• Another approach

• Based on the calculation of the frequency of the first mode of vibration
  
• Frequency of the ith mode of vibration can be calculated from the expression of the general form:

    where   B ...  flexural rigidity of the plate

                                                    a  ... diameter in the case of circular plate or side in the case of square plate

Values of coefficients C₁ for different forms of thin plates with different types of support are given in the following table:

• If plate deflection is comparable or bigger than a plate thickness, more general expression must be used.
• Relation between the plate displacement and the pressure difference P:

• This equation is nonlinear in u₀ and cannot be solved for u₀.
• First term represents the stiffness associated with the bending of the thin plate
• Second term represents the stiffness associated with the stretching of the plate that introduces nonlinearity
• For small-deflection cases, if u₀ < h, the second term can be neglected and simplified expression is:

More information

Related Reading

Malecki  I., Physical foundations of technical acoustics, Pergamon, 1969.

Merhaut  J.,  Theory of electroacoustics, McGraw-Hill, 1981.

Middelhoek, S., Audet, S. A., Silicon sensors, Academic Press, 1989.

Olson  H. F.,  Acoustical engineering, D. van Nostrand Comp.,Inc., 1957.

Vibrations, Techniques de l ingenieur, traite Sciences fondamentales, Doc. A 410, 1999.

Rossi  M.,  Electroacoustique, Presses Polytechniques Romandes, 1986.

Senturia S. D., Microsystem Design, Kluwer Academic Publishers, 2001.

Skvor Z., Vibrating systems and their equivalent circuits, Prague, Academia, 1991.

Smee, S. A., Gaitan, M, Novotny, D. B., Joshi, Y., Blackburn, D. L., "IC test structures for multilayer interconnect stress determination", IEEE Electron Device Letter, vol. 21, No. 1, pp. 12-14, 2000.

Tilmans H.A.C., "Equivalent circuit representation of electromechanical transducers: I. Lumped-parameter systems", J. Micromech. Microeng., vol. 6, pp. 157-176, 1996.

Tilmans H.A.C., "Equivalent circuit representation of electromechanical transducers: II. Distributed-parameter systems", J. Micromech. Microeng., vol. 7, pp. 285-309, 1997.

Timoshenko S., Woinowsky-Krieger S., Theorie des plaques et coques, Librairie Polytechnique Ch. Beranger, 1961.