Novel Hilbert Huang transform techniques for bearing fault detection

Bearings are commonly used in rotary machinery; while up to half of machinery malfunctions could be related to bearing defects. A reliable bearing fault detection technique becomes vital to a wide array of industries to recognize an incipient bearing defect to prevent machinery performance degradation, malfunction, and unexpected breakdown. Many signal processing techniques have been suggested in literature to extract fault-related signatures for bearing fault detection, but most of them are not robust in real-world bearing health condition monitoring when signal properties vary with time. Vibration signals generated from bearings can be either stationary or nonstationary. If bearing defect-related signature is stationary, it is relatively easy to analyze using these classical data analysis techniques. However, bearing nonstationary signals are much more complex to analyze using these classical signal processing techniques, especially when slippage has occurred. Reliable fault detection still remains a challenging task, especially when bearing defect-related features are nonstationary. Two alternative approaches are proposed in this work for bearing fault detection: The first technique is based on analytical normality test, named Normalized Hilbert Haung Transform (NHHT). The second technique is based on information domain analysis, named enhanced Hilbert Haung Transform (eHHT). In the proposed NHHT technique, a novel strategy based on d’Agostino-Pearson normality analysis is suggested to demodulate feature functions and highlight feature characteristics for bearing fault detection. In the proposed eHHT, a novel strategy is proposed to enhance feature extraction based on the analysis of correlation and mutual information. The effectiveness of the proposed techniques is verified by a series of experimental tests corresponding to different bearing health conditions. Their robustness in bearing fault diagnostic is examined by the use of data sets from a different experimental setup

Inverse Dynamics Problems

The inverse dynamics problem was developed in order to provide researchers with the state of the art in inverse problems for dynamic and vibrational systems. Contrasted with a forward problem, which solves for the system output in a straightforward manner, an inverse problem searches for the system input through a procedure contaminated with errors and uncertainties. An inverse problem, with a focus on structural dynamics, determines the changes made to the system and estimates the inputs, including forces and moments, to the system, utilizing measurements of structural vibration responses only. With its complex mathematical structure and need for more reliable input estimations, the inverse problem is still a fundamental subject of research among mathematicians and engineering scientists. This book contains 11 articles that touch upon various aspects of inverse dynamic problems

Sparse modelling and estimation for nonstationary time series and high-dimensional data

Sparse modelling has attracted great attention as an efficient way of handling statistical problems in high dimensions. This thesis considers sparse modelling and estimation in a selection of problems such as breakpoint detection in nonstationary time series, nonparametric regression using piecewise constant functions and variable selection in high-dimensional linear regression. We first propose a method for detecting breakpoints in the secondorder structure of piecewise stationary time series, assuming that those structural breakpoints are sufficiently scattered over time. Our choice of time series model is the locally stationary wavelet process (Nason et al., 2000), under which the entire second-order structure of a time series is described by wavelet-based local periodogram sequences. As the initial stage of breakpoint detection, we apply a binary segmentation procedure to wavelet periodogram sequences at each scale separately, which is followed by within-scale and across-scales postprocessing steps. We show that the combined methodology achieves consistent estimation of the breakpoints in terms of their total number and locations, and investigate its practical performance using both simulated and real data. Next, we study the problem of nonparametric regression by means of piecewise constant functions, which are known to be flexible in approximating a wide range of function spaces. Among many approaches developed for this purpose, we focus on comparing two well-performing techniques, the taut string (Davies & Kovac, 2001) and the Unbalanced Haar (Fryzlewicz, 2007) methods. While the multiscale nature of the latter is easily observed, it is not so obvious that the former can also be interpreted as multiscale. We provide a unified, multiscale representation for both methods, which offers an insight into the relationship between them as well as suggesting some lessons that both methods can learn from each other. Lastly, one of the most widely-studied applications of sparse modelling and estimation is considered, variable selection in high-dimensional linear regression. High dimensionality of the data brings in many complications including (possibly spurious) non-negligible correlations among the variables, which may result in marginal correlation being unreliable as a measure of association between the variables and the response. We propose a new way of measuring the contribution of each variable to the response, which adaptively takes into account high correlations among the variables. A key ingredient of the proposed tilting procedure is hard-thresholding sample correlation of the design matrix, which enables a data-driven switch between the use of marginal correlation and tilted correlation for each variable. We study the conditions under which this measure can discriminate between relevant and irrelevant variables, and thus be used as a tool for variable selection. In order to exploit these theoretical properties of tilted correlation, we construct an iterative variable screening algorithm and examine its practical performance in a comparative simulation study