Instability Analysis of Rotating Convex and Concave Circular Nanoplate by Considering Surface Residual Stress
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摘要: 根据纳米材料表面残余应力和弹性板小挠度理论，建立旋转凸凹型纳米圆板横向振动微分方程，采用微分求积法得到不同条件下凸凹型纳米圆板的无量纲复频率随无量纲角速度和无量纲表面残余应力参数变化情况。研究结果显示：在周边固支和简支条件下，旋转凸凹型纳米圆板均发生第1阶发散失稳现象。在其他条件一定的情况下，周边固支凹型纳米圆板的临界失稳角速度小于凸型纳米圆板，周边简支凹型纳米圆板的临界失稳角速度大于凸型纳米圆板。临界失稳角速度随着表面残余应力的增加而增大，临界表面残余应力随着无量纲角速度的增加而增大。Abstract: Based on the theory of the surface residual stress of nanomaterials and small deflection of elastic plate, the transverse vibration differential equation of rotating convex-concave circular nanoplate was established. The variation of dimensionless complex frequency of convex-concave circular nanoplate with dimensionless angular speed and surface residual stress under different conditions was obtained by using the differential quadrature method. The results show that the first-order divergence instability of rotating convex-concave circular nanoplate was observed in both clamped and simply supported conditions. When other parameters are constant, the critical instability angular speed of the clamped concave circular nanoplate is smaller than that of the convex circular nanoplate, and the critical instability angular speed of the simply supported concave circular nanoplate is greater than that of the convex circular nanoplate. The critical instability angular speed increases with the increasing of surface residual stress, and the critical surface residual stress increases with the increasing of dimensionless angular speed.
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图 2 第1阶无量纲复频率$ \omega $与无量纲角速度$ c $的关系曲线（固支，$ g = {\text{3}} $）
Figure 2. Relationship curves for the first-order dimensionless complex frequency ω versus dimensionless angular speed c (clamped edge, g=3)
图 3 第1阶无量纲复频率$ \omega $与无量纲角速度$ c $的关系曲线（固支，m = −0.05）
Figure 3. Relationship curves for the first-order dimensionless complex frequency ω versus dimensionless angular speed c (clamped edge, m = −0.05)
图 4 第1阶无量纲复频率$ \omega $与无量纲表面残余应力$ g $的关系曲线（固支，m = −0.05）
Figure 4. Relationship curves for the first-order dimensionless complex frequency ω versus dimensionless surface residual stress g (clamped edge, m = −0.05)
图 5 第1阶无量纲复频率$ \omega $与无量纲角速度$ c $的关系曲线（简支，g = 3）
Figure 5. Relationship curves for the first-order dimensionless complex frequency ω versus dimensionless angular speed c (simply supported edge, g = 3)
图 6 第1阶无量纲复频率$ \omega $与无量纲角速度$ c $的关系曲线（简支，m = −0.05）
Figure 6. Relationship curves for the first-order dimensionless complex frequency ω versus dimensionless angular speed c (simply supported edge, m = −0.05)
图 7 第1阶无量纲复频率$ \omega $与无量纲表面残余应力$ g $的关系曲线（简支，m = −0.05）
Figure 7. Relationship curves for the first-order dimensionless complex frequency ω versus dimensionless surface residual stress g (simply supported edge, m = −0.05)
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