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    Liubov Klimina

    Michurinsky prosp. 1, Moscow, 119192, Russia
    Lomonosov Moscow State University, Institute of Mechanics,


    Vasiukova O. E., Klimina L. A.
    The paper presents a mathematical model of a controlled pendulum under the assumption that friction in a joint is a sum of Coulomb and viscous friction. Moreover, it is taken into account that the Coulomb friction torque depends on the value of normal reaction force in a joint. The control torque is chosen as a function that depends only on the sign of the angular speed of the pendulum. Via the Pontryagin approach for near-Hamiltonian systems, the program law is constructed for test self-oscillations. Test self-oscillations are to be used for identification of friction coefficients. Bifurcation diagrams are constructed that describe the dependence between amplitudes of self-oscillations and values of the control torque. The proposed approach to the identification of parameters of the friction requires information about amplitudes of test selfoscillations but does not require information about the trajectory of motion as a function of time. Numerical simulation of the motion of the system is carried out. The range of parameter values is described for which the method proposed in the paper is quite accurate.
    Keywords: identification of friction, small parameter, steady motions, controlled dynamic system, bifurcation diagrams
    Citation: Vasiukova O. E., Klimina L. A.,  Modelling of self-oscillations of a controlled pendulum with respect to a friction torque depending on a normal reaction in a joint, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp.  33-44
    Klimina L. A., Lokshin B. Y.
    An autonomous dynamical system with one degree of freedom with a cylindrical phase space is studied. The mathematical model of the system is given by a second-order differential equation that contains terms responsible for nonconservative forces. A coefficient $\alpha$ at these terms is supposed to be a small parameter of the model. So the system is close to a Hamiltonian one.
    In the first part of the paper, it is additionally supposed that one of nonconservative terms corresponds to dissipative or to antidissipative forces, and coefficient $b$ at this term is a varied parameter. The Poincaré – Pontryagin approach is used to construct a bifurcation diagram of periodic trajectories with respect to the parameter b for sufficiently small values of $\alpha$.
    In the second part of the paper, a system with nonconservative terms of general form is studied. Two supplementary systems of special form are constructed. Results of the first part of the paper are applied to these systems. Comparison of bifurcation diagrams for these supplementary systems has allowed deriving necessary conditions for the existence of periodic trajectories in the initial system for sufficiently small $\alpha$.
    The third part of the paper contains an example of the study of periodic trajectories of one system, which, for zero value of the small parameter, coincides with a Hamiltonian system $H_0$. It is proved that there exist periodic trajectories which do not satisfy the Poincaré – Pontryagin sufficient conditions for emergence of periodic trajectories from trajectories of the system $H_0$.
    Keywords: autonomous dynamical system, Poincaré – Pontryagin approach, sufficient conditions for the existence of periodic trajectories, bifurcation diagram, necessary conditions for the existence of periodic trajectories
    Citation: Klimina L. A., Lokshin B. Y.,  On a constructive method of search for rotary and oscillatory modes in autonomous dynamical systems, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 1, pp.  25-40

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