Hemispherical Shell Resonators Research Articles

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.

Optimization of Hemispherical Shell Resonator Structure Based on Thermoelastic Dissipation

Optimization of Hemispherical Shell Resonator Structure Based on Thermoelastic Dissipation

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.

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.

High-precision angular rate detection based on an optomechanical micro hemispherical shell resonator gyroscope.

Cavity optomechanics with picometer displacement measurement resolution has shown vital applications in high-precision sensing areas. In this paper, an optomechanical micro hemispherical shell resonator gyroscope (MHSRG) is proposed, for the first time. The MHSRG is driven by the strong opto-mechanical coupling effect based on the established whispering gallery mode (WGM). And the angular rate is characterized by measuring the transmission amplitude changing of laser coupled in and out from the optomechanical MHSRG based on the dispersive resonance wavelength shift and/or dissipative losses varying. The detailed operating principle of high-precision angular rate detection is theoretically explored and the fully characteristic parameters are numerically investigated. Simulation results show that the optomechanical MHSRG can achieve scale factor of 414.8 mV/ (°/ s) and angular random walk of 0.0555 °/ h1/2 when the input laser power is 3 mW and resonator mass is just 98 ng. Such proposed optomechanical MHSRG can be widely used for chip-scale inertial navigation, attitude measurement, and stabilization.

Geometric Error Analysis and Compensation in Spherical Generating Grinding of Hemispherical Shell Resonators.

The geometric accuracy of a hemispherical shell resonator (HSR) affects the assembly accuracy and final performance of a hemispherical resonant gyroscope in many ways. During the precision grinding of a resonator, the tool-setting error and wear error affect the form and positional accuracy of the inner and outer spherical surfaces. In this study, a compensation method for generating grinding of the HSR is proposed to address this problem. The geometric errors of the inner and outer spherical surfaces are systemically analyzed and a geometric model of the tool setting and wheel wear is established for generating grinding of the HSR. According to this model, a mapping relationship between the wheel pose and size, form, and positional error of the HSR was proposed. Experiments regarding machining, on-machine measurements, and error compensation were performed using the mapping relationship. The results demonstrate that the proposed method can reduce the radius error of the inner and outer spherical surfaces from 10 μm to 1 μm, sphericity from 5 μm to 1.5 μm, and concentricity from 15 μm to 3 μm following grinding. The form and positional errors are simultaneously improved, verifying the effectiveness of the proposed method.

Areal measurement of vibration modes of a hemispherical shell resonator by deflectometry.

The hemispherical shell resonator (HSR) is the core and sensitive part of a hemispherical resonator gyro. The geometrical accuracy and vibration properties of HSR determine the navigation performance of the system. A lack of areal measurement methods of vibration modes limits investigation of the kinetic mechanism and improvement in navigation performance. Consequently, an areal measurement method is developed based on deflectometry. The blurry spots on the image plane reflected from the vibrating HSR are extracted, and the blurring trajectories are obtained by the Wiener deconvolution method. The vibrating amplitude distribution of a standing wave mode is transformed into the swing angle distribution of normal vectors. The parameters of the vibration mode are fitted by the Levenberg-Marquardt method. This method can find widespread applications in the areal inspection of vibration modes.

An accurate thermoelastic model and thermal output error analysis of a hemispherical resonator gyroscope under varying temperatures

This paper proposes an accurate thermoelastic model of a hemispherical shell resonator (TM-HSR) taking into consideration thermal stress and reveals the effect of its thermal deformation on the hemispherical resonator gyroscope (HRG) output error for the first time. First, a set of thermoelastic dynamic equations of the TM-HSR are derived by leveraging the thermoelastic shell theory, and the relationship between temperature and the equivalent thermal load is determined. Next, an approximate analytical solution of thermal deformation of HSRs is derived using Bubnov-Galerkin method and its accuracy is evaluated by comparing its result with ANSYS simulation results of HSRs with different structures. The approximate analytical solution is found to be very accurate, which is important for temperature testing of HRGs and application to various vibratory sensors with hemispherical shell structures. Subsequently, the effects of thermal deformations under uniform and nonuniform temperature distributions on the HRG output error are revealed by using a thermal output error model. Comparative simulation and experimental results demonstrate the effectiveness and practicability of the proposed model. The proposed model can be used for accurate error analysis and developing a compensation scheme for the thermal output error of HRGs.

Design and Characterization of a Planar Motor Drive Platform Based on Piezoelectric Hemispherical Shell Resonators

In this study, a planar ultrasonic motor platform is presented that uses three half-side excited piezoelectric hemispherical shell resonators. To understand the working principle and the harmonic vibration behavior of the piezoelectric resonator, the trajectory of the friction contact was measured in free-oscillating mode at varying excitation frequencies and voltages. The driving performance of the platform was characterized with transport loads up to 5 kg that also serve as an influencing downforce for the friction motor. The working range for various transport loads and electrical voltages up to 30 V is presented. Undesirable noise and parasitic oscillations occur above the detected excitation voltage ranges, depending on the downforce. Therefore, minimum and maximum values of the excitation voltage are reported, in which the propulsion force and the speed of the planar motor can be adjusted, and noiseless motion applies. The multidimensional driving capacity of the platform is demonstrated in two orthogonal axes and one rotary axis in open-loop driving mode, by measuring forces and velocities to confirm its suitability as a planar motor concept. The maximum measured propulsion force of the motor was 7 N with a transport load of 5 kg, and its maximum measured velocity was 77 mm/s with a transport load of 3 kg.

Design and Fabrication of Hemispherical Shell Resonator by Glass Blowing Method

Background and Objectives: The Hemispherical Resonator Gyroscope (HRG) has been a valuable choice for the aerospace industry due to its low noise, good performance, and long lifetime. Its main part consists of a hemispherical shell resonator (HSR). Recently, with the idea of using MEMS technology and using materials such as Pyrex in the construction of HSR, a significant reduction in the size, weight, and power consumption of this gyroscope along with the special gyroscopic characteristics of HRG, Today, the Micro-HRG has been introduced as a sensor in the strategic class.Methods: Micro-HSR can be implemented in three ways: micromachining, blowtorching, and glass blowing. In this paper, after a brief introduction and comparison of possible approaches to make an HSR, the glass blowing method is selected and the sub-processes of this method are introduced and the results of its implementation are presented.Results: The proposed sensor is made by performing the glass blowing method, with a radius of 1.536 mm, a middle base radius of 252 µm, and a shell height of 355 µm. Also, the most important fabrication parameters for achieving the desired geometrical shell were as follows: The appropriate initial thickness of Pyrex is 200 µm and the depth of the cavity under the silicon layer is 532 µm. The bonding process to create a stable connection between Si and Pyrex must be performed at a voltage of 800 V at a temperature of 550 °C for the 30s. And the blowing process with a heating rate of 4 °C/s and a cooling rate of 9.5 °C/s has been evaluated as a suitable thermal profile. By optimally controlling the temperature process in the blowing sub-process, the crystallization phenomenon is prevented, which will lead to the improvement of the HSR quality factor.Conclusion: Achieving the technology of making this hemispherical glass shell in sub-millimeter dimensions, which plays the role of the resonator in an HRG, due to the superior features of this gyroscope such as low noise, good performance, and long lifetime and its application in strategic industries, like navigation and smart weapons, it is of particular importance.

Dynamic modeling and output error analysis of an imperfect hemispherical shell resonator

A hemispherical resonator gyroscope (HRG) is very useful for inertial sensors in the 21st century. This paper proposes a novel dynamic model of an imperfect hemispherical shell resonator (DM-IHSR) with thickness nonuniformity and an output error model of the HRG (OEM-HRG) associated with frequency split. First, the DM-IHSR is established based on the thin shell theory of elasticity, and the thickness nonuniformity is described. The differential equations of vibratory motion of the IHSR are derived by leveraging Bubnov-Galerkin method, which reveals that frequency split is mainly dominated by the fourth harmonic of thickness nonuniformity. Then, by considering the characteristics of standing waves, the OEM-HRG is proposed, which discovers the effects of frequency split and the initial orientation angle on the HRG output error. Finally, the effect of thickness nonuniformity on the HRG output error is clarified. The quantitative relationship among the HRG output error, the fourth harmonic of thickness nonuniformity, and the initial orientation angle is determined, which gives machining accuracy ranges of these variables. The above dynamic models and machining accuracy ranges are of great significance for producing and processing high-precision HSRs and HRGs.

Thermoelastic Dissipation in Diamond Micro Hemispherical Shell Resonators

Maximizing quality factor (Q) is essential to improve the performance of micro hemispherical shell resonators (μHSRs) which can be used in microelectromechanical system (MEMS) gyroscopes to measure angular rotation. Several energy dissipation mechanisms limit Q, where thermoelastic dissipation (TED) is the major one and studied in this paper. Fully coupled thermo-mechanical equations for calculating TED are formulated, and then temperature distribution in a deformed μHSR and its quality factor related to TED (QTED) are obtained by solving the equations through a finite-element method (FEM). It has been found that different fabrication process conditions can obtain various geometrical parameters in our previous studies. In order to provide guidelines for the design and fabrication of μHSRs, the effects of their geometry on resonant frequency (f0) and QTED are studied. The change of anchor height and small enough anchor radius have no effect on both f0 and QTED, but the shell size including its radius, thickness and height has significant impact on f0 and QTED. It is found that whether a μHSR has lower f0 and higher QTED or higher f0 and higher QTED can be achieved by changing these geometrical parameters. The results presented in this paper can also be applied to other similar resonators.

Fabrication and characterisation of microscale hemispherical shell resonator with diamond electrodes on the Si substrate

A self-aligned fabrication method is employed for microscale hemispherical shell resonator with integrated diamond electrodes on a silicon (Si) substrate. The millimetre-scale three-dimensional (3D) hemispherical shell resonator, with integrated electrostatic excitation and capacitive detection transducers for full control of the resonator, is fabricated on the Si substrate by employing a microelectromechanical systems process. The eight integrated diamond electrodes evenly distributed outside the hemispherical shell are fabricated using a self-aligned method, which ensures uniform capacitive gap along the whole circumference. The Boron-doped polycrystalline diamond, a material with low thermo-elastic damping and low surface losses to reach the high quality, is used as the millimetre-scale 3D hemispherical shell resonator with integrated electrodes. The fabrication process enables the production of almost perfect hemisphere mould (<2% roundness variation) with an average surface roughness of 3 nm. Vibration modes and frequencies are carried out by a scanning laser Doppler vibrometer; the elliptical mode frequency is determined to be at 81.3 kHz, with the two M = 2 degenerate modes having a relative frequency mismatch of 0.86%. This process can be further optimised to fabricate high-quality and full-symmetric microscale hemispherical resonator gyroscopes in batch.

The Synthesis Model of Flat-Electrode Hemispherical Resonator Gyro.

The Hemispherical Resonator Gyro (HRG) is a solid-state and widely used vibrating gyroscope, especially in the field of deep space exploration. The flat-electrode HRG is a new promising type of gyroscope with simpler structure that is easier to be fabricated. In this paper, to cover the shortage of a classical generalized Coriolis Vibration Gyroscope model whose parameters are hard to obtain, the model of flat-electrode HRG is established by the equivalent mechanical model, the motion equations of unideal hemispherical shell resonator are deduced, and the calculation results of parameters in the equations are verified to be reliable and believable by comparing with finite element simulation and the reported experimental data. In order to more truthfully reveal the input and output characteristics of HRG, the excitation and detection models with assemble errors and parameters are established based on the model of flat-electrode capacitor, and they convert both the input and output forms of the HRG model to voltage changes across the electrodes rather than changes in force and capacitance. An identification method of assemble errors and parameters is proposed to evaluate and improve the HRG manufacturing technology and adjust the performance of HRG. The average gap could be identified with the average capacitance of all excitation and detection capacitors; fitting the approximate static capacitor model could identify the inclination angle and direction angle. With the obtained model, a firm and tight connection between the real HRG system and theoretical model is established, which makes it possible to build a fully functional simulation model to study the control and detection methods of standing wave on hemispherical shell resonator.

Microsacle PolySilicon Hemispherical Shell Resonating Gyroscopes with Integrated Three-dimensional Curved Electrodes

A novel approach for fabrication of polysilicon hemispherical resonator gyroscopes with integrated 3-D curved electrodes is developed and experimentally demonstrated. The 3-D polysilicon electrodes are integrated as a part of the hemispherical shell resonator’s fabrication process, and no extra assembly process are needed, ensuring the symmetry of the shell resonator. The fabrication process and materials used are compatible with the traditional semiconductor process, indicating the gyroscope has a high potential for mass production and commercial development. Without any trimming or tuning of the n=2 wineglass frequencies, a 28 kHz shell resonator demonstrates a 0.009% frequency mismatch between two degenerate wineglass modes, and a 13.6 kHz resonator shows a frequency split of 0.03%. The ring-down time of a fabricated resonator is 0.51 s, corresponding to a Q of 22000, at 0.01 Pa vacuum and room temperature. The prototype of the gyroscope is experimentally analyzed, and the scale factor of the gyro is 1.15 mV/°/s, the bias instability is 80 °/h.