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Applied Mathematics and Mechanics (English Edition) ›› 2010, Vol. 31 ›› Issue (5): 635-642.doi: https://doi.org/10.1007/s10483-010-0511-7

• Articles • 上一篇    下一篇

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

李善倾 袁鸿   

  1. Key Laboratory of Disaster Forecast and Control in Engineering, Ministry of Education of China, Institute of Applied Mechanics, Jinan University, Guangzhou 510632, P. R. China
  • 收稿日期:2009-12-02 修回日期:2010-03-25 出版日期:2010-05-20 发布日期:2010-05-01

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

 LI Shan-Qing, YUAN Hong   

  1. Key Laboratory of Disaster Forecast and Control in Engineering, Ministry of Education of China, Institute of Applied Mechanics, Jinan University, Guangzhou 510632, P. R. China
  • Received:2009-12-02 Revised:2010-03-25 Online:2010-05-20 Published:2010-05-01

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摘要： 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 simplysupported 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.

关键词: Green function, integral equation, shallow spherical shell, free vibration

Abstract: 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 simplysupported 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

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