UNIVERSITY OF SOUTHAMPTON FACULTY OF ENGINEERING AND THE ENVIRONMENT Institute of Sound and Vibration Research Fatigue Life for non-Gaussian Random Excitations by Frédéric Kihm Thesis for the degree of Doctor of Philosophy December 2017 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF ENGINEERING AND THE ENVIRONMENT Institute of Sound and Vibration Research Thesis for the degree of Doctor of Philosophy FATIGUE LIFE FOR NON-GAUSSIAN RANDOM EXCITATIONS Frédéric Kihm This thesis presents several developments in analysing the fatigue life of mechanical components subjected to various kinds of random excitations and the subsequent mechanical vibration. Typically components must be designed in such a way that they can withstand the effects of the exposure to environmental conditions without being damaged. Their design must be verified using laboratory testing or by Finite Element (FE) calculations. Often, design and testing are performed on the basis of specifications taken from internal, national or international standards, with the implicit assumption that if the equipment survived the particular environment it would also survive the vibrations it will see in service. Previous work in the area has developed computer-based models to predict the fatigue damage witnessed by components under random loads, but most of these are limited to only stationary Gaussian random excitations. A few have concentrated on non-Gaussian responses, not excitation, resulting from some non-linearities in the structure being activated. This thesis describes the development of original statistical analysis methods with the ability to determine extreme responses and fatigue life estimates for linear structures when subjected to non-Gaussian random excitations. The emphasis is mainly on two sources of non-Gaussian random excitations namely clipped random excitations and random excitations with a high kurtosis value. Fatigue damage from specific sine-on-random excitations is also studied. In all cases, theoretical formulations were derived to obtain fatigue life estimates without the need for long time domain realisations. Such a statistical approach is particularly suited for simulating the fatigue damage induced during a random test on a shaker system. One of the main benefits of being able to assess the fatigue life in the case of a structure subjected to a leptokurtic random excitation is to create an accelerated test definition. The idea is to associate a specific kurtosis value to a given Power Spectral Density (PSD) in order to reduce the exposure duration, while encompassing the same fatigue damage potential as the original stationary and Gaussian random test. In practice, the engineer will be able to simulate the effects of the kurtosis control capability of some commercial vibration control systems on the fatigue damage experienced by in the device under test. This process will be achieved using a FE-based fatigue analysis tool, where the user specifies the excitation PSD, the kurtosis value and an FE results file representing the frequency response function linking the excitation and the stress response at each node or element of the FE model of the test article. The stress response PSD with the associated response kurtosis are obtained and the statistical rainflow histogram is extracted. Fatigue life estimates are then derived by associating the statistical rainflow histogram with the material fatigue curve. The theoretical formulations derived are applied to examples coming from numerical simulations. The estimate of the probability density functions obtained for the response stress and fatigue life correlate well with the results obtained from time domain simulated data, showing the robustness and the accuracy of the theoretical expressions. Table of Contents Table of Contents ................................................................................................................... i List of Tables........................................................................................................................ iii List of Figures ........................................................................................................................ v DECLARATION OF AUTHORSHIP ............................................................................... xiii Acknowledgements .............................................................................................................. xv Definitions and Abbreviations .......................................................................................... xvii 1 Introduction .................................................................................................................. 20 1.1 Introduction to environmental testing ................................................................... 20 1.2 Virtual vibration endurance tests ........................................................................... 25 1.3 Motivations ............................................................................................................ 27 1.4 Aims of the study .................................................................................................. 27 1.5 Objectives of the research ..................................................................................... 27 1.6 Contributions presented in the thesis ..................................................................... 28 1.7 Overview of the thesis contents ............................................................................. 31 2 Review of random vibration and fatigue analysis ........................................................ 34 2.1 Random variables - definitions .............................................................................. 34 2.2 Stationary random processes ................................................................................. 42 2.3 Nonstationary random processes ........................................................................... 61 2.4 Dynamic characteristics of physical systems ........................................................ 67 2.5 Static failure analysis ............................................................................................. 69 2.6 Fatigue failure analysis .......................................................................................... 71 2.7 Accelerated vibration endurance tests ................................................................... 85 2.8 Final conclusions ................................................................................................... 87 3 Non-Gaussian signal generation .................................................................................. 89 3.1 Introduction and assumptions ................................................................................ 89 3.2 Stationary random Gaussian noise generation ...................................................... 90 3.3 Clipped signal generation ...................................................................................... 94 3.4 Steady leptokurtic signal generation ..................................................................... 99 3.5 Nonstationary leptokurtic signal generation ........................................................ 105 3.6 Conclusions ......................................................................................................... 119 4 Relationship between the input and output statistics ................................................. 120 4.1 Introduction and assumptions .............................................................................. 120 4.2 Output statistics from a stationary random Gaussian noise excitation ................ 121 4.3 Output statistics from a stationary, non-Gaussian random excitation ................. 121 4.4 Output statistics from an amplitude modulated random excitation ..................... 127 4.5 Comparison of the output kurtosis from the two leptokurtic signal generation techniques ....................................................................................................................... 133 4.6 Conclusions ......................................................................................................... 135 5 Distributions of the peaks and rainflow ranges for filtered non-Gaussian processes 136 5.1 Introduction and assumptions .............................................................................. 136 5.2 PDF of instantaneous values in the response resulting from non-Gaussian excitations ...................................................................................................................... 137 5.3 Peak response distribution in the response resulting from non-Gaussian excitations ...................................................................................................................... 146 5.4 Maximum peak in the response resulting from non-Gaussian excitations .......... 151 5.5 Rainflow cycle count histogram .......................................................................... 153 5.6 Fatigue life predictions ........................................................................................ 157 5.7 Conclusions ......................................................................................................... 162 6 Overall conclusions .................................................................................................... 163 6.1 Conclusions of the research ................................................................................. 163 6.2 Perspective