Quasi-Green’s Function Method for Free Vibration of Simply-Supported Trapezoidal Shallow Spherical Shell
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摘要: 以简支梯形底扁球壳的自由振动问题为例，详细阐明了准Green函数方法的思想．即利用问题的基本解和边界方程构造一个准Green函数，此函数满足了问题的齐次边界条件，采用Green公式，将简支梯形底扁球壳自由振动问题的振形控制微分方程化为两个耦合的第二类Fredholm积分方程．边界方程有多种选择，在选定一种边界方程的基础上，可以通过建立一个新的边界方程来表示问题的边界，以克服积分核的奇异性．最后由积分方程的离散化方程组有非平凡解的条件，求得固有频率．数值结果表明，该方法具有较高的精度．Abstract: The idea of quasi Green's function method was clarified in detail by considering a free vibration problem of smiply-supported trapezoidal shallow spherical shell. A quasi-Green's function was established by using the fundamental solution and boundary equation of the problem. This function satis fies the homogeneous boundary condition of the problem. The mode shape differential equation of the free vibration problem of simply-supported trapezoidal shallow spherical shell is reduced to two smiultaneous Fredholm in tegral 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 in tegral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a non trivial solution in the numerically discrete algebraic equations derived from the in tegral equations. Numerical results show highaccuracy of the quasi-Green's function method.
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Key words:
- Green function /
- in tegral equation /
- shallow spherical shell /
- free vibration
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