In order to solve coupled fractional differential-integral equations more effectively and to deal with the problem that the huge algebraic equations lead to considerable computational complexity and large data storage requirements in the calculation process, this paper approximates the function of the unknown solution based on the Chebyshev wavelet of the second kind and then combines the collocation method to solve the numerical solution of nonlinear fractional Fredholm integral-differential equations. By using the proposed method, the original problem can be reduced to a system of linear algebraic equations, which can be easily solved by some mathematical techniques. In addition, the convergence analysis of the system based on the second kind of Chebyshev wavelet is studied. Several numerical test problems are presented, and the absolute error values under different fractional orders are given, which proves the superiority and effectiveness of the proposed method. It provides support for improving the precision and reliability of the system.
Grain size has an essential influence on the serviceability of metallic materials. In this paper, a noncontact laser ultrasonic testing platform is built to study the effect of copper grain size on the laser ultrasonic backscattered signal. According to the correlation between grain size and ultrasonic wavelength, the ultrasonic scattering by copper grains in the experiment contains not only Rayleigh scattering but also the transition region from Rayleigh scattering to stochastic scattering. Using time–frequency analysis, the influence of copper grain size on the characteristic parameters of backscattering was explored, and a prediction model of grain size was established, which was compared with the prediction model based on the attenuation method to verify the accuracy of the backscattering model. The results show that the backscattered signal can adequately characterize the grain size information and laser ultrasonics is a method that can realize on-line detection of grain size.
Metallic cylinders are widely used in various fields of industrial production, and the automatic detection of surface microcracks is of great significance to the subsequent grinding process. In this paper, laser-excited surface acoustic waves (SAW) are used to detect surface microcracks. Due to the dispersion of SAWs on the cylinder surface, the SAWs exhibit different polarities at different positions. In order to improve the consistency of signals and the accuracy of the modeling, the angle at which the polarity is completely reversed is selected as the detection point. A laser ultrasonic automatic detection system is established to obtain signals, and the B-scan image is drawn to determine the location of the microcrack. By comparing the time–frequency diagrams of the reflected SAWs and transmitted SAWs, the transmitted wave is chosen to establish the microcrack depth prediction model. In addition, according to the trajectory of the grinding wheel, a prediction model based on the absolute depth of the microcracks is established, and the influence of the orientation of the microcracks on the signal energy is considered. The method proposed in this paper can provide a reference for the rapid grinding of microcracks on the surface of metallic cylinders; it has the characteristics of visualization and high efficiency, and overcomes the shortcomings of the currently used eddy current testing that provides information on the depth of microcracks with difficulty.
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