Higher Frequencies of Square Plates with Symmetrically Mixed of Simply Supported-Clamped Edge Conditions
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Abstract
Since the vibration problem of plates with mixed edge conditions is of academic and technological importance, the need of knowing natural frequencies is then required. This paper attempts to deal with an accurate value for some higher frequencies expressed in terms of frequency parameters of square plates having mixed edges between the simple and clamped supports. Two plate configurations are considered. The first is of the clamped plate with varying corner simply supported lengths; while the second is the simply supported plate having an equal angle type of clamped support placed at all corners. The analysis is made by means of finite element code with a dense net of element mesh. The first twenty frequencies and their vibratory modes are presented, which could serve as a benchmark for comparison with other methods.
Article Details
Copyright @2021 Engineering Transactions
Faculty of Engineering and Technology
Mahanakorn University of Technology
References
R. Szilard, Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods, John Wiley & Sons, Inc., New Jersey, 2004.
S.P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed., McGraw-Hill, Singapore, 1959.
S.P. Timoshenko and D.H. Young, Vibration Problems in Engineering, 3rd ed., D. Van Nostrand Company, Inc., Princeton, 1955.
W. Nowacki, Dynamics of Elastic System, Chapman & Hall Ltd., London, 1963.
L. Meirovitch, Analytical Methods in Vibrations, Macmillan Publishing Co., Inc., New York, 1967.
G.B. Warburton, “The vibration of rectangular plates”, Proceedings of the Institution of Mechanical Engineers, vol. 168, pp. 371-384, 1954.
A.W. Leissa, “Free vibrations of elastic plates”, Proceedings of the AIAA 7th Aerospace Science Meeting, 20-22 January 1963, New York City, New York, AIAA Paper No. 69-24, 1963.
A.W. Leissa, Vibration of Plates, Reprinted ed., Acoustical Society of America, Washington, D.C., 1993.
A.W. Leissa, “The free vibration of rectangular plates”, Journal of Sound and Vibration, vol. 31, pp. 257-293, 1973.
D.J. Gorman, Free Vibration Analysis of Rectangular Plates, Elsevier North Holland, Inc., New York, 1982.
J.J. Tuma, Engineering Mathematics Handbook, 2nd enlarged and revised ed., McGraw-Hill, New York, 1979.
A.D. Polyanin and V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed., Chapman & Hall/CRC, Boca Raton, 2002.
T. Ota and M. Hamada, “Bending and vibration of a simply supported but partially clamped rectangular plate”, Transactions of the Japan Society of Mechanical Engineers, vol. 25, pp. 668-674, 1959 (in Japanese).
T. Ota and M. Hamada, “Fundamental frequencies of simply supported but partially clamped square plates”, Bulletin of JSME, vol. 6, pp. 397-403, 1963.
H.P. Lee and S.P. Lim, “Free vibration of isotropic and orthotropic rectangular plates with partially clamped edges”, Applied Acoustics, vol. 35, pp. 91-104, 1992.
Y. Narita, “Application of a series-type method to vibration of orthotropic rectangular plates with mixed boundary conditions”, Journal of Sound and Vibration, vol. 77, pp. 345-355, 1981.
Z.-Q. Cheng, “The application of Weinstein-Chien’s method-the upper and lower limits of fundamental frequency of rectangular plates with edges are the mixture of simply supported portions and clamped portions”, Applied Mathematics and Mechanics, vol. 5, pp. 1399-1408, 1984.
V.M. Alexsandrov and V.B. Zelentsov, “Dynamic problems on the bending of a rectangular plate with mixed fixing conditions on the outline”, Journal of Applied Mathematics and Mechanics, vol. 43, pp. 124-131, 1979.
L.M. Keer and B. Stahl, “Eigenvalue problems of rectangular plates with mixed edge conditions”, Journal of Applied Mechanics, vol. 39, pp. 513-520, 1972.
T. Sakiyama and H. Matsuda, “Free vibration of rectangular Mindlin plate with mixed boundary conditions”, Journal of Sound and Vibration, vol. 113, pp. 208-214, 1987.
S. Chaiyat and Y. Sompornjaroensuk, “Integral equation for symmetrical free vibration of Levy-plate having discontinuous simple supports”, Procedia Engineering, vol. 14, pp. 2933-2930, 2011.
D.J. Gorman, “An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions”, Journal of Sound and Vibration, vol. 93, pp. 235-247, 1984.
R.K. Singhal and D.J. Gorman, “Free vibration of partially clamped rectangular plates with and without rigid point supports”, Journal of Sound and Vibration, vol. 203, pp. 181-192, 1997.
K.M. Liew, K.C. Hung and K.Y. Lam, “On the use of the substructure method for vibration analysis of rectangular plates with discontinuous boundary conditions”, Journal of Sound and Vibration, vol. 163, pp. 451-462, 1993.
T.-P. Chang and M.-H. Wu, “On the use of characteristic orthogonal polynomials in the free vibration analysis of rectangular anisotropic plates with mixed boundaries and concentrated masses”, Computers and Structures, vol. 62, pp. 699-713, 1997.
G.H. Su and Y. Xiang, “A non-discrete approach for analysis of plates with multiple subdomains”, Engineering Structures, vol. 24, pp. 563-575, 2002.
D. Zhou, Y.K. Cheung, S.H. Lo and F.T.K. Au, “Three-dimensional vibration analysis of rectangular plates with mixed boundary conditions”, Journal of Applied Mechanics, vol. 72, pp. 227-236, 2005.
G. Venkateswara Rao, I.S. Raju and T.V.G.K. Murthy, “Vibration of rectangular plates with mixed boundary conditions”, Journal of Sound and Vibration, vol. 30, pp. 257-260, 1973.
A.Y.T. Leung and F.T.K. Au, “Spline finite elements for beam and plate”, Computers and Structures, vol. 37, pp. 717-729, 1990.
Y.K. Cheung and J. Kong, “The application of a new finite strip to the free vibration of rectangular plates of varying complexity”, Journal of Sound and Vibration, vol. 181, pp. 341-353, 1995.
M.-H. Huang and D.P. Thambiratnam, “Free vibration analysis of rectangular plates on elastic intermediate supports”, Journal of Sound and Vibration, vol. 240, pp. 567-580, 2001.
P.A.A. Laura and R.H. Gutierrez, “Analysis of vibrating rectangular plates with non-uniform boundary conditions by using the differential quadrature method”, Journal of Sound and Vibration, vol. 173, pp. 702-706, 1994.
C. Shu and C.M. Wang, “Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates”, Engineering Structures, vol. 21, pp. 125-134, 1999.
A.G. Striz, W.L. Chen and C.W. Bert, “Free vibration of plates by the high accuracy quadrature element method”, Journal of Sound and Vibration, vol. 202, pp. 689-702, 1997.
F.-L. Liu and K.M. Liew, “Analysis of vibrating thick rectangular plates with mixed boundary constraints using differential quadrature element method”, Journal of Sound and Vibration, vol. 225, pp. 915-934, 1999.
G.W. Wei, Y.B. Zhao and Y. Xiang, “The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution”, International Journal of Mechanical Sciences, vol. 43, pp. 1731-1746, 2001.
C.H.W. Ng, Y.B. Zhao and G.W. Wei, “Comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates”, Computer Methods in Applied Mechanics and Engineering, vol. 193, pp. 2483-2506, 2004.
Y. Xiang, S.K. Lai, L. Zhou and C.W. Lim, “DSC-Ritz element method for vibration analysis of rectangular Mindlin plates with mixed edge supports”, European Journal of Mechanics A/Solids, vol. 29, pp. 619-628, 2010.
H. Zhang, Y.P. Zhang and C.M. Wang, “Hencky bar-net model for vibration of rectangular plates with mixed boundary conditions and point supports”, International Journal of Structural Stability and Dynamics, vol. 18, 1850046, 2018.
T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc., New Jersey, 1987.
K.J. Bathe, Finite Element Procedures, Prentice-Hall, Inc., New Jersey, 1996.
ANSYS, Inc., ANSYS Mechanical APDL Theory Reference, Release 14.5, 2012.
ANSYS, Inc., ANSYS Mechanical APDL Element Reference, Release 14.5, 2012.
N. Apaipong and Y. Sompornjaroensuk, “On the free-vibration frequencies of square plates with different edge conditions”, Engineering Transactions, vol. 18, pp. 21-38, 2015.
Y. Sompornajroensuk, “Vibration of circular plates with mixed edge conditions. Part II: Numerical determination for higher frequencies”, UTK Research Journal, vol. 14, 2020. (Accepted for publication).
C.S. Huang and C.W. Chan, “Vibration analyses of cracked plates by the Ritz method with moving least-squares interpolation functions”, International Journal of Structural Stability and Dynamics, vol. 14, 1350060, 2014.
H.C. Zeng, C.S. Huang, A.W. Leissa and M.J. Chang, “Vibrations and stability of a loaded side-cracked rectangular plate via the MLS-Ritz method”, Thin-Walled Structures, vol. 106, pp. 459-470, 2016.
C.S. Huang and Y.J. Lin, “Fourier series solutions for vibrations of a rectangular plate with a straight through crack”, Applied Mathematical Modelling, vol. 40, pp. 10389-10403, 2016.
C.S. Huang, M.C. Lee and M.J. Chang, “Vibration and buckling analysis of internally cracked square plates by the MLS-Ritz approach”, International Journal of Structural Stability and Dynamics, vol. 18, 1850105, 2018.
C.S. Huang, H.T. Lee, P.Y. Li, K.C. Hu, C.W. Lan and M.J. Chang, “Three-dimensional buckling analyses of cracked functionally graded material plates via the MLS-Ritz method”, Thin-Walled Structures, vol. 134, pp. 189-202, 2019.
L. Sun and X. Wei, “A frequency domain formulation of the singular boundary method for dynamic analysis of thin elastic plate”, Engineering Analysis with Boundary Elements, vol. 98, pp. 77-87, 2019.
M.R. Ayatollahi, M. Nejati, and S. Ghouli, “The finite element over-deterministic method to calculate the coefficients of crack tip asymptotic fields in anisotropic planes”, Engineering Fracture Mechanics, vol. 231, 106982, 2020.