Mathematical model of the motion of a spherical rotor during finishing with cup laps and free abrasive

Due to the increasing requirements for the performance characteristics of gyroscopes with electrostatic non-contact rotor suspension, there is a need to improve the technology for manufacturing parts and assembling devices. The most important component of the sensitive element of an electrostatic gyroscope is a spherical beryllium rotor. The disturbing moments from the suspension forces are proportional to the voltage supplied to the electrodes and the deviation of the rotor surface from the spherical shape. For this reason, the technology of finishing the rotor surface must ensure that high requirements for the sphericity of the rotor are met. In the manufacture of rotors of all known types of electrostatic gyroscopes, the technology of centerless finishing with cup laps with free abrasive is used. One of the key factors influencing the resulting sphericity is the parameters of the rotor motion in the finishing machine. The article presents a mathematical model that allows one to determine the parameters of the rotor motion in the finishing machine under the action of friction forces from the rotation of the cup laps. The method of mathematical modeling was used in the work. The process of centerless finishing with cup laps is considered as a type of friction drive. The rotor motion is considered as the motion of an absolutely rigid body. To determine the motion parameters, the Euler differential equations for rotational motion are used the solution of which is carried out numerically using the MATLAB software package. The pressure distribution in the lap-rotor pairs is considered by analogy with the expression of effects in a ball joint. The result of the work is a mathematical model of the rotor motion during finishing with cup laps, which made it possible to identify the main patterns of rotor motion during centerless finishing. The model made it possible to reveal that the difference in the moments of inertia of the rotor can have a significant effect on the rotor motion during processing, in particular, during polishing. Boundary conditions were determined under which the rotor motion can be permissibly considered as the motion of a ball with equal moments of inertia. The proposed model of rotor motion can be used in designing algorithms and control systems for machines for centerless finishing of spheres with free abrasive as well as a component of mathematical and physical models describing the processing of the rotor surface by finishing with cup laps.

1. Landau B.E., Belash A.A., Gurevich S.S., Levin S.L., Romanenko S.G., Tsvetkov V.N. Electrostatic gyroscope in spacecraft attitude reference systems. Gyroscopy and Navigation, 2021, vol. 12, no. 3, pp. 247–253. https://doi.org/10.1134/s2075108721030056

2. Martynenko Y.G. Motion of a Rigid Body in Electric and Magnetic Fields. Moscow, Nauka Publ., 1988, 368 p. (in Russian)

3. Fedorovich S.N. Current state and perspectives for development of the technology of lapping of precision spherical system elements. Metalworking, 2018, no. 1 (103), pp. 27–32. (in Russian)

4. Angele W. Finishing high precision quartz balls. Precision Engineering, 1980, vol. 2, no. 3, pp. 119–122. https://doi.org/10.1016/0141-6359(80)90025-2

5. Marcelja F., DeBra D.B., Keiser G.M., Turneaure J.P. Precision spheres for the Gravity Probe B experiment. Classical and Quantum Gravity, 2015, vol. 32, no. 22, pp. 224007. https://doi.org/10.1088/0264-9381/32/22/224007

6. Becker P., Schiel D. The Avogadro constant and a new definition of the kilogram. International Journal of Mass Spectrometry, 2013, vol. 349-350, pp. 219–226. https://doi.org/10.1016/j.ijms.2013.03.015

7. Fedorovich S. N. Modeling the process of finishing the spherical rotor of a ball gyroscope. Journal of Instrument Engineering, 2021, vol. 64, no. 4, pp. 307–315. (in Russian). https://doi.org/10.17586/0021-3454-2021-64-4-307-315

8. Orlov P.N. Technological Quality Assurance of Parts by Finishing Methods. Moscow, Mashinostroenie Publ., 1988, 383 p. (in Russian)

9. Babaev S.G., Sadygov P.G. Lapping and Finishing of Machine Parts Surfaces. Moscow, Mashinostroenie Publ., 1976, pp. 6–15. (in Russian)

10. Markeev A.P. Theoretical Mechanics. Moscow, Nauka Publ., 1990. 414 p. (in Russian)

11. Amelkin N.I. Kinematics and Dynamics of a Rigid Body. Moscow, MIPT Publ., 2000, 63 p. (in Russian)

12. Farkas Z., Bartels G., Unger T., Wolf D. Frictional coupling between sliding and spinning motion. Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 139–146. (in Russian). https://doi.org/10.20537/nd1101007

13. Khala M.J., Hare C., Wu C., Martin M.J., Venugopal N., Freeman T. The importance of a velocity-dependent friction coefficient in representing the flow behaviour of a blade-driven powder bed. Powder Technology, 2021, vol. 385, pp. 264–272. https://doi.org/10.1016/j.powtec.2021.02.060

14. Aublin M. Systèmes Mécaniques: Théorie et Dimensionnement. Dunod, 1993, 662 p. (in French)

15. Anfinogenov A.S., Parfenov O.I., Method to reduce the deformations of the outer surface of thin-walled spherical rotors in gyroscopes. Morskoe priborostroenie, 1969, no. 1, pp. 114–119. (in Russian)

16. Yulmetova, O.S. Ion-plasma and laser technologies in gyroscopic instrumentation. Dissertation for the degree of doctor of technical sciences. St. Petersburg, 2019, 220 p. (in Russian)