纵弯复合换能器阶梯辐射圆板的振动和辐射性能研究
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摘要
纵弯复合空气换能器由纵向振动夹心式换能器与弯曲振动薄圆盘组成，具有纵向振动换能器的高效大功率和弯曲振动圆盘的低辐射阻抗、大辐射面积等特点。在超声除尘、超声雾化、超声检测等领域有广泛的应用。然而对于辐射面积较大的弯曲振动圆盘，一般激励的振动阶数比较高，因此存在弯曲振动的反相区，导致换能器的辐射阻抗下降，电声效率和声功率降低以及指向性变差等问题。为此，采用弯曲振动阶梯圆盘，通过阶梯高度补偿相位是一种比较常用的办法。但是对于这种阶梯圆盘的理论研究还比较少，已有的一般近似程度较大，设计上也存在一些问题。本文从薄圆板弯曲振动理论出发，推导了这种阶梯圆盘的频率方程、等效质量、等效弹性和辐射阻抗，并进而提出了这种阶梯圆盘的一般设计方法。考虑到大功率超声应用中的非线性和吸收作用，本文从KZK方程和准线性条件出发，对于工作在共振频率下的这种阶梯圆盘，推导了它的一阶和二阶谐波声场计算方法。并对其声场远场采用瑞利积分推导了一般表达式和指向性公式。经过实验验证，与理论符合较好。
具体的，主要有以下几个方面：
1．对于圆形薄板，计算了它的振形方程，根据厚度变化处的位移和位移导数连续条件、剪切力连续条件、弯矩连续条件，通过数值方法求解方程组获得弯曲阶梯圆板的频率方程的求解。经过对于不同边界问题的频率根值进行比较，得出了不同于弯曲平圆板的结论。即由简支到自由到固定频率依次升高的结论。在求得频率的基础上，进一步求解了与之对应的归一化了的振形曲线系数，从而为后面的计算铺平了道路。
2．对于工作在谐振频率下做轴对称运动的阶梯圆板，采用解析法求得它的等效质量，并由C=1/ω~2M求得等效弹性系数。采用解析法求得它在谐振频率下工作时的辐射阻抗为，在此基础上可以得到阶梯圆板的等效电路。
3．在给出设计频率和节圆数目的条件下，可以根据节圆处振幅为零可以设
计出任一个节圆的阶梯板的尺寸参数。这些阶梯板克服了弯曲圆扳振动时的相为
相消的问题。因而具有良好的辐射和指向性能。根据这一设计思想，设计了两个
两节圆的阶梯圆板，经试验符合设计要求。
4．对KZK方程在准线性条件下采用微扰法进行求解，将阶梯板辐射的声源条
件带入后，可以获得它的一阶和二阶声场的积分计算方法，可用于数值计算。这
些公式可以计算出一个波长以外的整个声场，另外，也可以获得远场指向性公式。
这些公式的适用条件是切＞＞1，或者说，他们是高频近似条件下的声场解。
5．利用瑞利积分，可以得到阶梯辐射圆板的远场解析解：
几＝ Re｛：exP（j。r）D 0冈＋ 11＊十 Ill ＊H厂）exP（jk。H冈川／ A；，并且对于任意轴对
厂 二7
称声场，按照瑞利积分办法，可以得到其远场指向性公式为：D仔）＝一一二二二二一二7一一。
The compound transducer consisting of a longitudinal vibrator and a flexural circular plate combines the high energy and efficiency of the former and the low impendence and wide spreading scale of the latter. It is now widely used in aerosol particle precipitation, ultrasonic atomization and ultrasonic detection. But to circular plate with large surface area, the vibrate modes generated is often so high that there is reverse phases existing, which result in a decrease in radiation impedance, electric-acoustic efficiency and the electric power, as well as poor directivity. So a stepped circular plate is adopted which can compensate the phase by different thickness. However, the theory about it is still scarce. Or if there is some, they usually introduce large approximation so it is difficult to be used in practical design. In this article based on vibration theory of the flexural plate, the frequency equation, the equivalent mass and radiation impedance are derived and further, design theory is introduced. After the nonlinear and absorption effect is taken into account in the applications where high power ultrasonic is involved, the first and second harmonic components of the sound field of stepped circular plate is analyzed, based on the KZK equation and quasi-linear condition. Moreover, taking on the Rayleigh integral the far field acoustic pressure and the directivity formula is obtained. The measured results are in good agreement with this theoretical value.
In detail, the context include several aspects:
1. According to several boundary conditions, the frequency equation of flexural circular plate is calculated. Similarly some coefficients in the displacement distribution can also be acquired, which provide a preparation for further study.
2.To the circular plate working in axial symmetry vibration at resonant frequency, the equivalent mass in analytical form is derived, that's is,
, and further, the equivalent compliance is obtained using the formula C = 1l 2 M . The radiation impedance at the resonant frequency, in
analytical form, is Consequently, the
equivalent circuit of the plate is obtained.
3. When the number of the nodal circles and the design frequency are given, the stepped circular plate satisfying the requirement can be designed. The stepped plate designed in this method can overcome the limitation of the sound field generated by circular flat.
4.According the Rayleigh integral, the far field of this kind of plate is resolved,
namely, . And the
formula as a far field directivity function is assumed to be useful in analyzing the axis-symmetry sound field.
具体的，主要有以下几个方面：
1．对于圆形薄板，计算了它的振形方程，根据厚度变化处的位移和位移导数连续条件、剪切力连续条件、弯矩连续条件，通过数值方法求解方程组获得弯曲阶梯圆板的频率方程的求解。经过对于不同边界问题的频率根值进行比较，得出了不同于弯曲平圆板的结论。即由简支到自由到固定频率依次升高的结论。在求得频率的基础上，进一步求解了与之对应的归一化了的振形曲线系数，从而为后面的计算铺平了道路。
2．对于工作在谐振频率下做轴对称运动的阶梯圆板，采用解析法求得它的等效质量，并由C=1/ω~2M求得等效弹性系数。采用解析法求得它在谐振频率下工作时的辐射阻抗为，在此基础上可以得到阶梯圆板的等效电路。
3．在给出设计频率和节圆数目的条件下，可以根据节圆处振幅为零可以设
计出任一个节圆的阶梯板的尺寸参数。这些阶梯板克服了弯曲圆扳振动时的相为
相消的问题。因而具有良好的辐射和指向性能。根据这一设计思想，设计了两个
两节圆的阶梯圆板，经试验符合设计要求。
4．对KZK方程在准线性条件下采用微扰法进行求解，将阶梯板辐射的声源条
件带入后，可以获得它的一阶和二阶声场的积分计算方法，可用于数值计算。这
些公式可以计算出一个波长以外的整个声场，另外，也可以获得远场指向性公式。
这些公式的适用条件是切＞＞1，或者说，他们是高频近似条件下的声场解。
5．利用瑞利积分，可以得到阶梯辐射圆板的远场解析解：
几＝ Re｛：exP（j。r）D 0冈＋ 11＊十 Ill ＊H厂）exP（jk。H冈川／ A；，并且对于任意轴对
厂 二7
称声场，按照瑞利积分办法，可以得到其远场指向性公式为：D仔）＝一一二二二二一二7一一。
The compound transducer consisting of a longitudinal vibrator and a flexural circular plate combines the high energy and efficiency of the former and the low impendence and wide spreading scale of the latter. It is now widely used in aerosol particle precipitation, ultrasonic atomization and ultrasonic detection. But to circular plate with large surface area, the vibrate modes generated is often so high that there is reverse phases existing, which result in a decrease in radiation impedance, electric-acoustic efficiency and the electric power, as well as poor directivity. So a stepped circular plate is adopted which can compensate the phase by different thickness. However, the theory about it is still scarce. Or if there is some, they usually introduce large approximation so it is difficult to be used in practical design. In this article based on vibration theory of the flexural plate, the frequency equation, the equivalent mass and radiation impedance are derived and further, design theory is introduced. After the nonlinear and absorption effect is taken into account in the applications where high power ultrasonic is involved, the first and second harmonic components of the sound field of stepped circular plate is analyzed, based on the KZK equation and quasi-linear condition. Moreover, taking on the Rayleigh integral the far field acoustic pressure and the directivity formula is obtained. The measured results are in good agreement with this theoretical value.
In detail, the context include several aspects:
1. According to several boundary conditions, the frequency equation of flexural circular plate is calculated. Similarly some coefficients in the displacement distribution can also be acquired, which provide a preparation for further study.
2.To the circular plate working in axial symmetry vibration at resonant frequency, the equivalent mass in analytical form is derived, that's is,
, and further, the equivalent compliance is obtained using the formula C = 1l 2 M . The radiation impedance at the resonant frequency, in
analytical form, is Consequently, the
equivalent circuit of the plate is obtained.
3. When the number of the nodal circles and the design frequency are given, the stepped circular plate satisfying the requirement can be designed. The stepped plate designed in this method can overcome the limitation of the sound field generated by circular flat.
4.According the Rayleigh integral, the far field of this kind of plate is resolved,
namely, . And the
formula as a far field directivity function is assumed to be useful in analyzing the axis-symmetry sound field.
引文
[1] 张光斌，纵弯模式转换空气换能器的圆形弯曲振动辐射器研究，陕西师范大学研究生学位论文，2000．4．
[2] R. Hickling and S. Martin, The use of Ultrasonics for Guanging and Proximity Sensingin in Air, J. Acoust. Soc. Am, 1986,9(4),1151.
[3] S. H Yang, et al. Noncontact thickness measurement of wet/dry paint coating using air coupled transducer, Mat. Eval, 1990, 48-471.
[4] T. Yano, M. Tone and A. Fukumoto, Range finding and surface characterization using high frequency air transducer, IEEE Trans. UFFC, 1987, 34-232.
[5] R. W. Smith et al. High frequency acoustic system for robotic applications, Mat. Eval. 1988, 46-1262.
[6] W. S. H. Munor et al., Ultrasonic vehicle guidance transducer, Ultrasonic, 1990, 28-350.
[7] B. T. Khuri-Yakub, J. H. Kim, G. S. Kino, A new design for air transducers. IEEE Ultrasonics Symposium proceeding, 1988, 503
[8] L. Mautitzson, et al. Two dimensional airbore focused ultrasound. Ultrasonic, 1990, 28-350.
[9] 张小凤，尚志远，静电式超声检测换能器的设计，陕西师范大学学报，2001．29(4)，47-50．
[10] 王兆球，张淑仪，高频空气换能器研究现状与展望，应用声学，1992，11(2)，1．
[11] V. Magori, H. Walker, Ultrasonic presence sensor with wide range and high local resolution. IEEE Trans. UFFC, 1987(34), 202.
[12] 宋宙模等，175kHz 高频空气换能器研制及特性测量，应用声学，1992，13(1)，33-37．
[13] 尚志远，压电超声换能器的性能分析及应用领域，压电与声光，1994，16(1)29-33．
[14] 林书玉，复频超声换能器的研究，声学与电子工程，1995，1，9-13．
[15] J. A. Gallego Juarez. Axisymmetric vibration of circular plate with stepped thickness. Journal of Sound and Vibraton, 1973, 26(3), 411-416.
[16] A. Barone and J. A. Gallego Juarez. On a modification of vibration flat plates in order to obtain phase-coherent radiation. Acustica, 1969, 70, 22, 187-188.
[17] J. A. Gallego Juarez, G. Rodriguez Corral,L. Garreton. An ultrasonic transducer of high applications in gases. Ultrasonics, 1978, 12, 267-271.
[18] 马志敏等，气介超声换能器的振动模式与反相区的补偿办法，武汉水利电力大学学报，2000，33(3)，104-107．
[19] R. Beeuwer, Ultrasonic scanning, imaging and recognition of bottles, Ultasonics Symposium proceeding, 1989,635.
[20] 程存弟，超声技术-功率超声及其应用，陕西师范大学出版社，1992．
[21] Sigurd Aanonson et al. Distortion and harmonic generation in the nearfied of a finite amplitude sound beam, J. Acoust. Soc. Am., 1984, 75(3), 749-768.
[22] Robin O. Cleveland, Mark F. Hamilton and David T. Blackstock, Time-domain modeling of finite-amplitude sound in relaxing fluids, J. Acoust. Soc. Am., 1996, 99(6), 3312-3318.
[23] Yang-Sub Lee and Mark F. Hamilton, Time-domain modeling of pulsed finite-amplitude sound beams, J. Acoust. Soc. Am.,1995, 97(2), 906-917.
[24] Jahangir Tavakkoli, Dominique Cathignoland Remi Souchon, Modeling of pulsed finite-amplitude focused sound beams in time domain, J. Acoust. Soc. Am, 1998, 104(4), 2061-2072.
[25] 夏荣民，寿文德，孙俊霞，严家勇，自聚焦换能器的声场研究，应用声学，2001，20(5)，16-20．
[26] Desheng Ding and Zuhong Lu, The second harmonic component in the Bessel beam, Appl. Phys. Lett. 1996, 68(5), 608-724.
[27] J. J. Wen and M. A. Breazeale, A diffraction beam field expressed as the superposition of Guassian Beams, J. Acoust. Soc. Am. 1988, 83(5), 1752-1756.
[28] Desheng Ding, Yonggan Shui, Jingbo Lin, and De Zhang, A simle calculation approach for the second harmonic sound field generated by an arbitrary axia-symmetric source, J. Acoust. Soc. Am.,1996, 100(2).
[29] Dong Zhang, Xiu-fen Gong, and Bo Zhang, Second harmonic sound field after insertion of a biological tissue sample, J. Acoust. Soc. Am. 2002, 111(1), 45-48.
[30] 张波，章东，龚秀芬，活塞聚焦换能器的二次谐波声场，声学技术，2000，19(3)．
[31] 何祚镛，赵玉芳，声学理论基础，国防工业出版社，1981．
[32] Mark. F. Hamilton and David. T. Blackstock, Nonlinear acoustics, Acdemic Press, 1998.
[33] G. N. Watson, Sc. D, F. R. S, Theory of Bessel Function, The Cambridge University Press, 1952.
[34] 林书玉，张福成，郭孝武，一种新型的功率超声辐射器的研究，陕西师范大学学报，1993，21(1)，83-85．
[35] 吴永芬，冯永芳，向空气辐射的弯曲振动型超声换能器的研究，应用声学，1982，1(2)，37-39．
[36] 林书玉，张福成，郭孝武等，大功率气介声波换能器的研究，陕西师范大学学报，1995，23(4)，43-45．
[37] 张光斌，林书玉，气介式弯曲振动换能器的辐射声压及其指向特性，陕西
师范大学学报，1999，27(3)，45-49．
[38] Youichi, Ito and Masatada Kavamura, Sound field between two parallel reflection plates by vibration plate with stripes mode, Japanese Journal of Applied Physics, 1984, 23, 218-220.
[39] Tesuro Otsuka, Yoshiyuki Kamishima and Koishiro Seva, Japanese Journal of Applied Physics, 1983, 22, 108-110.
[2] R. Hickling and S. Martin, The use of Ultrasonics for Guanging and Proximity Sensingin in Air, J. Acoust. Soc. Am, 1986,9(4),1151.
[3] S. H Yang, et al. Noncontact thickness measurement of wet/dry paint coating using air coupled transducer, Mat. Eval, 1990, 48-471.
[4] T. Yano, M. Tone and A. Fukumoto, Range finding and surface characterization using high frequency air transducer, IEEE Trans. UFFC, 1987, 34-232.
[5] R. W. Smith et al. High frequency acoustic system for robotic applications, Mat. Eval. 1988, 46-1262.
[6] W. S. H. Munor et al., Ultrasonic vehicle guidance transducer, Ultrasonic, 1990, 28-350.
[7] B. T. Khuri-Yakub, J. H. Kim, G. S. Kino, A new design for air transducers. IEEE Ultrasonics Symposium proceeding, 1988, 503
[8] L. Mautitzson, et al. Two dimensional airbore focused ultrasound. Ultrasonic, 1990, 28-350.
[9] 张小凤，尚志远，静电式超声检测换能器的设计，陕西师范大学学报，2001．29(4)，47-50．
[10] 王兆球，张淑仪，高频空气换能器研究现状与展望，应用声学，1992，11(2)，1．
[11] V. Magori, H. Walker, Ultrasonic presence sensor with wide range and high local resolution. IEEE Trans. UFFC, 1987(34), 202.
[12] 宋宙模等，175kHz 高频空气换能器研制及特性测量，应用声学，1992，13(1)，33-37．
[13] 尚志远，压电超声换能器的性能分析及应用领域，压电与声光，1994，16(1)29-33．
[14] 林书玉，复频超声换能器的研究，声学与电子工程，1995，1，9-13．
[15] J. A. Gallego Juarez. Axisymmetric vibration of circular plate with stepped thickness. Journal of Sound and Vibraton, 1973, 26(3), 411-416.
[16] A. Barone and J. A. Gallego Juarez. On a modification of vibration flat plates in order to obtain phase-coherent radiation. Acustica, 1969, 70, 22, 187-188.
[17] J. A. Gallego Juarez, G. Rodriguez Corral,L. Garreton. An ultrasonic transducer of high applications in gases. Ultrasonics, 1978, 12, 267-271.
[18] 马志敏等，气介超声换能器的振动模式与反相区的补偿办法，武汉水利电力大学学报，2000，33(3)，104-107．
[19] R. Beeuwer, Ultrasonic scanning, imaging and recognition of bottles, Ultasonics Symposium proceeding, 1989,635.
[20] 程存弟，超声技术-功率超声及其应用，陕西师范大学出版社，1992．
[21] Sigurd Aanonson et al. Distortion and harmonic generation in the nearfied of a finite amplitude sound beam, J. Acoust. Soc. Am., 1984, 75(3), 749-768.
[22] Robin O. Cleveland, Mark F. Hamilton and David T. Blackstock, Time-domain modeling of finite-amplitude sound in relaxing fluids, J. Acoust. Soc. Am., 1996, 99(6), 3312-3318.
[23] Yang-Sub Lee and Mark F. Hamilton, Time-domain modeling of pulsed finite-amplitude sound beams, J. Acoust. Soc. Am.,1995, 97(2), 906-917.
[24] Jahangir Tavakkoli, Dominique Cathignoland Remi Souchon, Modeling of pulsed finite-amplitude focused sound beams in time domain, J. Acoust. Soc. Am, 1998, 104(4), 2061-2072.
[25] 夏荣民，寿文德，孙俊霞，严家勇，自聚焦换能器的声场研究，应用声学，2001，20(5)，16-20．
[26] Desheng Ding and Zuhong Lu, The second harmonic component in the Bessel beam, Appl. Phys. Lett. 1996, 68(5), 608-724.
[27] J. J. Wen and M. A. Breazeale, A diffraction beam field expressed as the superposition of Guassian Beams, J. Acoust. Soc. Am. 1988, 83(5), 1752-1756.
[28] Desheng Ding, Yonggan Shui, Jingbo Lin, and De Zhang, A simle calculation approach for the second harmonic sound field generated by an arbitrary axia-symmetric source, J. Acoust. Soc. Am.,1996, 100(2).
[29] Dong Zhang, Xiu-fen Gong, and Bo Zhang, Second harmonic sound field after insertion of a biological tissue sample, J. Acoust. Soc. Am. 2002, 111(1), 45-48.
[30] 张波，章东，龚秀芬，活塞聚焦换能器的二次谐波声场，声学技术，2000，19(3)．
[31] 何祚镛，赵玉芳，声学理论基础，国防工业出版社，1981．
[32] Mark. F. Hamilton and David. T. Blackstock, Nonlinear acoustics, Acdemic Press, 1998.
[33] G. N. Watson, Sc. D, F. R. S, Theory of Bessel Function, The Cambridge University Press, 1952.
[34] 林书玉，张福成，郭孝武，一种新型的功率超声辐射器的研究，陕西师范大学学报，1993，21(1)，83-85．
[35] 吴永芬，冯永芳，向空气辐射的弯曲振动型超声换能器的研究，应用声学，1982，1(2)，37-39．
[36] 林书玉，张福成，郭孝武等，大功率气介声波换能器的研究，陕西师范大学学报，1995，23(4)，43-45．
[37] 张光斌，林书玉，气介式弯曲振动换能器的辐射声压及其指向特性，陕西
师范大学学报，1999，27(3)，45-49．
[38] Youichi, Ito and Masatada Kavamura, Sound field between two parallel reflection plates by vibration plate with stripes mode, Japanese Journal of Applied Physics, 1984, 23, 218-220.
[39] Tesuro Otsuka, Yoshiyuki Kamishima and Koishiro Seva, Japanese Journal of Applied Physics, 1983, 22, 108-110.
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