Effects of Residual Stress on the Hydro-Elastic Vibration of Circular Diaphragm

ABSTRACT

The effects of residual stress on the hydro-elastic vibration of
circular diaphragm are theoretically investigated by using the added mass
approach. The Kirchhoff theory of plates is used to model the elastic thin circular
diaphragm on an aperture of an infinite rigid wall and in contact with a fluid
on one side. The fluid is assumed to be incompressible and inviscid and the
velocity potential is used to describe its irrotational motion. A
non-dimensional tension parameter is defined, and the effects of the tension
parameter on the frequency parameters and mode shapes of the diaphragm in the
air are presented. The Hankel transform is applied to solve the fluid-diaphragm
coupled system; boundary conditions are expressed by integral equations. Finally,
the effects of residual stress on the non-dimensional added virtual mass
incremental (NAVMI) factors of the diaphragm contact with a fluid on one side
are investigated. It is found that the effects of the residual stress cannot be
neglected when the edges of the circular diaphragm are clamped. The effects of
residual stress for NAVMI factors can be increases 11% when the non-dimensional
tension parameter is 1000.

KEYWORDS

Circular Diaphragm; Residual Stress; Hankel Transform; Frequency Parameter; Mode Shape; NAVMI

Circular Diaphragm; Residual Stress; Hankel Transform; Frequency Parameter; Mode Shape; NAVMI

Cite this paper

J. Zhao and S. Yu, "Effects of Residual Stress on the Hydro-Elastic Vibration of Circular Diaphragm,"*World Journal of Mechanics*, Vol. 2 No. 6, 2012, pp. 361-368. doi: 10.4236/wjm.2012.26041.

J. Zhao and S. Yu, "Effects of Residual Stress on the Hydro-Elastic Vibration of Circular Diaphragm,"

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[28] E. C. Wente, “A Condenser Transmitter as a Uniformly Sensitive Instrument for the Absolute Measurement of Sound Intensity,” Physics Review, Vol. 10, No. 1, 1917, pp. 39-63.

[1] P. Muralt, “Ferroelectric Thin Films for Microsensors and Actuators: A Review,” Journal of Micromechanics and Microengineering, Vol. 10, No. 2, 2000, pp. 136-146. doi:10.1088/0960-1317/10/2/307

[2] Y. Xin, Z. Li, L. V. Odum, Z. Y. Cheng and Z. Xu, “Piezoelectric Diaphragm as a High Performance Biosensor Platform,” Applied Physics Letters, Vol. 89, No. 22, 2006, Article ID: 223508. doi:10.1063/1.2396919

[3] A. K. Isamil, J. S. Burdess, A. J. Harris, C. J. McNeil and J. Hedley, et al., “The Principle of a MEMS Circular Diaphragm Mass Sensor,” Journal Micromechanics Microengineering, Vol. 16, No. 8, 2006, pp. 1487-1493. doi:10.1088/0960-1317/16/8/008

[4] A. K. Isamil, J. S. Burdess, A. J. Harris, G. Suarez and N. Keegan, et al., “The Fabrication, Characterization and Testing of a MEMS Circular Diaphragm Mass Sensor,” Journal Micromechanics Microengineering, Vol. 18, No. 2, 2006, Article ID: 25021. doi:10.1088/0960-1317/18/2/025021

[5] M. Olfatnia, T. Xu, J. M. Miao, L. S. Ong and X. M. Jing, et al., “Piezoelectric Circular Microdiaphragm Based Pressure Sensors,” Sensors and Actuators A: Physical, Vol. 163, No. 1, 2010, pp. 32-36. doi:10.1016/j.sna.2010.06.016

[6] W. D. Nix, “Mechanical Properties of Thin Films,” Metallurgical and Materials Transactions A, Vol. 20, No. 11, 1989, pp. 2217-2240.

[7] C. V. Thompson and R. Carel, “Stress and Grain Growth in Thin Films,” Journal of the Mechanics and Physics of Solids, Vol. 44, No. 5, 1996, pp. 657-673. doi:10.1016/0022-5096(96)00022-1

[8] E. Hong, R. Smith, S. V. Krishnaswamy, C. B. Freidholff, and S. Trolier-McKinstry, “Residual Stress Development in Pb(Zr,Ti)O3/ZrO2/SiO2 Stacks for Piezoelectric Microactuators,” Thin Solid Films, Vol. 510, No. 1-2, 2006, pp. 213-221. doi:10.1016/j.tsf.2005.12.300

[9] M. Olfatnia, T. Xu, L. S. Ong, J. M. Miao and Z. H. Wang, “Investigation of Residual Stress and its Effects on the Vibrational Characteristics of Piezoelectric Based Multilayered Microdiaphragms,” Journal of Micromechanics and Microengineering, Vol. 20, No. 1, 2010, Article ID: 15007. doi:10.1088/0960-1317/20/1/015007

[10] S. Lee, T. Tanaka, K. Inoue, J. M. Kim and Y. E. Shin, et al., “Stress Influences on the Ultrasonic Transducers,” Sensors Actuators A, Vol. 119, No. 2, 2005, pp. 405-411. doi:10.1016/j.sna.2004.10.026

[11] P. Muralt, A. Kholkin, M. Kohli and T. Maeder, “Piezoelectric Actuation of PZT Thin Film Diaphragms at Static and Resonant Conditions,” Sensors Actuators A, Vol. 53, No. 1-3, 1996, pp. 398-404. doi:10.1016/0924-4247(96)01139-9

[12] L. F. Ge, “Dynamic Behavior of Micro-diaphragms and Its Characterized Description,” Science China Series G, Vol. 52, No. 9, 2009, pp. 1345-1356.

[13] M. K. Kwak, “Hydroelastic Vibration of Circular Plates,” Journal of Sound and Vibration, Vol. 201, No. 3, 1997, pp. 293-303. doi:10.1006/jsvi.1996.0775

[14] M. Amabili, G. Dalpiaz and C. Santolini, “Free Edge Circular Plates Vibrating in Water,” Modal Analysis: The International Journal of Analytical and Experimental Modal Analysis, Vol. 10, No. 1, 1995, pp. 187-202.

[15] M. Amabili and M. K. Kwak, “Vibration of Circular Plates on a Free Fluid Surface: Effect of Surface Waves,” Journal of Sound and Vibration, Vol. 226, No. 3, 1999, pp. 407-424. doi:10.1006/jsvi.1998.2304

[16] M. Amabili, G. Frosali and M. K. Kwak, “Free Vibrations of Annular Plates Coupled with Fluids,” Journal of Sound and Vibration, Vol. 191, No. 5, 1996, pp. 825-846. doi:10.1006/jsvi.1996.0158

[17] C. C. Liang, Y. S. Tai and P. L. Li, “Natural Frequencies of Annular Plates Having Contact with Fluid,” Journal of Sound and Vibration, Vol. 228, No. 5, 1999, pp. 11671181. doi:10.1006/jsvi.1999.2463

[18] H. F. Bauer, “Coupled Frequencies of a Liquid in a Circular Cylindrical Container with Elastic Liquid Surface Cover,” Journal of Sound and Vibration, Vol. 180, No. 5, 1995, pp. 689-704. doi:10.1006/jsvi.1995.0109

[19] H. F. Bauer and M. Chiba, “Hydroelastic Viscous Oscillations in a Circular Cylindrical Container with an Elastic Cover,” Journal of Fluids Structures, Vol. 14, No. 6, 2000, pp. 917-936. doi:10.1006/jfls.2000.0296

[20] M. Amabili, “Vibrations of Circular Plates Resting on a Sloshing Liquid: Solution of the Fully Coupled Problem,” Journal of Sound and Vibration, Vol. 245, No. 2, 2001, pp. 261-283. doi:10.1006/jsvi.2000.3560

[21] W. P. Mason, “Electromechanical Transducers and Wave Filters,” 2nd Edition, Van Bistrabd D. Company Inc., New York, 1948.

[22] A. Park, F. R. Blom, M. Elwenspoek and S. J. Lammerink, “Q-Factor and Frequency Shift of Resonating Silicon Diaphragms in Air,” Sensors and Actuators A, Vol. 27, No. 1-3, 1991, pp. 691-698. doi:10.1016/0924-4247(91)87072-B

[23] M. Olfatnia, Z. Shen, J. M. Miao, L. S. Ong and T. Xu, et al., “Medium Damping Influences on the Resonant Frequency and Quality Factor of Piezoelectric Circular Microdiaphragm Sensors,” Journal Micromechanics Microengineering, Vol. 21, No. 4, 2011, Article ID: 45002. doi:10.1088/0960-1317/21/4/045002

[24] I. N. Sneddon, “Mixed Boundary Value Problems in Potential Theory,” North-Holland Pub. Co., Amsterdam, 1966.

[25] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, “Integrals and Series. Special Functions,” Gordon and Breach, New York, 1986.

[26] F. Zhu, “Rayleigh Quotients for Coupled Free Vibrations,” Journal of Sound and Vibration, Vol. 171, No. 5, 1994, pp. 641-649. doi:10.1006/jsvi.1994.1146

[27] S. M. Vogel and D. W. Skinner, “Natural Frequencies of Transversely Vibrating Uniform Annular Plates,” Journal of Applied Mechanics, Vol. 32, No. 4, 1965, pp. 926-931. doi:10.1115/1.3627337

[28] E. C. Wente, “A Condenser Transmitter as a Uniformly Sensitive Instrument for the Absolute Measurement of Sound Intensity,” Physics Review, Vol. 10, No. 1, 1917, pp. 39-63.