On Stabilization of a Triple Inverted Pendulum via Vibration of a Support Point with an Arbitrary Frequency

  • I. M. ArkhipovaEmail author


The stabilization of the upper statically unstable position of a triple inverted pendulum via parametric excitation of the support has been studied. The presented results were obtained by the multiple scale method and the Floquet theory. The stability diagrams in the excitation parameters space (amplitude and frequency of the support excitation) are plotted. It is shown that stabilization is possible for low, medium, and high excitation frequencies. The influence of system parameters on stabilization zones of the upper unstable position of the pendulum is analyzed.


vibrational stabilization triple pendulum 


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  1. 1.
    A. Stephenson, “On a new type of dynamical stability,” Mem. Proc. — Manchester Lit. Philos. Soc. 52 (8), 1–10 (1908).zbMATHGoogle Scholar
  2. 2.
    A. Stephenson, “On induced stability,” Philos. Mag. 17, 765–766 (1909).CrossRefzbMATHGoogle Scholar
  3. 3.
    A. P. Seyranian and A. A. Mailybaev, “Vibrational stabilization of statically unstable systems,” Dokl. Phys. 54, 294–300 (2009).CrossRefGoogle Scholar
  4. 4.
    I. M. Arkhipova, A. Luongo, and A. P. Seyranian, “Vibrational stabilization of the upright statically unstable position of a double pendulum,” J. Sound Vib. 331, 457–469 (2012).CrossRefGoogle Scholar
  5. 5.
    I. M. Arkhipova and A. Luongo, “Stabilization via parametric excitation of multi-dof statically unstable systems,” Commun. Nonlinear Sci. Numer. Simul. 19, 3913–3926 (2014).MathSciNetCrossRefGoogle Scholar
  6. 6.
    I. M. Arkhipova and A. Luongo, “On the effect of damping on the stabilization of mechanical systems via parametric excitation,” Z. Angew. Math. Phys. 67, 69 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973).zbMATHGoogle Scholar
  8. 8.
    D. R. Merkin, Introduction to the Theory of Stability (Nauka, Moscow, 1987; Springer-Verlag, New York, 1997).zbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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