高次谐振微悬臂的有限元设计与分析

Design and analysis of higher harmonic cantilever based on finite element analysis

多频原子力显微术(MF-AFM)是近些年发展的包含一大类先进原子力显微术的新技术,具有很高的空间分辨率,不仅可以实现更多样品物性的表征,还具有时间分辨的非线性力的探测能力.MF-AFM的基本原理是对微悬臂探针的多个振动频率进行激励和探测,而这些振动频率通常与微悬臂的高次谐波振动或本征模式有关,因此,实现这一新技术的关键是设计特殊的微悬臂.传统AFM中应用较为广泛的颇具代表性的传统微悬臂有硅微悬臂与石英音叉微悬臂等.传统矩形硅微悬臂的二阶弯曲本征模式与基础模式的频率比为6.27,而石英音叉微悬臂的二阶弯曲本征模式与基础模式的频率之比无统一的值,但均为非整数,直接利用传统微悬臂去探测高次谐振信号是非常困难的.基于微悬臂的共振放大效应,本文通过改变微悬臂的质量分布,讨论了传统硅微悬臂和石英音叉微悬臂本征模式的变化,调谐其高阶本征模式与基础模式间频率的耦合关系.通过理论计算和有限元分析,设计了二阶弯曲本征模式与基础模式的频率比为整数倍关系的特殊硅微悬臂与石英高次谐振型微悬臂,部分特殊微悬臂还涉及了一阶扭转模式与基础模式的耦合.

Recently, the developing multi-frequency atomic force microscopy, not only can characterize more physical properties of samples with a remarkable spatial resolution, but also detect non-linear force with time-resolved. Its basic principle is that multiple vibrating frequencies of micro cantilever probe is motivated and detected, which is closely related to higher harmonics or eigenmodes of micro cantilever. In order to realize this new technology, it is very important and necessary to design special micro cantilever. Generally, silicon cantilever and quartz cantilever are widely used in traditional AFM. For rectangular silicon cantilever, the ratio of the second order eigenmode and basic frequency is 6.27, and for quartz tuning micro-cantilever, it also isn＇t an integral value, so it is almost impossible to study the MF-AFM using the conventional cantilevers directly. Based on the resonant amplification effect, in this article, we discussed the corresponding eigenvibration of traditional rectangular silicon micro-cantilever and quartz tuning micro-cantilever by the method of changing the mass distribution of micro cantilever, namely changing its geometric size and shape. Finally we designed some special cantilevers, which the ratio of second mode frequency and basic frequency are all integer times by theoretical calculation and Finite Element Analysis, including some discussions between flexural resonant and basic frequency.