摘要
半球谐振陀螺仪(Hemispherical Resonator Gyroscope，HRG)具有的长寿命、高精度、高稳定性、低噪声、低功耗、抗辐射等优点，使其在长寿命惯性导航中具有很大的应用前景。因此，对HRG进行误差机理分析及在现有技术条件下寻找进一步提高HRG精度的方法具有重要的意义。本文以半球谐振子的误差源为研究对象，建立了具有误差缺陷谐振子的动力学方程，据此深入研究了谐振子频率裂解与密度不均匀误差之间的关系。然后从以下三个方面重点研究了HRG误差抑制方法：控制谐振子不同的振动位移分量，调节激励电极相对于谐振子的位置，平衡谐振子参数的不均匀误差。此外还深入研究了陀螺仪角速率解算方法及解算误差补偿方法等，为提高HRG的使用精度提供了理论指导。本文的具体工作如下：
建立了半球谐振子的动力学方程。首先推导出谐振子薄壳上任一点的惯性力表达式，然后通过选取谐振子中曲面微元并分析其受力状态，建立了谐振子薄壳单元的力平衡方程。在此基础上，利用布勃诺夫-加廖尔金法建立了谐振子振动的动力学方程，并推导得出了谐振子振动振型精确的进动因子，为研究HRG误差机理及误差抑制方法奠定了基础。
谐振子频率裂解及HRG开环模式的误差分析。首先建立了环向密度分布不均匀谐振子的动力学方程，据此推导出谐振子固有频率极大值与极小值的表达式，进而得到了谐振子固有频率裂解与环向密度不均匀误差之间的关系。其次在开环位置激励模式下，推导波腹方位角的稳态解和角速率解算表达式，通过仿真得到开环控制模式下为保证HRG角速率解算精度的密度不均匀第四次谐波的允差。
研究了HRG力反馈控制模式下，通过不同控制模式控制谐振子不同的振动位移分量来抑制密度不均匀误差的方法，并给出了HRG角速率解算方法。首先推导了普通控制模式下角速率解算表达式，并通过仿真分析了密度不均匀第四次谐波对角速率解算精度的影响。其次推导了正交控制模式下角速率解算表达式。针对开环控制、普通控制及正交控制各自存在的问题，提出了拟正交控制模式，并推导了拟正交控制模式下角速率解算表达式。通过仿真分析，得出了拟正交控制模式在误差抑制方面的优势，并分析了其在工程中的可实行性。此外还详细推导了拟正交控制模式下波腹方位角表达式和振幅控制的稳态模型。
研究了HRG精度提高方法及密度不均匀误差的平衡。首先根据固有刚性轴的特性得到了在谐振子密度不均匀误差下固有刚性轴的方位。然后在陀螺仪开环控制模式及拟正交控制模式下，通过调节激励电极与固有刚性轴的夹角，得到了提高陀螺仪精度的方法，并通过仿真验证其有效性。为了消除密度不均匀误差，提高HRG精度，建立了含有谐振子密度不均匀1~4次谐波成分的动力学方程，根据动力学方程建立了辨识密度不均匀各次谐波的模型，进而依据辨识模型详细论述辨识密度不均匀各次谐波幅值及相位的方法。在对密度不均匀各次谐波的幅值及相位辨识后，详细论述并推导了通过在谐振子上去除质量点以平衡谐振子密度不均匀误差的方法。
HRG误差模型的建立与误差分析。首先建立了含谐振子固有参数不均匀、尺寸参数不均匀及激励参数误差的动力学方程，据此得到辨识各误差的方法及动力学方程中的非等弹性与非等阻尼误差项，并提出了平衡谐振子非阻尼类误差的方法。然后通过数值计算得到了谐振子频率裂解达到一定要求时谐振子各参数不均匀误差的最大值，并计算了当陀螺仪工作在开环控制模式与拟正交控制模式下满足惯导级陀螺仪要求时，谐振子各参数不均匀误差的允差。最后提出了提高谐振子阻尼不均匀误差允差的方法，从而降低了加工工艺难度。
HRG has a promising application in the inertial navigation of long-lifespacecraft due to its advantages, such as longevity, high precision, high reliability,low noise, low power and free of nuclear radiation. So it is meaningful to analyzethe HRG’s error and to explore the methods of raising HRG’s accuracy under currenttechnical conditions. In this paper, the error analysis for HRG is based on theresonator with defects in the circular direction. The dynamics equations of resonatorwith error are established, then the relationship between the frequency split and thedensity nonuniformity error is deeply analyzed. The error restraint method isresearched from the following three aspects: control different parts of theresonator’s vibration displacement, adjust the position of the excitation electroderelative to the resonator, balance the resonator’s nonuniformity error. Moreover, themethods of angular rate calculation and error compensation are deeply researched,which give the guidance for raising HRG’s accuracy. The specific tasks of thisdissertation are as follows:
The dynamics equations of resonator are established. At first, the inertia forcesof an arbitrary point in the hemispherical thin shell are deduced, then the forceequilibrium equations of the thin shell element which is sectioned from the mediumplane are gained. On the basis of it, the dynamics equations of the resonator areestablished by way of Bubonov–Galerkin method, and the accurate expression of theangular precession factor is deduced. All those lay a foundation for further analyzingthe error mechanism and error restraint method of HRG.
Analysis on the frequency split and the error analysis for HRG under open-loopcontrol mode are achieved. Firstly, the dynamics equations of resonator with densitydistributed nonuniformity are established. According to the dynamics equations, theexpressions of natural frequency of hemispherical resonator are solved. Thefrequency split mechanism caused by density distribution nonuniformity error of thehemispherical resonator is achieved. Secondly, under open-loop mode, thesteady-state solutions of the antinode’s azimuth and the expression of the angularrate calculation are given. By simulation calculating, the tolerance towards thefourth harmonic of density nonuniformity is determined to guarantee the accuracy ofHRG under open-loop mode.
Under force-rebalance mode, the method that different parts of the resonator’svibration displacement are controlled by different modes to restrain densitynonuniformity error is researched, and the method of angular rate calculation isgiven. Firstly, the expression of the angular rate calculation is given under general control. By simulation calculating, the influence of the fourth harmonic of densitynonuniformity on angular rate calculation is analyzed. Secondly, the expression ofthe angular rate calculation is given under quadrature control. The quasi quadraturecontrol is put forward to solve the problems of open-loop control, general controland quadrature control. The expression of the angular rate calculation is deducedunder quasi quadrature control. By simulation and analyzing, the quasi quadraturecontrol has more advantages in the aspects of error restraint and practicalapplication. In addition, under the quasi quadrature control, the expression of theazimuth of radial vibration antinode is deduced, and the steady state model ofamplitude-control is established.
The methods of enhancing HRG’s accuracy and balancing the densitynonuniformity error are put forward. At first, according to the characteristics of theinherent toughness axis, the azimuth of inherent toughness axis is given. Then themethods of raising HRG’s accuracy are given by adjusting the angle between theexcitation electrode and the azimuth of inherent toughness axis under open-loopcontrol and quasi quadrature control modes, and the method’s effectiveness isproved by simulating. To counteract density nonuniformity error and raise theaccuracy of HRG, the dynamics equations of resonator with the1st~4thharmonic ofdensity nonuniformity are established. The identification model of densitynonuniformity error is established through the dynamics equations, and then theamplitude and phase of each density nonuniformity harmonic are identified. Afterthat, the process of balancing the density nonuniformity error by removing the micromass out of resonator is analyzed and deduced.
The establishment of error model and error analysis for HRG are conducted.Firstly, the dynamics equations of resonator with the nonuniformity errors ofinherent parameters, dimension parameters and excitation parameters are established.The method of identifying the errors is given. The anisoelasticity and dampingnonuniformity are given through the dynamics equations. And the method ofbalancing the errors except damping nonuniformity error is also given. Secondly, themaximum values of errors are calculated under the requirement of frequency split,and the tolerances towards errors are determined to guarantee the accuracy of HRGunder open-loop control and quasi quadrature control modes. Finally, the method ofincreasing the tolerance towards damping nonuniformity error is given, so theprocessing and machining difficulty is reduced.
引文
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