航空航天學院關于普林斯頓大學Lee Chung- Yi教授學術報告的通知

發布日期：2010-12-01來源：航空航天學院作者：系統管理員訪問量：12433

時間：2010年12月3號下午15:00pm—17:00pm
地點：浙江大學玉泉校區教十二118室
題目：Theory of Vibrations of Plates:Its Evolution and Applications to Piezoelectric Crystals and Ceramics
報告人：Professor Emeritus of Civil and Environmental Engineering
主持人：陳偉球教授

Theory of Vibrations of Plates: Its Evolution and Applications to

Piezoelectric Crystals and Ceramics

P. C. Y. Lee (Lee Chung- Yi)

Professor Emeritus of Civil and Environmental Engineering

Princeton University

Princeton, NJ 08540

lee@princeton.edu

　　In 1809, the French Academy invited Chladni to give a demonstration of his experiments on nodal lines and frequencies of various modes of thin, vibrating plates. It was said that the emperor Napolean attended the meeting, was very impressed, and suggested the Academy to establish an extraordinary prize for the “ problem of deriving a mathematical theory of plate vibration and of comparing theoretical results with those obtained experimentally ”(1). Sophie Germain entered the competition and won the prize in 1816. Thus, the classical equation of flexural vibrations of elastic plates or the Germain- Lagrange plate equation (1811-1816) was born (2). The application of the theory was limited to waves which are long as compared to the thickness of the plate and to frequencies of low-order modes. These limitations are similar to those for the classical equation of flexural vibrations of beams or the Bernoulli-Euler beam equation (1725-1736).

In the case of beam theory, it was Timoshenko (1921) who made significant advancement by including transverse shear deformation and introducing a shear correction factor in his derivation. His equation gives satisfactory results for short waves and high modes and since is called the Timoshenko beam equation (3). Analogous to Timoshenko’s 1-D theory of beams, many 2-D equations were obtained by including shear deformation and correction factor (4,5).

　　In 1951, Mindlin deduced a 2-D theory of flexural motions of isotropic elastic plates from the 3-D equations of elasticity (6). It was shown that with a correction factor the predicted dispersion curves of straight-crested waves agree closely with those from the 3-D theory. These equations and the subsequent ones for crystal plates (1951) and piezoelectric plates (1952) have since become well known worldwide in applied mechanics, structures, frequency control, and ultrasonics, and are generally referred to as the Mindlin (first-order) plate equations (7,8).

　　By expanding displacement in a series of trigonometrical functions, which are the simple thickness modes of an infinite plate and by following a general method of deduction of Mindlin (9), 2-D equations were obtained by Lee and Nikodem for isotropic plates in 1972 (10), for anisotropic plates in 1974 (11), and by Lee, Syngellakis, and Hou for piezoelectric plates in1987 (12). Computed dispersion curves agree closely with the exact ones and attain the exact cut-off frequencies for each successive high-order approximation, except for the lowest frequency branch of flexural mode which is not as accurate as that obtained from Mindlin’s equations.

　　By adding to the afore mentioned series a term linear in the thickness coordinate to accommodate the in-plane displacements induced by the gradients of deflection in low-frequency flexural motions or static bending, a system of 2-D equations of flexural vibrations are obtained for isotropic, elastic plates (13). Although the form of coupled equations of thickness shear and flexural motions is different from that of Mindlin’s first –order equations (9), the single governing equation in plate deflection is shown to be identical to the corresponding one by Mindlin (6), and the dispersion relations from both systems are shown to be identical. Hence the present system of equations has been shown analytically to be equivalent to the Mindlin first- order equations without introducing any correction factors.

　　The same method of displacement expansion has been applied to piezoelectric crystals and ceramics and for higher- order approximations (14-16).

References

S. Timoshenko, History of Strength of Materials, McGraw-Hill , New York , 1953, p.119.
S. Germain, “Recherches sur la theorie des surfaces elastiques,” Courcier, Paris, 1821.
S. Timoshenko, D Young, and W Weaver, Jr., Vibration Problems in Engineering, John Wiley & Sons, New York , 1974, p. 432.
Ya. S. Uflyand, “The propagation of waves in transverse vibrations of bars and plates,” Akad. Nauk SSSR, Prikl. Mat. Meh., vol. 12, 1948, pp.287-300.
E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates,” J. Appl. Mech., vol. 67, 1945, p. A-69.
R.D. Mindlin,” Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates,” J. Appl. Mech., vol. 18, 1951, pp. 31-38.
R.D. Mindlin, “Thickness shear and flexural vibrations of crystal plates,” J. Appl. Phys., vol. 22, 1951, pp. 316-323.
R.D. Mindlin, “Forced thickness shear and flexural vibrations of piezoelectric crystal plates,” J. Appl. Phys., vol. 23, 1952, pp. 83-88.
R.D. Mindlin, “An introduction to the mathematical theory of vibrations of elastic plates, ” U.S. Army Signal Corps Engineering Laboratories, Fort Monmouth , NJ , 1955. The same monograph is available in book form, ed. by J. Yang, World Scientific, New Jersey , 2006.
P.C.Y. Lee and Z. Nikodem, “An approximate theory for high-frequency vibrations of elastic plates, Int. J. Solids Structures, vol. 8, 1972, pp581-612.
Z. Nikodem and P.C.Y. Lee, “Approximate theory of vibrations of crystal plates at high frequencies,” Int. J. Solids Structures, vol. 10, 1974, pp. 177-196.
P.C.Y. Lee, S. Syngellakis , and J.P. Hou, “A two-dimensional theory for vibrations of piezoelectric crystal plates with or without electrodes,” J. Appl. Phys., vol. 61, no.4, 1987, pp 1249-1262.
P.C.Y. Lee, “An accurate two-dimensional theory of vibrations of isotropic, elastic plates,” Proc. 2006 IEEE International Frequency Control Symposium. Also accepted in 2010 for publication in Acta Mechanica Solida Sinica.
P.C.Y. Lee, J.D. Yu, and W.S. Lin,” A two-dimensional theory for vibations of piezoelectric crystal plates with electrodes faces,” J. Appl Phys., vol. 83, no. 3 1998, pp1213-1223.
R. Huang, P.C.Y. Lee, W.S. Lin, and J.-D. Yu, “Extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoceramic diskes, “IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49, no. 11, 2002, pp. 1507-1515.
P.C.Y. Lee, N.P. Edwards, W.S. Lin, and S. Syngellakis, “Second-order theories for extensional vibrations of piezoelectric crystal plates and strips,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49, no. 11, 2002, pp. 1497-1506.