Why is there Mode Ordering in Disk Resonator Gyros?

We provide, a rational explanation for mode ordering observed in some specific disk resonating gyros (DRG). In DRGs that are made from (100) single crystalline silicon (SCS), it is well known that the frequencies of the two orthogonal conjugate 2^nd order (elliptic) wineglass modes, are distinct. This is because the material is anisotropic, and the elastic moduli in the [100] direction are different from those in the [110] direction. For DRGs that are made from (100) SCS, it is also well known that the frequencies of the two orthogonal conjugate 3^rd order (rounded triangular) wineglass modes, are identical. In essence, the 3^rd order wineglass mode is insensitive to material anisotropy. In most structures, it is often the case that higher order modes occur at a higher frequency. However, in a recent study of DRGs that are made from (100) SCS and are constructed from spoke-connected rings, it was surprisingly discovered that the frequency of the 3^rd order wineglass mode is lower than the frequency of the 2^nd order wineglass mode. In this work we analyze the problem and provide new insight, and for the first time provide a rational explanation of this phenomenon.