圆型薄板的临界转速分析

ANALYSIS ON THE CRITICAL SPEED FOR A SPINNING THIN ANNULAR PLATE

对内边夹支外边自由的一类薄圆板,利用卡门大挠度理论建立了其旋转时的横向振动方程.通过化简并应用伽辽金法对该偏微分方程离散化,计算出了一个刚度算子在不同的节径数n下的最低阶(节圆数m=0)特征值及简化方程相应模态的特征频率.给出了临界转速的计算方法,并讨论了其随板的内外半径比a/b及泊松比μ的变化关系.分析结果表明,对于节径n=0和n=1均不存在临界转速,对于其它节径n,临界转速随着内外半径比的增大而增大随着泊松比的增大而减小.这对于电脑硬盘、圆锯、涡轮机等高速旋转设备的设计有重要的参考意义.

Based on the von Karman plate theory, the transverse vibration equation of a spinning thin axisymmetrie annular plate with clamped inner - boundary and free outer - boundary was formulated. The diseretization of a simplified partial differential equation was obtained by the Galerkin method. For any nodal diameter , the smallest eigenvalue （ zero nodal circle） of a stiffness operator and the eigenfrequeney of the corresponding mode with regard to the simplified equation were calculated. The calculating method for the critical speed was given, and the critical speed versus inner - to - outer radius ratio and Poisson＇ s ratio was investigated. The analysis results indicate that there is no critical speed for nodal diameter and , but for other nodal diameter , the critical speed increases with the increase of the inner - to - outer radius ratio and decreases with the increase of the Poisson＇ s ratio. These observation is helpful to the design of annular plate for high - speed applications such as computer hard disks , circular sows , and turbines.