Study on free oscillations of a micromechanical gyroscope taking into account the nonorthogonality of the torsion axes

Introduction. The paper is devoted to the study on free oscillations of the sensing element of a micromechanical R-Rtype gyroscope of frame construction developed by the Kuznetsov Research Institute of Applied Mechanics, taking into account the nonorthogonality of the torsion axes. The influence of the instrumental manufacturing error on the accuracy of a gyroscope on a movable base in the case of free oscillations is studied. The work objective was to improve the device accuracy through developing a mathematical model of an R-R type micromechanical gyroscope, taking into account the nonorthogonality of the torsion axes, and to study the influence of this error on the device accuracy. The urgency of the problem of increasing the accuracy of micromechanical gyroscopes is associated with improving the accuracy of inertial navigation systems based on micromechanical sensors.

Materials and Methods. A new mathematical model that describes the gyroscope dynamics, taking into account the instrumental error of manufacturing the device, and a formula for estimating the error of a gyroscope, are proposed. The dependences of the state variables obtained from the results of modeling and on the basis of the experiment are presented. Methods of theoretical mechanics and asymptotic methods, including the Lagrange formalism and the Krylov-Bogolyubov averaging method, were used in the research.

Results. A new mathematical model of the gyroscope dynamics, taking into account the nonorthogonality of the torsion axes, is developed. The solution to the equations of small oscillations of the gyroscope sensing element and the estimate of the precession angle for the case of a movable base are obtained. A comparative analysis of the developed model and the experimental data obtained in the case of free oscillations of the gyroscope sensing element with a fixed base is carried out. The analysis has confirmed the adequacy of the constructed mathematical model. Analytical expressions are formed. They demonstrate the fact that the nonorthogonality of the torsion axes causes a cross-influence of the amplitudes of the primary vibrations on the amplitudes of the secondary vibrations of the sensing element, and the appearance of an additional error in the angular velocity readings when the gyroscope is operating in free mode.

Discussion and Conclusions. The results obtained can be used to improve the device accuracy using the algorithm for analytical compensation of the gyroscope error and the method for identifying the mathematical model parameters.

1. Peshekhonov VG. Gyroscopic navigation systems: Current status and prospects. Gyroscopy and Navigation. 2011;2(3):111–118.

2. Zhuravlev VF. Upravlyaemyi mayatnik Fuko kak model' odnogo klassa svobodnykh giroskopov. Izvestiya RAN. Mekhanika Tverdogo Tela = Mechanics of Solids. 1997;6:27–35. (In Russ.)

3. Raspopov VYa. Mikromekhanicheskie pribory. Moscow: Mashinostroenie; 2007. 400 p. (In Russ.)

4. Neapolitansky AS, Khromov BV. Mikromekhanicheskie vibratsionnye giroskopy. Moscow: Kogito-tsentr; 2002. 122 p. (In Russ.)

5. Merkuryev IV, Podalkov VV. Dinamika mikromekhanicheskogo i volnovogo tverdotel'nogo giroskopov. Moscow: Fizmatlit; 2009. 228 p. (In Russ.)

6. Yongmeng Zhang, Tongqiao Miao, Kechen Guo, et al. Electronic Gain Error Compensation for WholeAngle Coriolis Vibrating Gyroscopes With High Q Factor. In: Proc. IEEE Int. Symposium on Inertial Sensors and Systems (INERTIAL). 2020. P. 1–4. https://doi.org/10.1109/INERTIAL48129.2020.9090062

7. Apostolyuk V. Theory and Design of Micromechanical Vibratory Gyroscopes. MEMS/NEMS Handbook. 2006;1(6):173–195. 10.1007/0-387-25786-1_6

8. Wei Wang, Xiaoyong Lv, Feng Sun. Design of a Novel MEMS Gyroscope Array. Sensors. 2013;13(2):1651–1663. https://doi.org/10.3390/s130201651

9. Sina Askari, Mohammad H. Asadian, Andrei M. Shkel. High quality factor MEMS gyroscope with whole angle mode of operation. In: Proc. IEEE Int. Symposium on Inertial Sensors and Systems (INERTIAL). 2018;12(6):141-144. https://doi.org/10.1109/ISISS.2018.8358148

10. Markeev AP. Theoretical mechanics. Moscow, Izhevsk: RKhD; 2007. 592 p. (In Russ.)

11. Bolotin SV, Karapetyan AV, Kugushev EI, et al. Theoretical mechanics. Moscow: Akademiya; 2010. 432 p. (In Russ.)

12. Saipulaev MR, Merkuryev IV. Dynamics and accuracy of a micromechanical gyroscope with provision for the displacement of inertial mass. BSU Bulletin. Mathematics, Informatics. 2020;3:49–62. https://doi.org/10.18101/2304-5728-2020-3-49-62 (In Russ.)

13. Andrei L. Smirnov, Sergei Filippov, Petr E. Tovstik, et al. Asymptotic Methods in Mechanics of Solids. Birkhäuser Basel: International Series of Numerical Mathematics; 2015. 323 p. 10.1007/978-3-319-18311-4

14. Awrejcewicz J, Krys’ko VA. Introduction to asymptotic methods. Boca Raton Fl: Chapman & Hall/CRC; 2006. 242 p.

15. Burd V. Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications. Chapman & Hall/CRC; 2007. 343 p.

16. Medvedev GN. Lektsii po metodu usredneniya. Moscow: Faculty of Physics of Moscow State University; 2019. 109 p. (In Russ.)