We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary. We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection and torsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whose configuration influences the result of homogenizing the biharmonic equation by decreasing the size of the limiting system of differential equations or completely eliminating it. The boundary-layer phenomenon near the end faces of the plate is studied for various ways of fastening as well as for angular junctions of two long plates, possibly by point clamps (Sobolev conjugation conditions). We discuss full asymptotic series for solutions of static problems and the spectral problems of plate oscillations.

中文翻译：

---

具有振动边缘和点支撑的Kirchhoff板的均质化

我们研究具有周期性（快速振荡）边界的长（缩放后变窄）Kirchhoff板的变形。我们推导了一个极限系统，该系统由两个阶数为4和2的常微分方程组成，它们描述了二维板在前导阶中的挠度和扭转。我们还考虑了点支撑（Sobolev条件），其配置会通过减小微分方程极限系统的大小或完全消除它来影响双谐方程均质化的结果。研究了板端面附近的边界层现象，研究了各种固定方式以及两个长板的角接合，可能是通过点夹（Sobolev共轭条件）进行的。