Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Qualitative Analysis of the Equation of Motion of the "Mobile Platform with a Pendulum" System

Author(s):

Alexander V. Bratishchev

Don State Technical University,
Professor of Applied Mathematics Department ,
Rostov-on-Don, Gagarin sq.,1

avbratishchev@spark-mail.ru

Abstract:

In this paper, we study the free movement on the horizontal plane of a platform with a spherical pendulum fixed in the center of mass. This system is considered as a system of two stationarily connected material points, one of which moves in the plane. The subsystem of differential equations describing the motion of the platform is integrated explicitly and the solution is a function of the variables describing the motion of this pendulum relative to the platform. In turn, in the system of fourth-order equations of motion of the pendulum relative to the platform, an independent autonomous subsystem of third-order equations is distinguished. A phase portrait of the latter system is constructed. This allowed us to give a complete description of the movement of the "Мobile platform with a pendulum" system at any of its initial states. The relationship in quadratures between the angular phase coordinates of the points of the trajectories of the system is established.

Keywords

References:

  1. Appell, P. Traite de mecanique rationnelle. v. 1, 2. Paris: Gauthier-Villars, 1953, 559 p
  2. Smirnov, A. S., Smol’nikov, B. A. Mekhanika sfericheskogo mayatnika [Mechanics of a spherical pendulum]. S-Peterburg, POLYTECH-PRESS, 2019. 265 p
  3. Efimov, V. V. Investigation of oscillations of a physical pendulum with a movable point suspension as a simplified model of cargo on the external suspension of the helicopter. Scientific Bulletin of MSTU GA. Aeromechanics and Strength series. 2009; (138):126-133. (In Russ. )
  4. Korytov, M. S., Shcherbakov, V. S., Titenko, V. V., Belyakov, V. E. Model of spherical a pendulum with a movable suspension point in the problem of spatial displacement load lifting crane with limited vibrations. Dynamics of systems, mechanisms, and machines. 2019; v. 7 (1):104-110. (In Russ. )
  5. Bratchishev, A. V. Control of oscillation of the pendulum on a movable platform. Differencialnie Uravnenia i Protsesy Upravlenia, 2020 (4):75-86,. (In Russ. ) https://diffjournal.spbu.ru/pdf/20404-jdecp-bratischev.pdf
  6. Dabney, J.. Harman, T. Mastering SIMULINK 4. New Jersey 07458 : Prentice Hall, 2001. 412 p
  7. Ol’khovsky, I. I. Kurs teoreticheskoy mekhaniki dlya fizikov [Course of theoretical mechanics for physicists]. Moscow, Lan’, 2009. 576 p
  8. Bautin, N. N., Leontovich, E. A. Metodi i priomy kachestvennogo issledovaniya dinamicheskikh system na ploskosti [Methods and techniques of qualitative research dynamic systems on the plane]. Moscow, Nauka Publ., 1990. 496 p
  9. Fikhtengolts, G. M. Kurs differencial’nogo i integral’nogo ischisleniya [Course of differential and integral calculus]. V. 2. Moscow, Lan’, 2020. 800 p

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