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Technical Brief

Vibrations of Completely Free Rounded Rectangular Plates

[+] Author and Article Information
C. Y. Wang

Professor
Departments of Mathematics and
Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: cywang@mth.msu.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 12, 2013; final manuscript received November 13, 2014; published online January 20, 2015. Assoc. Editor: Olivier A. Bauchau.

J. Vib. Acoust 137(2), 024502 (Apr 01, 2015) (5 pages) Paper No: VIB-13-1049; doi: 10.1115/1.4029159 History: Received February 12, 2013; Revised November 13, 2014; Online January 20, 2015

The natural vibration of rectangular plates with rounded corners is studied by using a family of homotopy shapes and an efficient Ritz method.

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References

Leissa, A. W., 1969, “Vibration of Plates,” NASA, Washington, D.C., Report No. SP-160.
Leissa, A. W., 1973, “The Free Vibration of Rectangular Plates,” J. Sound Vib.31(8), pp. 257–293. [CrossRef]
Leissa, A. W., 2005, “The Historical Bases of Rayleigh and Ritz Methods,” J. Sound Vib.287(4–5), pp. 961–978. [CrossRef]
Gorman, D. J., 1978, “Free Vibration Analysis of the Completely Free Rectangular Plate by the Method of Superposition,” J. Sound Vib.57(3), pp. 437–447. [CrossRef]
Behnke, H., and Mertins, U., 1995, “Eigenwertschranken fur das problem der frei schwingenden rechteckigen platte und untersuchungen zum ausweichphanomen,” Z. Angew. Math. Mech.75(5), pp. 343–363. [CrossRef]
Mochida, Y., and Ilanko, S., 2008, “Bounded Natural Frequencies of Completely Free Rectangular Plates,” J. Sound Vib.311(1–2), pp. 1–8. [CrossRef]
Sato, K., 1973, “Free Flexural Vibrations of an Elliptical Plate With Free Edge,” J. Acoust. Soc. Am.54(2), pp. 547–550. [CrossRef]
Beres, D. P., 1974, “Vibration Analysis of a Completely Free Elliptical Plate,” J. Sound Vib.34(3), pp. 441–443. [CrossRef]
Singh, B., and Chakraverty, S., 1991, “Transverse Vibration of Completely Free Elliptic and Circular Plates Using Orthogonal Polynomials in the Rayleigh Ritz Method,” Int. J. Mech. Sci.33(9), pp. 741–751. [CrossRef]
Irie, T., Yamada, G., and Sonoda, M., 1983, “Natural Frequencies of Square Membrane and Square Plate With Rounded Corners,” J. Sound Vib.86(3), pp. 442–448. [CrossRef]
Wang, C. M., Wang, L., and Liew, K. M., 1994, “Vibration and Buckling of Super Elliptic Plates,” J. Sound Vib.171(3), pp. 301–314. [CrossRef]
Ceribasi, S., and Altay, G., 2009, “Free Vibration of Super Elliptical Plates With Constant and Variable Thickness by Ritz Method,” J. Sound Vib.319(1–2), pp. 668–680. [CrossRef]
Lim, C. W., and Liew, K. M., 1995, “Vibrations of Perforated Plates With Rounded Corners,” J. Eng. Mech.121(2), pp. 203–213. [CrossRef]
Timoshenko, S., and Woinowsky-Krieger, S., 1959, Theory of Plates and Shells, McGraw-Hill, New York.
Washizu, K., 1982, Variational Methods in Elasticity and Plasticity, Pergamon, Oxford, UK.

Figures

Grahic Jump Location
Fig. 1

The rounded rectangular plate. The width is 2L and the height is 2bL. From outside: α = 0, 0.01, 0.05, 0.15, 0.3, 0.6, and 1.

Grahic Jump Location
Fig. 2

First five mode shapes for b = 0.7. Lowest (fundamental) modes are on top. From left: ellipse (α = 1), rounded rectangle (α = 0.15), and rectangle (α = 0). Note the switching of modes.

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