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Russia Ufa
Year
2021
Volume
31
Issue
1
Pages
89-101
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Section Mechanics
Title Small nutation of a symmetic gyroscope: two viewpoints
Author(-s) Voytik V.V.a, Migranov N.G.a
Affiliations Bashkir State Medical Universitya
Abstract The paper is devoted to the small nutation of an axisymmetric gyroscope in the field of gravity. The expansion of the known solution of the nutation equation as a function of time in powers of the amplitude is obtained. In this case, the frequencies of third order Raman oscillations are both the tripled frequency and the frequency coinciding with the initial one. A formula is found for the nutation amplitude as a function of the integrals of the gyroscope motion. The frequency of zero nutation is also calculated. Another way to obtain the decomposition is to use the results of the general theory of free one-dimensional oscillations. This method is based on the ability to represent the gyro nutation as the movement of a material point of unit mass in a field that cubically-quadratically depends on the coordinate. In this case the only frequency of the third-order Raman oscillation is a triple of the original frequency. Thus, both methods give the same result only for oscillations no higher than second order. In the third approximation, the existing theory of oscillations is insufficient.
Keywords unbalanced gyroscope, pseudo-regular rotation, small nutation, anharmonic oscillations
UDC 531.381
MSC 42A10, 70E17
DOI 10.35634/vm210107
Received 18 June 2020
Language Russian
Citation Voytik V.V., Migranov N.G. Small nutation of a symmetic gyroscope: two viewpoints, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 1, pp. 89-101.
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