Instantaneous frequency (IF) of shaft rotation is pivotal for bearing fault diagnosis under variable speed operations. However, shaft IF cannot always be measured as tachometers are not allowed to be installed in every case due to design reasons and cost concerns. Extracting the shaft IF ridge from time frequency representation (TFR) of vibration signals, therefore, becomes an alternative. Linear transform (LT), such as short time Fourier transform (STFT), has been widely adopted for such a purpose. Nevertheless, the accuracy of extracted IF ridges relies on the readability of TFR. Unfortunately, readability of TFR from STFT is often impaired by the smearing effect caused by non-synchronous frequencies between bases and signal components and limited time frequency resolution capability, which in turn adversely influences the accuracy of IF ridge extraction. To accurately extract IF ridges from vibration signals, this paper focuses on the first factor, which causes the smearing problem, and proposes a method named frequency matching linear transform (FMLT) to enhance the TFR, where transforming bases with frequencies varying with the shaft IF are constructed to alleviate the smearing effects. To construct the transforming bases with frequencies synchronous with shaft IF, a fast path optimization (FPO) algorithm, which generates all possible optimization paths among amplitude peaks and thereby ensures the continuity of extracted IF ridges, is adopted for IF pre-estimation. The TFR with improved readability can be subsequently obtained via FMLT, paving the way for accurate IF ridge extraction. Then, multiple IF ridges can be iteratively extracted using the FPO algorithm. The accuracy of extracted IF ridges before and after TFR enhancement is compared, indicating that the proposed FMLT can enhance the readability of TFR and lead to more accurate IF ridge extraction for bearing condition monitoring.

Rolling element bearings are one of the key elements used in rotating machines. Their failure will result in system breakdowns and cause unexpected accidents. The health condition monitoring of bearings is, therefore, an emerging discipline to scientifically manage machine lifetime. Instantaneous frequency (IF) extraction and IF-based resampling consist of the major tasks used in conventional approaches for bearing fault diagnosis under variable speeds, where it is often desirable for the IF to be extracted based on the time-frequency analysis (TFA) of the vibration signal, instead of using a tachometer. However, an accurate IF extraction based on vibration signals from incipient bearing faults is often undermined by poor energy concentration and the weak readability of the TFA. Furthermore, the resampling procedure for bearing fault diagnosis is error-prone and vulnerable to noise. An approach based on multiple IF ridge integration is, therefore, proposed to address such problems. The proposed approach is dedicated to an accurate IF estimation and bearing fault diagnosis without tachometer utilization and resampling involvement. It is mainly comprised of four steps: (1) acquire multiple pre-IF ridges via a regional peak search algorithm (RPSA) from time frequency representations (TFRs); (2) integrate pre-IF ridges based on the probability density function (PDF) to gain an accurate IF estimation; (3) rectify the multiple pre-IF ridges; (4) diagnose bearing health condition according to the average ratios of any two rectified IF ridges, i.e. fault characteristic coefficient (FCC) or FCC-related numbers. Then, the IF can be accurately estimated based only on the collected vibration signals, and bearing fault diagnosis under variable speed conditions can be implemented. Numerical simulations and experimental signal analyses validate the effectiveness of the proposed method.

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