中国物理B ›› 2009, Vol. 18 ›› Issue (9): 3616-3623.doi: 10.1088/1674-1056/18/9/002

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The symplectic eigenfunction expansion theorem and its application to the plate bending equation

黄俊杰1, 阿拉坦仓1, 王华2   

  1. (1)School of Mathematical Sciences, Neimongol University, Hohhot 010021, China; (2)School of Mathematical Sciences, Neimongol University, Hohhot 010021, China;Department of Mathematics, College of Sciences, Neimongol University of Technology, Hohhot 010051, China
  • 收稿日期:2008-11-07 修回日期:2009-02-24 出版日期:2009-09-20 发布日期:2009-09-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No 10562002), the Natural Science Foundation of Inner Mongolia, China (Grants No 200508010103 and 200711020106), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No 20070126002).

The symplectic eigenfunction expansion theorem and its application to the plate bending equation

Huang Jun-Jie(黄俊杰)a), Alatancang(阿拉坦仓)a), and Wang Hua(王华)a)b)†   

  1. a School of Mathematical Sciences, Neimongol University, Hohhot 010021, China; b Department of Mathematics, College of Sciences, Neimongol University of Technology, Hohhot 010051, China
  • Received:2008-11-07 Revised:2009-02-24 Online:2009-09-20 Published:2009-09-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No 10562002), the Natural Science Foundation of Inner Mongolia, China (Grants No 200508010103 and 200711020106), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No 20070126002).

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摘要： This paper deals with the bending problem of rectangular plates with two opposite edges simply supported. It is proved that there exists no normed symplectic orthogonal eigenfunction system for the associated infinite-dimensional Hamiltonian operator H and that the two block operators belonging to Hamiltonian operator H possess two normed symplectic orthogonal eigenfunction systems in some space. It is demonstrated by using the properties of the block operators that the above bending problem can be solved by the symplectic eigenfunction expansion theorem, thereby obtaining analytical solutions of rectangular plates with two opposite edges simply supported and the other two edges supported in any manner.

Abstract: This paper deals with the bending problem of rectangular plates with two opposite edges simply supported. It is proved that there exists no normed symplectic orthogonal eigenfunction system for the associated infinite-dimensional Hamiltonian operator H and that the two block operators belonging to Hamiltonian operator H possess two normed symplectic orthogonal eigenfunction systems in some space. It is demonstrated by using the properties of the block operators that the above bending problem can be solved by the symplectic eigenfunction expansion theorem, thereby obtaining analytical solutions of rectangular plates with two opposite edges simply supported and the other two edges supported in any manner.

Key words: plate bending equation, symplectic eigenfunction expansion theorem, infinite dimensional Hamiltonian operator, analytical solution

中图分类号:  (Beams, plates, and shells)

  • 46.70.De
02.10.Ud (Linear algebra) 46.35.+z (Viscoelasticity, plasticity, viscoplasticity)