Vibration of Plates with Constrained V-Notches or Cracks
Publication: Journal of Engineering Mechanics
Volume 129, Issue 7
Abstract
This paper reports the first known free vibration solutions for thin circular plates with V-notches having various edge conditions. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include: (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained; and (2) corner functions which account for the bending moment singularities at the sharp re-entrant corner of the V-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondimensional frequencies for circular plates having a free circumferential edge and various combinations edge conditions of the V-notch. Accurate (five significant figures) frequencies are presented for clamped-free, clamped-hinged, and hinged-free notches for the spectra of notch angles (1°,5°,10°,30°,60°,90°), causing a re-entrant vertex corner of the radial edges. For very small notch angles, a clamped-free, clamped-hinged, or hinged-free radial crack ensues. One general observation is that, for the range of notch angles considered, there is a substantial increase in the first six frequencies as the notch depth increases. The frequency increase with increasing notch depth is more pronounced in the higher modes than the lower ones, and is quite substantial for segmental plates with notch angles equal to 180°. A large reduction in frequencies is also observed as the notch angle decreases at a constant notch depth. A new database of accurate frequencies and mode shapes for sectorial, semicircular and segmental plates is presented with which future solutions drawn from alternative numerical procedures and finite element and boundary element techniques may be compared. Normalized contours of the transverse vibratory displacement are shown for plates with various notch depths and notch angles of 5°, 30°, 60°, 90°, and 180°.
Get full access to this article
View all available purchase options and get full access to this article.
References
Eastep, F. E., and Hemming, F. G.(1978). “Estimation of fundamental frequency of noncircular plates with free, circular cutouts.” J. Sound Vib., 56, 155–165.
Huang, C. S. (1991). “Singularities in plate vibration problems.” PhD dissertation, Ohio State Univ., Columbus, Ohio.
Khurasia, H. B., and Rawtani, S.(1978). “Vibration analysis of circular plates with eccentric hole.” ASME J. Appl. Mech., 45(1), 215–217.
Leissa, A. W. (1969). Vibration of Plates, NASA SP-160. Washington, D.C., U.S. Government Printing Office (Reprinted by The Acoustical Society of America, 1993).
Leissa, A. W.(1977). “Recent research in plate vibrations: Classical theory.” The shock and vibration digest, Naval Research Laboratory Washington, D.C., 9(10), 13–24.
Leissa, A. W.(1981). “Plate vibration research, 1976–1980: Classical theory.” The shock and vibration digest, Naval Research Laboratory Washington, D.C., 13(9), 11–22.
Leissa, A. W.(1987). “Recent studies in plate vibrations: 1981–1985, part I, classical theory.” The shock and vibration digest, Naval Research Laboratory Washington, D.C., 19(2), 11–18.
Leissa, A. W., McGee, O. G., and Huang, C. S.(1993). “Vibrations of circular plates having V-notches or sharp radial cracks.” J. Sound Vib., 161(2), 227–239.
McGee, O. G., Leissa, A. W., Huang, C. S., and Kim, J. W.(1995). “Vibrations of circular plates with clamped V-notches or rigidly constrained radial cracks.” J. Sound Vib., 181(2), 185–201.
Nagaya, K.(1977). “Transverse vibration of a plate having an eccentric inner boundary.” ASME J. Appl. Mech., 44(1), 165–166.
Nagaya, K.(1979). “Vibration of a viscoelastic plate having a circular outer boundary and an eccentric inner boundary for various edge conditions.” J. Sound Vib., 63, 73–85.
Nagaya, K.(1981). “Simplified method for solving problems of vibrating plates of doubly-connected arbitrary shape. I: Derivation of the frequency equation. II: Applications and experiments.” J. Sound Vib., 74, 543–551, 553–564.
Nagaya, K.(1983). “Direct method for the determination of eigenfrequencies of arbitrarily-shaped plates.” J. Vibration, Acoustics, Stress and Reliability in Design,105(1), 132–136.
Stewart, G. W. (1970). Introduction to matrix computation, Academic Press, New York.
Stoer, J., and Bulirsch, R. (1980). Introduction to numerical analysis, Springer-Verlag, New York, Article 6.7.
Williams, M. L. (1951). “Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending.” Proc., 1st U.S. National Congress of Applied Mechanics, Chicago, 325–329.
Information & Authors
Information
Published In
Copyright
Copyright © 2003 American Society of Civil Engineers.
History
Received: Aug 13, 1998
Accepted: Dec 6, 2002
Published online: Jun 13, 2003
Published in print: Jul 2003
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.