The Shannon entropy measure, applied to the time frequency distribution of a signal, is a reliable indicator to quantitatively analyze the health status of a rolling element bearing. Usually, however, conventional time frequency representations rely on the selection of the best scale of a base function in order to deal with the noise components present in a vibration signal. In this context, the time frequency manifold is a relatively new time frequency analysis method, capable of cancelling out or suppressing the majority of noise components by correlating the deterministic information of a faulty signal in its multidimensional phase space representation, rather than depending on the scale selection of a certain base function. With the aim of examining the merits of the time frequency manifold, in this research, the concepts of the time frequency manifold and Shannon entropy are integrated, so as to construct an accurate bearing health monitoring index. The effects of embedding dimension and time delay on the calculation of the proposed health index are studied. Simulated signals are employed in order to study the characteristic properties of the proposed index. Two experimental datasets are used to validate the effectiveness of the proposed method in compare to some conventional and hybrid bearing health monitoring indexes. Our research shows that the proposed index demonstrates a consistently good performance for both inner and outer race failure, in cases of run to failure bearing testing.

The entropy-based feature extraction is a promising tool for extracting weak features from rotating machinery. However, the existing research has paid little attention to the state transition process, which brings the problem of accuracy and comprehensiveness in complexity estimation. To address this issue, this paper proposes link dispersion entropy (LDE) based on the theory of the Markov chain for weak feature extraction. By calculating the transition probability of symbol patterns, the LDE can extract the fault information contained in the transition, enabling it to capture the early weak fault. Furthermore, LDE is extended to a multiscale analysis by combining it with the coarse-gaining process for comprehensive feature extraction, termed multiscale LDE (MLDE). Finally, three simulated signals and two different experimental data are utilized to verify the advantage of MLDE in extracting the weak fault features. Results demonstrate that MLDE has the best performance in fault diagnosis of rotating machinery compared with the existing five methods, namely sample entropy, fuzzy entropy, permutation entropy, dispersion entropy and symbolic dynamic entropy.

As a nonlinear measure, Lempel–Ziv complexity (LZC) can be considered as a suitable parameter for characterizing bearing health status by measuring the complexity of vibration signals. However, in continuous monitoring scenario under noisy condition, all components of a multicomponent bearing signal are not equally sensitive toward a change of LZC value. As a result, a direct application of LZC for bearing health monitoring not only suffers from its inefficient early fault warning but also fails to infer the fault progression. In this article, instead of direct utilization of a whole vibration signal, its fundamental component (FC) sensitive to LZC calculation is separated with the help of continuously adjustable parameterized tunable $Q$ factor wavelet transform (TQWT). In this context, a study based on sparsity indices has been done for $Q$ factor selection of TQWT. Since TQWT uses an oscillation-based bearing FC separation scheme for LZC calculation, the proposed measure is termed as oscillatory Lempel–Ziv complexity (OLZC). Two experimental cases are used for validation. Performance of OLZC is compared with original LZC, representative sparsity indices and recently proposed multiscale symbolic Lempel–Ziv complexity. Results demonstrate that the proposed OLZC can not only overcome the limitations of the original LZC but also performs better than other indices in comparison to continuous monitoring of bearing health.

As a nonlinear measure, sample entropy (SE) can be considered a suitable parameter for characterizing rolling element bearing health status by measuring complexity of vibration signals. However, in continuous monitoring scenario under noisy condition, all components of a multicomponent bearing signal are not equally sensitive toward change of SE value. As a consequence, a direct application of SE results in inefficient early fault warning and inability to differentiate among different fault types. To deal with this problem, instead of direct utilization of a whole vibration signal, its principal component (PC) sensitive to SE calculation is separated with the help of continuously adjustable parameterized tunable $Q$ factor wavelet transform (TQWT). Since TQWT uses an oscillation-based bearing PC separation scheme for SE calculation, the newly proposed measure is termed as oscillatory sample entropy (OSE). Due to the biasness of SE algorithm toward bearing PC, the proposed OSE can anticipate theoretical concept of complexity change more efficiently with the change of bearing health. Two experimental case studies have shown that proposed OSE can not only overcome the limitations of SE algorithm but also demonstrate superiority over approximate entropy (AE) and fuzzy entropy (FE) for continuous monitoring of bearing health.