Dynamics of precession of the Thompson top

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Abstract

In this work, the dynamics of precession of the Thompson top was studied. Test benches have been developed and experimental work has been carried out. Based on the data obtained, the dependences of the angular velocity of precession at high initial values of up to 340 rad/s are shown. Analytical studies were carried out and the dependences of energy changes during precession were obtained. The relationship between the roughness coefficient and the resulting friction moment is obtained. The relationships between the work of friction and the roughness coefficient were obtained, which gives an understanding of the dynamics of energy indicators during the entire process. The question remains open about the use of a viscous friction model and a more detailed study of the proportionality coefficient to establish refined dependencies for precession. The experimental data obtained show the influence of the roughness coefficient on precession and on the occurrence of the friction force, leading to the capsizing of the top. Further research in this matter should show the possibility of applicability of the viscous model and the study of critical points, namely the rise to the leg and the passage of the highest value of the moment of inertia.

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About the authors

А. I. Andreev

Federal State Budgetary Educational Institution of Higher Education, Astrakhan State Technical University

Author for correspondence.
Email: aresut79@mail.ru
Russian Federation, Astrakhan

А. Е. Semenov

Federal State Budgetary Educational Institution of Higher Education, Astrakhan State Technical University

Email: aresut79@mail.ru
Russian Federation, Astrakhan

B. М. Slavin

Federal State Budgetary Educational Institution of Higher Education, Astrakhan State Technical University

Email: aresut79@mail.ru
Russian Federation, Astrakhan

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Supplementary files

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2. Fig. 1. Rig for studying the precession of a rigid body in suspension: (a) – diagram, (b) – view of the working part: 1 – top connected to an electric motor, 2 – substrate, 3 – lifting table, 4 – strain gauge, 5 – damper, 6 – setting spring.

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3. Fig. 2. Rig for studying the precession of a rigid body in a free state: 1 – housing, 2 – substrate, 3 – plates, 4 – coupling, 5 – electric motor, 6 – disengagement, 7 – tachometer, 8 – guides, 9 – bearing, 10 – strain gauges, 11 – computer (HX711 + Arduino Uno + PC).

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4. Fig. 3. Change in angular velocity ω (rad/s) over time t (s) during free precession on surfaces with different roughness at an initial angular velocity from 250 to 340 rad/s: theoretical solution (○), friction coefficient ξ = 0.22 (◊), friction coefficient ξ = 0.31 (∆), friction coefficient ξ = 0.51 ( ).

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5. Fig. 4. Change in angular velocity ω (rad/s) over time t (s) during free precession on surfaces with different roughness at an initial angular velocity from 35 to 180 rad/s: steel top on steel (○), steel top on concrete ( ), plastic top on steel (◊), plastic top on concrete (∆).

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6. Fig. 5. Dependence of the resulting forces (from the friction torque) F (N) on the roughness coefficient ξ (the angular velocity in this experiment is 340 rad/s).

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7. Fig. 6. Movement of the point of contact of the top with the surface relative to the center of gravity: A is the center of gravity, B is the top leg, B1–B6 are the set of contact points, G is the zone of impossibility of contact with the surface.

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8. Fig. 7. Movement of the center of gravity relative to the initial position of the center of gravity: 1 is the body of the top, 2 is the leg, 3 is the trajectory of the center of gravity, A1–A8 are the initial position of the center of gravity and its movement over time.

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9. Fig. 8. Change in kinetic energy T (J) versus time t (s) at an initial angular velocity of 340 rad/s (the black line shows the exponential dependence, the blue line is the approximation taking into account the rise of the points at the moment of maximum moment of inertia).

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10. Fig. 9. Dependence of the work of the friction force A (J) on the coefficient ξ.

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