Shell Resonator Research Articles

Thermoelastic damping properties in hemi-ellipsoidal shells with variable thickness

Thermoelastic damping (TED) is a fundamental dissipation mechanism that inevitably exists in shell resonators with high quality factors. Based on the thermal energy method, this paper demonstrates an effective method for the TED characterization of the hemi-ellipsoidal shells with variable thickness which manifest lower TED compared with the hemispherical shells. The equation of motion of the hemi-ellipsoidal shell under clamped-free boundary conditions is established by Hamilton's principle and the assumed mode method, and the natural frequencies and mode shape functions of the hemi-ellipsoidal shell with variable thickness are obtained by solving the eigenvalue problem. The temperature field is acquired by solving the heat conduction equation along the radial direction, and an analytical model for the TED of the hemi-ellipsoidal shell with variable thickness is presented by calculating the maximum elastic potential energy and the work lost per cycle of vibration due to irreversible heat conduction. Analysis on TED at the vibration patterns of meridional wave number m = 1 and the circumferential wave number n = 2 or 3 where the shell resonators typically operate is carried out. The analytically calculated TED results are compared with those of the finite element method (FEM) to verify the feasibility and correctness of the present method. The influences of the geometrical parameters on the TED characteristics of the hemi-ellipsoidal shells with variable thickness are analyzed in detail. A meaningful discovery is that compared with the hemispherical shell, the hemi-ellipsoidal shell with variable thickness has a smaller TED when its semiminor axis is shorter than the semimajor axis, which is particularly significant for optimizing the design of the shell resonators with high quality factors.

Mitigating rim bending in fused silica micro shell resonators by a chamfered mold

In this work, we addressed the bending of a micro shell resonator’s rim toward the stem during glass-blowing process. This phenomenon occurs due to increased heat loss at the mold’s edge and reduced heat flux at the side of the Gaussian heat source. To mitigate rim bending and the subsequent reduction in capacitance between the rim and 3D electrodes, which can degrade performance in micro glass-blown shell resonators for gyroscopes, we propose two potential mold solutions, using short-stem mold and chamfered mold. To quantitatively compare resonators created by these two types of molds, we calculated the noise factor for predicting its gyroscope performance. From these calculations, we find out that using the chamfered mold enables us to increase in the resonator’s effective mass, eigenfrequency, angular gain, and quality factor, consequently reducing the noise factor compared to using the short-stem mold. We also demonstrated these results by manufacturing shell resonators and measuring its vibrational characteristics. After metalization of the glass shell resonator, Q factor and eigen frequency of n = 2 vibration mode along the primary (secondary) axis were measured to 622,466 ± 6,427 (619,793 ± 6,402) and 4,343 ± 40 Hz (4,350 ± 40 Hz), respectively. This simulation process will assist us in getting insight into understanding a shell resonator as a gyroscope, designing a mold and determining experimental conditions.

Optomechanical Gyroscope Based on Micro-Hemispherical Shell and Optical Ring Resonators

Optomechanical Gyroscope Based on Micro-Hemispherical Shell and Optical Ring Resonators

Identification and trimming of the eccentric mass in shell resonators

This paper presents a novel method to identify and suppress the undesired spurious mode caused by the 1st and 3rd harmonics of eccentric mass defects in shell resonators. This approach eliminates the need for pre-trimming of frequency split, simplifying the identification and trimming process, with the advantages of high efficiency and minimal structural damage. Firstly, the influence of the mass defects on the dynamics of resonators is analyzed using the mass ring model. The shear force caused by the 1st and 3rd harmonics of the eccentric mass, as well as the axial force caused by the 2nd harmonic, result in forced bending vibration and axial vibration of resonators. These two types of vibrations are superimposed as spurious modes with the operating mode. The bending vibration caused by eccentric mass is decoupled from the mixed vibration signal by analyzing the dynamic characteristics. An analytic scheme is proposed to identify the 1st and 3rd harmonics without eliminating frequency split. To validate the proposed scheme, the identification and trimming process for eccentric mass is simulated by the finite element method based on the geometric model with preset first four harmonics. Finally, experimental identification and trimming of the eccentric mass for the hemispherical resonator are performed based on the laser Doppler vibrometer and ion beam etching process. After multiple iterations, the 1st and 3rd harmonics are reduced by 87.2% and 49.8%, and the quality factors of the operating modes increase by 26.7% and 40.7%, respectively, demonstrating the effectiveness of the proposed method.

Frequency mismatch analysis of hemispherical shell resonators with both radius and thickness imperfections

The hemispherical shell resonator (HSR) is the core element of the hemispherical resonator gyroscope (HRG), and its frequency mismatch is a key property influencing gyroscope accuracy. Investigating the mechanism of frequency mismatch is vital for improving the quality of HSRs and performance of HRGs. Midsurface radius imperfections and thickness imperfections are two principal causes of frequency mismatch, but their combined effects have rarely been discussed. This paper develops a model to comprehensively analyze the frequency mismatch of HSRs with both radius and thickness imperfections. The model derives a quantitative relation between the frequency mismatch and the two imperfections, and provides principles for evaluating dominant imperfections. To validate the model, we conduct experiments on a batch of 12 HSRs with random geometric imperfections. We apply the model in calculating the theoretical frequency mismatches of these HSRs, and the results agree well with the experimental data. The experiment also confirms that for macro-HSRs, thickness imperfections have a larger impact than radius imperfections, and the frequency mismatch is approximately linear to the fourth thickness harmonic. Our research can be a useful reference for the design and fabrication of HSRs and may open new possibilities for high-precision manufacture of HRGs.

Vibrations and thermoelastic quality factors of hemispherical shells with fillets

In engineering applications, hemispherical shell resonators are typically machined with fillets to reduce stress concentration and enhance structure strength. The fillets will inevitably affect the dynamic properties and the mechanical quality factor of hemispherical shell resonators, which has been seldom investigated before. In this paper, an effective analytical method is developed to explore the free vibration and thermoelastic damping (TED) characteristics of the hemispherical shell with fillets. The fillets are characterized by the variation in the thickness of the hemispherical shell during modelling. The first-order shear deformation theory (FSDT) is used to describe the theoretical formulas of the hemispherical shell with fillets. By employing the unified Jacobi polynomials and Fourier series as the assumed mode shape functions, the equation of motion of the structure is established by Hamilton's principle and the assumed mode method. The analytical model for thermoelastic quality factor (QTED) which is determined by TED is obtained by computing the dissipated energy and the maximum elastic potential energy of the hemispherical shell with fillets. The validity and accuracy of the present method are confirmed by comparing the present solutions with the published results and those obtained from the finite element method (FEM). The influences of fillets on the vibration behaviors and QTED characteristics of the hemispherical shells are analyzed in detail. The present model can be used to optimize the design of the fillets of the hemispherical shell resonators with high quality factors.

A confocal measurement method for surface profile variations and shell thickness uniformity of hemispherical shell resonator

Here we present a confocal measurement method for the precise measurements of variations in the inner and outer surface profiles (IOSPs) and shell thickness uniformity (STU) of the hemispherical shell resonator (HSR). In this method, the unilateral-shift-subtracting confocal technique is applied to determine the focus of the sensor on the IOSPs of the HSR. The normal tracking technique is employed for measuring the outer surface, while the normal direction of the outer surface serves as the reference for measuring the inner surface, thereby enhancing the accuracy of relative position measurements for the HSR. Furthermore, we applied a trajectory-prediction method to narrow down the scanning range of the confocal sensor, which significantly improved the measurement efficiency. The experimental results showed that the repeated accuracies of the proposed methods are better than 40 nm for IOSP variations measurement and better than 50 nm for HSR STU measurement.

Achieving Mode Detection and Precise Mechanical Trimming of Uncoated Micro Shell Resonator Using Interdigital Electrode

Achieving Mode Detection and Precise Mechanical Trimming of Uncoated Micro Shell Resonator Using Interdigital Electrode

Six million Q factor micro fused silica shell resonator with teeth-like tines released by femtosecond laser-assisted chemical etching

Six million Q factor micro fused silica shell resonator with teeth-like tines released by femtosecond laser-assisted chemical etching

Optimization of Hemispherical Shell Resonator Structure Based on Thermoelastic Dissipation

Optimization of Hemispherical Shell Resonator Structure Based on Thermoelastic Dissipation

A 3D-printed microhemispherical shell resonator with electrostatic tuning for a Coriolis vibratory gyroscope

The emergence of microhemispherical resonant gyroscopes, which integrate the advantages of exceptional stability and long lifetime with miniaturization, has afforded new possibilities for the development of whole-angle gyroscopes. However, existing methods used for manufacturing microhemispherical resonant gyroscopes based on MEMS technology face the primary drawback of intricate and costly processing. Here, we report the design, fabrication, and characterization of the first 3D-printable microhemispherical shell resonator for a Coriolis vibrating gyroscope. We remarkably achieve fabrication in just two steps bypassing the dozen or so steps required in traditional micromachining. By utilizing the intricate shaping capability and ultrahigh precision offered by projection microstereolithography, we fabricate 3D high-aspect-ratio resonant structures and controllable capacitive air gaps, both of which are extremely difficult to obtain via MEMS technology. In addition, the resonance frequency of the fabricated resonators can be tuned by electrostatic forces, and the fabricated resonators exhibit a higher quality factor in air than do typical MEMS microhemispherical resonators. This work demonstrates the feasibility of rapidly batch-manufacturing microhemispherical shell resonators, paving the way for the development of microhemispherical resonator gyroscopes for portable inertial navigation. Moreover, this particular design concept could be further applied to increase uptake of resonator tools in the MEMS community.

An Adaptive Multi-Population Approach for Sphericity Error Evaluation in the Manufacture of Hemispherical Shell Resonators.

The performance of a hemispherical resonant gyroscope (HRG) is directly affected by the sphericity error of the thin-walled spherical shell of the hemispherical shell resonator (HSR). In the production process of the HSRs, high-speed, high-accuracy, and high-robustness requirements are necessary for evaluating sphericity errors. We designed a sphericity error evaluation method based on the minimum zone criterion with an adaptive number of subpopulations. The method utilizes the global optimal solution and the subpopulations' optimal solution to guide the search, initializes the subpopulations through clustering, and dynamically eliminates inferior subpopulations. Simulation experiments demonstrate that the algorithm exhibits excellent evaluation accuracy when processing simulation datasets with different sphericity errors, radii, and numbers of sampling points. The uncertainty of the results reached the order of 10-9 mm. When processing up to 6000 simulation datasets, the algorithm's solution deviation from the ideal sphericity error remained around -3 × 10-9 mm. And the sphericity error evaluation was completed within 1 s on average. Additionally, comparison experiments further confirmed the evaluation accuracy of the algorithm. In the HSR sample measurement experiments, our algorithm improved the sphericity error assessment accuracy of the HSR's inner and outer contour sampling datasets by 17% and 4%, compared with the results given by the coordinate measuring machine. The experiment results demonstrated that the algorithm meets the requirements of sphericity error assessment in the manufacturing process of the HSRs and has the potential to be widely used in the future.

A normal tracking differential confocal measurement method for multiple geometric parameters of hemispherical shell resonator with a common reference

This paper proposes a normal tracking differential confocal measurement method for the inside and outside surface profiles, shell thickness uniformity, and central asymmetry of inside and outside surfaces of the hemispherical shell resonator (HSR). A differential confocal technique with high-transmittance focusing ability is used to measure a single point on the inside and outside surfaces of the HSR. The normal alignment measurement technique is used to accurately measure the inside and outside surfaces and shell thickness of the HSR with a common reference in one measurement process. The HSR is step-rotated to synchronously collect information on the inside and outside surfaces, and using the differential confocal sensor to measure the different normal-section profiles. The experimental results indicate successful measurement of HSR central asymmetry. The repeated measurement accuracy for the inside and outside surface profiles and thickness uniformity is better than 30 nm.

Fused Silica Micro Shell Resonators by a Wafer-Level Thermal Reflow Process

Fused Silica Micro Shell Resonators by a Wafer-Level Thermal Reflow Process

Stiffness Tuning of Hemispherical Shell Resonator Based on Electrostatic Force Applied to Discrete Electrodes

Stiffness Tuning of Hemispherical Shell Resonator Based on Electrostatic Force Applied to Discrete Electrodes

Quality Factors in Borosilicate Glass Shell Resonators

Quality Factors in Borosilicate Glass Shell Resonators

A 3D TLM code for the study of the ELF electromagnetic wave propagation in the Earth's atmosphere

The interest in the study of electromagnetic propagation through planetary atmospheres is briefly discussed. Special attention is devoted to extremely-low-frequency fields in the Earth's atmosphere for its global nature and possible applications to climate monitoring studies among others. In the Earth's case, the system can be considered as a spherical electromagnetic shell resonator in which two concentric and large conducting spheres with a radius around 6300 km are separated by a very small distance of around 100 km, the atmosphere height. A numerical solution using the Transmission Line Method is proposed. The classical spherical-coordinate description is easy to use, however, the important difference in the dimensions along the three coordinate directions causes high numerical dispersion in the results. A Cartesian scheme with equal node size for all directions is used to reduce this undesired dispersion. A pre-processing stage is the key point introduced to lessen the resulting high demand of memory and time calculation and make the solution feasible. A parallelized Fortran code together with pre- and post-processing Python programs to ease the user interface are provided with this work. Details on the Fortran code and the Python modules are included both in the paper and the source codes to allow the use and modifications by other researchers interested in electromagnetic propagation through planetary atmospheres. The program allows calculation of the time evolution of the electromagnetic field at any point in the atmosphere. It includes the possibility of considering multiple time-dependent sources and different homogeneous and inhomogeneous conductivity profiles to model different situations. Profiles to study day-night asymmetries or locally perturbed profiles which have been attributed to earthquakes in the literature are implemented, for instance.

Dynamic modelling and quality factor evaluation of hemispherical shell resonators

A novel strategy is proposed for the dynamic modelling and quality factor evaluation of the hemispherical shell resonators (HSR) considering the coupling of hemispherical shell and support rod. The artificial spring technique is introduced to realize the coupling connection of substructures and simulate the arbitrary boundary conditions. Hamilton's principle with the Rayleigh-Ritz method is employed to establish the equation of motion of the HSR, and the natural frequencies of the HSR are obtained by solving the eigenvalue problem of the equation of motion. The thermal energy method is applied to evaluate the thermoelastic damping (TED) of the HSR. The model of anchor loss is based on a separation and transfer method, in which the HSR and its substrate are first separated for analysis and then the stress from the clamped region of the support rod is transferred to the substrate for determining the vibration displacement across the clamped region. The energy losses to the substrate are formulated as the integral of the product of the stress and the vibration displacement across the clamped region per cycle of vibration. The results calculated from the present theoretical method are compared with those from the published literature and the finite element method (FEM), which validates the correctness and feasibility of the present theoretical model. The effects of the boundary conditions and geometrical parameters on the vibration behaviors of the HSR are analyzed, and the influences of the boundary conditions, geometrical parameters and material properties of the HSR on the quality factors related to the TED and anchor loss are investigated. The present model can be used to optimize the design of the HSR with a high quality factor.

A novel method to characterize the residual stress on the fused silica surface based on the evolution of the atomic point defects

The residual stress would have a significant impact on the service performance and fatigue life of the fused silica (non-crystal material) components (e.g., Hemispherical Shell Resonator). However, there is still no effective method for characterizing the residual stress on their working surfaces. Therefore, it is essential to develop a novel method for characterizing the residual stress on fused silica surfaces. Herein, the residual stress is found to significantly affect the mechanical properties of fused silica surfaces. On the basis of the study on the photoluminescence (PL) properties of residual stress zones, atomic point defects are found to be closely related to the residual stress. Based on an investigation of the PL properties of different types of mechanically processed surfaces, four types of point defects (ODCII, STE, Si clusters and NBOHCI) have been determined. On the basis of the molecular dynamics (MD) simulation results and the PL experiments on different types of mechanically processed surfaces, STE point defects at the atomic scale are found to be closely related to the residual stress for the first time. Based on this, PL-detection method is proposed to characterize the residual stress on the material surface for the first time. Besides, the relative amount of STE point defect (K = S/S0) is first proposed to quantitatively characterize the residual stress. In sum, this work determines the structure at the atomic scale which is closely related to the residual stress and opens a convenient door to quantitatively characterize the residual stress with PL-detection method. It could also provide a reference for the characterization and even the alleviation of the residual stress on the surfaces of other non-crystal materials.

A novel method of water bath heating assisted small ball-end magnetorheological polishing for hemispherical shell resonators

Hemispherical shell resonator (HSR) is the core component of hemispherical resonator gyro. It is a ψ-shaped small-bore complex component with minimum curvature radius less than 3 mm. Thus, traditional polishing methods are difficult to polish it. Small ball-end magnetorheological polishing method can polish the small components with complicated three-dimensional surface and obtain non-destructive surface. Therefore, this method is suitable for polishing HSR. However, the material removal rate of the ordinary small ball-end magnetorheological polishing is low, leading to long polishing time and low output of HSR. To solve this problem, a water bath heating assisted small ball-end magnetorheological polishing method is proposed in this research. The influence rule of processing parameters on the material removal rate is studied experimentally. A set of optimal processing parameters is obtained to maximize the material removal rate. Compared with the ordinary method, the material removal rate of the new method can be improved by 143%. Subsequently, an HSR is polished by the new method. The results show that the polishing time can be reduced by 55%, and the polished surface roughness can reach 7.7 nm. The new method has the great potential to be used in actual production to improve the polishing efficiency of HSR.