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Streszczenie:
This paper deals with the dynamic stability and near-critical vibrations of Beck׳s column with electromagnetic actuators. The governing equations of the actuators are derived, and the continuous dynamics problem of the column is discretised with the multimodal Galerkin procedure. Firstly, the linear analysis of the stability is carried out. An application of the electromagnetic actuators leads to increase in the critical load and considerably affects the initial flutter frequency. Then the Iooss and Joseph method of construction of the bifurcating solution is used to obtain analytical approximations of the near-critical periodic vibrations. It is shown that the electromagnetic effects lead to a supercritical Hopf bifurcation of the equilibrium state of the column into an orbitally stable limit cycle. Moreover, the actuators allow one to decrease significantly the amplitude of the post-critical vibrations. The effect of magnetic damping strongly depends on the position at which the actuators are attached to the column. Theoretical predictions are verified by numerical simulations of the system.

Słowa kluczowe:
Beck׳s column, Electromagnetic actuators, Follower load, Hopf bifurcation, Flutter, Dynamic stability

Streszczenie:
We study the influence of electromagnetic damping on the dynamics of a pipe conveying fluid. Pipes supported at both ends as well as cantilever ones (both discharging and aspirating the fluid) are considered. We assume physical parameters of the systems which allow an experimental verification of results. We develop simple methods of calculation of the internal and external damping coefficients which are based on known models, and do not require experiments. The governing partial equation of the pipe is discretised with Galerkin’s procedure, and the stability of the resultant dynamical system is determined with eigenvalues of its linearization. The actuators destabilise the pipe supported at both ends, but can remarkably improve stability of cantilever ones. The effect of magnetic damping strongly depends on the position at which actuators are attached to the pipe.

Słowa kluczowe:
pipe conveying fluid, dynamic stability, flutter, electromagnetic actuators

Streszczenie:
We study the application of electromagnetic actuators for the active stabilization of the Leipholz column. The cases of the compressive and tensional load of the column placed in air and in water are considered. The partial differential equation of the column is discretized by Galerkin’s procedure, and the stability of the obtained control system is evaluated by the eigenvalues of its linearization. Four different methods of active stabilization are investigated. They incorporate control systems based on feedback proportional to the transverse displacement of the column, its velocity and the current in the electromagnets. Conditions in which these strategies are effective in securing safe operation of the column are discussed in detail.

Słowa kluczowe:
follower load, flutter, closed-loop control, active stabilization, electromagnetic actuators

Streszczenie:
The paper presents analysis of nonlinear response of a classical mechanical oscillator placed within a magnetic field and driven by a harmonic force. With an appropriate choice of control parameters, the system vibrates chaotically between different equilibrium positions. To prove this result, Lyapunov exponents have been calculated using the algorithm proposed by Rangarajan G., Habib S. and Ryne R. [18]. Moreover, the appropriate time series, phase portrait, Poincaré cross-section and power spectrum are given to support the conclusion.

Słowa kluczowe:
nonlinear vibrations, oscillator, chaos, stability, magnetic damping

Streszczenie:
The paper presents a method of damping of mechanical vibrations by application of electromagnetic actuators. The physical phenomena that take place in such elements are related to magnetic induction and hysteresis. Induction, which consists in generation of electric current in a conductor placed within a variable magnetic field, has a precise quantitative description in form of the Faraday law. At the same time, magnetic hysteresis in ferromagnetic materials reflects the relation between magnetization of a given sample and magnitude of the external magnetic field that had been active during its whole history. Unfortunately, there is no at present a single commonly accepted model of this phenomenon. In this paper, some considerations by Bertotti (1998) are incorporated to this subject. The authors use a bilinear magnetization curve, which depicts the phenomenon of magnetic saturation. Analysis of the derived equations of motion of the coupled electromagneto-mechanical model revealed that the effect of hysteresis on reduction of vibration amplitude is rather of minor impact, however noticeable. At the same time, the electromagnetic damping itself is powerful. To support the conclusions, appropriate time histories, graphs of the logarithmic damping decrement as well as resonant characteristics are given here as in the previous authors’ paper.

Słowa kluczowe:
damping of vibration, electromagnetic actuator, magnetic hysteresis, saturation

Streszczenie:
Oscylator harmoniczny zawieszony pomiędzy dwoma elektromagnesami może być skutecznie tłumiony magnetycznie. Postać drgań, charakter tłumienia, liczba i stateczność punktów równowagi zależą od napięcia załączonego do elektromagnesów. Drgania wymuszone w pewnych warunkach przypominają drgania chaotyczne. Wpływ histerezy magnetycznej jest pominięty.

Słowa kluczowe:
siłowniki elektromagnetyczne, tłumienie drgań, oscylator