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In this paper, the stability of the split doublet mode pair in a disc brake squeal is analytically investigated. A reduced-order doublet mode model is derived based on the formulation of annular plate vibration, sliding contact kinematics, and the assumed mode method. The solution is approximated using a two-mode expansion for the doublet mode pair. The time-varying motion of the slightly asymmetric rotating disc is described through the moving mode shape function method. A stability analysis of the frequency split between the doublet mode pair is performed by calculating the characteristic multipliers using the Floquet theory. The stability boundary is determined within the domain of frequency split variations and other parameters. Finally, the influence of mode splitting on the squeal behaviour of the asymmetric disc is analysed and discussed. [ABSTRACT FROM AUTHOR]

The paper presents the engineering estimate of the dispersion level of the values of such fused quartz glass physical and mechanical properties as density and elasticity modulus. The value scatter of the considered parameters over material volume leads to negative effect appearance — working natural frequency splitting of gyroscope resonator. Evaluation is carried out on the basis of the consideration of the fused quartz glass micro-structure and variability of its parameters. As a result, resonator frequency splitting values from calculated scattering levels of the structural material properties are presented.

Loosely coupled transformer is an important device for realizing electric energy transfer in inductively coupled power transfer (ICPT) system. However, the low transmission power and low transmission efficiency of loosely coupled transformer greatly limit its application. Three-terminal shaft loosely coupled transformer structure was proposed, and a mathematical model was established respectively for the two cases where adjacent coils of the new transformer have the same direction and reverse end. Then, the frequency splitting phenomenon of the ICPT system based on the three-terminal shaft loosely coupled transformer under the LCC-S compensation method was analyzed. The simulation verifies that the three-terminal shaft loosely coupled transformer can improve the transmission characteristics of the system through the mutual inductance between different coil pairs, and compares the transmission power and transmission efficiency of the same direction and the reverse of the same end. Finally, the experimental results show that the ICPT system based on the three-terminal shaft loosely coupled transformer can improve the transmission power and transmission efficiency, and the transmission efficiency is higher when the same end is in the same direction, which increases by 16. 4% . When the same end is reversed, the transmission power increases by 33. 2% . [ABSTRACT FROM AUTHOR]

In this paper, the stability of the split doublet mode pair in a disc brake squeal is analytically investigated. A reduced-order doublet mode model is derived based on the formulation of annular plate vibration, sliding contact kinematics, and the assumed mode method. The solution is approximated using a two-mode expansion for the doublet mode pair. The time-varying motion of the slightly asymmetric rotating disc is described through the moving mode shape function method. A stability analysis of the frequency split between the doublet mode pair is performed by calculating the characteristic multipliers using the Floquet theory. The stability boundary is determined within the domain of frequency split variations and other parameters. Finally, the influence of mode splitting on the squeal behaviour of the asymmetric disc is analysed and discussed.

In capacitive microelectromechanical system (MEMS) devices, the application of dielectric materials causes long-term charging problems in the dielectric layers or substrates, which especially affect the repeatability and stability of high-performance devices. Due to the difficulties of observation and characterization of charge accumulation, an accurate characterization method is needed to study the effect of charge and propose suppression methods. In this paper, we analyze the influence of charge accumulation on the MSRG and propose a characterization method for charge accumulation based on stiffness variation. Experiments are carried out to characterize the charge accumulation in MSRG, and the effect of temperature on the process is also investigated. In the experiment, the charge accumulation is characterized accurately by the variation of the frequency split and stiffness axes. Furthermore, the acceleration of the charge accumulation is observed at high temperatures, as is the higher additional voltage from the charge accumulation. [ABSTRACT FROM AUTHOR]

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This paper presents a study on dynamics of a dual-mass MEMS vibratory gyroscope in presence of fabrication imperfections and reports a method for precision electrostatic frequency tuning of the operational modes. A number of multi-mass MEMS gyroscopes have emerged in recent years pursuing different goals, such as dynamically balanced structure, increased bandwidth, and dynamic amplification. Along with many perceived advantages of multi-mass devices, several challenges associated with mode-matching in a system with increased number of degrees-of-freedom (DOF) have to be considered. This work shows that it is possible to apply the DC tuning techniques, similar to tuning a conventional single-mass gyroscope, to achieve the precision tuning in a dual-mass sensor, without losing advantages of increased DOF of the system. The presented frequency trimming technique is based on assessing the modes mismatch and cross-coupling between modes by means of fitting the experimental frequency response curves to the analytical solutions of the dual-mass system in presence of imperfections. The tuning algorithm involves two steps. First, the stiffness mismatch along the two axes and the anisoelasticity angles α and β are identified, then the tuning DC voltages for modification of diagonal, off-diagonal, and coupling terms in the stiffness matrix are chosen. The method of electrostatic tuning was validated through the experimental characterization of a dual-mass dynamically amplified gyroscope, where the coupling between the two operational modes was minimized and frequency split was reduced from 26 Hz down to 50 mHz, resulting in 17.5× increase in the gyroscope scale factor and significantly improved noise characteristics. The presented electrostatic compensation method is suitable for both off-line and on-line calibration.

This paper presents a study on dynamics of a dual-mass MEMS vibratory gyroscope in presence of fabrication imperfections and reports a method for precision electrostatic frequency tuning of the operational modes. A number of multi-mass MEMS gyroscopes have emerged in recent years pursuing different goals, such as dynamically balanced structure, increased bandwidth, and dynamic amplification. Along with many perceived advantages of multi-mass devices, several challenges associated with mode-matching in a system with increased number of degrees-of-freedom (DOF) have to be considered. This work shows that it is possible to apply the DC tuning techniques, similar to tuning a conventional single-mass gyroscope, to achieve the precision tuning in a dual-mass sensor, without losing advantages of increased DOF of the system. The presented frequency trimming technique is based on assessing the modes mismatch and cross-coupling between modes by means of fitting the experimental frequency response curves to the analytical solutions of the dual-mass system in presence of imperfections. The tuning algorithm involves two steps. First, the stiffness mismatch along the two axes and the anisoelasticity angles α and β are identified, then the tuning DC voltages for modification of diagonal, off-diagonal, and coupling terms in the stiffness matrix are chosen. The method of electrostatic tuning was validated through the experimental characterization of a dual-mass dynamically amplified gyroscope, where the coupling between the two operational modes was minimized and frequency split was reduced from 26 Hz down to 50 mHz, resulting in 17.5× increase in the gyroscope scale factor and significantly improved noise characteristics. The presented electrostatic compensation method is suitable for both off-line and on-line calibration.

Axisymmetric resonators are key elements of Coriolis vibratory gyroscopes (CVGs). The performance of a CVG is closely related to the stiffness and damping symmetry of its resonator. The stiffness symmetry of a resonator can be effectively improved by electrostatic tuning or mechanical trimming, both of which need an accurate knowledge of the azimuth angles of the two stiffness axes of the resonator. Considering that the motion of a non-ideal axisymmetric resonator can be decomposed as two principal oscillations with two different natural frequencies along two orthogonal stiffness axes, this paper introduces a novel high-precision method of stiffness axes identification. The method is based on measurements of the phase difference between the signals detected at two orthogonal sensing electrodes when an axisymmetric resonator is released from all the control forces of the force-to-rebalance mode and from different initial pattern angles. Except for simplicity, our method works with the eight-electrodes configuration, in no need of additional electrodes or detectors. Furthermore, the method is insensitive to the variation of natural frequencies and operates properly in the cases of either large or small frequency splits. The introduced method is tested on a resonator gyroscope, and two stiffness axes azimuth angles are obtained with a resolution better than 0.1°. A comparison of the experimental results and theoretical model simulations confirmed the validity of our method. [ABSTRACT FROM AUTHOR]

A cylindrical resonator gyroscope is a kind of Coriolis gyroscope, which measures angular velocity or angle via processing of the standing wave. The symmetry of a cylindrical resonator is destroyed by different degrees of geometric nonuniformity and structural damage in the machining process. The uneven mass distribution caused by the asymmetry of the resonator can be expressed in the form of a Fourier series. The first three harmonics will reduce the anti-interference ability of the resonator to the external vibration, as well as increase the angular random walk and zero-bias drift of the gyroscope. In this paper, the frequency split of different modes caused by the first three harmonic errors and the displacement of the center of the cylindrical resonator bottom plate are obtained by simulation, and the relationship between them is explored. The experimental results on five fused silica cylindrical resonators are consistent with the simulation, confirming the linear relationship between the n = 1 frequency split and second harmonic error. A method for evaluating the first three harmonic errors of fused silica cylindrical resonators is provided. [ABSTRACT FROM AUTHOR]

In capacitive microelectromechanical system (MEMS) devices, the application of dielectric materials causes long-term charging problems in the dielectric layers or substrates, which especially affect the repeatability and stability of high-performance devices. Due to the difficulties of observation and characterization of charge accumulation, an accurate characterization method is needed to study the effect of charge and propose suppression methods. In this paper, we analyze the influence of charge accumulation on the MSRG and propose a characterization method for charge accumulation based on stiffness variation. Experiments are carried out to characterize the charge accumulation in MSRG, and the effect of temperature on the process is also investigated. In the experiment, the charge accumulation is characterized accurately by the variation of the frequency split and stiffness axes. Furthermore, the acceleration of the charge accumulation is observed at high temperatures, as is the higher additional voltage from the charge accumulation.

• The first three harmonics of mass defects induce adverse vibrations in shell resonators. • Extracts eccentricity-induced vibration from composite signals leveraging dynamic properties. • Presents a streamlined, low-damage approach to identify eccentric mass, bypassing the need for initial frequency split trimming. • Establishes an integrated platform combining harmonic identification with ion beam etching for mass balancing. • The quality factors are improved by suppressing the undesired mode, demonstrating its potential for engineering applications. 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. [Display omitted] [ABSTRACT FROM AUTHOR]

We present a new mathematical model for a vibrating ring gyroscope (VRG) in the presence of imperfections, namely, structural defects and material anisotropy. As a novelty, we calculate the mode shapes of the internal suspension structure to enable a more accurate and modular analysis of the VRG’s mass and stiffness distributions. Solving the associated eigenvalue problem shows that imperfections result in the frequency split between the gyroscope’s operating mode shapes, rotating their orientation with respect to the nominal drive and sense axes. We then use perturbation analysis to solve the VRG’s equations of motion and analyze the quadrature error that arises from frequency/damping mismatch between the mode shapes. We use our model to detail the various effects of the etching-related undercuts, structural uncertainties, and Young’s modulus anisotropy–in the form of suitable space-dependent functions–on the mode shapes and the quadrature error for the first time. The results reveal that rings are robust against imperfection, while the straight beams used in the suspension system are most likely responsible for the frequency split and quadrature error. For example, 50 nm (0.5%) width variation in a beam that connects the VRG’s suspension to an anchored internal structure leads to 4700°/s quadrature error. To validate our modeling, using the experimental data from a fabricated 59 kHz VRG, we provide rigorous, comparative simulations against the finite element method (FEM). [2022-0035] [ABSTRACT FROM AUTHOR]

This paper proposes a novel gear-like disk resonator (GDR). The design, fabrication, and characterization of GDR are presented. In comparison with a ring-like disk resonator (RDR), a GDR replaces the circular rings with meander-shaped rings consisting of linear beams. The finite element method (FEM) is implemented, and the simulation results show that the GDR has a much lower frequency and effective stiffness, higher quality factor (Q), and better immunity to crystal orientation error. Affected by high Q and small frequency splits, the mechanical sensitivity (Smech) is shown to increase greatly. GDR and RDR with the same structure parameters are built side-by-side on the same wafer, and prototypes are fabricated through the SOI fabrication technique. The frequency response test and ring-down test are implemented using a readout circuit under a vacuum condition (5 Pa) at room temperature. The frequency split (9.1 Hz) of the GDR is about 2.8 times smaller than that (25.8 Hz) of the RDR without electrostatic tuning. Compared with the RDR, the Q (19.2 k) and decay time constant (0.59 s) of the GDR are improved by 145% and 236%, respectively. The experimental results show great promise for the GDR being used as a gear-like disk resonator gyroscope (GDRG). [ABSTRACT FROM AUTHOR]

MEMS gyroscopes play an important role in inertial navigation measurements, which mainly works in n = 2 mode. However, mode matching is the basis for high-precision detection, which can improve the sensitivity, resolution, and signal-to-noise ratio of the gyroscopes. An initial frequency split is inevitably generated during the manufacturing process. There are two methods to eliminate the frequency split and to achieve mode matching for the gyroscopes, which are electrostatic tuning and mechanical trimming, respectively. In this paper, we report a novel ring MEMS resonator and a novel method of mechanical frequency tuning. The most prominent characteristic of the resonator is that 16 raised mass blocks are increased in the circumferential positions of the ring uniformly. This structural design can achieve mass-stiffness decoupling, which means that punching holes on the mass blocks only affects the mass distribution but the stiffness is almost unchanged for the resonator. We verify the mass-stiffness decoupling by way of comparing the simulation with the conventional resonator. In addition, we put up an online tuning platform based on a femtosecond laser and reduce a resonator's frequency split from 23.3 Hz to 0.4 Hz, which reveals that the frequency split is linearly related to the removed mass. These findings will have a referential significance for other transducers. [ABSTRACT FROM AUTHOR]

A cylindrical resonator gyroscope is a kind of Coriolis gyroscope, which measures angular velocity or angle via processing of the standing wave. The symmetry of a cylindrical resonator is destroyed by different degrees of geometric nonuniformity and structural damage in the machining process. The uneven mass distribution caused by the asymmetry of the resonator can be expressed in the form of a Fourier series. The first three harmonics will reduce the anti-interference ability of the resonator to the external vibration, as well as increase the angular random walk and zero-bias drift of the gyroscope. In this paper, the frequency split of different modes caused by the first three harmonic errors and the displacement of the center of the cylindrical resonator bottom plate are obtained by simulation, and the relationship between them is explored. The experimental results on five fused silica cylindrical resonators are consistent with the simulation, confirming the linear relationship between the n = 1 frequency split and second harmonic error. A method for evaluating the first three harmonic errors of fused silica cylindrical resonators is provided.

Axisymmetric resonators are key elements of Coriolis vibratory gyroscopes (CVGs). The performance of a CVG is closely related to the stiffness and damping symmetry of its resonator. The stiffness symmetry of a resonator can be effectively improved by electrostatic tuning or mechanical trimming, both of which need an accurate knowledge of the azimuth angles of the two stiffness axes of the resonator. Considering that the motion of a non-ideal axisymmetric resonator can be decomposed as two principal oscillations with two different natural frequencies along two orthogonal stiffness axes, this paper introduces a novel high-precision method of stiffness axes identification. The method is based on measurements of the phase difference between the signals detected at two orthogonal sensing electrodes when an axisymmetric resonator is released from all the control forces of the force-to-rebalance mode and from different initial pattern angles. Except for simplicity, our method works with the eight-electrodes configuration, in no need of additional electrodes or detectors. Furthermore, the method is insensitive to the variation of natural frequencies and operates properly in the cases of either large or small frequency splits. The introduced method is tested on a resonator gyroscope, and two stiffness axes azimuth angles are obtained with a resolution better than 0.1°. A comparison of the experimental results and theoretical model simulations confirmed the validity of our method.

This paper proposes a novel gear-like disk resonator (GDR). The design, fabrication, and characterization of GDR are presented. In comparison with a ring-like disk resonator (RDR), a GDR replaces the circular rings with meander-shaped rings consisting of linear beams. The finite element method (FEM) is implemented, and the simulation results show that the GDR has a much lower frequency and effective stiffness, higher quality factor (Q), and better immunity to crystal orientation error. Affected by high Q and small frequency splits, the mechanical sensitivity (Smech) is shown to increase greatly. GDR and RDR with the same structure parameters are built side-by-side on the same wafer, and prototypes are fabricated through the SOI fabrication technique. The frequency response test and ring-down test are implemented using a readout circuit under a vacuum condition (5 Pa) at room temperature. The frequency split (9.1 Hz) of the GDR is about 2.8 times smaller than that (25.8 Hz) of the RDR without electrostatic tuning. Compared with the RDR, the Q (19.2 k) and decay time constant (0.59 s) of the GDR are improved by 145% and 236%, respectively. The experimental results show great promise for the GDR being used as a gear-like disk resonator gyroscope (GDRG).

MEMS gyroscopes play an important role in inertial navigation measurements, which mainly works in n = 2 mode. However, mode matching is the basis for high-precision detection, which can improve the sensitivity, resolution, and signal-to-noise ratio of the gyroscopes. An initial frequency split is inevitably generated during the manufacturing process. There are two methods to eliminate the frequency split and to achieve mode matching for the gyroscopes, which are electrostatic tuning and mechanical trimming, respectively. In this paper, we report a novel ring MEMS resonator and a novel method of mechanical frequency tuning. The most prominent characteristic of the resonator is that 16 raised mass blocks are increased in the circumferential positions of the ring uniformly. This structural design can achieve mass-stiffness decoupling, which means that punching holes on the mass blocks only affects the mass distribution but the stiffness is almost unchanged for the resonator. We verify the mass-stiffness decoupling by way of comparing the simulation with the conventional resonator. In addition, we put up an online tuning platform based on a femtosecond laser and reduce a resonator’s frequency split from 23.3 Hz to 0.4 Hz, which reveals that the frequency split is linearly related to the removed mass. These findings will have a referential significance for other transducers.

This study investigated the effects of structural dimension variation arising from fabrication imperfections or active structural design on the vibration characteristics of a (100) single crystal silicon (SCS) ring-based Coriolis vibratory gyroscope. A mathematical model considering the geometrical irregularities and the anisotropy of Young’s modulus was developed via Lagrange’s equations for simulating the dynamical behavior of an imperfect ring-based gyroscope. The dynamical analyses are focused on the effects on the frequency split between two vibration modes of interest as well as the rotation of the principal axis of the 2θ mode pair, leading to modal coupling and the degradation of gyroscopic sensitivity. While both anisotropic Young’s modulus and nonideal deep trench verticality affect the frequency difference between two vibration modes, they have little contribution to deflecting the principal axis of the 2θ mode pair. However, the 4θ variations in the width of both the ring and the supporting beams cause modal coupling to occur and the degenerate 2θ mode pair to split in frequency. To aid the optimal design of MEMS ring-based gyroscopic sensors that has relatively high robustness to fabrication tolerance, a geometrical compensation based on the developed model is demonstrated to identify the geometries of the ring and the suspension.

Due to the blur information and content information entanglement in the blind deblurring task, it is very challenging to directly recover the sharp latent image from the blurred image. Considering that in the high-dimensional feature map, blur information mainly exists in the low-frequency region, and content information exists in the high-frequency region. In this paper, we propose a encoder–decoder model to realize disentanglement from the perspective of frequency, and we named it as frequency disentanglement distillation image deblurring network (FDDN). First, we modified the traditional distillation block by embedding the frequency split block (FSB) in the distillation block to separate the low-frequency and high-frequency region. Second, the modified distillation block, we named frequency distillation block (FDB), can recursively distill the low-frequency feature to disentangle the blurry information from the content information, so as to improve the restored image quality. Furthermore, to reduce the complexity of the network and ensure the high-dimension of the feature map, the frequency distillation block (FDB) is placed on the end of encoder to edit the feature map on the latent space. Quantitative and qualitative experimental evaluations indicate that the FDDN can remove the blur effect and improve the image quality of actual and simulated images.

• We propose a systematic frequency trimming process including stiffness modification and mass modification. • We apply FEM simulation to calculate the change of mode eigenfrequency and verify the trimming process. • We set up frequency trimming experiments on different MSRs with T-shape masses to demonstrate the effectiveness of this method. This paper presents a novel method to achieve permanent frequency trimming between wineglass modes of micro shell resonator (MSR) with T-shape masses (T-masses) using femtosecond laser ablation. Trimming effects on stiffness and mass distribution around the rim are analyzed by finite element method (FEM). Stiffness and mass trimming can be achieved by grooves ablating in different positions. A systematic frequency trimming process including stiffness and mass modification is proposed. FEM simulation is further applied to estimate the variation of cross coupling and frequency split during the trimming process. High resolution trimming without surface deterioration is achieved by using a low-power femtosecond laser. Finally, frequency trimming experiments were implemented on imperfect resonators. Test results show that frequency split can be effectively reduced to less than 0.5Hz within several steps, demonstrating the effectiveness of this method. [ABSTRACT FROM AUTHOR]

The cylindrical resonator gyroscope (CRG) is a type of Coriolis vibratory gyroscope which measures the angular velocity or angle through the precession of the elastic wave of the cylindrical resonator. The cylindrical fused silica resonator is an essential component of the CRG, the symmetry of which determines the bias drift and vibration stability of the gyroscope. The manufacturing errors breaking the symmetry of the resonator are usually described by Fourier series, and most studies are only focusing on analyzing and reducing the fourth harmonic error, the main error source of bias drift. The second harmonic error also is one of the obstacles for CRG towards high precision. Therefore, this paper provides a chemical method to evaluate and balance the second harmonic error of cylindrical fused silica resonators. The relation between the frequency split of the n = 1 mode and the second harmonic error of the resonator is obtained. Simulations are performed to analyze the effects of the first three harmonic errors on the frequency splits. The relation between the location of the low-frequency axis of n = 1 mode and the heavy axis of the second harmonic error is also analyzed by simulation. Chemical balancing experiments on two fused silica resonators demonstrate the feasibility of this balancing procedure, and show good consistency with theoretical and simulation analysis. The second harmonic error of the two resonators is reduced by 86.6% and 79.8%, respectively.

The resonator of a solid-state vibratory gyro is responsible for sensing angular motion. Frequency splitting of an axisymmetric-shell resonator is a common problem caused by manufacturing defects. The defect causes a frequency difference between two working modes which consist of two nodes and two antinodes. The difference leads to the loss of gyroscopic effect, and thus the resonator cannot sense angular motion. In this paper, the resonator based on an axisymmetric multi-curved surface shell structure is investigated and an approach to eliminate frequency splits is proposed. Since axisymmetric multi-curved surface shell resonators are too complex to be modeled, this paper proposes a simplified model by focusing on a common property of the axisymmetric shell. The resonator with stochastic imperfections is made equivalent to a perfect shell with an imperfect mass point. Rayleigh’s energy method is used in the theoretical analysis. Finite element modeling is used to demonstrate the effectiveness of the elimination approach. In real cases, a resonator’s frequency split is eliminated by the proposed approach. In this paper, errors in the theoretical analysis are discussed and steps to be taken when the deviation between assumptions and the real situation is large are figured out. The resonator has good performance after processing. The elimination approach can be applied to any kind of solid-state vibratory gyro resonators with an axisymmetric shell structure.

An automatic mode-matching method for MEMS (Micro-electromechanical Systems) disk resonator gyroscopes (DRGs) based on virtual Coriolis force is presented in this paper. For this mode-matching method, the additional tuning electrodes are not required to be designed, which simplifies the structure design. By using the quadratic relationship between the driving voltage and the electrostatic force, the virtual Coriolis force is obtained by applying an AC voltage whose frequency is half of the driving mode resonant frequency to the sense electrode. The phase difference between the virtual Coriolis force and the sense output signal is used for mode-matching. The structural characteristics and electrode distribution of the DRG are briefly introduced. Moreover, the mode-matching theories of the DRG are studied in detail. The scheme of the mode-matching control system is proposed. Simultaneously, the feasibility and effectiveness of the mode-matching method are verified by system simulation. The experimental results show that under the control of mode-matching at room temperature, the bias instability is reduced from 30.7575 ° /h to 2.8331 ° /h, and the Angle Random Walk (ARW) decreases from 1.0208 ° / h to 0.0524 ° / h . Compared with the mode mismatch condition, the ARW is improved by 19.48 times.

The cylindrical resonator gyroscope (CRG) is a kind of solid-state gyroscope with a wide application market. The cylindrical resonator is the key component of CRG, whose quality factor and symmetry will directly affect the performance of the gyroscope. Due to the material properties and fabrication limitations, the actual resonator always has some defects. Therefore, frequency trimming, i.e., altering the local mass or stiffness distribution by certain methods, is needed to improve the overall symmetry of the resonator. In this paper, we made further derivation based on the chemical trimming theory proposed by Basarab et al. We built up the relation between the frequency split and the balanced mass to determine the mass to be removed. Chemical trimming experiments were conducted on three cylindrical fused silica resonators. The frequency splits of the three resonators were around 0.05 Hz after chemical trimming. The relation between frequency split and balanced mass established from experimental data was consistent with the theoretical calculation. Therefore, frequency split can be reduced to lower than 0.05 Hz under rigorous theoretical calculation and optimized chemical trimming parameters.

Fused silica cylindrical resonant gyroscope (CRG) is a novel high-precision solid-wave gyroscope, whose performance is primarily determined by the cylindrical resonator’s frequency split and quality factor (Q factor). The laser Doppler vibrometer (LDV) is extensively used to measure the dynamic behavior of fused silica cylindrical resonators. An electrical method was proposed to characterize the dynamic behavior of the cylindrical resonator to enhance the measurement efficiency and decrease the equipment cost. With the data acquisition system and the designed signal analysis program based on LabVIEW software, the dynamic behavior of the fused silica cylindrical resonator can be analyzed automatically and quickly. We compared all the electrical measurement results with the optical detection by LDV, demonstrating that the fast Fourier transform (FFT) result of the resonant frequency measured by the electrical method was 0.12 Hz higher than that with the optical method. Thus, the frequency split measured by the electrical and optical methods was the same in 0.18 Hz, and the measurement of the Q factor was basically the same in 730,000. We conducted all measurements under the same operation condition, and the optical method was used as a reference, demonstrating that the electrical method could characterize the dynamic behavior of the fused silica cylindrical resonator and enhance the measurement efficiency.

Vibratory cylinder gyroscope (VCG) is an important type of vibratory gyroscopes with a lot of advantages. Manufacturing errors and material inhomogeneity are regarded as the main cause of the zero bias drift of the VCG. In this paper, the nonlinear vibration of the cylindrical resonator is considered. It is analyzed that the nonlinear vibration can cause additional drift of the VCG’s output. Experiments are implemented in this research. According to the testing results, the metal cylindrical resonator is a soft nonlinear vibration system. To increase the stability of the VCG, the cylindrical resonator should be trimmed to a specific frequency split. [ABSTRACT FROM AUTHOR]

In this paper, we present two design methods to improve the performance of disk resonator gyroscope (DRG), including decreasing the frequency split and increasing the quality factor (Q). The structure parameters, which can affect the frequency split and Q value were concluded with the help of the FEM software. Meanwhile, devices with different parameters were designed, fabricated, and tested, and the experimental result was in accordance with the simulation. With the proposed methods, the DRG was selected with a high Q value and a low frequency split to satisfy the demand of high performance. The weakness and future works were pointed at last. [ABSTRACT FROM AUTHOR]

This paper introduces a utility algorithm to reduce the permanent frequency mismatch of the primary wineglass modes in a disk resonator gyroscope via stiffness perturbation. In this algorithm, each tuning electrode is regarded as two rows of radial springs with negative stiffness. On the basis of the tuning model for a ring with radial springs, we can obtain the functional relation between equivalent stiffness or voltage of each tuning electrode and the frequency split. Corresponding simulations are conducted to demonstrate the performance of the algorithm. Finally, the electrostatic tuning experiment is presented to further verify the algorithm and illustrate the use of the tuning procedure. Results of experiment show that the frequency split of resonator is decreased to smaller than 0.03 Hz from original about 15 Hz. [ABSTRACT FROM PUBLISHER]

This paper analyzes the effect of the anisotropy of single crystal silicon on the frequency split of the vibrating ring gyroscope, operated in the n = 2 wineglass mode. Firstly, the elastic properties including elastic matrices and orthotropic elasticity values of (100) and (111) silicon wafers were calculated using the direction cosines of transformed coordinate systems. The (111) wafer was found to be in-plane isotropic. Then, the frequency splits of the n = 2 mode ring gyroscopes of two wafers were simulated using the calculated elastic properties. The simulation results show that the frequency split of the (100) ring gyroscope is far larger than that of the (111) ring gyroscope. Finally, experimental verifications were carried out on the micro-gyroscopes fabricated using deep dry silicon on glass technology. The experimental results are sufficiently in agreement with those of the simulation. Although the single crystal silicon is anisotropic, all the results show that compared with the (100) ring gyroscope, the frequency split of the ring gyroscope fabricated using the (111) wafer is less affected by the crystal direction, which demonstrates that the (111) wafer is more suitable for use in silicon ring gyroscopes as it is possible to get a lower frequency split.

The hemispherical resonator gyroscope (HRG) has attracted the interest of the world inertial navigation community because of its exceptional performance, ultra-high reliability and its potential to be miniaturized. These devices achieve their best performance when the differences in the frequencies of the two degenerate working modes are eliminated. Mechanical treatment, laser ablation, ion-beams etching, etc., have all been applied for the frequency tuning of resonators, however, they either require costly equipment and procedures, or alter the quality factors of the resonators significantly. In this paper, we experimentally investigated for the first time the use of a chemical etching procedure to decrease the frequency splits of hemispherical resonators. We provide a theoretical analysis of the chemical etching procedure, as well as the relation between frequency splits and mass errors. Then we demonstrate that the frequency split could be decreased to below 0.05 Hz by the proposed chemical etching procedure. Results also showed that the chemical etching method caused no damage to the quality factors. Compared with other tuning methods, the chemical etching method is convenient to implement, requiring less time and labor input. It can be regarded as an effective trimming method for obtaining medium accuracy hemispherical resonator gyroscopes.

A mathematical model is developed to describe the 2? in-plane flexural response of supported ring-based Coriolis Vibrating Gyroscopes (CVGs) as the ring is driven into large amplitude vibration. Whilst the 2? degenerate modes have same resonance frequency in the linear regime, mechanical non-linearity within the support structure induces a frequency split as the vibration amplitude increases. The origins and effects of geometrical non-linearity are investigated using the proposed analytical model.

In order to eliminate the frequency mismatch of MEMS (Microelectromechanical Systems) gyroscopes, this paper proposes a frequency tuning technology based on a quadrature modulation signal. A sinusoidal signal having a frequency greater the gyroscope operating bandwidth is applied to the quadrature stiffness correction combs, and the modulation signal containing the frequency split information is then excited at the gyroscope output. The effects of quadrature correction combs and frequency tuning combs on the resonant frequency of gyroscope are analyzed. The tuning principle based on low frequency input excitation is analyzed, and the tuning system adopting this principle is designed and simulated. The experiments are arranged to verify the theoretical analysis. The wide temperature range test (-20 ∘ C –60 ∘ C ) demonstrates the reliability of the tuning system with a maximum mismatch frequency of less than 0.3 Hz. The scale factor test and static test were carried out at three temperature conditions (−20 ∘ C, room temperature, 60 ∘ C), and the scale factor, zero-bias instability, and angle random walk are improved. Moreover, the closed-loop detection method is adopted, which improves the scale factor nonlinearity and bandwidth under the premise of maintaining the same static performances compared with the open-loop detection by tuning.

In this paper, vibration of imperfect resonators for vibratory cylinder gyroscopes is investigated. A model of the vibration based on the wave superposition is established. For an imperfect resonator, the principal stiffness axis does not coincide with the driving electrodes, so the primary mode and the secondary mode of the resonator are both excited. Superposition of these two pattern components produces the resultant pattern, which leads to a complicated vibration near the nodes and the pattern angle error. The influence of the frequency split and the driving vector orientation on the nodal vibration and the pattern angle error is particularly analyzed. Theoretical results show that the amplitude and phase of the nodal vibration are affected by the frequency split and the driving vector orientation. The pattern angle error has a positive correlation with the frequency split, and this angle also has a positive proportion with sin(8α), where α is the angle between the drive electrodes and the principal stiffness axis of the primary mode. Experiments are also included in this work to validate these influence factors. The theoretical and measured results are in close agreement. These results are useful for the mechanical balance, the electrodes assembly and the control circuit design of the cylindrical gyroscopes. [ABSTRACT FROM AUTHOR]

This paper presents, for the first time, the analysis, design, fabrication, and characterization of a microhemispherical resonating gyroscope ( \mu HRG). A 3D high aspect-ratio poly- and single-crystalline silicon (HARPSS) process is developed to fabricate a stem-supported hemispherical shell with self-aligned tall capacitive electrodes intended for driving, sensing, and tuning of the gyroscope. The monolithic process consists of seven lithographic steps combining the HARPSS process with fabrication of a 3D curved microstructure to create electrodes with scalable capacitive gaps. The fabrication is successfully demonstrated with polysilicon \mu HRG prototypes with a diameter of 1.2 mm and a shell thickness of 1 \mu \textm . The frequency response is measured and gyro operation is examined using the integrated electrodes. Mode matching and mode alignment are successfully performed by applying tuning and quadrature nulling voltages. A closed-loop rate sensitivity scale factor of 8.57 mV/°/s was measured. Furthermore, design and process optimization of the support structure resulted in an improved quality factor ( Q \sim 40\,000 ). [ABSTRACT FROM PUBLISHER]

In this paper, we present four design methods to overcome (100) silicon crystalline anisotropy and achieve mode-matching in wineglass-mode disk resonator gyroscope (DRG). These methods were validated through experimental characterization of more than 145 different devices that arose from simulations. With the proposed methods, the frequency split of the 250-kHz DRG wineglass modes in (100) silicon was reduced from >10 kHz to as low as 96 Hz (<0.04% of 250-kHz resonant frequency) without any electrostatic tuning. Perfect mode-matching is then achieved using electrostatic tuning. Mode-matching was maintained within ±10 Hz over a temperature range from −20 °C to 80 °C. The temperature dependence of quality factor is also discussed in this paper. These results allow for the development of high-performance miniature DRGs tuned for degenerate wineglass mode operation from high-quality crystalline silicon material. [2013-0303] [ABSTRACT FROM AUTHOR]

The bell-shaped vibratory angular rate gyro (abbreviated as BVG) is a novel type of solid-state vibratory gyroscope (SVG), which has a three-dimensional sensitive structure as its resonator. Strict symmetry of the 3-D sensitive structure is the key factor to ensure the gyro performance. However, mass imperfection always exists in a physical structure. The aim of this study is to investigate the influence of mass imperfection on gyro performance by analyzing the vibration of standing wave in imperfect resonator, as well as methods to eliminate the influence. Two important influence factors of frequency split and vibration mode offset caused by mass imperfection and the resulting gyro error are analyzed by building the imperfect ring model. The vibration process of imperfect BVG resonator and the quantitative relationship between mass deviation and the two factors above are studied using FEM simulation. Finally, a practical balance method for BVG based on addition of mass is proposed and validated by specific engineering experiments. The results demonstrate that the method is both feasible and easy to implement. Conclusions of this study will contribute to theoretical research and practical development work of BVG. [ABSTRACT FROM AUTHOR]

• Performance parameters testing of a resonator is independent of different excitation methods. • All performance parameters of the resonator can be simultaneously measured in a single test. • Low-pass filtering of vibration signals does not affect the results of Q factor measurement. • The calculation method of the rigid axes angle proposed in this paper can achieve an accuracy of 0.01°. The hemispherical resonator is an important component of hemispherical resonator gyro (HRG). The accuracy of resonator performance detection determines HRG service level. In this paper, a method for calculating the rigid axes angle of the resonator is proposed. The reliability of different test methods with shock and harmonic excitations is compared and analyzed. Firstly, a simplified two-dimension spring-mass equivalent modal of the resonator is developed. Secondly, theoretical analysis of the resonator vibration characteristics with different excitations is conducted. The calculation equation of rigid axes angle is next derived. Furthermore, simulation analysis of resonator performance parameters is carried out based on the state space method. Finally, the vibration characteristic testing device of a resonator is established, and the feasibility of the theoretical analysis is verified. Results indicate that the resonator performance parameters are unaffected by different excitation methods. And the proposed calculation method can reach an angle measurement accuracy of 0.01°. [ABSTRACT FROM AUTHOR]

We demonstrate, for the first time, sub-1 Hz frequency symmetry in micro-glassblown wineglass resonators with integrated electrode structures. A new fabrication process based on deep glass dry etching was developed to fabricate micro-wineglasses with self-aligned stem structures and integrated electrodes. The wineglass modes were identified by electrostatic excitation and mapping the velocity of motion along the perimeter using laser Doppler interferometry. A frequency split (Δf) of 0.15 and 0.2 Hz was demonstrated for n=2 and n=3 wineglass modes, respectively. To verify the repeatability of the results, a total of five devices were tested, three out of five devices showed . Frequency split stayed below 1 Hz for dc bias voltages up to 100 V, confirming that the low frequency split is attributed to high structural symmetry and not to capacitive tuning. High structural symmetry and atomically smooth surfaces (0.23 nm Sa) of the resonators may enable new classes of high performance 3-D MEMS devices, such as rate-integrating MEMS gyroscopes. [ABSTRACT FROM AUTHOR]