Peculiarities of Motion of Pendulum on Mechanical System Engine Rotating Shaft

  • A. I. Artyunin
  • S. V. BarsukovEmail author
  • O. Yu. Sumenkov
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


The authors carried out the mechanical and mathematical modeling of the motion of a pendulum mounted with the possibility of free rotation on the rotor shaft of an electric motor fixed in housing on elastic supports. The peculiarities of the motion of the pendulum within the range from zero to a given value of the angular velocity of the rotor of the electric motor with the presence in this range of natural frequencies of oscillations of the mechanical system are revealed. It is established that when changing the friction moment in the pendulum support or the pendulum mass, a mode of movement is possible and when the rotor rotates with a given angular velocity, the angular velocity of the pendulum rotation is equal to one of the natural frequencies of the mechanical system. The results of a numerical experiment conducted for the first time in a mechanical and mathematical modeling of the motion of a pendulum on a rotating shaft confirmed the results of a previous full-scale experiment on the possible emergence of a pendulum “sticking” effect for such mechanical systems of a general form.


Mechanical system Natural frequency Angular velocity Pendulum “Sticking” effect 


  1. 1.
    Newton I (2017) Matematicheskie nachala natural`noy filosofii (Mathematical principles of natural philosophy). LENAND, MoscowGoogle Scholar
  2. 2.
    Huygens H (1951) Tri memuara po mekhanike (Three memoirs on mechanics). AS of Lith. SSR, MoscowGoogle Scholar
  3. 3.
    Kapitsa PA (1951) Dinamicheskaya ustoychivost` mayatnika pri koleblyushcheysya tochke podvesa (The dynamic stability of the pendulum with an oscillating point of suspension). J Exper Theor Phys 21(5):588–597MathSciNetGoogle Scholar
  4. 4.
    Ragulskis KM (1963) Mekhanizmy na vibriruyushchem osnovanii (Mechanisms on a vibrating base). AS of Lith. SSR, KaunasGoogle Scholar
  5. 5.
    Valeev KG, Dolya VV (1974) O dinamicheskoy stabilizatsii mayatnika (On the dynamic stabilization of the pendulum). Appl Mech 10(2):88–99Google Scholar
  6. 6.
    Chelomey VN (1989) Izbrannye trudy (Selected Works). Mashinostroenie, MoscowGoogle Scholar
  7. 7.
    Strizhak TG (1991) Metody issledovaniya dinamicheskikh system tipa “mayatnik” (Research methods for dynamic systems of the “pendulum” type). Nauka, Alma-AtaGoogle Scholar
  8. 8.
    Bardin BS, Markeev AP (1995) Ob ustoychivosti ravnovesiya mayatnika pri vertikal`nykh kolebaniyakh tochki podvesa (On the stability of the pendulum equilibrium with vertical oscillations of the suspension point). Appl Math Mech 9(6):922–929Google Scholar
  9. 9.
    Morozov AD (1995) K zadache o mayatnike s vibriruyushchei tochkoi podvesa (To the problem of a pendulum with a vibrating point of suspension). Appl Math Mechs 59(4):590–598Google Scholar
  10. 10.
    Seyranyan AA, Seyranyan AM (2006) Ob ustoychivosti perevyornutogo mayatnika s vibriruiushchei tochkoi podvesa (On the stability of an inverted pendulum with a vibrating point of suspension). Appl Math Mech 70(5):835–843MathSciNetGoogle Scholar
  11. 11.
    Malkin IA (1951) Nekotorye zadachi teorii nelineinykh kolebaniy (Some problems of the theory of nonlinear oscillations). Gostekhizdat, MoscowGoogle Scholar
  12. 12.
    Bogolyubov NN, Mitropolsky YuA (1974) Asimptoticheskie metody v teorii nelineinykh kolebaniy (Asymptotic methods in the theory of nonlinear oscillations). Nauka, MoscowGoogle Scholar
  13. 13.
    Moiseev NN (1969) Asimptoticheskie metody nelineinoi mekhaniki (Asymptotic methods of nonlinear mechanics). Nauka, MoscowGoogle Scholar
  14. 14.
    Volosov VM, Morgunov BI (1971) Metod osredneniya v teorii nelineynykh sistem (Averaging method in the theory of nonlinear systems). Moscow University Press, MoscowzbMATHGoogle Scholar
  15. 15.
    Grebennikov EA (1986) Metod usredneniya v prikladnykh zadachakh (Averaging method in applied problems). Nauka, MoscowzbMATHGoogle Scholar
  16. 16.
    Blekhman II (1994) Vibratsionnaya mekhanika (Vibration mechanics). Fizmatlit, MoscowzbMATHGoogle Scholar
  17. 17.
    Artyunin AI, Khomenko AP, Eliseev SV, Ermoshenko YuV (2015) Obobshchyonnaya model` vibratsionnoi nelineinoi mekhaniki i effekt “zastrevaniya” mayatnika na rezonansnykh chastotakh mekhanicheskoi sistemy (The generalized model of vibration nonlinear mechanics and the effect of “sticking” of the pendulum at the resonant frequencies of the mechanical system). Sci J. Eng and Eng Educ 1:61–67Google Scholar
  18. 18.
    Artyunin AI (1993) Issledovanie dvizheniya rotora s avtobalansirom (Study of the movement of the rotor with autobalance). Proceedings of Higher Educational Institutions. Machi Build 1:15–19Google Scholar
  19. 19.
    Artyunin AI, Ermoshenko YuV, Popov SI (2015) Eksperimental`nye issledovaniia effekta “zastrevaniya” mayatnika na rezonansnykh chastotakh mekhanicheskoi sistemy (Experimental studies of the effect of “sticking” of the pendulum at the resonant frequencies of the mechanical system). Mod Techn Sys Analys Model 2(46):20–25Google Scholar
  20. 20.
    Artyunin AI, Eliseev SV, Sumenkov OYu (2018) Experimental studies on influence of natural frequencies of oscillations of mechanical system on angular velocity of pendulum of rotating shaft. Dynamics of Machines and Lecture Notes. Working Processes in Mechanical Engineering ICIE 2018, pp 159–166Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • A. I. Artyunin
    • 1
  • S. V. Barsukov
    • 1
    Email author
  • O. Yu. Sumenkov
    • 2
  1. 1.Irkutsk State Transport UniversityIrkutskRussia
  2. 2.Tomsk Polytechnic UniversityTomskRussia

Personalised recommendations