考虑表面效应的双层圆板的自由振动分析

考虑微生化传感器中谐振器的结构特点,基于Kirchhoff薄板理论与表面弹性理论推导了考虑表面效应的双层圆板的自由振动方程.使用伽辽金法得到了近似解.分析了硬化与软化表面效应与表面残余应力对双层圆板固有频率的影响.结果表明,与已有简化的单层圆板模型相比,现有考虑表面效应的双层板模型会得到与之不同的固有频率.随着板厚与上下两层板厚度比的变化,单层板模型预测的结果会比实际值偏大或者偏小,将无法为传感器膜片的研究与设计提供准确的理论值.对具有双层结构的传感器膜片的理论研究,使用现有模型得到的结果更加准确.

Vibration Analysis of Bilayered Circular Plate Considering Surface Effects

With the development of the micro-electric-mechanical system (MEMS) technology, MEMS devices, such as micro resonators, sensors and actuators, have been applied more and more extensively. Micro-plate is one of the key components of these MEMS devices. The dynamic characteristics of micro-plate have gained more and more attention from a growing number of scholars. At present, the existing theoretical models including surface effects are mainly developed for the single-layer thin plates. However, the micro-plates are usually bilayered or multilayered structures in practical applications. Therefore, considering the structure of resonators in micro biochemical sensors, the governing equations of the bilayered circular plate including surface effects are developed based on the Kirchhoff plate theory and continuum surface elasticity theory in this paper. The displacement field of the bilayered circular micro-plate with a neutral surface unknown beforehand is given. The Galerkin method is employed to obtain the approximate results. The influences of surface effects and surface residual stress on the natural frequency of the bilayered circular plate are discussed. The comparison shows that the natural frequency of the presented model is obviously different form that of the existed circular plate model. For different thicknesses and thickness ratios, the natural frequencies obtained by the existed model are either larger or smaller than the actual value. For the bilayered circular micro-plate including both stiffened and softened surface effects, the critical value of the thickness ratio is 0.6. Considering the surface residual stress, when the normalized tension parameter k is smaller than 4, the natural frequency of the bilayered micro-plate increases slowly. When k is larger than 4, the natural frequency increases rapidly with the increase of surface residual stress. For the study and design of resonators with bilayered structure, the results of the presented model are more accurate than the existing theoretical models.