In the current investigation, a novel mathematical formulation is established for the size-dependent thin doubly curved microshells of revolution with arbitrary meridian curve based on the modified couple stress theory and the Kirchhoff-Love's hypothesis. The impact of size-dependency on both the governing equations and reduced boundary conditions is taken into account. Afterwards, based on the weak formulation, a size-dependent four-node quadrilateral doubly curved microshell element with nine degrees of freedom at each node is developed. The new microshell element is also capable of capturing the mechanical behavior of microshells of revolution based on the classical theory. In the finite element analysis, the role of size-dependency in the free vibration characteristics of the paraboloidal, elliptical, and hyperbolical microshells of revolution under different boundary conditions and geometrical parameters are examined. The mode shifting and the configuration change of mode shapes due to the size-dependency are also analyzed. Eventually, the comparative, convergence and stability studies are conducted to demonstrate the accuracy and reliability of numerical results presented in this study.

在当前的研究中，基于改进的耦合应力理论和基尔霍夫-洛夫（Kirchhoff-Love）的假设，为具有任意子午线曲线的尺寸相关的薄的双曲线旋转微壳建立了新的数学公式。考虑了大小相关性对控制方程和简化边界条件的影响。然后，基于弱公式，开发了在每个节点具有九个自由度的尺寸相关的四节点四边形双曲面微壳元件。基于经典理论，新的微壳元件还能够捕获旋转微壳的机械性能。在有限元分析中，尺寸相关性在抛物面，椭圆，并研究了在不同边界条件和几何参数下旋转的双曲微壳。还分析了由于尺寸依赖性而引起的模式偏移和模式形状的配置变化。最终，进行了比较，收敛和稳定性研究，以证明本研究中给出的数值结果的准确性和可靠性。