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                                                 Resonant Micromechanical Systems

                              278   Chapter Five

                                                          drive direction

                              sense direction
















                              Figure 5.51 Vibrating ring gyroscope.

                                                              c s
                              and                      ȟ =                              (5.141)
                                                        s  2J Ȧ
                                                              s s,r
                                The particular cases of drive resonance (Ȧ d  = Ȧ d,r ), sense resonance
                              (Ȧ  = Ȧ ), and full resonance (Ȧ  = Ȧ  = Ȧ ) can be formulated from
                                d
                                    s,r
                                                                      s,r
                                                                d,r
                                                            d
                              the generic Eq.  (4.138). The well-tuned case (Ȧ d  =  Ȧ d,r  =  Ȧ s,r ), for
                              instance, gives the following amplitude of the sense solution:
                                                              ȦM 0
                                                   Ĭ     =
                                                     y,ds          3                    (5.142)
                                                           2ȟ ȟ J Ȧ
                                                             d s x d,r
                                It should be mentioned that squeeze-film damping dominates in this
                              type of micro- and nanogyroscope, as opposed to slide-film damping,
                              which was the prevalent damping mechanism in  linear drive  gyro-
                              sensors.
                                Another microfabricated gyroscope is the vibrating ring gyroscope
                                               20
                              (Ayazi and Najafi ) which consists of an elastic ring symmetrically
                              supported  by eight  identical  semicircular springs, as  sketched in
                              Fig. 5.51. This structure exhibits two flexural modes of equal resonant
                              frequencies which are situated 45° apart, as sketched in Fig. 5.52. When
                              a driving force is applied along the drive axis, the primary flexural mode
                              is excited. Superposition of this motion to a rotary input about an axis
                              perpendicular to the planar structure results in the secondary mode
                              being excited, and this motion can be sensed capacitively.








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