0
Research Papers

Frequencies of Circular Plates Weakened Along an Internal Concentric Circle and Elastically Restrained Edge Against Translation

[+] Author and Article Information
Lokavarapu Bhaskara Rao

School of Mechanical and Building Sciences,
VIT University,
Chennai Campus,
Vandalur-Kelambakkam Road,
Chennai-600048, India
e-mail: bhaskarbabu_20@yahoo.com

Chellapilla Kameswara Rao

Department of Mechanical Engineering,
TKR College of Engineering and Technology,
Medbowli,
Meerpet, Saroornagar,
Hyderabad-500079, India
e-mail: chellapilla95@gmail.com

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 11, 2011; final manuscript received May 7, 2012; accepted manuscript posted June 6, 2012; published online October 29, 2012. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 80(1), 011005 (Oct 29, 2012) (7 pages) Paper No: JAM-11-1045; doi: 10.1115/1.4006938 History: Received February 11, 2011; Revised May 07, 2012; Accepted June 06, 2012

The present study deals with the derivation of an exact solution for the problem of obtaining the natural frequencies of the vibration of circular plates weakened along an internal concentric circle due to the presence of a radial crack and elastically restrained along the outer edge of the plate against translation. The frequencies of the circular plates are computed for varying values of the elastic translational restraint, the radius of the radial crack, and the extent of the weakening duly simulated by considering the radial crack as a radial elastic rotational restraint on the plate. The results for the first six modes of the plate vibrations are computed. The effects of the elastic edge restraint, the radius of the weakened circle, and the extent of the weakening represented by an elastic rotational restraint on the vibration behavior of thin circular plates are studied in detail. The internal weakening due to a crack resulted in decreasing the fundamental frequency of the plate. The exact method of solution and the results presented in this paper are expected to be of specific use in analyzing the effect of a radial crack on the fundamental natural frequency of the circular plate in the presence of a translational restraint existing along the outer edge of the plate. These exact solutions can be used to check the numerical or approximate results.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Timoshenko, S., and Woinowsky-Krieger, S., 1959, Theory of Plates and Shells, McGraw-Hill, New York.
Leissa, A. W., 1969, Vibration of Plates, NASA SP-160, Office of Technology Utilization, Washington, D.C.
Szilard, R., 1974, Theory and Analysis of Plates, Prentice-Hall, New Jersey.
Azimi, S., 1988, “Free Vibrations of Circular Plates With Elastic or Rigid Interior Support,” J. Sound Vib., 120, pp. 37–52. [CrossRef]
Weisensel, G. N., 1989, “Natural Frequency Information for Circular and Annular Plates,” J. Sound .Vib., 133(1), pp. 129–137. [CrossRef]
Ding, Z., 1994, “Free Vibration of Arbitrarily Shaped Plates With Concentric Ring Elastic and/or Rigid Supports,” Comput. Struct, 50(5), pp. 685–692. [CrossRef]
Wang, C.Y., andWang, C. M., 2003, “Fundamental Frequencies of Circular Plates With Internal Elastic Ring Support,” J. Sound Vib., 263(5), pp. 1071–1078. [CrossRef]
Dimarogonas, A. D., 1996, “Vibration of Cracked Structures: A State of the Art Review,” Eng. Fract. Mech., 55(5), pp. 831–857. [CrossRef]
Lynn, P.P., andKumbasar, N., 1967, “Free Vibrations of Thin Rectangular Plates Having Narrow Cracks With Simply Supported Edges,” Dev. Mech., 4, pp. 911–928.
Petyt, M., 1968, “The Vibration Characteristics of a Tensioned Plate Containing a Fatigue Crack,” J. Sound Vib., 8(3), pp. 377–389. [CrossRef]
Stahl, B., and Keer, L. M., 1972, “Vibration and Stability of Cracked Rectangular Plates,” Int. J. Solids Struct., 8(1), pp. 69–91. [CrossRef]
Hirano, Y., and Okazaki, K., 1980, “Vibration of Cracked Rectangular-Plates,” Bull. JSME, 23, pp. 732–740. [CrossRef]
Solecki, R., 1983, “Bending Vibration of a Simply Supported Rectangular Plate With a Crack Parallel to One Edge,” Eng. Fract. Mech., 18(6), pp. 1111–1118. [CrossRef]
Yuan, J., and Dickinson, S. M., 1992, “The Flexural Vibration of Rectangular Plate Systems Approached by Using Artificial Springs in the Rayleigh-Ritz Method,” J. Sound Vib., 159(1), pp. 39–55. [CrossRef]
Liew, K. M., Hung, K. C., and Lim, M. K., 1994, “A Solution Method for Analysis of Cracked Plates Under Vibration,” Eng. Fract. Mech., 48(3), pp. 393–404. [CrossRef]
Ma, C.C., andHuang, C. H., 2001, “Experimental and Numerical Analysis of Vibrating Cracked Plates at Resonant Frequencies,” Exp. Mech., 41(1), pp. 8–18. [CrossRef]
Qian, G. L., Gu, S. N., and Jiang, J. S., 1991, “A Finite Element Model of Cracked Plates and Applications to Vibration Problems,” Comput. Struct., 39, pp. 483–487. [CrossRef]
Lee, H. P., and Lim, S. P., “Vibration of Cracked Rectangular Plates Including Transverse Shear Deformation and Rotary Inertia,” Comput. Struct., 49, pp. 715–718. [CrossRef]
Krawczuk, M., 1993, “Natural Vibrations of Rectangular Plates With a Through Crack,” Arch. Appl. Mech., 63, pp. 491–504. [CrossRef]
Liew, K. M., Hung, K. C., and Lim, M. K., 1994, “A Solution Method for Analysis of Cracked Plates Under Vibration,” Eng. Fract. Mech., 48, pp. 393–404. [CrossRef]
Khadem, S.E., andRezaee, M., 2000, “An Analytical Approach for Obtaining the Location and Depth of an All-Over Part-Through Crack on Externally In-Plane Loaded Rectangular Plate Using Vibration Analysis,” J. Sound Vib., 230, pp. 291–308. [CrossRef]
Krawczuk, M., Zak, A., and Ostachowicz, W., 2001, “Finite Element Model of Plate With Elasto-Plastic Through Crack,” Comput. Struct., 79, pp. 519–532. [CrossRef]
Lee, P., 1992, “Fundamental Frequencies of Annular Plates With Internal Cracks,” Comput. Struct., 43, pp. 1085–1089. [CrossRef]
Ramesh, K., Chauhan, D. P. S., and Mallik, A. K., 1997, “Free Vibration of an Annular Plate With Periodic Radial Cracks,” J. Sound Vib., 206, pp. 266–274. [CrossRef]
Anifantis, N. K., Actis, R. L., and Dimarogonas, A. D., 1994, “Vibration of Cracked Annular Plates,” Eng. Fract. Mech., 49, pp. 371–379. [CrossRef]
Yuan, J., Young, P. G., and Dickinson, S. M., 1994, “Natural Frequencies of Circular and Annular Plates With Radial or Circumferential Cracks,” Comput. Struct., 53, pp. 327–334. [CrossRef]
Wang, C. Y., 2002, “Fundamental Frequency of a Circular Plate Weakened Along a Concentric Circle,” Z. Angew. Math. Mech., 82(1), pp. 70–72. [CrossRef]
Yu, L. H., 2009, “Frequencies of Circular Plate Weakened Along an Internal Concentric Circle,” Int. J. Struct. Stab. Dyn., 9(1), pp. 179–185. [CrossRef]
Kim, C.S., andDickinson, S. M., 1990, “The Flexural Vibration of the Isotropic and Polar Orthotropic Annular and Circular Plates With Elastically Restrained Peripheries,” J. Sound Vib., 143(1), pp. 171–179. [CrossRef]
Wang, C.Y., andWang, C. M., 2001, “Buckling of Circular Plates With an Internal Ring Support and Elastically Restrained Edges,” Thin-Walled Struct., 39(5), pp. 821–825. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Circular plate with elastically restrained outer edge against translation and weakened along an internal concentric circle

Grahic Jump Location
Fig. 2

The fundamental frequency parameter k versus the internal concentric weakened radius parameter b for various values of R22 and T11=10 [ν=0.3 and n=0]

Grahic Jump Location
Fig. 3

The fundamental frequency parameter k versus the internal concentric weakened radius parameter b for various values of T11 and R22=10 [ν=0.3 and n=0]

Grahic Jump Location
Fig. 4

The fundamental frequency parameter k versus the internal concentric weakened radius parameter b for various values of T11 and R22 [ν=0.3 and n=3]

Grahic Jump Location
Fig. 5

The fundamental frequency parameter k versus the internal concentric weakened radius parameter b for various values of T11 and R22 [ν=0.3 and n=4]

Grahic Jump Location
Fig. 6

The fundamental frequency parameter k versus the internal concentric weakened radius parameter b for various values of T11 and R22 [ν=0.3 and n=5]

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In