Most previous studies on nonlinear Lamb waves are conducted using mode pairs that satisfying strict phase velocity matching and non-zero power flux criteria. However, there are some limitations in existence. First, strict phase velocity matching is not existed in the whole frequency bandwidth; Second, excited center frequency is not always exactly equal to the true phase-velocity-matching frequency; Third, mode pairs are isolated and quite limited in number; Fourth, exciting a single desired primary mode is extremely difficult in practice and the received signal is quite difficult to process and interpret. And few attention has been paid to solving these shortcomings. In this paper, nonlinear S0 mode Lamb waves at low-frequency range satisfying approximate phase velocity matching is proposed for the purpose of overcoming these limitations. In analytical studies, the secondary amplitudes with the propagation distance considering the fundamental frequency, the maximum cumulative propagation distance (MCPD) with the fundamental frequency and the maximum linear cumulative propagation distance (MLCPD) using linear regression analysis are investigated. Based on analytical results, approximate phase velocity matching is quantitatively characterized as the relative phase velocity deviation less than a threshold value of 1%. Numerical studies are also conducted using tone burst as the excitation signal. The influences of center frequency and frequency bandwidth on the secondary amplitudes and MCPD are investigated. S1–S2 mode with the fundamental frequency at 1.8 MHz, the primary S0 mode at the center frequencies of 100 and 200 kHz are used respectively to calculate the ratios of nonlinear parameter of Al 6061-T6 to Al 7075-T651. The close agreement of the computed ratios to the actual value verifies the effectiveness of nonlinear S0 mode Lamb waves satisfying approximate phase velocity matching for characterizing the material nonlinearity. Moreover, the ratios derived from the primary and secondary horizontal displacements generated from nonlinear S0 mode Lamb waves are closest to the real value, which indicates that using horizontal displacements is more suitable for detecting evenly distributed microstructural changes in large thin plate-like structure. Successful application to evaluating material at different levels of evenly distributed fatigue damage is also numerically conducted.

Compared with conventional linear ultrasonic inspection methods, which are sensitive only to severe defects, nonlinear ultrasonic inspection methods are better for revealing micro-cracks in thin plates. However, most nonlinear ultrasonic inspection methods have only been experimentally investigated using bulk or Rayleigh waves. Numerical studies, especially numerical simulations of Lamb ultrasonic waves, have seldom been reported. In this paper, the interaction between nonlinear S0 mode Lamb waves and micro-cracks of various lengths and widths buried in a thin metallic plate was simulated using the finite element method (FEM). The numerical results indicate that after interacting with a micro-crack, a new wave-packet was generated in addition to the S0 mode wave-packet. The second harmonics of the S0 mode Lamb waves and the new wave-packet were caused by nonlinear acoustic effects at the micro-crack. An amplitude ratio indicator is thus proposed for the early detection of buried micro-cracks.

Bearings, as widely employed supporting components, frequently work in challenging working conditions, leading to diverse fault types. Traditional methods for diagnosing bearing faults primarily center on time–frequency analysis, but this often requires expert experience for accurate fault identification. Conversely, intelligent fault recognition and classification methods frequently lack interpretability. To address this challenge, this paper introduces a convolutional neural network with an attention mechanism method, denoted as CBAM-CNN, for bearing fault diagnosis. This approach incorporates an attention mechanism, creating a Convolutional Block Attention Module (CBAM), to enhance the fault feature extraction capability of the network in the time–frequency domain. In addition, the proposed method integrates a weight visualization module known as the Gradient-Weighted Class Activation Map (Grad-CAM), enhancing the interpretability of the convolutional neural network by generating visual heatmaps on fault time–frequency graphs. The experimental results demonstrate that utilizing the dataset employed in this study, the CBAM-CNN achieves an accuracy of 99.81%, outperforming the Base-CNN with enhanced convergence speed. Furthermore, the analysis of attention weights reveals that this method exhibits distinct focus of attention under various fault types and degrees. The interpretability experiments indicate that the CBAM module balances the weight allocation, emphasizing signal frequency distribution rather than amplitude distribution. Consequently, this mitigates the impact of the signal amplitude on the diagnostic model to some extent.