-Gaussian random loadings. The welds in the specimens are common locations for initiation and propagation of fatigue cracks in actual structures. The configuration and dimensions of the test specimens are shown in Fig. 5.16. The yield stress and ultimate stress of the steel palate are 638 and 683 MPa, respectively. During the fatigue tests, the cruciforms were loaded axially in the vertical direction with loads applied to the ends of the vertical legs by means of hydraulic grips. Owing to the presence of stress concentrations and residual stresses at the weld toe, the fatigue cracks normally began at the toe of the welds, as shown in Fig. 5.16. The S–N curve of the structural detail is, N S 3.210 = 1.7811 × 1012 .

(5.60)

The expression in Eq. (5.60) was fitted based on the results of constant-amplitude fatigue tests in four different stress levels where the lowest and highest levels are 83 and 310 MPa, respectively. The non-Gaussian random loadings are generated using the standard Gaussian simulation technique [36] combined with nonlinear transformation [20]. The kurtosis value of the non-Gaussian loadings is five. The three RMS stress levels used in the experiments are 52, 69 and 103 MPa. The sample time histories and the PSDs of the broadband non-Gaussian random loadings in different RMS stress levels are shown in Fig. 5.17. Each simulated load was used as input in the fatigue test and, treated as a loading block, was repeated many times until failure [20]. Four specimens were tested for

116

5 Fatigue Life Analysis Under Non-Gaussian Random…

Fig. 5.17 Sample time histories and the PSDs of the three broadband non-Gaussian random loadings in different RMS stress levels: a 52 MPa, b 69 MPa, c 103 MPa, and d PSDs

each loading process. The broadband non-Gaussian fatigue test results are shown in Table 5.6. Also presented in this table are the mean values of fatigue lives, in applied cycles, for each stress level. The critical fatigue damage is assumed to be Dcr = 1, and then the predicted number of cycles to failure based on the proposed method Eq. (5.59) is, Table 5.6 A comparison of fatigue damage estimations for the non-Gaussian process (T = 100 s) using different methods b

Dobs

DNew

DKihl

DWH

{2, 1.15}

{3, 0.135}

{4, 0.033}

2

1.187 × 10–10

1.264 × 10–10

1.364 × 10–10

1.328 × 10–10

1.293 × 10–10

1.339 × 10–10

4

6.672 × 10–8

6.817 × 10–8

7.604 × 10–8

7.978 × 10–8

8.360 × 10–8

8.995 × 10–8

6

1.187 × 10–4

1.203 × 10–4

0.960 × 10–4

1.476 × 10–4

2.4189 × 10–4

1.941 × 10–4

8

4.775 × 10–1

4.935 × 10–1

2.150 × 10–1

6.237 × 10–1

17.927 × 10–1

9.512 × 10–1

10

3.175 × 103 3.384 × 103 0.748 × 103 4.646 × 103 20.743 × 103 7.940 × 103

5.3 A Spectral Method to Estimate Fatigue Life …

NGMM = vc T =  ∞ 0

117

A sb

(5.61)

f GMM (s)ds

In this example, A = 1.7811 × 1012 , b = 3.210. The rain-flow cycle distribution f GMM (s) is derived based on the proposed method. For simplicity, we shall just demonstrate the application of the proposed method to the case that the RMS stress level is 52 MPa. The procedures for other cases are similar, and we will just list the results. Based on Eq. (5.42), we get the estimations of the second-, fourth-, and sixth-order moments of the non-Gaussian random loading shown in Fig. 5.17a, m 2 = 2704, m 4 = 3.8564 × 107 , and m 6 = 1.2044 × 1012 . By substituting these values into Eq. (5.9), we get the parameters of GMM are, α = 0.7560, σ 1 = 36.9662, and σ 2 = 82.7539. And then according to Eq. (5.52), the two parameters for p-PSD are, #

#

#

σ2 η1 = 12 = σX



36.9662 52

2

σ2 = 0.5054; η2 = 22 = σX



82.7539 52

2 = 2.5326. (5.62)

Based on these two parameters and Eq. (5.50), the p-PSD of the non-Gaussian loading is obtained, as shown in Fig. 5.18. By substituting S 1 (f ) and S 2 (f ) into the Dirlik formula (Eqs. 5.55 and 5.56), we can get two Gaussian rain-flow cycle distributions, p1 (s) and p2 (s). And then we can get the non-Gaussian rain-flow cycle distribution according to Eq. (5.57), as shown in Fig. 5.19. Also illustrated in this figure are the empirical distribution and Gaussian rain-flow cycle distribution. The empirical distribution is estimated based on rain-flow cycles counted from the sample time history with time duration T = 4000 s. There are 1425 rain-flow cycles in the sample time history. The comparison shows the accuracy of the proposed in describing the rain-flow cycle distribution of broadband non-Gaussian random loading. The full range comparisons are presented in Fig. 5.19a and b in the linear and semi-log coordinates, respectively. And we can see that the proposed methodology can give a reasonable description of the

Fig. 5.18 p-PSD of the non-Gaussian random loading with RMS stress level 52 MPa

118

5 Fatigue Life Analysis Under Non-Gaussian Random…

Fig. 5.19 Comparison of rain-flow cycle distributions based on Gaussian assumption and GMM with empirical distribution: a linear scale, b semi-log scale, c close up view of large cycle amplitude in linear scale, d close up view of large cycle amplitude in semi-log scale

non-Gaussian rain-flow cycle distribution, especially in the larger range of the rainflow cycles. Normally, the larger cycles will dominate the fatigue damage process of mechanical component, so we have given a close up view of the distribution curves when cycle amplitude is above 83 MPa in both linear and semi-log scales in Fig. 5.19c and d, respectively. Furthermore, we can see that the empirical distribution curve fluctuates severely when the PDF value is close to or below 0.1% in semi-log scale. The reason for this phenomenon is that the sample size of the rain-flow cycles is 1425, which is too small to give a stable prediction in that order of magnitude. But the proposed method can give a stable prediction, as shown in Fig. 5.19b and d. The predicted fatigue life of the specimen is derived by substituting the nonGaussian rain-flow cycle distribution into Eq. (5.61). The mean values of the test fatigue data together with the predicted results based on the proposed method (Eq. 5.61), the nonlinear transformation model in [20], and Dirlik formula, in three RMS stress levels are shown in Table 5.7, where −

N exp indicates the mean values of fatigue data in Table 5.7, N GMM indicates the results from the proposed method, N Kihl indicates the results from the nonlinear transformation model [20], N G indicates the results based on Gaussian assumption, namely Dirlik solution. The predicted results of the proposed method seem to agree very well with the mean values of the experimental fatigue data for broadband non-Gaussian random

5.3 A Spectral Method to Estimate Fatigue Life …

119

Table 5.7 Broadband non-Gaussian fatigue test results RMS stress level (MPa)

Cycle to failure, N exp −

Exp. 1

Exp. 2

Exp. 3

Exp. 4

Mean value, N exp

52

951,800

742,900

1,067,900

703,000

866 400

69

373,800

326,300

273,000

301,000

318,525

103

47,900

45,100

39,500

44,200

44,175

Table 5.8 Comparison of fatigue lives for broadband non-Gaussian loadings (γ 4 = 5) RMS stress level (MPa)

Cycles to failure −

N exp

N GMM

N Kihl

NG

1,085,800 (25.32%)

1,580,338 (82.40%)

52

866,400

891,600 (2.91%)

69

318,525

359,833 (12.97%)

431,200 (35.37%)

743,634 (133.46%)

103

44,175

91,388 (106.88%)

117,300 (165.53%)

216,792 (390.75%)

loadings except the condition that the RMS stress level is 103 MPa. Compared to the mean values of the experimental fatigue data in the second column of Tables 5.7 and 5.8, the relative deviations of results of the proposed methods are 2.91, 12.97, and 106.88% for RMS stress levels, 52, 69 and 103 MPa, respectively. While the relative deviations of the results based on nonlinear transformation model are 25.32, 35.37, and 165.53%. The relative deviations for Gaussian assumption (Dirlik solution) based results are 82, 133.46, and 390.75%. Large deviations between the experimental results and the predicted ones present for all the methods when RMS stress level is 103 MPa. There are two reasons for this phenomenon. First, the S–N curve of the structure is fitted based on constantamplitude tests where the lowest and highest stress levels are 83 and 310 MPa, respectively. But in the non-Gaussian random loading, some extrema are much greater than 310 MPa, as shown in Fig. 5.17c. Second, some extrema in the loading process have approached the yield stress (638 MPa) of the material of the specimen, as shown in Fig. 5.17c. These higher extrema cause significant fatigue damage in the structures changing the fatigue mechanism, and the linear damage summation rule is not applicable herein. Maybe one can refer to the strain-life methodology [17] in this condition.

5.3.7 Conclusions This study has focused on the rain-flow cycle distribution and fatigue life estimation of broadband non-Gaussian random loading. A two-term Gaussian mixture model

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5 Fatigue Life Analysis Under Non-Gaussian Random…

has been proposed to decompose the underlying non-Gaussian loadings into Gaussian components with different variances. Then the Gaussian mixture model was transferred from the time domain to the frequency domain. Based on the assumption that the PSDs of the Gaussian components are proportional to the PSD of the underlying non-Gaussian process, a definition of probabilistic PSD for non-Gaussian loading has been proposed. Dirlik’s empirical method was then used on the PSDs of the Gaussian components to obtain their loading-cycle distributions. By substituting the cycle distributions of the Gaussian components into the Gaussian mixture model, the cycle distribution of the non-Gaussian loading was obtained. Fatigue life was predicted based on the proposed method combining with Palmgren–Miner rule. Comparison between the results of the proposed method with the experimental results shows good agreement, indicating the capability and reasonable accuracy of the proposed method. During the error analysis, the proposed method has resulted in smaller relative deviations. This verified the advantage of the proposed method to deal with broadband non-Gaussian random vibration loadings. Acknowledgements The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Nos. 51875570 and 50905181).

References 1. Anthes RJ (1997) Modified rainflow counting keeping the load sequence. Int J Fatigue 19:529– 535 2. Benasciutti D, Tovo R et al (2006) Comparison of spectral methods for fatigue analysis of broad-band Gaussian random processes. Probabilistic Eng Mech 3. Benasciutti D, Tovo R (2005) Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random processes. Probab Eng Mech 20:115–127 4. Benasciutti D, Tovo R (2006) Fatigue life assessment in non-Gaussian random loadings. Int J Fatigue 28:733–746 5. Benasciutti D, Tovo R (2005) Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random pro-cesses. Prob Eng Mech 20:115–127 6. Bendat JS (1971) Random data analysis and measurement procedures. Willey and Sons, Inc. 7. Bishop N (1988) The use of frequency domain parameters to predict structural fatigue. University of Warwick 8. Blum RS, Kozick RJ (1999) An adaptive spatial diversity receiver for non-Gaussian interference and noise. IEEE Trans Signal Process 47:2100–2111 9. Brodtkorb PA, Johannesson P, Lindgren G et al (2000) WAFO—a Matlab toolbox for analysis of random waves and loads. In: The proceedings of the 10th international offshore and polar engineering conference, vol 3 10. Butler RW, Machado UB, Rychlik I (2009) Distribution of wave crests in a non-gaussian sea. Appl Ocean Res 31(1):57–64 11. Dirlik T (1985) Application of computers in fatigue analysis. University of Warwick 12. Dowling NE (1971) Fatigue failure predictions for complicated stress-strain histories. J Mater 13. Fatemi A, Young L (1998) Cumulative fatigue damage and live prediction theories: a survey of the state of the art for homogenous materials. Int J Fatigue 20:3–34

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14. Han, Sangbo, Jin WC (2002) Retrieving displacement signal from measured acceleration signal. Spie Proc 15. Jan MM, Gaenser HP, Eichls Ed Er W (2012) Prediction of the low cycle fatigue regime of the s–n curve with application to an aluminium alloy. In: ARCHIVE Proc Inst Mech Eng Part C J Mech Eng Sci 1989–1996 (vols 203–210) 226(5):1198–1209 16. Jin W, Lutes LD (1997) Analytical methods for non-Gaussian stochastic response of offshore structures. Int J Offshore Polar Eng 7:11–17 17. Johannesson P (1999) Rainflow analysis of switching markov loads. Lund University, Sweden 18. Kihl DP, Sarkani S, Beach JE (1995) Stochastic fatigue damage accumulation under broadband loadings. Int J Fatigue 17:321–329 19. Lee YL, Pan J, Hathaway R et al. (2004) Fatigue testing and analysis: theory and practice. Mech Eng 126:59 20. Lindgren G, Rychlik I, Prevosto M (1999) Stochastic doppler shift and encountered wave period distributions in Gaussian waves. Ocean Eng 26:507–518 21. Matsuishi M, Endo T (1968) Fatigue of metals subject to varying stress. Jpn Soc Mech Eng 22. Mcinnes CH, Meehan PA (2008) Equivalence of four-point and three-point rainflow cycle counting algorithms. Int J Fatigue 30:547–559 23. Mendel JM (1991) Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications. IEEE Proc 79(3):278–305 24. Middleton D (1999) Non-Gaussian noise models in signal processing for telecommunications: new methods an results for class A and class B noise models. Information Theory IEEE Transactions on 45:1129–1149 25. Miner MA (1945) Cumulative damage in fatigue. Journal of Applied Mechanics 12 26. Niesłony A, Böhm M (2012) Application of spectral method in fatigue life assessment— determination of crack initiation. J Theor Appl Mech 50:819–829 27. Nikias CL, Petropulu AP (1993) Higher-order spectra analysis. Prentice-Hall, Englewood Cliffs, NJ 28. Ochi MK, Ahn K (1994) Probability distribution applicable to non-Gaussian random processes. Probab Eng Mech 9:255–264 29. Park Y-S, Kang D-H (2013) Fatigue reliability evaluation technique using probabilistic stresslife method for stress range frequency distribution of a steel welding member. J Vibroeng 15:77–89 30. Rambabu DV, Ranganath VR, Ramamurty U et al (2010) Variable stress ratio in cumulative fatigue damage: experiments and comparison of three models. ARCHIVE Proc Inst Mech Eng Part C J Mech Eng Sci 1989–1996 (vols 203–210) 224:271–282 31. Rouillard V (2007) The synthesis of road vehicle vibrations based on the statistical distribution of segment lengths. Engineers Australia 32. Rychlik I (1994) On the ‘narrow-band’ approximation for expected fatigue damage. Int J Fatigue 33. Rychlik I et al (2007) Rain-flow fatigue damage for transformed gaussian loads. Int J Fatigue 29:406–420 34. Rychlik I (1987) A new definition of the rainflow cycle counting method. International Journal of Fatigue 9(2):119–121 35. Shannon CE (1949) Communication in the presence of noise. Proc IRE 86:10–21 36. Shinozuka M, Jan CM (1972) Digital simulation of random processes and its applications. J Sound Vib 25:111–128 37. Steinwolf A, Ibrahim RA (1999) Numerical and experimental studies of linear systems subjected to non-Gaussian random ex-citations. Prob Eng Mech 14:289–299 38. Tovo R (2002) Cycle distribution and fatigue damage under broad-band random loading. Int J Fatigue 24:1137–1147 39. Vincent R (2007) On the non-Gaussian nature of random vehicle vibrations. Lect Notes Eng Comput Sci 2166 40. Wang J (1992) Non-Gaussian stochastic dynamic response and fatigue of offshore structures. Texas A&M University

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41. Wang MZ (2009) Research on structural vibration fatigue life analysis method. Nanjing University of Aeronautics and Astronautics 42. Wang X, Sun JQ (2005) Multistage regression fatigue analysis of non-Gaussian stress processes. J Sound Vibr 280:455–465 43. Winerstein SR (1988) Nonlinear vibration models for extremes and fatigue. J Eng Mech 114:1772–1790 44. Yu L, Da S PK, Barltrop N (2010) A new look at the effect of bandwidth and non-normality on fatigue damage. Fatigue Fract Eng Mater Struct 27 45. Yu BY, Feng QK, Yu XL (2013) Dynamic simulation and stress analysis for reciprocating compressor crankshaft. Proc Inst Mech Eng Part C J Mech Eng Sci 227:845–851 46. Zheng L, Yao H (2013) Effects of the dynamic vehicle-road interaction on the pavement vibration due to road traffic. J Vibroeng. 15:1291–1301

Chapter 6

Fatigue Reliability Evaluation of Structural Components Under Random Loadings

Abstract An efficient method for time-dependent fatigue reliability assessment of mechanical components under random loadings is proposed. The randomness of fatigue damage is treated in two aspects. The first one is the uncertainty quantification from the external random loading. The second one is the uncertainty quantification of the fatigue property of the structural component. The former is characterized by Gaussian distribution derived from the rainflow cycle distribution, medium stresslife (S–N) curve, and the linear damage accumulation rule. The latter is described with the probabilistic stress-life (P–S–N) curve based on log-normal distribution. The proposed method has colligated these two aspects together to evaluate the expectation and confidence interval of fatigue reliability. Finally, a numerical example is provided to verify the effectiveness of the developed approach. A comparison with bootstrap method is also carried out. The comparative result has shown rational accuracy of the proposed method [1–11].

6.1 Introduction Normally, the service loadings for many mechanical components are random [12]. The fatigue life of mechanical component under random loading is stochastic in nature [15]. Assessing fatigue reliability is one of the most frequently encountered problems in engineering practice [14–16]. Fatigue reliability evaluation under random loading is a challenging problem, despite extensive progress made in the past decades. Five causes of uncertainty of fatigue life were detected by Svensson [13]. They are (1) load variation, (2) material properties, (3) structural properties of components, (4) parameter estimations and, (5) model errors. It is clear that the first three items are actual random factors of the fatigue phenomenon itself, and the last two items are caused by a lack of knowledge of the fatigue damage mechanism for a specified structural component. The last two causes are ineluctable for any theoretical method. The remaining three causes can be divided into two categories. The first item indicates the uncertainty quantification from the external random loading. In essence, the following two items represent the random fatigue property of the

© National Defense Industry Press 2022 Y. Jiang et al., Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test, https://doi.org/10.1007/978-981-16-3694-3_6

123

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6 Fatigue Reliability Evaluation of Structural Components …

mechanical structure itself. Hence, for a theoretical method, these two aspects should be taken into account substantially. The stress time history induced by random loading is very complex and the stress cycle is not obvious. Hence, the first problem is how to extract loading cycles from the stress time history. Many loading cycle counting methods have been proposed in literature; however, rainflow algorithm has been the most widely accepted [9]. For detail of rainflow counting procedure, one can refer to [4, 9]. The complexity of rainflow counting algorithm makes the characterization of rainflow fatigue damage be very difficult. A lot of efforts have been made to describe the rainflow cycle distributions in literature [1, 3, 18]. Because the random loading is a stochastic process, and the number of rainflow cycles in a specified time period is a random variable. However, for high-cycle-fatigue (HCF) problem, this randomness is negligible [5]. Generally, the fatigue damage randomness from external random loading can be divided into two aspects: the number of rainflow cycles, and the rainflow cycle distribution. Often, the medium S–N curve is used to describe the average fatigue property of the structural components [6, 8]. But it cannot represent the randomness of fatigue property. Concerning the randomness of fatigue property is more valuable and necessary in engineering field. Hence, a lot of investigations about the expression of P–S–N curve are presented in literature [7, 10, 17]. Normally, the P–S–N curve is fitted based on fatigue test data in different stress levels, and it can characterize the randomness of fatigue property of the structure objectively. Based on the two aspects of fatigue damage randomness stated hereinbefore, a new methodology for fatigue reliability evaluation under random loading is developed. The non-parametric method [2] is employed to estimate the rainflow cycle distribution based on stress time history, and then the Gaussian distribution derived from the non-parametric rainflow cycle distribution, medium S–N curve and linear damage accumulation rule is used to characterize the randomness from the external loading aspect. The randomness from the fatigue property of structure is described by a log-normal distribution based P–S–N curve. The proposed method has colligated these two aspects together to evaluate the expectation and confidence interval of fatigue reliability. Finally, a numerical example is provided to verify the effectiveness of the newly developed method. The comparison with bootstrap estimations has shown the rational accuracy of the proposed method.

6.2 Uncertainty Quantification from External Loading 6.2.1 The Number of Rainflow Cycles For a specified time period t, the number of rainflow cycles included in the random loading is uncertain. This can be attributed to that the number of rainflow cycles, vc , in unit time is a random variable. And the total number of cycles M t , in time period t is,

6.2 Uncertainty Quantification from External Loading

Mt =

t 

125

vc(i)

(6.1)

i=1

where vc(i) is the number of rainflow cycles in the i-th unit time interval. There is no explicit closed-form expression for the probability density function (PDF) of vc presented in literature. The expectation of M t is E[Mt ] = v¯ c t

(6.2)

where E[·] denotes mathematical expectation, v¯ c is the expectation of vc , and it can be derived from the power spectral density (PSD) of the random loading [3]. Fatigue failures related to random loadings are always HCF problems. Johannesson [5] assumed that the randomness of M t is practically negligible and it can be treated as a deterministic value for HCF problem. The theoretical argument of this assumption is provided herein below, based on large number hypothesis. According to the research of Bishop [1], the numbers of rainflow cycles in different unit time intervals can be recognized as independent identically distributed (i.i.d.) random variables. The practical time duration t will be very long, for HCF problem. t  vc(i) can give an accurate When t tends to infinity, the sample average vˆ c = 1t i=1

estimation of v¯ c in Eq. (6.2). According to Chebyshev theorem, for any positive number ε, we have    lim P vˆ c − v¯ c  < ε = 1

t→∞

where P(·) is the probability operator, and then

t  i=1

(6.3)

vc(i) = Mt ≈ E[Mt ] = v¯ c t.

Hence, the total number of rainflow cycles, M t can be referred to as a constant for the specified time duration t during HCF problems. And it is determined by the spectral moments of PSD [3] and time duration t.

6.2.2 The Rainflow Cycle Distribution The amplitude of rainflow cycle from random loading is a random variable, and it follows the so-called rainflow cycle distribution which is related to the PSD, higherorder statistics, etc. It is very difficult to determine the mathematical expression of rainflow cycle distribution in most case in spite of that there are many methods proposed in the past decades [1, 3, 5, 18]. Although some methods based on spectral data, namely spectral method could provide elegant expressions, the accuracy and applicable range are limited. Here, we begin with the time-domain data, and use

126

6 Fatigue Reliability Evaluation of Structural Components …

Fig. 6.1 The non-parametric rainflow cycle distribution from stress time history

non-parametric statistical method to fit the rainflow cycle distribution based on the stress time history, as shown in Fig. 6.1. Rainflow counting method combined with the linear damage accumulation rule, namely Miner rule, can give reasonable estimation of fatigue damage for stationary random loading [5]. When studying the uncertainty quantification from external loading, the fatigue property of the structure is assumed to be deterministic. And the medium S–N curve, namely the S–N curve corresponding to the failure probability of 50% is selected. Normally, it is expressed as follows, N Sb = A

(6.4)

where b and A are the stress-life exponent and fatigue strength coefficient, respectively. S denotes the stress amplitude of rainflow cycle, and N is the medium number of cycles to failure at amplitude S. Based on Miner rule, the cumulative fatigue damage for time duration t, is, Mt  Sib D(t) = A i=1

(6.5)

where M t denotes the number of rainflow cycles corresponding to time t. It can be recognized as a constant according to the analysis in Sect. 6.2.1. Hence, the statistical property of b-th power of rainflow cycle amplitude, S b , will determine the randomness of fatigue damge D(t). The amplitudes of different rainflow cycles can assumed to be i.i.d. random variables [1], then the expectation and variance of fatigue damage can be expressed as follows, Mt  b  E S A

(6.6)

 

Mt  2b  2 b E S − E S A2

(6.7)

μ D (t) = σ D2 (t) =

6.2 Uncertainty Quantification from External Loading

127

Hence, based on the non-parametric distribution f NP (S) (Fig. 6.1), the mean and variance can be rewritten as follows, Mt μ D (t) = A

∞ S b f RFC (S)dS

(6.8)

⎛∞ ⎞2 ⎤  S 2b f RFC (S)dS − ⎝ S b f RFC (S)dS ⎠ ⎦

(6.9)

0

⎡ σ D2 (t) =

Mt ⎣ A2

∞ 0

0

Referring back to Eq. (6.6), the amplitudes of rainflow cycles are i.i.d. random variables. And from the viewpoint of statistics, the fatigue damage induced by each cycle, D is a random variable. M t is very  large for HCF problem. Based on central limit theorem, the total damage D(t) = 1Mt D will follow Gaussian distribution. Thus, the randomness of fatigue damage from the external loading aspect can be characterized by the following Gaussian distribution,   [D(t) − μ D (t)]2 exp − f D (D(t)) = √ 2σ D2 (t) 2π σ D (t) 1

(6.10)

where μ D (t) and σ D (t) are defined in Eqs. (6.8) and (6.9). The coefficient of variation referred hereinbelow is ς D = σ D (t)/μ D (t). The distribution function will change with time t. One should note that the expression in Eq. (6.10) is not the distribution function of the actual fatigue damage, and it is just represent one respect of the randomness.

6.3 The Uncertainty Quantification of the Fatigue Property The external loading should be constant when describing the fatigue property of mechanical structure only. Often, the log-normal distribution is used to describe the randomness of fatigue property in one specified stress level [7, 10],   1 (ln N − ln N50% )2 exp − f N (N ) = √ 2σ 2 2π σ N

(6.11)

where N is the random number of cycles to failure at stress level S, log(N 50% ) = E[log(N)] is the logarithm to the base 10 of medium fatigue life, σ 2 is the variance of log(N). Note that log(N) follows Gaussian distribution with mean log(N 50% ) and standard σ,

128

6 Fatigue Reliability Evaluation of Structural Components …

Fig. 6.2 A schematic plot of P–S–N curve based on log-normal distribution

 

L N − L 50% N f L N (L N ) = √ exp − 2σ 2 2π σ 1

2  (6.12)

where L N denotes log(N) and, L 50% N is log(N 50 ). The coefficient of variation referred  50% hereinbelow is δ = σ L N . Normally, P–S–N curve is used to describe the randomness of fatigue property in different stress levels. It can be evaluated from the constant-amplitude fatigue data in three to four different stress levels [6, 7], as show in Fig. 6.2. We have used many simulated fatigue data in every stress level in Fig. 6.2; however, four to six specimens are tested in each stress level in practice, for saving time and cost. Many investigations have shown that the variances of fatigue lives in different stress levels are different. Shimizu has assumed that the coefficient of variation δ is constant for different levels [10]. This assumption has been validated by some materials and structures [10]. Let k β be the β (0 < β < 1) lower quantile of standardized normal distribution. Based on Eq. (6.12), the fatigue life corresponding to lower β fractile in the stress level S satisfying the following expression,   β 1 + kβ δ L N = L 50% N

(6.13)

Taking the exponents in both sides of Eq. (6.13), results in, Nβ = (N50 )1+kβ δ

(6.14)

where N β denotes the number of cycles to failure with probability β. According to Eq. (6.4), N 50 = A/S b , then we have, Nβ S b(1+kβ δ) = A1+kβ δ

(6.15)

6.3 The Uncertainty Quantification of the Fatigue Property

129

Equation (6.15) is the mathematical expression of the P–S–N curve based on log-normal distribution. It is clear that S–N curves corresponding to different failure probabilities β 1 and β 2 have the similar expressions. Figure 6.2 illustrates a schematic plot of the P–S–N curve, and the 50%-S–N curve is the so-called medium S–N curve.

6.4 Fatigue Reliability Analysis The two aspects of fatigue damage randomness are analyzed hereinbefore. They are the uncertainty quantification from the external random loading and that from the fatigue property of the structure. Quantitative models have been established to characterize them in Sects. 6.2 and 6.3. Based on these models, a methodology is developed to evaluate the expectation and confidence interval of the time-dependent fatigue reliability.

6.4.1 Fatigue Reliability Expectation Based on Miner rule, the critical fatigue damage is assumed to be DCR = 1. The fatigue damage expectation μD (t) is calculated with medium S–N curve (Eq. (6.8)). From the definition of medium S–N curve, the equation μD (t) = 1 means that the expectation of fatigue reliability R(t) = 0.5. But in most cases, μD (t) is not equal to unity, and the results are not explicit. Generally, this problem can be treated in three cases: Case 1: μD (t) = 1, means R(t) = 0.5. Case 2: μD (t) < 1, means R(t) > 0.5. Referring back to the definition of P–S–N curve, there will be a β-S–N curve (β < 50%) on the left-side of medium S–N curve (Fig. 6.2) β β satisfying μ D (t) = 1, where μ D (t) is the fatigue damage expectation calculated from the β-S–N curve. Now, R(t) = 1 – β. Case 3: μD (t) > 1, means R(t) < 0.5, and this is contrary to case 2. There will be a β-S–N curve β (β > 50%) on the right-side of medium S–N curve (Fig. 6.2) satisfying μ D (t) = 1. And R(t) = 1 – β. We can obtain the result directly in case 1. But a method is needed to determine β for cases 2 and 3. According to Eqs. (6.8) and (6.15), the fatigue damage expectations from medium S–N curve and β-S–N curve are,

130

6 Fatigue Reliability Evaluation of Structural Components …

Fig. 6.3 The computer program algorithm for fatigue reliability expectation

Mt μ D (t) = A

∞ S 0

b

β f NP (S)dSμ D (t)

=

Mt

∞

A1+δkβ

S b(1+δkβ ) f NP (S)dS;

(6.16)

0

The non-parametric distribution f NP (S) does not have explicit closed-form expression. So it is impossible to determine β based on mathematical derivation. But the computer program is a tool to solve this problem. The flowchart of the algorithm is provided in Fig. 6.3.

6.4.2 Fatigue Reliability Confidence Interval The fatigue damage distribution from medium S–N curve is described using Gaussian distribution (Eq. (6.10)). Therefore, the 1−2α confidence interval for the fatigue damage can be expressed as follows,   [Dα (t), D1−α (t)] = μ D (t) + kα σ D (t), μ D (t) + k1−α σ D (t)

(6.17)

where μ D (t) and σ D (t) are defined in Eqs. (6.8) and (6.9); k α and k 1−α are the α and 1−α lower quantiles of standardized normal distribution, respectively. We cannot obtain the 1−2α confidence interval of fatigue reliability from Eq. (6.17) directly. Furthermore, the problem about the confidence interval is complex than that for the expectation. The lower confidence limit of fatigue damage corresponds to the upper limit of fatigue reliability, and vice versa. Taking the upper confidence limit of fatigue reliability R1−α (t) for example, there is no direct transformation from Dα (t) to Dαβ1 (t)

6.4 Fatigue Reliability Analysis

131

= 1. So we should solve this problem indirectly. Imagine that if Dαβ1 (t) = 1 for β1 β 1 -S–N curve,  damage expectation  μ D (t) from β 1 -S–N will be  then the fatigue β

β

β

β

β

μ D1 (t) = 1 1 + kα ς D1 (t) , where ς D1 = σ D1 μ D1 is defined after Eq. (6.12). Hence, we can build the algorithm for R1-α (t) according to the following cases, Case 1: Dα (t) = 1, means R1-α (t) = 0.5. Case 2:

  β β Dα (t) < 1, calculate μ D1 (t) = 1/ 1 + kα ς D1 (t) , then search on the left-side of medium S–N curve a β 1 -S–N curve satisfying this equation, and R1−α (t) = 1−β 1 . Case 3:

  β β Dα (t) > 1, calculate μ D1 = 1/ 1 + kα ς D1 (t) , then search on the right-side of medium S–N curve a β 1 -S–N curve satisfying this equation, and R1−α (t) = 1−β 1 . For the lower confidence limit of fatigue reliability Rα (t), the algorithm is similar; we will not list the cases herein for the sake of simplicity. The target probability S–N curve is assumed to be β 2 -S–N, then Rα (t) = 1−β 2 . The algorithms for R1-α (t) and Rα (t) are similar with that illustrated in Fig. 6.3. Finally, the 1−2α confidence interval of fatigue reliability is, 

 Rα (t), R1−α (t) = [1 − β2 , 1 − β1 ]

(6.18)

6.5 Numerical Example A numerical example is provided to validate the capability of the proposed method. The P–S–N curve of the structure is assumed to be,  1+0.1kβ Nβ S 4(1+0.1kβ ) = 2.5 × 1014

(6.19)

We can see that b = 4, A = 2.5 × 1014 , δ = 0.1. The random loading is a Gaussian process, as shown in Fig. 6.4. It is a zero mean process, and the RMS value is 120 MPa. A sample time history and the PSD are shown in Fig. 6.4a and b, respectively. The PSD can be used to calculate the rate of rainflow occurrence, v¯ c in Eq. (6.2). To validate the accuracy of the proposed method, Monte-Carlo simulation and bootstrap method are used with a great deal of sample time histories and bootstrap replications. Comparisons are carried out extensively.

132

6 Fatigue Reliability Evaluation of Structural Components …

Fig. 6.4 The Gaussian random loading: a sample time history; b PSD

6.5.1 Monte-Carlo Simulation and Bootstrap Method As shown in Fig. 6.4b, when the PSD is specified, we can generate different sample time histories based on the standard Gaussian simulation method [11]. We have generated 5000 sample time histories. Rainflow cycle counting is applied to each time history. An S–N curve with random fatigue probability is selected from the P–S–N curve (Eq. (6.19)) for each counted rainflow cycle series. When t = 500 s, the fatigue damage series corresponding to the 5000 sample time histories is shown in Fig. 6.5. It is clear that magnitudes of the fatigue damages are very different. Let the critical fatigue damage Dcr = 1, then for the i-th fatigue damage Di > 1 means that a failure occurs. It is assumed that the failure number in time t is (t), then the observed fatigue reliability based on Monte-Carlo simulation is, RMC (t) = 1 −

(t) 5000

(6.20)

The observed result from Eq. (6.20) is based on single fatigue damage series. It cannot represent the uncertainty of the fatigue reliability. To improve the estimation Fig. 6.5 The observed fatigue damage series (t = 500 s) from Monte-Carlo simulation

6.5 Numerical Example

133

Fig. 6.6 The expectation and 98% confidence interval of fatigue reliability from bootstrap method

accuracy and give a description of the uncertainty of the fatigue reliability, bootstrap method is introduced. The replicated bootstrap sample size is 10,000, and the lowest empirical requirement is 200. The expectation and 98% confidence interval of fatigue reliability with respect to time t are shown in Fig. 6.6, from bootstrap method.

6.5.2 Theoretical Results from the Proposed Method First, we have obtained the non-parametric rainflow cycle distribution (Fig. 6.7) based on the stress time history as shown in Fig. 6.4a. The non-parametric distribution can describe the distribution behavior of the rainflow cycles properly, as shown in Fig. 6.7. It gives a smooth PDF curve especially for the larger amplitude range which will dominate the fatigue damage process. According to the algorithms developed in Sects. 6.4.1 and 6.4.2, we have derived the theoretical results of the expectation and 98% confidence interval of fatigue reliability, as shown in Fig. 6.8. A comparison with the bootstrap results is shown in Fig. 6.8 as well. It is clear that the theoretical method can give stable estimations. Fig. 6.7 The histogram and non-parametric distribution of rainflow cycles

134

6 Fatigue Reliability Evaluation of Structural Components …

Fig. 6.8 Comparison between the theoretical estimations and the bootstrap estimations

Normally, the concerned range of reliability in engineering practice is 0.5 ≤ R(t) ≤ 1. From the upper panel in Fig. 6.8, we can see that the theoretical results are very consistent with the bootstrap results in this range. The bootstrap estimations fluctuate severely below the 10–3 order of magnitude. This is mainly because in this order of magnitude, the bootstrap sample size 10,000 is relative small to give a stable estimation. Furthermore, based on the bootstrap results and theoretical results, we can see that the randomness of the fatigue reliability is very small in the concerning range.

6.5.3 Results Analysis The results from Monte-Carlo simulation and the proposed method are provided in the former subsections, and a qualitative comparison is illustrated in Fig. 6.8. Furthermore, we would like to analyze these results quantitatively herein. The computational errors of the theoretical method are investigated with reference to bootstrap estimations from large sample size. The uncertainty of the fatigue reliability is analyzed as well. It is characterized by the confidence interval-width coefficient, which is defined as follows, ϑ(t) =

R1−α (t) − Rα (t) R(t)

(6.21)

6.5 Numerical Example

135

where Rα (t) and R1−α (t) are the lower and upper limits of the 1−2α confidence interval, R(t) is the expectation of fatigue reliability. The fatigue reliability values at some representative time points are tabulated in Table 6.1. The first column lists the time points. The second column lists the reliability expectations from bootstrap method. In the third column, there are the 98% confidence intervals from bootstrap method. The fourth column lists the confidence interval-width coefficients (Eq. (6.21)). The fifth column lists the reliability expectations from the proposed method (the italic numbers behind indicate the relative deviations from the bootstrap estimations in second column). The sixth column lists the 98% confidence intervals from the proposed method (the italic numbers behind indicate the relative deviations from the bootstrap estimations in the third column). We can see that for the reliability expectation, the largest deviation of theoretical result from the bootstrap result is 14.55%, this is acceptable. Comparing to the bootstrap results, the confidence intervals from the theoretical method are rational as well. For the lower and upper confidence limits, the largest relative deviations are 13.05% and 16.28%, respectively. These deviations are acceptable in engineering field. Furthermore, these deviations are partially due to the unstableness of the bootstrap estimations. In addition, from the fourth column of Table 6.1, it is clear that the uncertainty quantification of the fatigue reliability increases with the reduction of the reliability itself. However, the values of confidence interval-width coefficient Table 6.1 The expectation and 98% confidence interval of fatigue reliability Time (s) Monte-Carlo

Proposed method

RB

[RB(0.01) , RB(0.99) ] ϑ

500

0.9983

[0.9968, 0.9994]

0.0026 0.9836 (−1.47%) [0.9826, 0.9846] (−1.42%, −1.48%)

3000

0.9049

[0.8950, 0.9146]

0.0217 0.9017 (−0.35%) [0.8958, 0.9079] (0.09%, 0.73%)

5000

0.8070

[0.7940, 0.8196]

0.0317 0.8361 (3.61%)

[0.8263, 0.8465] (4.07%, 3.28%)

7500

0.6926

[0.6773, 0.7074]

0.0435 0.7542 (8.89%)

[0.7394, 0.7697] (9.17%, 8.81%)

10,000

0.5934

[0.5772, 0.6098]

0.0549 0.6723 (13.30%)

[0.6525, 0.6930] (13.05%, 13.64%)

13,000

0.5011

[0.4850, 0.5168]

0.0435 0.5740 (14.55%)

[0.5483, 0.6008] (13.05%, 16.25%)

17,000

0.4048

[0.3886, 0.4210]

0.0800 0.4429 (9.41%)

[0.4093, 0.4780] (5.33%, 13.54%)

23,000

0.3004

[0.2858, 0.3152]

0.0979 0.3146 (4.73%)

[0.2774, 0.3541] (−2.94%, 12.34%)

33,000

0.2001

[0.1870, 0.2132]

0.1309 0.2146 (7.25%)

[0.1854, 0.2479] (−0.9%, 16.28%)

54,000

0.0982

[0.0882, 0.1082]

0.2037 0.0948 (−3.46%) [0.0779, 0.1163] (−11.68%, 7.49%)

R

[R0.01 , R0.99 ]

136

6 Fatigue Reliability Evaluation of Structural Components …

are always much less than unity. This means that for HCF problem, neglecting the uncertainty quantification from external loading will not result in great computational errors.

6.6 Conclusions The randomness of fatigue damage of structural component is divided into two aspects: the uncertainty quantification from the external random loading; and that from the fatigue property of the mechanical component. The Gaussian distribution function is employed to characterize the first aspect. The P–S–N curve based on lognormal distribution is adopted to characterize the second aspect. A theoretical method by taking these two aspects into consideration sufficiently is proposed to evaluate the expectation and confidence interval of time-dependent fatigue reliability. A numerical example is provided to validate the applicability of the proposed method. First, Monte-Carlo simulations and bootstrap method are used to evaluate the expectation and confidence interval of fatigue reliability based on a large amount of samples. And these results can be recognized credible in some sense. Second, the proposed method is applied. The theoretical estimations are compared extensively with those from bootstrap method. The comparative result has verified the capability and accuracy of the proposed approach; and it is much less computationally intensive than the bootstrap method. Furthermore, for HCF issues, the uncertainty quantification from the stationary random loading does not have significant effect on the computational accuracy of the fatigue reliability in the normally concerned reliability range (0.5 ≤ R(t) ≤ 1). The uncertainty quantification of fatigue reliability increases with the reduction of the reliability itself. When the reliability is in the 10–3 or 10–4 order of magnitude, the bootstrap method can not give a stable estimation even based on a very large sample size. But the proposed method could provide stable estimations relatively. Acknowledgements The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (No. 51875570).

References 1. Bishop NWM (1988) The use of frequency domain parameters to predict structural fatigue. PhD thesis, University of Warwick, UK 2. Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis-the kernel approach with S-plus illustrations. Oxford Science Publications 3. Dirlik T (1985). Application of computers in fatigue analysis. PhD thesis, University of Warwick, UK 4. Johannesson P (2002) On rainflow cycles and the distribution of the number of interval crossings by a Markov chain. Probab Eng Mech 17(2):123–130

References

137

5. Johannesson P (1999) Rainflow analysis of switching Markov loads. PhD dissertation, Lund University, Sweden 6. Kihl DP, Sarkani S, Beach JE (1995) Stochastic fatigue damage accumulation under broadband loadings. Int J Fatigue 17(5):321–329 7. Ling J, Pan J (1997) A maximum likelihood method for estimating P-S-N curves. Int J Fatigue 19(5):415–419 8. Rambabu DV, Ranganath VR, Ramamurty U, Chatterjee A (2010) Variable stress ration in cumulative fatigue damage: experiments and comparison of three models. Proc I Mech E Part C: J Mech Eng Sci 224(2):271–282 9. Rychlik I (1987) A new definition of the rainflow cycle counting method. Int J Fatigue 9(2):119– 121 10. Shimizu S, Tosha K, Tsuchiya K (2010) New data analysis of probabilistic stress-life (P-S-N) curve and its application for structural materials. Int J Fatigue 32(3):565–575 11. Shinozuka M, Jan CM (1972) Digital simulation of random processes and its applications. J Sound Vib 25(1):111–128 12. Smith CL, Chang JH, Rogers MH (2007) Fatigue reliability analysis of dynamic components with variable loadings without Monte-Carlo simulation. American Helicopter Society 63rd Annual Forum, Virginia, US, pp 1–11 13. Svensson T (1997) Prediction uncertainties at variable amplitude fatigue. Int J Fatigue 17(Suppl. 1):S295-302 14. Szerszen MM, Nowak AS, Laman JA (1999) Fatigue reliability of steel bridges. J Constr Steel Res 52(1):83–92 15. Tomaszek H, Jasztal M, Zieja M (2011). A simplified method to assess fatigue life of selected structural components of an aircraft for a variable load spectrum. Eksploatacja i Niezawodnosc – Maintenance and Reliability 13(4):29–34 16. Tomaszek H, Jasztal M, Zieja M (2013) Application of the Paris formula with m=2 and the variable load spectrum to a simplified method for evaluation of reliability and fatigue life demonstrated by aircraft components. Eksploatacja i Niezawodnosc – Maintenance and Reliability 15(4):297–303 17. Tosha K, Ueda D, Shimoda H, Shimizu A (2008) A study on P-S-N curve for rotating bending fatigue test for bearing steel. STLE Tribol Trans 51(2):166–172 18. Tovo R (2002) Cycle distribution and fatigue damage under broad-band random loading. Int J Fatigue 24(11):1137–1147

Chapter 7

Non-Gaussian Random Vibration Accelerated Test

Abstract A novel non-Gaussian random vibration accelerated test methodology was proposed, which can significantly reduce the test time and the sample size. First, fatigue life prediction models of Gaussian and non-Gaussian random vibration were proposed based on random vibration and fatigue theory. These mathematical models comprehensively associated the vibration fatigue life of structures, the characteristics of vibration excitations (such as the root mean square, power spectral density, spectral bandwidth and kurtosis value) and the dynamic transfer characteristics of structures (such as the natural frequency and damping ratio) together. Meanwhile a detailed solving method was also presented for determining the unknown parameters in the models. Second, a non-Gaussian random vibration accelerated test system including test specimens and fixtures was designed. Third, several groups of random vibration fatigue tests were designed and conducted with the aim of investigating effects of both Gaussian and non-Gaussian random excitation on the vibration fatigue. The stress response at a weak point of a notched specimen structure was measured under different base random excitations. According to the measured stress responses, the structural fatigue lives corresponding to the different vibrational excitations were predicted by using the WAFO simulation technique. Then a couple of destructive vibration fatigue tests were carried out to validate the ac-curacy of the WAFO fatigue life prediction method. After applying the pro-posed experimental and numerical simulation methods, various factors that affect the vibration fatigue life of structures were systematically studied, including root mean squares of acceleration, power spectral density, power spectral bandwidth and kurtosis. Finally, an application case for the fatigue life prediction of electronic product structures verified the effectiveness of the above non-Gaussian random vibration accelerated test method.

7.1 Introduction Many engineering structures usually undergo vibration loading. Fatigue is the most commonly encountered type of failure for structures operating under dynamic loading. Vibration fatigue is more complicated than general cyclic fatigue. To ensure the reliability and safety of structures during the operation, we need timely validation © National Defense Industry Press 2022 Y. Jiang et al., Non-Gaussian Random Vibration Fatigue Analysis and Accelerated Test, https://doi.org/10.1007/978-981-16-3694-3_7

139

140

7 Non-Gaussian Random Vibration Accelerated Test

of long-term durability of engineering structures under their service vibration environment. This is usually done by the laboratory vibration tests. However, operational life under normal vibration conditions could be too long that the laboratory vibration tests at those levels would not be possible for many structures and materials. Accelerated testing provides a savings in time and cost compared with testing at normal conditions. Therefore it is necessary to perform accelerated vibration testing in a laboratory environment, which in terms of loading is considerably more severe than the operative one. Then the operative life duration is estimated by relating the structural fatigue life tested in the laboratory condition by a proper scaling factor. In recent years, experimental methods for accelerated vibration fatigue testing are continuously under development. The work by Allegri and Zhang [1] addressed the usage of inverse power laws in accelerated fatigue testing under wide-band Gaussian random loading. The aim was not at predicting an absolute value of fatigue life but assessing the relative accumulation of fatigue damage. Özsoy et al. [2] proposed an accelerated life testing approach for aerospace structural components. A closedloop system driven by power spectral density profiles was employed to run the constant amplitude resonance test. By changing the test durations and accordingly the mission profile amplitudes, a simple equation was proposed which relates accelerated test durations with the equivalent alternating stresses. Shires [3] discussed the time compression (test acceleration) of broadband random vibration tests. Conventionally, the test level is accelerated from the root mean acceleration and an assumed power constant (k = 2) is applied. The Miner-Palmgren hypothesis of accumulated fatigue is used to re-assess the potential error in test severity, which shows a substantially reduced sensitivity to the value of k depending on the distribution of actual vibration intensities around the time-compressed test intensity. Xu et al. [4] developed a method for extracting the information on the frequency of the events expected in the service life from a time series based on wavelet analysis, clustering, and Fourier analysis. The identified events and their corresponding data are used to generate the accelerated durability testing PSD profiles, which can be directly applied as the driven profile in the lab test. Yun et al. [5] developed a vibration-based closed-loop high-cycle resonant fatigue testing system. To minimize the testing duration, the test setup was designed for a base-excited multiple-specimen arrangement driven in a high-frequency resonant mode, which allows completion of fatigue testing in an accelerated period. Cesnik et al. [6] proposed an improved accelerated fatiguetesting methodology based on the dynamic response of the test specimen to the harmonic excitation in the near-resonant area with simultaneous monitoring of the modal parameters. The measurements of the phase angle and the stress amplitude in the fatigue zone were used for the real-time adjustment of the excitation signal according to the changes in the specimen’s modal parameters. Ashwini et al. [7] addressed applicability of various theories for estimation of failure time in normal usage and accelerated condition for Gaussian random vibration testing. Another issue addressed was whether application of damping material makes any difference. Experimentally observed failure time in random vibration for both bare and damped beams under random vibration were compared, and the exponent in terms of both G2/Hz and Grms for accelerated testing was obtained for bare and damped beams. Vibration

7.1 Introduction

141

fatigue analysis of a cantilever beam under white noise random input using several vibration fatigue theories was performed by Eldogan and Cigeroglu [8]. Fatigue life calculations by utilizing time domain (Rain-flow counting method) and frequency domain methods were repeated for different damping ratios and the effect of damping ratio was studied. Fatigue tests were performed on cantilever beam specimens and fatigue life results obtained experimentally were compared with that of in-house numerical codes. It was observed that the fatigue life result obtained from Dirlik method is considerably similar to that of the rain-flow counting method. The vibration loading in the above studies is limited to sinusoidal loading with constant amplitude or Gaussian random loading with alternating amplitude, and the random loading fatigue damage calculation is based on the assumption of Gaussian distribution. However, the dynamic loading shows non-Gaussianity in some practical applications, such as the ground vibration generated by wheeled vehicles travelling over irregular terrain, atmospheric turbulence for the aerospace sector or wind pressure fluctuations on building envelopes [9]. Because traditional Gaussian random vibration test signals cannot accurately represent the non-Gaussian vibration signal with high-peak characteristics seen in the real-life use of many structures, the latest MIL-STD-810G standard also requires test engineers to “ensure that test and analysis hardware and software are appropriate when non-Gaussian distributions are encountered” (refer to Method 525 on Page 514.6A-5 in literature [10]). For this purpose, the non-Gaussian vibration controller has just been developed by few manufacturers (such as Econ Corporation and Vibration Research Corporation) in recent years. Since the basic purpose of the non-Gaussian vibration controller is to simulate the real non-Gaussian vibration environment of some products, it is meaningful to carry out further experimental research on how to use non-Gaussian vibration in accelerated fatigue testing. Generally, conservative or incorrect results will be obtained if non-Gaussianity is ignored during fatigue damage estimation and fatigue life prediction. Although in recent years there have been few studies on the non-Gaussian random vibration fatigue theory, the studies described in literature [11–14] are directly based on non-Gaussian response without considering non-Gaussian excitation and structural dynamics characteristic. However, in the laboratory vibration test, the main consideration is the vibration excitation profile. Therefore, there is a “gap” between theoretical research and engineering applications and it is necessary to establish a link between the Gaussian or non-Gaussian vibration excitation and structural vibration fatigue life, which will facilitate the design and statistical analysis of the accelerated vibration test. Fatigue damage as a result of random loading can be assessed in either the time or frequency domain. In the time domain, the Rain-flow counting method is universally accepted for random vibration fatigue analysis. WAFO (Wave Analysis for Fatigue and Oceanography) is a toolbox of MATLAB routines for statistical analysis and simulation of random waves and random loads [15]. The main purpose of WAFO is for scientific research, and thus the aim of WAFO is not to contain all the features of the commercial software. Nevertheless, it is also widely used in industry. So far, there has been no report on WAFO applied to non-Gaussian vibration fatigue analysis.

142

7 Non-Gaussian Random Vibration Accelerated Test

Therefore, this book is not about proposing another method for predicting fatigue life. Rather, the objectives of this study are as follows: (1) develop a hybrid test strategy for the accelerated random vibration fatigue test, which can generate a design of the experimental test plan, significantly reduce test times and costs, and avoid complex finite element modeling and verification process and the risk of inaccuracies caused by modeling; (2) verify the feasibility of WAFO for non-Gaussian vibration fatigue life prediction; (3) experimentally and numerically investigate all the factors affecting the structure random vibration fatigue life; (4) explore the possibility of non-Gaussian vibration for accelerated fatigue testing.

7.2 Model of Random Vibration Accelerated Test 7.2.1 Model of Gaussian Random Vibration Accelerated Test Firstly the analysis starts from the most basic description of the material fatigue in terms of S–N curve. Typically ideal mathematical expression for S–N curve is written as follows: N = cS −b

(7.1)

where S denotes the stress amplitude, N denotes the stress cycles resulting in the failure, b and c are the constant fatigue parameters that depend on the material. On the base of the famous Miner cumulative fatigue damage criterion, the fatigue damage under a joint action of different amplitudes of stress is: D=

 ni Ni i

(7.2)

where n i : the number of cycles applied at a fixed stress amplitude Si , Ni : the number of cycles the material can withstand at applied fixed stress amplitude Si , and D: Cumulative fatigue damage (fatigue failure generally considered to occur at D = 1). Substituting Eq. (7.1) into Eq. (7.2), it is obtained: D=

 ni i

cSi−b

=

 ni i

c

Sib

(7.3)

For continuous time histories of random stress, Eq. (7.3) can be written in the form of the following integral [16]:

7.2 Model of Random Vibration Accelerated Test

D=

v0+ T

∞ [ p(S)/(cS

−b

v+ T )]d S = 0 c

0

143

∞ p(S)S b d S

(7.4)

0

where T is the total time of exposure to the random vibration excitation, p(S) denotes the probability density function of random stress response on the specimen, v0+ is the average number of the zero up-crossing per unit time in the stress time history. The specimen under the vibration test generally can be approximated as a linear system and the excitation generated by the vibration test equipment can be regarded as the input of the system. As the frequency–response characteristics of the specimen is similar to a narrow-band filter, it can be considered that under stationary Gaussian random excitation (either broad-band or narrow-band), the stress response of the specimen is close to the stationary narrow-band Gaussian distribution. When the random stress response approximates a stationary narrow-band Gaussian distribution, according to the random-process theory, the amplitude probability density function of the stress p(S) has the following Rayleigh form [16]: p(S) =

S −S 2 /2σS2 e σs2

(7.5)

where σ S is the RMS value of the stress (i.e. standard deviation). Substituting Eq. (7.5) into Eq. (7.4) and doing the integration, the following equation could be obtained: D=

b v0+ T √ 2σ S [1 + b/2] c

(7.6)

Herein  represents the Gamma function. Engineering practice shows that the damping ratio ξ of a general structure is usually much 3, λ > 1, the super-Gaussian stress response will accelerate the process of the fatigue damage. The influence factors for K s can be further analysed based on the random vibration theory. As the first-order mode of electronic assembly plays a decisive role in the structural response, a single degree of freedom model under basic excitation is established for analysis as shown in Fig. 7.1. The transfer function between the acceleration response y and basic acceleration excitation x can be derived as: cs + k Y (s) = 2 X (s) ms + cs + k  k c Given ω1 = , ξ= √ m 2 mk H (s) =

(7.15)

(7.16)

Fig. 7.1 Dynamic model of a base-excited SDOF system

m k

y c x

146

7 Non-Gaussian Random Vibration Accelerated Test

Hence H (s) =

2ξ ω1 s + ω12 s 2 + 2ξ ω1 s + ω12

(7.17)

where ω1 = 2π f 1 . f 1 denotes the first-order natural frequency, and ξ is damping ratio. These two parameters characterise the structural dynamics of an electronic assembly and the pass-band width of an electronic assembly is also determined by these two parameters, namely BW H = 2ξ f 1

(7.18)

Then the amplitude distribution of the structural response subject to the nonGaussian excitation is discussed. The output of a stochastic process x(t) through a linear system H ( f ) can be represented in the time domain: +∞ x(τ )h(t − τ )dτ y(t) =

(7.19)

−∞

where h(t) is the impulse response function of the system H ( f ), the integral above can be formulated in terms of the sum-limit: y(t) = lim

n 

n→∞

τ →0 k=1

x(τk )h(t − τk ) τk

(7.20)

where x(τk ) is an input random variable, τk is the sampling interval. According to the central-limit theorem, the distribution of the sum of a large number of statistically independent random variables tends to be a Gaussian distribution. If the output of the random process at any time y(t) is the sum of a large number of independent random variables, then y(t) is close to a Gaussian distribution. Obviously it requires two conditions: one is that random variables must be independent of each other, and the other is that independent random variables are added to solve a sum. 1 of the input random process is inversely Since the correlation time τx = 2BW X proportional to the effective frequency bandwidth BW X , τx will become very small when BW X is very large. If τx is small enough to meet τx > τk , y(t) tends to be a Gaussian distribution. On the contrary, if a non-Gaussian random process x(t) is applied to a linear system and the system pass-band BW H is relatively wide, ts is small. If ts