Rotating machines have been widely used in industrial engineering. The fault diagnosis of rotating machines plays a vital important role to reduce the catastrophic failures and heavy economic loss. However, the measured vibration signal of rotating machinery often represents non-linear and non-stationary characteristics, resulting in difficulty in the fault feature extraction. As a statistical measure, entropy can quantify the complexity and detect dynamic change through taking into account non-linear behavior of time series. Therefore, entropy can be served as a promising tool to extract the dynamic characteristics of rotating machines. Recently, many studies have applied entropy in fault diagnosis of rotating machinery. This paper aims to investigate the applications of entropy for the fault characteristics extraction of rotating machines. First, various entropy methods are briefly introduced. Its foundation, application and some improvements are described and discussed. The review is divided into eight parts: Shannon entropy, Rényi entropy, approximate entropy, sample entropy, fuzzy entropy, permutation entropy, and other entropy methods. In each part, we will review the applications using the original entropy method and the improved entropy methods, respectively. In the end, a summary and some research prospects are given.

This paper presents a fault diagnosis scheme based on multiscale diversity entropy (MDE) and extreme learning machine (ELM). First, a novel entropy method called diversity entropy (DE) is proposed to quantify the dynamical complexity. DE utilizes the distribution of cosine similarity between adjacent orbits to track the inside pattern change, resulting into better performance in complexity estimation. Then, the proposed DE is extended to multiscale analysis called MDE for comprehensive feature description by combining with the coarse gaining process. Third, the obtained features using MDE are fed into ELM classifier for pattern identification of rotating machinery. The effectiveness of the proposed MDE method is verified using simulated signals and two experimental signals collected from the bearing test and the dual-rotator of aero-engine test. The analysis results show that our proposed method has the highest classification accuracy compared with three existing approaches: sample entropy, fuzzy entropy, and permutation entropy.

Fault diagnosis of bearings using classification techniques plays an important role in industrial applications, and, hence, has received increasing attention. Recently, significant efforts have been made to develop various methods for bearing fault classification and the application of Case Western Reserve University (CWRU) data for validation has become a standard reference to test the fault classification algorithms. However, a systematic research for evaluating bearing fault classification performance using the CWRU data is still lacking. This paper aims to provide a comprehensive benchmark analysis of the CWRU data using various entropy and classification methods. The main contribution of this paper is applying entropy-based fault classification methods to establish a benchmark analysis of entire CWRU datasets, aiming to provide a proper assessment of any new classification methods. Recommendations are provided for the selection of the CWRU data to aid in testing new fault classification algorithms, which will enable the researches to develop and evaluate various diagnostic algorithms. In the end, the comparison results and discussion are reported as a useful baseline for future research.

Significance: Catastrophic events affecting technological or critical infrastructures are often originated by a cascading failure triggered by marginal perturbations, which are on their turn localized in one of the many interdependent graphs describing the systems. Understanding the robustness of these graphs is therefore of utmost importance for preventing crashes and/or for engineering more efficient and stalwart networked systems. Here we give a fresh framework by means of which cascading failures can be described in a very rich variety of dynamical models and/or topological network structures and which provides a series of quantitative answers able to predict the extent of the system’s failure.

Importance measure can be used to identify the most vulnerable components with respect to system functionality or failure. Traditional importance measures may not be effective to evaluate the contribution of an individual component if its reliability value does not fall in the full range between 0 and 1. Based on the cost–reliability relation, this paper proposes a generalized Birnbaum importance measure (GBIM) to quantify the contribution of individual components to system reliability improvement by considering reliability range, manufacturing complexity, and technology feasibility. Since GBIM possesses several unique features in terms of guiding system reliability optimization, in this paper, we further develop a GBIM-based genetic algorithm to solve a type of optimal reliability allocation problem. The numerical studies show that both the computational efficiency and the near global optimality based on GBIM outperforms the methods using the traditional importance measures.