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Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation

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Abstract

The quasi-Green’s function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green’s function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of the problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green’s formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method.

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References

  1. Zeng, X. Y., Zhu, A. J., and Deng, A. F. Application of natural element method to solution of elastic thin plate bending on Winkler soil foundation (in Chinese). Chinese Journal of Computational Mechanics, 25(4), 547–551 (2008)

    Google Scholar 

  2. Ma, L. H., Qiu, Z. P., Wang, X. J., and Zhang, J. H. Interval element-free Galerkin method for plates on Winkler foundation (in Chinese). Chinese Journal of Geotechnical Engineering, 30(3), 384–389 (2008)

    Google Scholar 

  3. Zhong, Y., Zhou, F. L., and Zhang, Y. S. Vibration analysis of a thin plate on Winkler foundation with completely free boundary by Kantorovich method (in Chinese). Journal of Vibration and Shock, 26(3), 33–36 (2007)

    Google Scholar 

  4. Xiong, Y. B., Long, S. Y., and Li, G. Y. A local Petrov-Galerkin method for analysis of plates on elastic foundation (in Chinese). China Civil Engineering Journal, 38(11), 79–83 (2005)

    Google Scholar 

  5. Zhang, W. X. and Pang, H. The element-free method for the bending problem of plates on elastic foundation (in Chinese). Engineering Mechanics, 17(3), 138–144 (2000)

    MathSciNet  Google Scholar 

  6. Li, N. and Wu, P. D. Elastic plate resting on Winkler foundation by finite difference method (in Chinese). Journal of PLA University of Science and Technology, 5(5), 64–66 (2004)

    MATH  Google Scholar 

  7. Wang, Y. H., Qin, X. M., and Cheung, Y. K. Plates on an elastic foundation calculated by isoparametric element methods (in Chinese). Chinese Journal of Geotechnical Engineering, 20(4), 7–11 (1998)

    Google Scholar 

  8. She, Y. H. and Zhu, W. N. A boundary element solution for the bending problem of the thin plates on elastic foundation (in Chinese). Computational Structural Mechanics and Applications, 8(3), 267–274 (1991)

    Google Scholar 

  9. Zhao, L. Analysis of free vibration problems of plates on elastic foundation by the spline function (in Chinese). Journal of Southwest Jiaotong University, 91(3), 99–104 (1993)

    Google Scholar 

  10. Qin, R. and He, C. R. Static and dynamic analysis of thin plates on elastic foundation (in Chinese). China Civil Engineering Journal, 21(3), 71–80 (1988)

    Google Scholar 

  11. Wang, Y. C. and Zhang, W. X. Spline boundary element method for plates on Winkler foundation (in Chinese). Journal of Hefei Polytechnic University, 4, 15–28 (1984)

    MATH  Google Scholar 

  12. Wen, P. H. Point intensity method of solving circular plate resting on elastic foundation (in Chinese). Engineering Mechanics, 4(2), 18–26 (1987)

    Google Scholar 

  13. Huang, Y. A general analytical solution for elastic vibration of rectangular thin plate. Applied Mathematics andMechanics (English Edition), 9(11), 1057–1065 (1988) DOI 10.1007/BF02454509

    Article  Google Scholar 

  14. Selvadurai, A. P. S. Elastic Analysis of Sail-Foundation Interaction, Elsevier, Amsterdam (1979)

    Google Scholar 

  15. Katsikadelis, J. T. and Kallivokas, L. F. Plates on biparametric elastic foundation by BDIE method. Journal for Engineering Mechanics, 114(5), 847–875 (1988)

    Article  Google Scholar 

  16. Xia, S. Q. and Feng, Z. N. A boundary element method for the stability and vibration problems of thin plates (in Chinese). Shanghai Journal of Mechanics, 13(1), 62–67 (1992)

    Google Scholar 

  17. Jari, P. and Pentti, V. Boundary element analysis of a plate on elastic foundation. International Journal for Numerical Methods in Engineering, 23(2), 287–305 (1986)

    Article  MATH  Google Scholar 

  18. Rvachev, V. L. Theory of R-function and Some of Its Application (in Russian), Naukova Dumka, Kiev, 415–421 (1982)

    Google Scholar 

  19. Yuan, H. Quasi-Green’s function method for thin plates on Winkler foundation (in Chinese). Chinese Journal of Computational Mechanics, 16(4), 478–482 (1999)

    Google Scholar 

  20. Wang, H. and Yuan, H. Application of quasi-Green’s function method in elastic torsion (in Chinese). Journal of South China University of Technology (Natural Science Edition), 32(11), 86–88 (2004)

    Google Scholar 

  21. Wang, H. and Yuan, H. Application of R-function theory to the problem of elastic torsion with trapezium sections (in Chinese). Journal of Huazhong University of Science and Technology (Natural Science Edition), 33(11), 99–101 (2005)

    Google Scholar 

  22. Yuan, H., Li, S. Q., and Liu, R. H. Quasi-Green’s function method for vibration of simplysupported thin polygonic plates on Pasternak foundation. Applied Mathematics and Mechanics (English Edition), 28(7), 847–853 (2007) DOI 10.1007/s10483-007-0701-y

    Article  MATH  Google Scholar 

  23. Li, S. Q. and Yuan, H. Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell. Applied Mathematics and Mechanics (English Edition), 31(5), 635–642 (2010) DOI 10.1007/s10483-010-0511-7

    Article  MathSciNet  MATH  Google Scholar 

  24. Cao, Z. Y. Vibration Theory of Plates and Shells (in Chinese), China Railway Press, Beijing (1989)

    Google Scholar 

  25. Ortner, V. N. Regularisierte faltung von distributionen, teil 2: eine tabelle von fundamentall ösungen. Zeitschrift für Angewandte Mathematik und Physik, 31(1), 155–173 (1980)

    Article  MathSciNet  Google Scholar 

  26. Kurpa, L. V. Solution of the problem of deflection and vibration of plates by the R-function method. International Applied Mechanics, 20(5), 470–473 (1984)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mechanics, Key Laboratory of Disaster Forecast and Control in Engineering of Ministry of Education of China, Jinan University, Guangzhou, 510632, P. R. China

    Shan-qing Li  (李善倾) & Hong Yuan  (袁 鸿)

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  1. Shan-qing Li  (李善倾)
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  2. Hong Yuan  (袁 鸿)
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Correspondence to Hong Yuan  (袁 鸿).

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Communicated by Ren-huai LIU

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Li, Sq., Yuan, H. Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation. Appl. Math. Mech.-Engl. Ed. 32, 265–276 (2011). https://doi.org/10.1007/s10483-011-1412-x

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  • DOI: https://doi.org/10.1007/s10483-011-1412-x

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Chinese Library Classification

2010 Mathematics Subject Classification

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