压电圆柱曲面板中的周向导波复频散特性
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摘要
提出了一种基于Legendre正交多项式级数计算导波复频散的解析方法,该方法可将复杂的波动方程求解问题转换为特征值为波数的特征计算问题,无须迭代便可得到完整解,避免了迭代法循环次数多、运算量大、精度不足等缺点。应用提出的方法计算了正交各向异性压电圆柱曲面板中的周向导波复频散曲线,并与已有文献应用回传射线矩阵法的计算结果进行了比较,两者计算结果非常吻合,验证了本文方法的正确性和有效性。计算给出了压电圆柱曲面板的频率-复波数三维频散曲线,研究了材料的压电性、力学与电学边界条件对导波复频散特性的影响,分析了复波数衰逝波的位移和电势分布,揭示了衰逝导波的传播特性。研究结果表明:本文提出的计算方法能求解得到超越频散方程的纯实数波数、纯虚数波数和复数波数解;压电圆柱曲面板中周向类Lamb衰逝波有纯虚数波数和复数波数两种模态,复波数衰逝波模态呈阻尼正弦式衰减,随着频率的增大而转变为传播波模态;材料的压电性和边界条件对导波复频散特性都有显著影响。
In this paper, an analytical method based on Legendre orthogonal polynomial series is presented to deal with complex dispersion problem, which can transform the complicated wave propagation problem into an eigenvalue matrix equation whose eigenvalues are wave numbers and can avoid the shortcomings of the iterative methods such as many times of circulation, huge quantity of calculation, and lower accuracy. A comparison of the obtained results of an orthotropic piezoelectric cylindrical curved plate with those from the reverberation-ray matrix method is provided to verify the correctness and effectiveness of the presented method. Three dimensional dispersion curves are plotted in frequency-complex wave number space for various piezoelectric cylindrical curved plates, and the effects of material piezoelectricity and the mechanical and electrical boundary conditions on complex dispersion properties are investigated. The displacement and electric potential distributions are also discussed to reveal the characteristics of the evanescent guided waves. The results show that the presented method can obtain the complete solutions of the transcendental dispersion equation in the real, imaginary and complex wave number ranges. For circumferential Lamb-like evanescent wave, it has purely imaginary modes and complex modes. The complex branches exhibit a damped sinusoidal attenuation and they will turn into the propagating modes with increasing frequency. Both the material piezoelectricity and the boundary conditions have significant effects on the complex dispersion properties.
In this paper, an analytical method based on Legendre orthogonal polynomial series is presented to deal with complex dispersion problem, which can transform the complicated wave propagation problem into an eigenvalue matrix equation whose eigenvalues are wave numbers and can avoid the shortcomings of the iterative methods such as many times of circulation, huge quantity of calculation, and lower accuracy. A comparison of the obtained results of an orthotropic piezoelectric cylindrical curved plate with those from the reverberation-ray matrix method is provided to verify the correctness and effectiveness of the presented method. Three dimensional dispersion curves are plotted in frequency-complex wave number space for various piezoelectric cylindrical curved plates, and the effects of material piezoelectricity and the mechanical and electrical boundary conditions on complex dispersion properties are investigated. The displacement and electric potential distributions are also discussed to reveal the characteristics of the evanescent guided waves. The results show that the presented method can obtain the complete solutions of the transcendental dispersion equation in the real, imaginary and complex wave number ranges. For circumferential Lamb-like evanescent wave, it has purely imaginary modes and complex modes. The complex branches exhibit a damped sinusoidal attenuation and they will turn into the propagating modes with increasing frequency. Both the material piezoelectricity and the boundary conditions have significant effects on the complex dispersion properties.
引文
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[8]郭永强,陈伟球,张永亮.层合压电结构中的导波[J].中国科学G辑,2009,39(7):1018-1028.(GUO Yongqiang,CHEN Weiqiu,ZHANG Yongliang.Guided wave propagation in multilayered piezoelectric structures[J].Science in China series G,2009,39(7):1018-1028(in Chinese)).
[9]QUINTANILLA F H,LOWE M J S,CRASTER R V.Full 3Ddispersion curve solutions for guided waves in generally anisotropic media[J].Journal of sound and vibration,2016,363:545-559.
[10]李念,钱征华,王彬.复波数域弥散曲线的求解方法研究[J].应用力学学报,2016,33(3):365-370.(LI Nian,QIAN Zhenghua,WANG Bin.Study on computational methods of dispersion curves in complex wave number range[J].Chinese journal of applied mechanics,2016,33(3):365-370(in Chinese)).
[11]SIMONETTI F,LOWE M J.On the meaning of Lamb mode nonpropagating branches[J].Journal of the acoustical society of America,2005,118(1):186-192.
[2]魏建萍,苏先樾.压电空心圆柱中波的传播[J].力学学报,2004,36(4):484-490.(WEI Jianping,SU Xianyue.Wave propagation in the piezoelectric hollow cylinder[J].Acta mechanica sinica,2004,36(4):484-490(in Chinese)).
[3]PONNUSAMY P.Wave propagation in a piezoelectric solid bar of circular cross-section immersed in fluid[J].International journal of pressure vessels and piping,2013,105/106:12-18.
[4]PUZYREV V,STOROZHEV V.Wave propagation in axially polarized piezoelectric hollow cylinders of sector cross section[J].Journal of sound and vibration,2011,330(18/19):4508-4518.
[5]YU J G,MA Q J.Circumferential waves in piezoelectric cylindrical curved plate[J].Acta acustica united with acustica,2008,94(5):725-733.
[6]HUANG K H,DONG S B.Propagating waves and edge vibrations in anisotropic composite cylinders[J].Journal of sound and vibration,1984,96(3):363-379.
[7]陈江义,潘尔年.任意梯度分布板中的导波频散特性分析[J].应用基础与工程科学学报,2014(6):1230-1237.(CHEN Jiangyi,PANErnian.Dispersion relations for guided wave in functionally graded plates with arbitrary grade[J].Journal of basic science and engineering,2014(6):1230-1237(in Chinese)).
[8]郭永强,陈伟球,张永亮.层合压电结构中的导波[J].中国科学G辑,2009,39(7):1018-1028.(GUO Yongqiang,CHEN Weiqiu,ZHANG Yongliang.Guided wave propagation in multilayered piezoelectric structures[J].Science in China series G,2009,39(7):1018-1028(in Chinese)).
[9]QUINTANILLA F H,LOWE M J S,CRASTER R V.Full 3Ddispersion curve solutions for guided waves in generally anisotropic media[J].Journal of sound and vibration,2016,363:545-559.
[10]李念,钱征华,王彬.复波数域弥散曲线的求解方法研究[J].应用力学学报,2016,33(3):365-370.(LI Nian,QIAN Zhenghua,WANG Bin.Study on computational methods of dispersion curves in complex wave number range[J].Chinese journal of applied mechanics,2016,33(3):365-370(in Chinese)).
[11]SIMONETTI F,LOWE M J.On the meaning of Lamb mode nonpropagating branches[J].Journal of the acoustical society of America,2005,118(1):186-192.