基于广义弹性理论的微梁固有频率及模态的尺寸效应  

Scale Effects on Natural Frequencies and Vibration Modes of Micro Cantilever Beams Based on Generalized Elasticity

作　　者：沈暗明 陈锐[2] 杜丘美 SHEN Anming;CHEN Rui;DU Qiumei(College of Aerospace Engineering,Chongqing University,Chongqing 400044,P.R.China;State Key Laboratory of Mechanical Transmissions（Chongqing University）,Chongqing 400044,P.R.China)

机构地区：[1]重庆大学航空航天学院,重庆400044 [2]机械传动国家重点实验室(重庆大学),重庆400044

出　　处：《应用数学和力学》2018年第9期999-1008,共10页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(51505044)~~

摘　　要：经典弹性理论在近代工程技术中得到广泛应用,但其本构关系中不包含任何与尺寸相关的参数,因此不适用于微观结构,不能预测和解释尺寸效应.广义弹性理论增加了偶应力及其对应的曲率张量,完善了对小变形的几何描述,适用于微结构的尺寸效应研究.该文采用广义弹性理论,并结合Hamilton变分原理推导了悬臂微梁的振动微分方程,对微梁的固有频率及其模态进行了分析.结果表明,随着微梁厚度的不断减小,固有频率的尺寸效应与其对应的模态密切相关.扭转和弯曲模态包含了旋转变形,其对应的固有频率显著提高,表现出了显著的尺寸效应;而拉压模态不涉及旋转变形,固有频率未产生明显变化,没有尺寸效应.Classical elasticity has been widely applied in engineering technologies. But the length scale parameter is not included in the classical elasticity,which leads to the scale effects on mechanical characteristics and no longer satisfies the micro scale. Generalized elasticity is especially applicable to microstructures with scale effects,where both the rotational deformation and the couple stress are taken into account,and the measurement of deformation is improved. By means of Hamilton＇s variation principle and generalized elasticity,the vibration differential equations for the micro cantilever beam in different motion states were derived. Then natural frequencies and vibration modes of the micro cantilever beam were analyzed. The results show that,with the decreasing of the micro beam height,the scale effect on the natural frequency is closely related to the mode. The corresponding natural frequencies of torsional and bending modes have significant increment and scale effect compared with those according to the classical elasticity,for the rotational deformation is considered. However,little variation of the natural frequency of the tensile mode is found because there is no rotational deformation involved.

关 键 词：尺寸效应 广义弹性理论 HAMILTON原理 固有频率 模态 悬臂微梁 

分 类 号：O343[理学—固体力学]