Pasternak地基上简支板振动问题的准格林函数方法

Green Quasifunction Method for Vibration of Simply-Supported Thin Polygonic Plates on Pasternak Foundation

提出一种新的数值方法——准格林函数方法.以Pasternak地基上简支多边形薄板的振动问题为例,详细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,这个函数满足了问题的齐次边界条件,采用格林公式将Pasternak地基上薄板自由振动问题的振型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.数值方法表明,该方法具有较高的精度.

A new numerical method--Green quasifunction method is proposed. The idea of Green quasifunction method was clarified in detail by considering vibration problem of simply-supported thin polygonic plates on Pastemak foundation. A Green quasifunction was 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 equation of vibrration problem of simply-supported thin plates on Pastemak foundation was 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 was overcome. Finally, natural frequency was obtained by the condition that there exists a nontrivial solution in the numerical- ly discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the Green quasifunction method.