A numerical study of an application of magnetorheological (MR) damper for semi-active control is presented in this paper. The damper is mounted in the suspension of a Duffing oscillator with an attached pendulum. The MR damper with properties modelled by a hysteretic loop, is applied in order to control of the system response. Two methods for the dynamics control in the closed-loop algorithm based on the amplitude and velocity of the pendulum and the impulse on-off activation of MR damper are proposed. These concepts allow the system maintaining on a desirable attractor or, if necessary, to change a position from one attractor to another. Additionally, the detailed bifurcation analysis of the influence of MR damping on the number of periodic solutions and their stability is shown by continuation method. The influence of MR damping on the chaotic behavior is studied, as well.

In this paper the dynamical interactions of a double pendulum arm and an electromechanical shaker is investigated. The double pendulum is a three degree of freedom system coupled to an RLC circuit based nonlinear shaker through a magnetic field, and the capacitor voltage is a nonlinear function of the instantaneous electric charge. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space and this behaviour is characterized by power spectral density and Lyapunov exponents. The bifurcation diagram is constructed to explore the qualitative behaviour of the system. This kind of electromechanical system is frequently found in robotic systems, and in order to suppress the chaotic motion, the State-Dependent Riccati Equation (SDRE) control and the Nonlinear Saturation control (NSC) techniques are analyzed. The robustness of these two controllers is tested by a sensitivity analysis to parametric uncertainties.

In the paper, the dynamics of a three degree of freedom vibratory system with a spherical pendulum in the neighbourhood of internal and external resonance is considered. It has been assumed that the spherical pendulum is suspended to the main body which is then suspended to the element characterized by some elasticity and damping. The system is excited harmonically in the vertical direction. The equation of motion has been solved numerically. The influence of initial conditions on the behaviour of the spherical pendulum is investigated. In this type of the system, one mode of vibration may excite or damp another one, and for different kinds of periodic vibrations there may also appear chaotic vibrations. For characterization of an irregular chaotic response, time histories, bifurcation diagrams, power spectral densities, Poincaré maps and the maximum Lyapunov exponents have been calculated.

The presented work deals with nonlinear dynamics of a three degree of freedom system with a spherical pendulum and a damper of the fractional type. Vibrations in the vicinity of the internal and external resonance are considered. The system consists of a block suspended from a linear spring and a fractional damper, and a spherical pendulum suspended from the block. The viscoelastic properties of the damper are described using the Caputo fractional derivative. The fractional derivative of an order of $$0 < \alpha \le 1$$ 0 < α ≤ 1 is assumed. The impact of a fractional order derivative on the system with a spherical pendulum is studied. Time histories, the internal and external resonance, bifurcation diagrams, Poincaré maps and the Lyapunov exponents have been calculated for various orders of a fractional derivative. Chaotic motion has been found for some system parameters.

The nonlinear response of a three degree of freedom autoparametric system with a double pendulum, including the magnetorheological (MR) damper when the excitation comes from a DC motor which works with limited power supply, has been examined. The non-ideal source of power adds one degree of freedom which makes the system have four degrees of freedom. The influence of damping force in MR damper on the phenomenon of energy transfer has been studied numerically. Near the internal and external resonance region, except periodic vibration also chaotic vibration has been observed. Results show that MR damper can be used to change the dynamic behavior of the autoparametric system.

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