Applied Mathematics and Mechanics

, Volume 31, Issue 5, pp 635–642 | Cite as

Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

  • Shan-qing Li (李善倾)
  • Hong Yuan (袁 鸿)Email author


The idea of quasi-Green’s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi-Green’s function 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 equations of the free vibration problem of a simply-supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the 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 to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green’s function method.

Key words

Green function integral equation shallow spherical shell free vibration 

Chinese Library Classification


2000 Mathematics Subject Classification

34B27 74K20 


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Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Key Laboratory of Disaster Forecast and Control in Engineering, Ministry of Education of China, Institute of Applied MechanicsJinan UniversityGuangzhouP. R. China

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