Self-Oscillation of a Gyroscopic System with Dry Friction in Gimbal Axes during Angular Motion of the Support

The dry (Coulomb) friction is one of the causes of self-oscillation arising in gyroscopic systems, where the mechanical energy loss for self-oscillation is continuously replenished by the energy inflow (from the source that has no intrinsic oscillation properties), which is controlled and transformed by the oscillation system using a feedback. Unlike these systems, a gyro system without feedback is considered here (a three-degree-of-freedom gyroscope with dry (non-Coulomb) friction in gimbal axes) while a source of self-oscillation energy is the angular motion of the support. In this case the dry-friction characteristic depending on the relative speed of sliding has the falling and rising parts and is presented as a sum of three constituents, the first of which is the Coulomb friction, the second (negative) one is proportional to the sliding speed, and the third one is proportional to the cube of the sliding speed. Based on the harmonic linearization method, it is shown that in the system that is stable on the immobile support, self-oscillation inevitably occurs when the angular motion of the support takes place at a speed corresponding to the falling part of the dry-friction curve. Conditions for self-oscillation arising, domains of existence and parameters of self-oscillation are obtained. The study results are confirmed by the computer simulation ofthe initial nonlinear gyro system. The mechanical analogy is found of the three-degree-of-freedom gyroscope with dry (non-Coulomb) friction in the gimbal axis during the angular motion of the support with the system known from the oscillation theory under the name of "oscillator with dry friction on the endless ribbon moving at a constant speed".

References

[1] Ishlinskiy A.Yu. Mekhanika spetsial’nykh giroskopicheskikh sistem [Mechanics of special gyroscopic systems]. Kiev, AN USSR Publ., 1952. 432p.

[2] Chernikov S.A. Symmetrical self-oscillations of a gyroscopic stabilizer. Izv. Akad. NaukSSSR, Otd. Tekh. Nauk, Energ. Avtom. [Bull. Acad. Sci. USSR, Tech. Sci. Sec., Power Eng. Autom.], 1960, no. 6, pp. 133-142 (in Russ.).

[3] Bronovets M.A., Zhuravlev V.F. On self-excited vibrations in friction force measurement systems. Mech. Solids, 2012, vol. 47, no. 3, pp. 261-268. doi: 10.3103/S0025654412030016.

[4] Kragel’skiy I.V., Gitis N.V. Friktsionnye avtokolebaniya [The friction oscillations]. Moscow, Nauka Publ., 1987. 181 p.

[5] Panovko Ya.G. Vvedenie v teoriyu mekhanicheskikh kolebaniy [Introduction to mechanical vibrations]. Moscow, Nauka Publ., 1971. 240 p.

[6] Zhuravlev V.F., Klimov D.M. Prikladnye metody v teorii kolebaniy [Applied methods in the theory of oscillations]. Moscow, Nauka Publ., 1988. 325 p.

[7] Popov E.P. Prikladnaya teoriya protsessov upravleniya v nelineynykh sistemakh [Applied theory of control processes in nonlinear systems]. Moscow, Nauka Publ., 1973. 583 p.