Gyroscopically coupled in-plane and transverse vibrations of an annular free-clamped microplate
DOI:
https://doi.org/10.21638/spbu01.2024.209Abstract
In this work, we construct and study a model of coupled plane-transverse oscillations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertia forces caused by the rotation of the system along an axis located in the plane of the plate. Equations of vibrations in partial derivatives are obtained using the variational principle of Hamilton - Ostrogradsky. Assuming the smallness of the angular velocity of rotation with respect to the frequency of the working skew-symmetric flexural form of the plate oscillations, an approximate analytical solution is found for both the radial and circumferential, and transverse components of the displacement field in the free oscillation mode. Using the Galerkin projection method, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric modes of the plate vibrations. It is found that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into the regime of amplitude-modulated beats. Analytical expressions are found both for the frequency of the slow beat envelope and for the relative depth of their amplitude modulation. The fundamental possibility of determining the modulus of the projection of the angular velocity vector onto the plane of the plate from the measured value of the envelope frequency is shown. The problem of choosing the optimal geometric shape of the resonator from the point of view of maximizing the sensitivity of the system to changes in the value of the angular velocity of rotation is studied. The question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of the amplitude modulation of the beat regime is considered.Keywords:
micromechanical gyroscope, rate-integrating gyroscope, triaxial MEMS gyroscope, inertial navigation, parametric ocsillations
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Литература
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18. Kang J. H. Axisymmetric Vibration of Rotating Annular Plate with Variable Thickness Subjected to Tensile Centrifugal Body Force. International Journal of Structural Stability and Dynamics 17 (9), 1750101 (2017). https://doi.org/10.1142/S0219455417501012
19. Vasiliev G. P., Smirnov A. L. Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 21 (2), 227-237 (2021). https://doi.org/10.18500/1816-9791-2021-21-2-227-237
20. Asher A., Gilat R., Krylov S. Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates. Applied Sciences 12 (5), 2704 (2022). http://dx.doi.org/10.3390/app12052704
21. Touz´e C., Thomas O., Chaigne A. Asymmetric non-linear forced vibrations of free-edge circular plates. Part 1: Theory. Journal of Sound and Vibration 258 (4), 649-676 (2002). https://doi.org/10.1006/jsvi.2002.5143
22. Thomas O., Touz´e C., Chaigne A. Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: Experiments. Journal of Sound and Vibration 265 (5), 1075-1101 (2003). https://doi.org/10.1016/S0022-460X(02)01564-X
23. Nayfeh T. A., Vakakis A. F. Subharmonic travelling waves in a geometrically non-linear circular plate. Pergamon Int. J. Non-Linear Mechanics 29 (2), 233-245 (1994). https://doi.org/10.1016/0020-7462(94)90042-6
24. Eley R., Fox C. H. J., Mcwilliam S. The dynamics of a vibrating-ring multi-axis rate gyroscope. Proceedings of The Institution of Mechanical Engineers Part C-journal of Mechanical Engineering Science 214, 1503-1513 (2000). https://doi.org/10.1243/0954406001523443
25. Nayfeh A. H., Pai P. F. Linear and Nonlinear Structural Mechanics. Wiley Series in Nonlinear Science. Wiley-Interscience (2004).
References
1. Peshekhonov V. G. The Outlook for Gyroscopy. Gyroscopy Navig. 11, 193-197 (2020). https://doi.org/10.1134/S2075108720030062
2. Indeitsev D. A., Belyaev Y. V., Lukin A. V., Popov I. A., Igumnova V. S., Mozhgova N. V. Analysis of imperfections sensitivity and vibration immunity of MEMS vibrating wheel gyroscope. Nonlinear Dynamics 105 (2), 1273-1296 (2021). https://doi.org/10.1007/s11071-021-06664-0
3. Tsai D. H., Fang W. Design and simulation of a dual-axis sensing decoupled vibratory wheel gyroscope. Sensors and Actuators, A: Physical 126 (1), 33-40 (2006). https://doi.org/10.1016/j.sna.2005.09.004
4. Johari H., Shah J., Ayazi F. High frequency xyz-axis single-disk silicon gyroscope. Proceedings of the IEEE International Conference on Micro Electro Mechanical Systems, 856-859 (2008). https://doi.org/10.1109/MEMSYS.2008.4443791
5. Vigna B. Tri-axial MEMS gyroscopes and six degree-of-freedom motion sensors. Technical Digest - International Electron Devices Meeting IEDM) (2011). https://doi.org/10.1109/IEDM.2011.6131635
6. Cao H., Zhao R., Cai Q., Shi Y., Liu L. Structural Design and Simulation Analysis of Silicon Micro Triaxial Wheel-ring Gyroscope. International Conference on Sensing, Measurement and Data Analytics in the Era of Artificial Intelligence ICSMD 2020 - Proceedings, 130-132 (2020). https://doi.org/10.1109/ICSMD50554.2020.9261699
7. Takahashi H., Abe K., Takahata T., Shimoyama I. MEMS triaxial gyroscope using surface and sidewall doping piezoresistors. Journal of Micromechanics and Microengineering 30 (10) (2020). https://doi.org/10.1088/1361-6439/ab9e4d
8. Sedebo G. T., Shatalov M. Y., Joubert S. V., Shafi A. A. The Dynamics of a ThreeDimensional Tuning Functionally Graded Plate Gyroscope. Mechanics of Solids 57 (6), 1577-1589 (2022). https://doi.org/10.3103/S0025654422060267
9. Smirnov A. L. Free Vibrations of the Rotating Shells of Revolution. ASME. J. Appl. Mech. 56 (2), 423-429 (June 1989). https://doi.org/10.1115/1.3176100
10. Hu Z., Gallacher B. J. Effects of Nonlinearity on the Angular Drift Error of an Electrostatic MEMS Rate Integrating Gyroscope. IEEE Sensors Journal 19 (22), 10271-10280 (2019). https://doi.org/10.1109/JSEN.2019.2929352
11. Indeitsev D. A., Zavorotneva E. V., Lukin A. V., Popov I. A., Igumnova V. S. Nonlinear Dynamics of a Microscale Rate Integrating Gyroscope with a Disk Resonator under Parametric Excitation. Rus. J. Nonlin. Dyn. 19 (1), 59-89 (2023).
12. Maslov A. A., Merkuryev I. V., Maslov D. A., Podalkov V. V. Scale Factor of the Wave Solid-State Gyroscope Operating in the Angular Velocity Sensor Mode. 29th Saint Petersburg International Conference on Integrated Navigation Systems, ICINS (2022). https://doi.org/10.23919/ICINS51784.2022.9815350
13. Luo A. C. J., Mote C. D., Jr. Nonlinear Vibration of Rotating Thin Disks. ASME. J. Vib. Acoust. October 2000 122 (4), 376-383 (2000). https://doi.org/10.1115/1.1310363
14. Nayfeh A. H., Jilani A., Manzione P. Transverse Vibrations of a Centrally Clamped Rotating Circular Disk. Nonlinear Dynamics 26, 163-178 (2001). https://doi.org/10.1023/A:1012957024898
15. Hamidzadeh H. R. Non-linear free transverse vibration of thin rotating discs. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-Body Dynamics 221 (3), 467-473 (2007). https://doi.org/10.1243/14644193JMBD46
16. Angoshtari A., Jalali M. A. On the existence of chaotic circumferential waves in spinning disks. Chaos 17 (2) (2007). https://doi.org/10.1063/1.2735813
17. Hu Y., Wang T. Nonlinear resonance of the rotating circular plate under static loads in magnetic field. Chinese Journal of Mechanical Engineering (English Edition) 28 (6), 1277-1284 (2015). https://doi.org/10.3901/CJME.2015.0720.097
18. Kang J. H. Axisymmetric Vibration of Rotating Annular Plate with Variable Thickness Subjected to Tensile Centrifugal Body Force. International Journal of Structural Stability and Dynamics 17 (9), 1750101 (2017). https://doi.org/10.1142/S0219455417501012
19. Vasiliev G. P., Smirnov A. L. Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 21 (2), 227-237 (2021). https://doi.org/10.18500/1816-9791-2021-21-2-227-237
20. Asher A., Gilat R., Krylov S. Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates. Applied Sciences 12 (5), 2704 (2022). http://dx.doi.org/10.3390/app12052704
21. Touz´e C., Thomas O., Chaigne A. Asymmetric non-linear forced vibrations of free-edge circular plates. Part 1: Theory. Journal of Sound and Vibration 258 (4), 649-676 (2002). https://doi.org/10.1006/jsvi.2002.5143
22. Thomas O., Touz´e C., Chaigne A. Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: Experiments. Journal of Sound and Vibration 265 (5), 1075-1101 (2003). https://doi.org/10.1016/S0022-460X(02)01564-X
23. Nayfeh T. A., Vakakis A. F. Subharmonic travelling waves in a geometrically non-linear circular plate. Pergamon Int. J. Non-Linear Mechanics 29 (2), 233-245 (1994). https://doi.org/10.1016/0020-7462(94)90042-6
24. Eley R., Fox C. H. J., Mcwilliam S. The dynamics of a vibrating-ring multi-axis rate gyroscope. Proceedings of The Institution of Mechanical Engineers Part C-journal of Mechanical Engineering Science 214, 1503-1513 (2000). https://doi.org/10.1243/0954406001523443
25. Nayfeh A. H., Pai P. F. Linear and Nonlinear Structural Mechanics. Wiley Series in Nonlinear Science. Wiley-Interscience (2004).
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Published
2024-08-10
How to Cite
Morozov, N. F., Lukin, A. V., & Popov, I. A. (2024). Gyroscopically coupled in-plane and transverse vibrations of an annular free-clamped microplate. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 11(2), 354–370. https://doi.org/10.21638/spbu01.2024.209
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
