ANISOTROPY EFFECTS ON THE VIBRATION OF CIRCULAR RINGS MADE FROM CRYSTALLINE SILICON

Abstract

Micro-engineered transducers, such as vibratory gyroscopes, accelerometers and pressure transducers made from wafers of crystalline silicon, are becoming increasingly common. These often contain ring-like components, the vibration properties of which are crucial to the operation of the transducer. The stability of the material properties of crystalline silicon is highly beneficial to the performance of these devices. However, the material has significant anisotropy, which must be properly accounted for in the design of the structure. Crystalline silicon has a cubic structure with three principal planes. Two of these, the (111) and (100) planes, are of particular interest since many devices are manufactured from silicon wafers that are nominally cut parallel to these planes. The precise cut of silicon with respect to the principal planes determines the form and degree of anisotropy in the material properties around a structure. This paper focuses on the effects of anisotropy on the natural frequencies and directional properties of the modes of circular rings of rectangular cross-section, made from crystalline silicon. Both in-plane and out-of-plane flexural modes are investigated using Lagrange's equations. The effects of anisotropy are accounted for in the strain energy formulation. General equations are given for directional variations in the elastic moduli. These equations have been simplified and linearized to allow analytical expressions for the natural frequencies to be obtained for a number of special cases. Results are presented for rings made from wafers that are cut nominally in the (100) and (111) principal planes. The effects of small departures in the plane of the wafer cut from these principal planes is investigated. Frequency splitting is predicted between pairs of similar modes which would be degenerate with equal natural frequencies, if the ring were made from an isotropic material. Differences are found between the frequency predictions obtained using general and simplified expressions for the elasitic moduli. These are explained on the basis of a Fourier analysis of the variation in the elastic properties around the ring.