张角为直角两半径边简支时扇形薄板二维驻波的研究

Research on two-dimensional standing waves on right-angled thin sector plate with two simply-supported radial edges

本文在小挠度理论下对张角为直角、两半径边简单支承、圆弧边悬空时水平放置的扇形薄金属板的竖向小振动问题进行了研究.通过求解薄板的小振动方程,得出了薄板在不同本征频率(自由频率)下的解析解的简正模式,求出了通解,并计算了相应本征频率下薄板上的圆弧状驻波波节线的半径及方程本征值所遵从的规律,给出了驻波图及薄金属板的弹性模量,得到的简正模式波节线的分布有3种:分别是仅有辐射状波节线、仅有圆弧状波节线(不含两半径边)及辐射状与圆弧状波节线同时存在,并与实验观察到的驻波图形(即克拉尼图形)的相应实测值进行了比较,理论与实验符合得很好.

Under small flexivity theory,the vertical small vibrations on right-angled sectorial thin plate which placed horizontally with two simply-supported radial edges and cantilevered arc edge is investigated theoretically and experimentally.The normal modes of analytical solutions on thin plate in different eigen frequencies are studied,and the radii of arc-shaped standing waves nodes on the plate are calculated.Furthermore,the laws of equation s eigen-values and the modulus of plate are obtained,and the patterns of standing waves are also observed.Three kinds of standing waves nodes on a metal board are presented:only having radical nodes,only having arc-shaped nodes(two radial edges excluded),and both having radical nodes and arc-shaped nodes.The theoretical values coincide with the experimental results.