Certificate of Registration Media number Эл #ФС77-53688 of 17 April 2013. ISSN 2308-6033. DOI 10.18698/2308-6033
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Using the Bubnov—Galerkin method to calculate the nonlinear oscillations of the variable-length mathematical pendulum at final movement from one state of rest into another

Published: 03.10.2018

Authors: Russkikh S.V., Shklyarchuk F.N.

Published in issue: #10(82)/2018

DOI: 10.18698/2308-6033-2018-10-1809

Category: Mechanics | Chapter: Mechanics of Deformable Solid Body

To assess the efficiency of the Bubnov — Galerkin method and its precision compared to the numerical technique, we consider the nonlinear oscillations of the variable-length mathematical pendulum at the moving suspension. The authors have set the following task: to move the pendulum attachment point for defined time at given distance with the simultaneous suppression of natural modes at the moment of stopping. The control function with the kinematic control of final movement is specified as a series of sines. The unknown required function representing the pendulum rotational angle in accordance with the Bubnov — Galerkin method is written as the resolution with the unknown coefficients by given time functions. The work forms a system of linear algebraic equations, the order of which depends on the number of approximating functions. We consider the calculation examples at various initial settings of the problem

[1] Chernousko F.L., Bolotnik N.N., Gradetskiy V.G. Manipulyatsionnye roboty: dinamika, upravlenie, optimizatsiya [Manipulating robots: dynamics, control, optimization]. Moscow, Nauka Publ., 1989, 363 p.
[2] Berbyuk V.B. Dinamika i optimizatsiya robototekhnicheskikh system [Dynamics and optimization of robotic systems]. Kyiv, Naukova Dumka Publ., 1989, 187 p.
[3] Banichuk N.V., Karpov I.I., Klimov D.M. Mekhanika bolshikh kosmicheskikh konstruktsiy [Mechanics of large space structures]. Moscow, Faktorial Publ., 1997, 302 p.
[4] Dokuchaev L.V. Nelineynaya dinamika letatelnykh apparatov s deformiruemymi elementami [Nonlinear dynamics of flying vehicles with crash elements]. Moscow, Mashinostroenie Publ., 1987, 232 p.
[5] Chernousko F.L., Akulenko L.D., Sokolov B.N. Upravlenie kolebaniyami [Oscillations control]. Moscow, Nauka Publ., 1976, 383 p.
[6] Chernousko F.L., Ananevskiy I.M., Reshmin S.A. Metody upravleniya nelineynymi mekhanicheskimi sistemami [The methods of controlling the nonlinear mechanical systems]. Moscow, Fizmatlit Publ., 2006, 326 p.
[7] Kubyshkin E.P. Prikladnaya matematika i mekhanika — Journal of Applied Mathematics and Mechanics, 2014, vol. 78, no. 5, pp. 656–670.
[8] Grishanina T.V., Shklyarchuk F.N. Dinamika uprugikh upravlyaemykh konstruktsiy [Dynamics of elastic controlled structures]. Moscow, Moscow Aviation Institute Publ., 2007, 328 p.
[9] Russkikh S.V. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroenie — Proceedings of Higher Educational Institutions. Machine Building, 2016, no. 12, pp. 97–105.
[10] Grishanina T.V., Russkikh S.V., Shklyarchuk F.N. Uchenye zapiski Kazanskogo universiteta. Seriya fiziko-matematicheskie nauki — Proceedings of Kazan University. Physics and Mathematics Series, 2017, vol. 159, no. 4, pp. 429–443.