Reflection/transmission characteristics based on Legendre orthogonal polynomial method
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摘要:
为研究薄层材料力学性能的声学检测方法，基于勒让德正交多项式法，利用液/固边界条件与波动控制方程，构建线性无关方程组并求解了界面处的反射/透射系数。分析截止项阶数对求解结果的影响，寻找不同频厚积下截止项阶数的临界值，并根据该临界值求解反射/透射系数的角度谱与频谱，与传递矩阵法进行了比较，以此验证了该理论模型的准确性。超声波斜入射薄层材料板并形成Lamb波，其达到稳态时的频散特性与反射特性存在内在联系，根据Snell定律并运用勒让德正交多项式法求解频率-相速度-反射系数三维曲面，与Disperse得到的频散曲线仿真结果进行比较，证明该理论模型对反射系数的求解结果符合Lamb波频散特性。通过采集无试样时的参考信号，降低了声波传播过程中衰减对实验结果的影响。搭建了反射与透射实验系统，对不同入射角度下反射与透射系数的频谱进行了测量并与相应的理论结果进行了对比，验证了该理论所得结果的准确性。实现了声反射/透射系数的非勘根求解，为薄层材料力学性能的无损检测提供了理论基础与实验指导。
Abstract:Study on acoustic testing method for mechanical properties of thin layer materials.Based on Legendre orthogonal polynomial method, linear independent equations are constructed and the reflection/transmission coefficient of acoustic waves at the interfaces is calculated using liquid/solid boundary conditions and wave control equations. The method analyzes the influence of the cut-off order on the solution result, and finds the critical value of the order under different frequency and thickness products, then calculate the reflection/transmission coefficient based on the cut-off order. The calculation results obtained by the Legendre orthogonal polynomial method are compared with those obtained by the transfer matrix method, and the accuracy of the theoretical model is verified. Ultrasonic waves' oblique incidence into a thin-layer material forms a Lamb wave. The dispersion characteristics at the steady state are intrinsically linked to the reflection features. According to Snell's law, the three-dimensional surface of the frequency-velocity-reflection coefficients is calculated using the Legendre orthogonal polynomial method. This result is compared with the dispersion curves simulated by Disperse. It is proved that the solving result is consistent with the Lamb wave dispersion characteristics. The influence of attenuation on the experimental results during the propagation of the acoustic wave is reduced by collecting the reference signal which is the directly obtained wave without a sample. A reflection and transmission experimental system was built to measure the frequency spectrum of reflection and transmission coefficients at different incident angles. The experimental results are compared with the theoretical results, and the accuracy of the results obtained by the theory is verified. A solving methodology without root-finding algorithm for acoustic reflection/transmission coefficients is realized. This method provides a theoretical basis and experimental guidance for the non-destructive testing of the mechanical properties of thin-layer materials.
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图 1 入射波、反射波和透射波传播示意图
Figure 1. Schematic diagram of incident wave, reflected wave and transmitted wave propagation
图 2 不同截止项阶数M下的反射系数频谱
Figure 2. Reflection coefficient spectrum underdifferent cut-off order M
图 3 铝板反射/透射系数角度谱
Figure 3. Angle spectrum of reflection/transmission coefficient of aluminum plate
图 4 入射波、反射波与Lamb波传播示意图
Figure 4. Schematic diagram of incident wave, reflected wave and Lamb wave propagation
图 5 1 mm薄铝板频率-角度-反射系数曲面
Figure 5. Frequency-angle-reflection coefficient surface of 1 mm thin aluminum plate
图 6 频率-波速-反射系数曲面与频散曲线对比
Figure 6. Comparison of frequency-wave velocity-reflection coefficient surface and dispersion curve
图 7 斜入射反射/透射系数检测系统示意图
Figure 7. Schematic diagram of obliquely incident reflection/transmission coefficient detection system
图 10 反射系数理论、实验频谱对比
Figure 10. Theoretical and experimental spectrum comparison of reflection coefficient
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