M.Sc Thesis

M.Sc StudentBenvenisty Elazar
SubjectVibration Mode Order in Mechanical Disk Resonators
DepartmentDepartment of Mechanical Engineering
Supervisor PROF. David Elata
Full Thesis textFull thesis text - English Version


The motivation for this work is a phenomenon that was identified in disk resonating gyros, by the research group of Prof. Andrei Shkel from U.C. Irvine. This discovery relates to wineglass modes of in-plane vibrations in circular disks. It is well understood that the natural frequencies of the wineglass modes, monotonically increase with mode order (e.g. the second-order mode has a natural frequency that is lower than that of the third-order mode). The third-order mode is of interest because unlike the second-order mode, it is known to be insensitive to the anisotropy of (100) single crystalline silicon (SCS). Surprisingly, it was discovered that in annular disks that are made from anisotropic SCS, the natural frequency of the third-order wineglass mode is lower than that of the second-order mode.

This phenomenon has not been explained yet, and the primary aim of the research presented here is to identify the causes of this reordering of mode frequencies, and provide physical insight.

The present study begins with the analysis of standing and traveling waves, in a straight elastic layer of finite thickness that is clamped over one of its surfaces. This analysis reveals, for the first time, that there are two types of waves associated with these waves, which we term R-wave and E-wave. Some special cases are discussed in this context.

Following this analysis, related standing waves are analyzed in an annular disk that is made from an isotropic material. The analysis demonstrates that as for R-waves, mode ordering does indeed occur and the mode with the lowest frequency may be one of the higher-order modes. In contrast, the analysis shows that no mode ordering occurs for E-waves. In any case, the analysis shows that the frequency of E-waves is considerably lower that the frequency of R-waves, and therefore the mode ordering of R-waves is essentially masked by the lower frequencies of E-waves. It is concluded that the source of mode ordering witnessed by Professor Shkel must be found elsewhere. Careful observation suggests that the cause of the mode-ordering phenomenon may be due to anisotropy: the annular disk designed by Professor Shkel's group was micro-structured. It was constructed from a collection of concentric rings connected together by a staggered arrangement of radial connectors. This made the structure softer in the radial direction relative to the circumferential direction. In essence, the structure responded as if it were made of an orthotropic material in which one symmetry direction is radial (i.e. axisymmetric orthotropic material). By analyzing an annular disk with the appropriate orthotropic elastic parameters, it was found, that mode ordering does occur as expected.

Finally, in this work the wineglass modes in (100) silicon were analyzed. From this analysis, it is found that for all even ordered wineglass modes (elliptical, square etc.) the two orthogonal modes may (and most probably will) have different frequencies. On the other hand, it is found that for all odd-ordered wineglass modes (triangular, pentagonal etc.) the two orthogonal modes necessarily have the same frequency.