# Inertial forces acting on a gyroscope

## Abstract

Gyroscopic devices for navigation and control systems are widely applied in various industries, such as shipping and aerospace. A remarkable property of gyroscopes is that their axes can be maintained within a particular space. This interesting property of a spinning disc mounted on an axle is represented by a mathematical model formulated based on L. Euler’s principle of change in angular momentum. Nevertheless, numerous publications and analytical approaches in known gyroscope theories do not correspond to practical tests on gyroscopes. A simple rotating disc creates problems that do not have long-term solutions. Recent investigations in this area have demonstrated that the origin of gyroscope properties is more sophisticated than that described in known hypotheses. Researchers have not considered the action of inertial forces produced by the mass elements and center mass of the spinning rotor that create internal resistance and precession torques. Resistance torque is established through the actions of centrifugal and Coriolis forces. Precession torque is established through the actions of common inertial forces and a change in angular momentum. These internal torques act simultaneously and interdependently on two axes and represent the fundamental principles of gyroscope theory. Equations for internal inertial torques of a spinning disc have been formulated through mathematical analysis and differential and integral equations. These calculus methods provide a basis for understanding the rates of change in inertial forces acting on a gyroscope and include the use of functions, their derivatives, and integrals in modeling the physical processes in gyroscopes. This paper presents mathematical models for several internal inertial torques generated by the load torque acting on a spinning rotor. These models can describe all gyroscope properties and represent their novelty for machine dynamics and engineering.

This is a preview of subscription content, access via your institution.

## References

1. M. N. Armenise, C. Ciminelli, F. V. Dell'Olio and M. N. Passaro, Advances in gyroscope technologies, Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, Berlin (2010).

2. R. F. Deimel, Mechanics of the gyroscope, Dover Publications Inc., New York (2003).

3. G. Greenhill, Report on gyroscopic theory, General Books LLC, London (2010).

4. J. B. Scarborough, The gyroscope theory and applications, Fairford, GLOS, U. K. (2015).

5. B. Neil, Gyroscope, The Charles Stark Draper Laboratory, Inc., Cambridge, Massachusetts (2014) Doi: http://dx.doi.org/10.1036/1097-8542.304100.

6. C. Acar and A. Shkel, MEMS vibratory gyroscopes: Structural approaches to improve robustness, Springer Science & Business Media, New York (2008).

7. H. Weinberg, Gyro mechanical performance: The most important parameter, Technical Article MS-2158, Analog Devices, Norwood, MA (2011) 1–5.

8. R. C. Hibbeler, Engineering mechanics-Statics and dynamics, 13th Ed., Prentice Hall, Pearson, Singapore (2015).

9. D. R. Gregory, Classical mechanics, Cambridge University Press, New York (2006).

10. J. Syngley and J. J. Uicker, Theory of machines and mechanisms, Third Ed., McGraw-Hill Book Company, New York (2002).

11. M. D. Aardema, Analytical dynamics. Theory and application, Academic/Plenum Publishers, New York (2005).

12. R. M. Jonsson, Gyroscope precession in special and general relativity from basic principles, American Journal of Physics, 75 (2007) 463, http://dx.doi.org/10.1119/1.2719202.

13. W. C. Liang and S. C. Lee, Vorticity, gyroscopic precession, and spin-curvature force, Physical Review D, 87 (2013) http://dx.doi.org/10.1103/PhysRevD.87.044024.

14. Y. W. Kim and H. H. Yoo, Design of a vibrating MEMS gyroscope considering design variable uncertainties, Journal of Mechanical Science and Technology, 24 (11) (2010) 2175–2180.

15. E. Butikov, Inertial rotation of a rigid body, Europien Journal of Physics, 27 (2006) 913–922.

16. L. Zyga, Gyroscope's unexplained acceleration may be due to modified inertia, PhysOrg.com, July 26 (2011).

17. M. Zareh and S. Soheili, A modified model reference adaptive control with application to MEMS gyroscope, Journal of Mechanical Science and Technology, 25 (8) (2011) 2061–2066.

18. J. A. Ferrari, Gyroscope's precession and the principle of equivalence, Annalen der Physik, 501 (5) (2006) 399–400, Doi: 10.1002/andp.19895010513.

19. D.-J. Jwo, J.-H. Shih, C.-S. Hsu and K.-L. You, Development of a strapdown inertial navigation system simulation platform, Journal of Mechanical Science and Technology, June (2014) Doi: 10.6119/JMST-013-0909-5.

20. J. Li, Z.-M. Lei, L.-Q. Sun and S.-W. Yan, Mechanism and model testing of pipelay vessel roll affected by large period swells, Journal of Marine Science and Technology (2016) Doi: 10.6119/JMST-016-0125-2.

21. F. Klein and A. Sommerfeld, The theory of the top, New York, NY: Springer, Birkhäuser, I-IV (2008-2014).

22. R. Usubamatov, K. A. Ismail and J. M. Sah, Analysis of a coriolis acceleration, Journal of Advanced Science and Engineering Research, 4 (1) March (2014) 1–8.

23. R. Usubamatov, Mathematical model for gyroscope effects, Proceedings of AIP Conference, 1660, 050018 (2015) Doi: 10.1063/1.4915651.

24. R. Usubamatov, Properties of gyroscope motion about one axis, International Journal of Advancements in Mechanical and Aeronautical Engineering, 2 (1) (2015) 39–44, ISSN: I2372-4153.

25. R. Usubamatov, A mathematical model for motions of gyroscope suspended from flexible cord, Cogent Engineering, 3 (2016) 1245901, http://dx.doi.org/10.1080/23311916.2016.1245901.

## Author information

Authors

### Corresponding author

Correspondence to Ryspek Usubamatov.

Recommended by Associate Editor Sungsoo Na

Ryspek Usubamatov graduated from Bauman Moscow State Technical University. He is a professional engineer in mechanical, manufacturing, and industrial engineering. He obtained his Ph.D. in 1972 and Doctor of Technical Sciences in 1993. He worked as an Engineer at a company and a lecturer in Kyrgyzstan and Malaysian universities. He has supervised around 100 professional engineer students, 15 Mc.S. students, and 7 Ph.D. students. His areas of research include productivity theory for industrial engineering, gyroscope theory, and wind turbines. He has published 7 books, 30 brochures, and more than 300 manuscripts and holds 60 patents of inventions.

## Rights and permissions

Reprints and Permissions

Usubamatov, R. Inertial forces acting on a gyroscope. J Mech Sci Technol 32, 101–108 (2018). https://doi.org/10.1007/s12206-017-1211-0

• Revised:

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1007/s12206-017-1211-0

• Theory
• Property
• Torque
• Force