In-plane free vibration of a single-crystal silicon ring
In-plane free vibration of a single-crystal silicon ring
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DOI：
10.1016/j.ijsolstr.2008.07.033
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发表时间：
2008-12-01
影响因子：
3.6
通讯作者：
Chen, Po-Chih
中科院分区：
文献类型：
--
作者：
Chang, Chia-Ou;Chang, Guo-En;Chen, Po-Chih
本文分析了单晶硅环面内自由振动的固有频率和相关振型。研究发现Si(111)环的弹性常数在(111)面上是二维各向同性的，但三维各向异性的，而Si(100)环是完全各向异性的。利用哈密顿原理推导了振动方程组，该方程组是一组系数在极坐标变量中具有周期性的偏微分方程组。将径向和切向位移以非预定幅值的正弦形式表示，通过对周向变量的积分，可以将原来的偏微分形式的控制方程转化为常微分形式的幅值方程。获得了频率和振型的精确表达式。研究发现，对于 Si(100) 环，各向同性环的一对模式的频率相等，仅由于第二面内振动模式的各向异性效应而分裂。频率分裂和简并模现象可以基于平均机械能守恒或晶体学对称群的概念来证明。当单晶硅被弹性常数各向同性的多晶硅取代时，推导的频率方程正确地预测了频率分裂现象的消失。 (C) 2008 Elsevier Ltd. 保留所有权利。
In this paper the natural frequencies and the associated mode shapes of in-plane free vibration of a single-crystal silicon ring are analyzed. It is found that the Si(111) ring is two-dimensionally isotropic in the (111) plane for elastic constants but three-dimensionally anisostropic, while the Si(100) ring is fully anisostropic. Hamilton's principle is used to derive the equations of vibration, which is a set of partial differential equations with coefficients being periodic in polar variable. Expressing the radial and tangential displacements in sinusoidal form with non-predetermined amplitudes, and through the integration with respect to the circumferential variable, the original governing equations in partial differential form can be converted into the amplitude equations in ordinary differential form. The exact expressions for frequencies and mode shapes are obtained. It is found that for Si(100) rings the frequencies of a pair of modes, which are equal for an isotropic ring, split due to the anisotropic effect only for the second in-plane vibration mode. The phenomena of frequency splitting and degenerate modes can be proved either based on the conservation of averaged mechanical energy or by the concept of crystallographic symmetry groups. When the single-crystal silicon is replaced by the polycrystalline silicon, which is isotropic in elastic constants, the derived equations for frequencies correctly predict the vanishing of the phenomenon of frequency splitting. (C) 2008 Elsevier Ltd. All rights reserved.