Tant Katherine M M, Mulholland Anthony J, Langer Matthias, Gachagan Anthony
Department of Mathematics and Statistics , University of Strathclyde , Glasgow, UK.
Centre for Ultrasonic Engineering , University of Strathclyde , Glasgow, UK.
Proc Math Phys Eng Sci. 2015 Mar 8;471(2175):20140958. doi: 10.1098/rspa.2014.0958.
Many safety critical structures, such as those found in nuclear plants, oil pipelines and in the aerospace industry, rely on key components that are constructed from heterogeneous materials. Ultrasonic non-destructive testing (NDT) uses high-frequency mechanical waves to inspect these parts, ensuring they operate reliably without compromising their integrity. It is possible to employ mathematical models to develop a deeper understanding of the acquired ultrasonic data and enhance defect imaging algorithms. In this paper, a model for the scattering of ultrasonic waves by a crack is derived in the time-frequency domain. The fractional Fourier transform (FrFT) is applied to an inhomogeneous wave equation where the forcing function is prescribed as a linear chirp, modulated by a Gaussian envelope. The homogeneous solution is found via the Born approximation which encapsulates information regarding the flaw geometry. The inhomogeneous solution is obtained via the inverse Fourier transform of a Gaussian-windowed linear chirp excitation. It is observed that, although the scattering profile of the flaw does not change, it is amplified. Thus, the theory demonstrates the enhanced signal-to-noise ratio permitted by the use of coded excitation, as well as establishing a time-frequency domain framework to assist in flaw identification and classification.
许多对安全至关重要的结构，比如核电站、石油管道以及航空航天工业中的结构，都依赖于由异质材料构成的关键部件。超声无损检测（NDT）利用高频机械波来检查这些部件，确保它们在不损害其完整性的情况下可靠运行。利用数学模型能够更深入地理解采集到的超声数据，并改进缺陷成像算法。本文在时频域中推导了裂纹对超声波散射的模型。分数阶傅里叶变换（FrFT）应用于一个非齐次波动方程，其中强迫函数规定为一个由高斯包络调制的线性调频信号。通过包含有关缺陷几何形状信息的玻恩近似找到齐次解。非齐次解通过高斯窗线性调频激励的逆傅里叶变换获得。可以观察到，尽管缺陷的散射轮廓不变，但它被放大了。因此，该理论证明了使用编码激励所允许的增强的信噪比，同时建立了一个时频域框架以协助缺陷识别和分类。
