Working Principle and Error Analysis for Hemispherical Resonator Gyro under Force-rebalance Mode

The two control methods, namely the general-control and the quadrature-control modes for HRG under force-rebalance mode were introduced firstly. Then the azimuth of antinode on the hemispherical resonator was deduced. The dynamics equations of resonator under the nonuniformity of density distribution were established by way of Bubonov-Galerkin method which is commonly used for solution of differential equations, and the state equation was established through the dynamics equations. The analytic solutions of the vibration displacement and the velocity were achieved by solving the state equation, and then the ratio of rebalance excitation to primary excitation was derived under the two working modes, thus the estimation of input angular rate of HRG were obtained. By comparing and calculating these two modes, the error caused by resonator's machining defects can be greatly inhibited under quadrature-control, and the fourth harmonic density error's tolerance were calculated to ensure the accuracy of HRG under these two modes.

Motion equations of hemispherical resonator and analysis of frequency split caused by slight mass non-uniformity

The mass non-uniformity of hemispherical resonator is one of reasons for frequency split,and frequency split can cause gyroscope to drift.Therefore,it is of great significance to analyze the relationship between mass non-uniformity and frequency split,which can provide a theoretical basis for mass balance of imperfect resonator.The starting point of error mechanism analysis for gyroscope is the motion equations of resonator.Firstly,based on the Kirchhoff-Love hypothesis in the elastic thin shell theory,the geometric deformation equations of resonator are deduced.Secondly,the deformation energy equation of resonator is derived according to the vibration mode and relationship between the stress and strain of hemispherical thin shell.Thirdly,the kinetic energy equation of resonator is deduced by the Coriolis theorem.Finally,the motion equations of resonator are established by the Lagrange mechanics principle.The theoretical values of precession factor and natural frequency are calculated by the motion equations,which are substantially consistent with the ones by the finite element method and practical measurement,the errors are within a reasonable range.Simultaneously,the varying trend of natural frequency with respect to the geometrical and physical parameters of resonator by the motion equations is consistent with that by the finite element analysis.The above conclusions prove the correctness and rationality of motion equations.Similarly,the motion equations of resonator with mass non-uniformity are established by the same modeling method in case of ignoring the input angular rate and damping,and the state equations with respect to the velocity and displacement of vibration system are derived,then twonatural frequencies are solved by the characteristic equation.It is concluded that one of reasons for frequency split is the 4 th harmonic of mass non-uniformity,and thus much attention should be paid to minimizing the 4 th harmonic of mass non-uniformity in the course of mass balancing for imperfect resonator.

Standing wave drift of hemispherical resonator with quality factor non-uniformity under a ring electrode excitation

Hemispherical Resonator Gyroscope(HRG)is a classical high precision Coriolis Vibration Gyroscope(CVG),which performs attitude estimation of carrier by detecting the precession of standing wave of resonator,thus,the drift of standing wave of resonator has a great influence on the output accuracy of gyroscope,where the quality factor non-uniformity of resonator is one of main error sources.Ring electrode is a classical excitation structure of HRG because the standing wave can precess freely under its excitation,which makes the gyroscope have more accurate scale factor,larger measurement range and better dynamic characteristics.In this paper,the equations of motion of an ideal resonator excited by a ring electrode are derived by the elastic thin shell theory and Lagrange mechanical principle,then the corresponding equivalent mechanical model is established.According to the“average method”,it can be seen that the ideal resonator excited by the ring electrode works in integral mode,and any position in the circumferential direction of resonator can be a working point,which means that the quality factor non-uniformity has a great effect on the drift of standing wave.Therefore,the equations of motion of resonator with quality factor non-uniformity under the ring electrode excitation are deduced by the equivalent mechanical model,and the drift model of standing wave is established by the“average method”,it can be found that both the amplitude of quality factor non-uniformity and angle between the“inherent damping axis”and antinode axis of standing wave can affect the drift rate of standing wave.Moreover,the drift model indicates that if the input angular rate does not reach the threshold,the precession angular rate of standing wave will appear“self-locking”phenomenon,that is,the gyroscope will lose the integral effect.

Hemispherical resonator with low subsurface damage machined by small ball-end fine diamond grinding wheel:A novel grinding technique

As for the ultra-precision grinding of the hemispherical fused silica resonator,due to the hard and brittle nature of fused silica,subsurface damage(SSD)is easily generated,which enormously influences the performance of such components.Hence,ultra-precision grinding experiments are carried out to investigate the surface/subsurface quality of the hemispherical resonator machined by the small ball-end fine diamond grinding wheel.The influence of grinding parameters on the surface roughness(SR)and SSD depth of fused silica samples is then analyzed.The experimental results indicate that the SR and SSD depth decreased with the increase of grinding speed and the decrease of feed rate and grinding depth.In addition,based on the material strain rate and the maximum undeformed chip thickness,the effect of grinding parameters on the subsurface damage mechanism of fused silica samples is analyzed.Furthermore,a multi-step ultra-precision grinding technique of the hemispherical resonator is proposed based on the interaction influence between grinding depth and feed rate.Finally,the hemispherical resonator is processed by the proposed grinding technique,and the SR is improved from 454.328 nm to 110.449 nm while the SSD depth is reduced by 94%from 40μm to 2.379μm.The multi-step grinding technique proposed in this paper can guide the fabrication of the hemispherical resonator.

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 aφ-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.

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.

Hybrid Process of Fabricating High-Quality Micro Wine-Glass Fused Silica Resonators

A new hybrid method, which combines improved glass-blown technology with wet etching, is reported to fabricate micro wine-glass resonators with high-quality fused silica. The optimum placement is compared to achieve the resonators with good shell shape. The typical shell diameter is about 4 mm and its thickness covers from dozens to hundreds of micrometers. The etching rates in corrosion solutions with different ratios and at different thicknesses of hemispherical shells are studied. We also conclude how to precisely control the thickness.The corrosion solutions with different ratios of HF solution to NH4 F solution make the spherical shells rougher in different degrees. The best roughness is 0.581 nm in the 1 : 8 ratio corrosion solution while the original roughness is 0.537 nm. This fact shows that the resonator remains atomically smooth surface. Based on the glassblowing spherical fused silica structure, the thickness of the resonator is effectively controlled by buffered oxide etch(BOE)technology according to the measured etching rate. The measured resonant frequency of the hemispherical shell at ambient pressure and room temperature is 1.75 k Hz of rocking mode which is close to the simulated frequency.Using such a low-cost hybrid approach, we can fabricate high-quality microscale resonators in batch.