变厚度圆环板/圆板横向自由振动的动刚度法求解

Dynamic stiffness method for the free transverse vibration of annular/circular plates with radial varying thickness

分别基于经典薄板理论和一阶剪切理论研究了沿半径方向变厚度的圆板及圆环板的横向自由振动,将结构离散为若干个等厚度同心圆环单元,在得出圆环单元的精确解后,通过动刚度法组装单元。应用该方法将变厚度圆板退化至等厚度板,与解析解对比验证了计算方法的正确性;用于计算线性或非线性变厚度板,也能与有限元三维解吻合。计算结果表明:基于一阶剪切理论和薄板理论的动刚度法计算等厚度薄板的振动均能取得与解析解完全吻合的数值解;而计算变厚度薄板时则采用基于一阶剪切理论的动刚度法更准确;与有限元法相比,本文采用的动刚度法划分单元少,具有较高的计算效率,尤其在工程中的大型板结构振动方面有较好的应用前景。

Free transverse vibrations of circular and annular plates with radial varying thickness based on the classical thin plate theory and first shear deformation theory are investigated. The model with variable thickness splits into several concentric annular elements of even thickness. Elements are assembled with dynamic stiffness method after the derivation of exact solution in every annular element. To validate the calculation method, the degenerative results of circular plates with even thickness are compared with analytical solutions. Meanwhile, the computational results of plates with linearly/nonlinearly varying thickness coincide with those derived by finite element software. It is shown that both the classical thin plate theory and first shear deformation theory applied for the thin plates with uniform thickness present high precision. Nevertheless, the shear deformation theory is more adequate to model thin plates with variable thickness. Compared with the finite element method, fewer elements are needed but higher efficiency can be achieved by the dynamic stiffness method, which may reveal application prospect for the vibration of large panel structures in engineering.