Skip to main navigation menu Skip to main content Skip to site footer


Vol. 9 (2022)

Corrected Mathematical Models for Motions of the Gyroscope with one Side Free Support

February 10, 2022


Abstract: The recent publications about gyroscope effects explained their physics and described them by mathematical models based on the action of forces and inertial torques of classical mechanics. This new analytical approach finally solved the old problem of the dynamic of rotating objects and showed their kinetic energy is the base of gyroscopic effects. Gyroscopic effects result from the action of the two sets of interrelated inertial torques acting about two axes. Each set contains torques generated by the centrifugal, Coriolis forces, and the change in the angular momentum. Detailed study of the inertial torque of the centrifugal force showed its expression derived with an error of mathematical processing. This error gives a less value for the angular velocity of the slow rotation of the gyroscope about one axis that cannot be measured. The angular velocity of the fast rotation of the gyroscope about the other axis is measured but remains of the same value that gives the expression of the torque with error. This manuscript presents the corrected mathematical model for the motion of the gyroscope suspended from the flexible cord.


  1. Cordeiro FJB (2015) The Gyroscope, Createspace, NV, USA.
  2. Greenhill G, (2015) Report on Gyroscopic Theory, Relink Books, Fallbrook, CA, USA.
  3. Scarborough JB, (2014) The Gyroscope Theory and Applications, Nabu Press, London.
  4. Weinberg H, (2011) Gyro Mechanical Performance: the most important parameter. Analog Devices, Technical Article MS-2158: 1-5.
  5. Hibbeler RC, Yap KB (2013) Mechanics for Engineers-Statics and Dynamics, 13th ed. Prentice Hall, Pearson, Singapore.
  6. Gregory DR, (2006) Classical Mechanics, Cambridge University Press, New York.
  7. Taylor JR. Classical Mechanics, University Science Books, California, USA, 2005.
  8. Aardema MD, (2005) Analytical Dynamics. Theory and Application. Academic/Plenum Publishers, New York.
  9. Liang WC and Lee SC, (2013) Vorticity, gyroscopic precession, and spin-curvature force. Physical Review D 87: 044024.
  10. Crassidis JL, Markley FL (2016) Three-Axis Attitude Estimation Using Rate-Integrating Gyroscopes. Journal of Guidance, Control, and Dynamics 39: 1513-1526.
  11. Nanamori Y, and Takahashi M. (2015) An Integrated Steering Law Considering Biased Loads and Singularity for Control Moment Gyroscopes. AIAA Guidance, Navigation, and Control Conference.
  12. Usubamatov R. (2018) Inertial Forces Acting on Gyroscope, Journal of Mechanical Science and Technology, 32 (1), pp. 101-108.
  13. Usubamatov R. (2020). Theory of gyroscope effects for rotating objects, Springer, Singapore.
  14. Usubamatov R. (2021). Physics of Gyroscopic Effects, Acta Scientific computer sciences, Vol. 3 Issue 12, pp. 22-25.
  15. Usubamatov R. and Bergander M. (2021), Physics of a Spinning Object Cyclic Inversion at an Orbital Flight, Journal of Modern Mechanical Engineering and Technology, 8, pp. 26-30.
  16. Usubamatov R. (2016), A mathematical model for motions of gyroscope suspended from flexible cord. Cogent Engineering, Vol.3, pp. 1-16.
  17. Peters CA. (2001), Statistics for Analysis of Experimental Data, Princeton University.