Natural frequencies of an inhomogeneous square thin plate
DOI:
https://doi.org/10.21638/spbu01.2021.202Abstract
Plates, which geometric and physical parameters slightly differ from constant and depend only on the radial coordinate, are analyzed. For free vibration frequencies of a plate, which thickness and/or Young’s modulus depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. As examples, free vibrations of a square plate with parameters linearly or parabolically depend on the radial coordinate, are examined. The double frequencies of square plates with similar edge support of all edges are of special interest, since any variation of the thickness or stiffness causes some loss of symmetry one may expect the split of double frequencies. The asymptotic formulas permit to determine, which of two equal unperturbed frequencies corresponding to wave numbers n and m increases faster with the small parameter. For a wide range of small parameter values, the asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4.Keywords:
free vibrations of plates, inhomogeneous circular plate, perturbation method
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References
Литература
1. Leissa A.W. Vibration of plates. Washington, US Government Printing Office (1969).
2. Roshan L., Yajuvindra K. Transverse Vibrations of Nonhomogeneous Rectangular Plates with Variable Thickness. Mechanics of Advanced Materials and Structures 20 (4), 264–275 (2013).
3. Grigorenko A.Ya., Tregubenko T.V. Numerical and experimental analysis of natural vibrations of rectangular plates with variable thickness. International Applied Mechanics 36 (2), 268–270 (2000).
4. Guti´errez R.H., Laura P.A.A., Grossi R.O. Vibrations of rectangular plates of bilinearly varying thickness and with general boundary conditions. Journal of Sound and Vibration 75 (3), 323–328 (1981).
5. Dawe D. J. Vibration of Rectangular Plates of Variable Thickness. Journal of Mechanical Engineering Science 8 (1), 42–51 (1966).
6. Bhat R.B., Laura P.A.A., Gutierrez R.G., Cortinez V.H., Sanzi H.C. Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness. Journal of Sound and Vibration 138 (2), 205–219 (1990).
7. Singha B., Saxena V. Transverse vibration of a rectangular plate with bidirectional thickness variation. Journal of Sound and Vibration 198 (1), 51–65 (1996).
8. Long-Yuan L. Vibration analysis of moderate-thick plates with slowly varying thickness. Applied Mathematics and Mechanics 7, 707–714 (1986).
9. Huang M., Xu Yu, Free B.C. Vibration Analysis of Cantilever Rectangular Plates with Variable Thickness. Applied Mechanics and Materials 130–134, 2774–2777 (2011). https://doi.org/10.4028/www.scientific.net/AMM.130-134.2774
10. Olson M.D., Hazell C.R. Vibrations of a square plate with parabolically varying thickness. Journal of Sound and Vibration 62 (3), 399–410 (1979).
11. Bauer S.M., Filippov S.B., Smirnov A. L., Tovstik P. E., Vaillancourt R. Asymptotic methods in mechanics of solids. Basel, Birkh¨auser (2015).
12. Vasiliev G. P., Smirnov A.L. Free Vibration Frequencies of a Circular Thin Plate with Variable Parameters. Vestnik St. Petersb. Univ., Math. 53 (3), 351–357 (2020).
References
1. Leissa A.W. Vibration of plates. Washington, US Government Printing Office (1969).
2. Roshan L., Yajuvindra K. Transverse Vibrations of Nonhomogeneous Rectangular Plates with Variable Thickness. Mechanics of Advanced Materials and Structures 20 (4), 264–275 (2013).
3. Grigorenko A.Ya., Tregubenko T.V. Numerical and experimental analysis of natural vibrations of rectangular plates with variable thickness. International Applied Mechanics 36 (2), 268–270 (2000).
4. Guti´errez R.H., Laura P.A.A., Grossi R.O. Vibrations of rectangular plates of bilinearly varying thickness and with general boundary conditions. Journal of Sound and Vibration 75 (3), 323–328 (1981).
5. Dawe D. J. Vibration of Rectangular Plates of Variable Thickness. Journal of Mechanical Engineering Science 8 (1), 42–51 (1966).
6. Bhat R.B., Laura P.A.A., Gutierrez R.G., Cortinez V.H., Sanzi H.C. Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness. Journal of Sound and Vibration 138 (2), 205–219 (1990).
7. Singha B., Saxena V. Transverse vibration of a rectangular plate with bidirectional thickness variation. Journal of Sound and Vibration 198 (1), 51–65 (1996).
8. Long-Yuan L. Vibration analysis of moderate-thick plates with slowly varying thickness. Applied Mathematics and Mechanics 7, 707–714 (1986).
9. Huang M., Xu Yu, Free B.C. Vibration Analysis of Cantilever Rectangular Plates with Variable Thickness. Applied Mechanics and Materials 130–134, 2774–2777 (2011). https://doi.org/10.4028/www.scientific.net/AMM.130-134.2774
10. Olson M.D., Hazell C.R. Vibrations of a square plate with parabolically varying thickness. Journal of Sound and Vibration 62 (3), 399–410 (1979).
11. Bauer S.M., Filippov S.B., Smirnov A. L., Tovstik P. E., Vaillancourt R. Asymptotic methods in mechanics of solids. Basel, Birkh¨auser (2015).
12. Vasiliev G. P., Smirnov A.L. Free Vibration Frequencies of a Circular Thin Plate with Variable Parameters. Vestnik St. Petersb. Univ., Math. 53 (3), 351–357 (2020).
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Published
2021-07-21
How to Cite
Smirnov, A. L., & Vasiliev, G. P. (2021). Natural frequencies of an inhomogeneous square thin plate. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(2), 212–219. https://doi.org/10.21638/spbu01.2021.202
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Section
In memoriam of P. E. Tovstik
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
