On stability of relative equilibria of a double pendulum with vibrating suspension point

Double Pendulum Induced Stability

“…The experimental results presented in this section are in good qualitative agreement with the results of the analysis performed by Kholostova [2009Kholostova [ , 2011 and with the experimental results of Sartorelli et al [2008Sartorelli et al [ , 2012 (Fig-7(a) configuration); v -¥>l, A -¥2 (Fig. 7(b)).…”

“…The velocity of vibration which corresponds to the value when one of the configurations becomes unstable -the procedure of searching fails (relative minimum of energy disappears on diagram <p\0<p2) -is taken as critical for this configuration to be stable. As seen, critical velocity depends on the potential energy of the configuration to be stabilized, which agrees with the conclusions of Landau [1976] and Kholostova [2009Kholostova [ , 2011.…”

“…This expression is similar to [Kholostova, 2009[Kholostova, , 2011, [Vishenkova and Kholostova, 2012] and [Bulanchuk and Petrov, 2012]. Keeping in mind that the goal is to find conditions for induced stability of the pendulum, we proceed to search local minima of the expression for U e , which can be done by different methods.…”

“…Most investigations are theoretical and are dedicated to various aspects of dynamics and stability of the different configurations of the pendulums and outer loads. Among the publications, dealing with theoretical research, we point out these paper - Acheson and Mullin [1993], where the necessary conditions to maintain the vertical position of n-link upside-down pendulum were found: Kholostova [2009Kholostova [ , 2011, Vishenkova and Kholostova [2012], whose detailed analysis of the stability and dynamics of the double pendulum have been performed by an asymptotic method; Balanchuk and Petrov [2012], who studied the stabilization of the mathematical pendulum by the method of Kapitsa. We have used the theoretical results, which are presented in these publications, in order to compare and to interpret the results of our experiments.…”

On the stability of the specific motions of a heavy rigid body due to fast vertical vibrations of one of its points

“…В рамках этой приближенной системы был исследован ряд частных движений твердого тела. Для случая вертикальных вибраций точки подвеса решена задача о существовании и устойчивости относительных равновесий тела с произвольной геометри-ей масс, для которых прямая, содержащая центр масс и точку подвеса, вертикальна или наклонена к вертикали [15,16]. Исследовано влияние быстрых периодических и условно-периодических вибраций точки подвеса на существование и устойчивость известных част-ных движений волчка Лагранжа -стационарных вращений вокруг вертикали и регулярных прецессий [17].…”