The bearing system of an alternating current (AC) motor is a nonlinear dynamics system. The working state of rolling bearings directly determines whether the machine is in reliable operation. Therefore, it is very meaningful to study the fault diagnosis and prediction of rolling bearings. In this paper, a new fault diagnosis method based on variational mode decomposition (VMD), Hilbert transform (HT), and broad learning model (BLM), called VHBLFD is proposed for rolling bearings. In the VHBLFD method, the VMD is used to decompose the vibration signals to obtain intrinsic mode functions (IMFs). The HT is used to process the IMFs to obtain Hilbert envelope spectra, which are transformed into the mapped features and the enhancement nodes of BLM according to the complexity of the modeling tasks, and the nonlinear transformation mean according to the characteristics of input data. The BLM is used to classify faults of the rolling bearings of the AC motor. Next, the pseudo-inverse operation is used to obtain the fault diagnosis results. Finally, the VHBLFD is validated by actual vibration data. The experiment results show that the BLM can quickly and accurately be trained. The VHBLFD method can achieve higher identification accuracy for multi-states of rolling bearings and takes on fast operation speed and strong generalization ability.

The variational mode decomposition (VMD) method for signal decomposition is severely affected by the number of components of the VMD method. In order to determine the decomposition modal number, K, in the VMD method, a new center frequency method of the multi-threshold is proposed in this paper. Then, an improved VMD (MTCFVMD) algorithm based on the center frequency method of the multi-threshold is obtained to decompose the vibration signal into a series of intrinsic modal functions (IMFs). The Hilbert transformation is used to calculate the envelope signal of each IMF component, and the maximum frequency value of the power spectral density is obtained in order to effectively and accurately extract the fault characteristic frequency and realize the fault diagnosis. The rolling element vibration data of the motor bearing is used to test the effectiveness of proposed methods. The experiment results show that the center frequency method of the multi-threshold can effectively determine the number, K, of decomposed modes. The proposed fault diagnosis method based on MTCFVMD and Hilbert transformation can effectively and accurately extract the fault characteristic frequency, rotation frequency, and frequency doubling, and can obtain higher diagnostic accuracy.

Broad learning system (BLS) is an effective and efficient incremental learning system without the deep architecture. It has strong feature extraction ability and high computational efficiency. However, it is greatly limited in the applicability of supervised learning. For the collected actual data, more data are unlabeled data and less data are labeled data. To overcome these problems, Fick’s law assisted propagation (FLAP) is introduced into the BLS to propose a new semi-supervised classification algorithm, namely FLAP-BLS in this paper. In the FLAP-BLS, the FLAP has the labeled ability from the labeled examples to unlabeled examples, it is used to mark plenty of unlabeled samples by few labeled samples in order to obtain a large number of labeled samples and build the sample data matrix. Then an efficient incremental BLS without deep structure can effectively extract features from large-scale data, it is used to effectively classify the sample matrix. Finally, USPS, MNIST and NORB datasets are selected to validate the effectiveness of the FLAP-BLS. The experiment results show that the FLAP-BLS can effectively classify the few labeled samples and a large of unlabeled samples and obtain classification results with high accuracy, and it has faster classification speed, stronger generalization ability and better stability. The proposed method provides a new method for image classification.

Aiming at the traditional stochastic resonance system’s difficulty in extracting the weak signal fault eigenfrequency in the strong noise background, an asymmetric hybrid bistable stochastic resonance system model (AHBSR system) is proposed based on the hybrid bistable system model. Firstly, the asymmetric factor is introduced to improve the hybrid bistable model, the asymmetric hybrid bistable stochastic resonance system model is constructed, and the system is solved numerically based on the fourth-order Lunger-Kutta algorithm to analyze the effect of noise intensity on the system. Secondly, the output signal-to-noise ratio (SNR) is used as a measure of the stochastic resonance effect, and the particle swarm algorithm is used to optimize the system parameters in combination with the quadratic sampling technique, and the system outputs are compared and analyzed with those of the hybrid bistable model and the classical asymmetric bistable model, which proves that the extraction of the characteristic frequencies of the weak-signal faults by the AHBSR model is more accurate, and the detection deviations are all within 0.4Hz, which is lower than those of the other two kinds of bistable systems. Compared with the original signals, the output signal-to-noise ratios are increased by more than 30 dB, which is higher than the other two bistable systems, and the amplitudes corresponding to the eigenfrequencies are amplified by more than seven hundred times, moreover, the AHBSR system is capable of extracting the composite fault signature frequency that cannot be recognized by the other two bistable systems. Finally, given the problem that the signal-to-noise ratio index needs to predict the exact fault characteristic frequency when calculating, the weighted kurtosis coefficient is proposed as a measure of the stochastic resonance effect, which is verified by signal simulation and experimental datasets of bearings with different fault types, to prove the accuracy of the AHBSR system in identifying and extracting the fault characteristics of the bearing signals, as well as the validity of the weighted kurtosis coefficient index.

For the problem of inconsistent quantitative standards for running status analysis of rolling bearings, this paper uses principal component analysis (PCA) to extract a new index F, which is the joint parameters of time domain and frequency domain, and by establishing the value of F to analyze the running states of the rolling bearings. Firstly, the acceleration sensors are used to collect the vibration signal of the whole life cycle of the rolling bearings. Secondly, empirical mode decomposition (EMD) method is used to denoise the acquired vibration signal. Then, the main components of the denoised vibration signal are used to propose the characteristic parameters and synthesized into new parameter indicators. Finally, envelope analysis spectrum is used to analyze the fault classification under the new parameter index. The exepriment results show that the whole life cycle of the rolling bearings can be classified into five different operating periods by using the new parameter index, and each period represents a different bearing operating state.

The fault features of rolling bearings in a time-varying rotational speed condition are often submerged in strong noise. In this paper, to address the issue that traditional stochastic resonance systems’ physical effects struggle to accurately extract weak fault features from strong noise backgrounds, an asymmetric unsaturated Gaussian stochastic resonance (AUGSR) mathematical model is proposed. The AUGSR model innovatively adopts a Gaussian decay function to construct an asymmetric term, whose core advantage lies in its simultaneous possession of functional global smoothness and strong localized decay. This combination ensures that the potential function remains continuously differentiable, physically realizable, and numerically stable, realizing the unsaturated nature of the model and thus breaking through the limitations of traditional symmetric models. First, the Signal-to-Noise Ratio (SNR) is utilized as an evaluation index for the stochastic resonance effect, and the Particle Swarm Optimization (PSO) algorithm is employed to optimize the model parameters, thereby verifying the accuracy of the AUGSR model. Next, for scenarios where SNR calculation requires a priori knowledge of the fault frequency, the Mainband Energy Ratio (MER) is proposed as an alternative evaluation index. The Hybrid Genetic Algorithm and Particle Swarm Optimization (HGAPSO) algorithm is employed to optimize the parameters under time-varying rotational speeds, which verifies the general applicability of the AUGSR model in fault feature extraction and the effectiveness of the MER index. Experimental results on rolling bearings confirm the superiority of the proposed mathematical model over conventional stochastic resonance systems.