中文摘要：

用分离变量法对极坐标下垂直板面方向的金属薄板的小振动方程求解,解出扇形薄板在全部边界均自由的条件下的解析解的简正模式及通解,并计算了不同频率下对应的简正振动模式下薄板上的圆弧形驻波波节线的半径和方程的本征值所满足的规律及薄板的弹性模量,给出了驻波图,与实验上观察到的仅有辐射状波节线或辐射状波节线与圆弧形波节线同时存在这两种简正模式（即克拉尼图形）相比较,发现理论结果与实验符合得很好.

英文摘要：

Using the variables- separated method under the polar coordinate system,the normal modes of analytical solution of small flexivity vibration equation on a sectorial thin plate along the vertical direction is studied experimentally and theoretically when the boundary is free.The radii of arc-shaped standing wave nodes on the thin plate are calculated exactly for different frequencies,the laws of equation＇s eigen-values and the modulus of plate are obtained.The figures of standing waves are also plotted. Two kinds of standing wave nodes are observed on the board： only radical nodes and both of arc- shaped and radical shape nodes. The theoretical results are in well agreement with the experimental measurements.