In-plane free vibration of a single-crystal silicon ring

Abstract

a b s t r a c tIn 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(1 1 1) ring is two-dimensionally isotropic in the (1 1 1) plane for elastic constants but three-dimensionally anisotropic, while the Si(1 0 0) ring is fully anisotropic. 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(1 0 0) 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.

Effect of size dependency on in-plane vibration of circular micro-rings

“…By sensing the lateral vibration of the ring, the angular velocity of the device is calculated. Many researchers have studied the vibration and natural frequencies of circular rings [5,6]. For instance, Chang et al [5] derived the equations of motion of a circular ring in polar coordi-nate using energy method.…”

“…Many researchers have studied the vibration and natural frequencies of circular rings [5,6]. For instance, Chang et al [5] derived the equations of motion of a circular ring in polar coordi-nate using energy method. They used their formulation to calculate the natural frequency and mode shapes of a silicon micro-scale circular ring.…”

Tactical grade MEMS vibrating ring gyroscope with high shock reliability

“…(2) It is possible to increase its sensitivity proportional to the quality factor (Q-factor) because the sensing and driving modes have approximately the same frequency [4,7,8]. (3) It is less sensitive to temperature because the structure is uniformly affected by the thermal environment [9]. (4) The frequency mismatch caused by the asymmetric distribution of mass and stiffness can be electrically compensated using balancing electrodes [10].…”

A mode-matching 130-kHz ring-coupled gyroscope with 225 ppm initial driving/sensing mode frequency splitting

“…For the n=2 mode, the resonance frequencies of the drive/sense modes in crystalline silicon are inherently different so that the mode-matching is difficult to realize, which necessitates (i) the change of ring width and truss location, or (ii) the fabrication in specific crystalline directions [2] [3]. In contrast, the resonance frequencies of the drive/sense modes are inherently identical in the n=3 mode [4] [5]; however the displacement of the n=3 mode is smaller than that of the n=2 mode so that the sensitivity becomes lower. Although the [8] have been adopted to alleviate the aforementioned trade-off, they introduce several new issues, such as bulky size of the device or low yield concern due to challenging fabrication processes.…”