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Evolution of a heavy rigid body rotation under the action of unsteady restoring and perturbation torques

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Abstract

The paper develops an approximate solution to the system of Euler’s equations with additional perturbation term for dynamically symmetric rotating rigid body. The perturbed motions of a rigid body, close to Lagrange’s case, under the action of restoring and perturbation torques that are slowly varying in time are investigated. We describe an averaging procedure for slow variables of a rigid body perturbed motion, similar to Lagrange top. Conditions for the possibility of averaging the equations of motion with respect to the nutation phase angle are presented. The averaging technique reduces the system order from 6 to 3 and does not contain fast oscillations. An example of motion of the body using linearly dissipative torques is worked out to demonstrate the use of general equations. The numerical integration of the averaged system of equations is conducted of the body motion. The graphical presentations of the solutions are represented and discussed. A new class of rotations of a dynamically symmetric rigid body about a fixed point with account for a nonstationary perturbation torque, as well as for a restoring torque that slowly varies with time, is studied. The main objective of this paper is to extend the previous results for problem of the dynamic motion of a symmetric rigid body subjected to perturbation and restoring torques. The proposed averaging method is implemented to receive the averaging system of equations of motion. The graphical representations of the solutions are presented and discussed. The attained results are a generalization of our former works where µ and Mi are independent of the slow time τ and Mi depend on the slow time only.

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Acknowledgements

In this research, Dmytro Leshchenko is responsible for the general ansatz and solving procedure and survey in the literature on the problem under consideration. Sergey Ershkov is responsible for theoretical investigations and for analysis of obtained results. Tetiana Kozachenko is responsible for receiving the averaged system of equations of motion and for the plots and numerical solutions for test example.

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Authors and Affiliations

  1. Odessa State Academy of Civil Engineering and Architecture, Scopus Number 66029960055, 4 Didrikhsona st., Odessa, 65029, Ukraine

    Dmytro Leshchenko

  2. Plekhanov Russian University of Economics, Scopus Number 60030998, 36 Stremyanny Lane, Moscow, Russia, 117997

    Sergey Ershkov

  3. Odessa State Academy of Civil Engineering and Architecture, 4 Didrikhsona st., Odessa, 65029, Ukraine

    Tetiana Kozachenko

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  1. Dmytro Leshchenko
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  2. Sergey Ershkov
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  3. Tetiana Kozachenko
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Correspondence to Dmytro Leshchenko.

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Leshchenko, D., Ershkov, S. & Kozachenko, T. Evolution of a heavy rigid body rotation under the action of unsteady restoring and perturbation torques. Nonlinear Dyn 103, 1517–1528 (2021). https://doi.org/10.1007/s11071-020-06195-0

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