Applied Mathematics and Mechanics

, Volume 28, Issue 7, pp 847–853 | Cite as

Green quasifunction method for vibration of simply-supported thin polygonic plates on Pasternak foundation

  • Yuan Hong  (袁鸿)Email author
  • Li Shan-qing  (李善倾)
  • Liu Ren-huai  (刘人怀)


A new numerical method—Green quasifunction is proposed. The idea of Green quasifunction method is clarified in detail by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation. A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The mode shape differential equation of the vibration problem of simply-supported thin plates on Pasternak foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equation, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the Green quasifunction method.

Key words

Green function integral equation vibration of thin plates Pasternak foundation 

Chinese Library Classification

O241.8 TU471.2 

2000 Mathematics Subject Classification

34B27 74K20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Winker E. Die Lehre von der elastigit at und festigkeit[M]. Dominicus Prague, 1867.Google Scholar
  2. [2]
    Selvadurai A D S. Elastic analysis of soil foundation interaction[M]. London: Elsevier Scientific Publishing Co, 1979.Google Scholar
  3. [3]
    Rvachev V L. Theory of R-function and some of its application[M]. Kiev: Nauk Dumka, 1982, 415–421 (in Russian).Google Scholar
  4. [4]
    Yuan Hong. Green quasifunction method for thin plates on Winkler foundation[J]. Chinese Journal Computational Mechanics, 1999, 16(4):478–482 (in Chinese).Google Scholar
  5. [5]
    Lu Wei, Yuan Hong. Green quasifunction method for simply-supported parallelogrammic thin plates on Winkler foundation[J]. Journal of Jinan University (Natural Science Edition), 2006, 27(1):81–86 (in Chinese).Google Scholar
  6. [6]
    Wang Hong, Yuan Hong. Application of Green quasifunction method in elastic torsion[J]. Journal of South China University of Technology (Nature Science Edition), 2004, 32(11):86–88 (in Chinese).Google Scholar
  7. [7]
    Wang Hong, Yuan Hong. Application of R-function theory to the problem of elastic torsion with trapezium sections[J]. Journal of Huazhong University of Science & Technology (Nature Science Edition), 2005, 33(11):99–101 (in Chinese).Google Scholar
  8. [8]
    Zheng Jianjun, Fan Chengmou. The theorem of meanvalue of vibration for plates on foundation of two parameters[J]. Shanghai Journal of Mechanics, 1995, 16(2):166–171 (in Chinese).Google Scholar
  9. [9]
    Ortner V N. Regularisierte faltung von distributionen. Teil 2: Eine tabelle von fundamentallocunngen[J]. ZAMP, 1980, 31:155–173.CrossRefGoogle Scholar
  10. [10]
    Kurpa L V. Solution of the problem of deflection and vibration of plates by the R-function method[J]. Sov Appl Mech, 1984, 20(5):470–473.zbMATHCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. 2007

Authors and Affiliations

  • Yuan Hong  (袁鸿)
    • 1
    Email author
  • Li Shan-qing  (李善倾)
    • 1
  • Liu Ren-huai  (刘人怀)
    • 1
  1. 1.Institute of Applied MechanicsJinan UniversityGuangzhouP. R. China

Personalised recommendations