准格林函数方法在Winkler地基上简支多边形薄板振动问题中的应用

Application of Green quasifunction method in vibration of simply-supported thin polygonic plates on Winkler foundation

提出了一种新的数值方法——准格林函数方法.以Winkler地基上简支多边形薄板振动问题为例,阐明了准格林函数方法的思想.即利用问题的基本解和边界方程构造一个准格林函数,该函数满足问题的齐次边界条件,采用格林公式将Winkler地基上薄板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.数值算例表明,该方法具有较高的精度.

A new numerical method-Green quasifunction method is proposed. The idea of this method is clarified by taking vibration problem of simply-supported thin polygonic plate on Winkler foundation. A Green quasifunction is established by using the fundamental solution and it satisfies the homogeneous boundary condition of the problem. The mode shape differential equation of vibration problem of simplysupported thin plate 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 existence condition of nontrivial solution of the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the method.