2.1. Resonator Structure

The structure of the monolithic cylindrical fused silica resonator is shown in Figure 1a. The resonator comprises a cylindrical resonant shell, a vibration-conducting shell, a bottom plate with eight holes equi-angularly distributed on, and a rigid stem on the bottom plate inside the cylindrical shell. The cylindrical resonator works at the n = 2 wineglass mode, with n referring to the number of circumferential node lines with respect to the axis of symmetry. For an ideal cylindrical resonator, the n = 2 wineglass mode actually includes two degenerate modes with the same natural frequencies and an angular interval of 45°, as depicted in Figure 1b, and they do not have a determined angular location around the axis of symmetry. These modes are often referred to as primary and secondary mode for excitation and detection mode, respectively. The finite element model was built using Ansys Multiphysics 14.0 software, as depicted in Figure 2a, and the simulated n = 2 wineglass mode was shown in Figure 2b. The cylindrical shell is the main vibrating body of the resonator, thus it demands extremely high machining precision. The vibration-conducting shell is considerably thinner than the cylindrical shell and the requirement on its machining precision is less stringent. The holes on the bottom plate ensure that the excitation and detection elements excite and detect the vibration mode of the resonator instead of membrane vibration of the bottom plate [19]. Furthermore, they may block the heat flow of the bottom plate while it deforms, thus decreasing the overall thermoelastic damping [20]. The rigid stem is used to fix the resonator, usually by gluing with epoxy or by indium bonding. In this design, the resonator can be excited and detected either by piezoelectric effect or by electrostatic force depending on the electrode design.

Schematic of: (a) the monolithic cylindrical fused silica resonator structure; (b) the n = 2 wineglass mode of an ideal cylindrical resonator.

The finite element model and n = 2 wineglass mode: (a) The sectional view of the finite element model of the monolithic fused silica cylindrical resonator built with Ansys Multiphysics 14.0; (b) The simulated n = 2 wineglass mode of the resonator.

It should be pointed out that, our design was inspired by the structure first proposed by Chikovani et al. [6,7,19], based on which we fabricated the monolithic fused silica resonator, which is different from the work reported by Xiang et al. [13,14,21], in which they investigated resonators made of metallic alloys and resonators with fused silica shells glued onto metallic disks, which will hardly result in high Q resonators for high-accuracy applications because of the material used and gluing procedure. In addition, our stem design was different from reported resonators for the simplicity and better precision of manufacturing.