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Construction of the Lyapunov function reflecting the physical properties of the model
Náprstek, Jiří ; Fischer, Cyril
Practical experience shows that the random excitation component can affect the system response and its dynamic stability not only negatively but also positively. Such mechanisms are usually developed heuristically and are often not sufficiently justified theoretically. The paper presents a possibility of using the properties of first integrals for the construction of a Lyapunov function for the analysis of a dynamic system stability in the stochastic domain. In such case, the Lyapunov function itself contains information on the examined system and, consequently, it is able to provide a more detailed insight into the system stability properties. The procedure is illustrated by a nonlinear SDOF example.
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Approximation of the movement of the spherical pendulum
Fischer, Cyril ; Náprstek, Jiří
The horizontally driven two-degrees of freedom spherical pendulum is an auto-parametric system which exhibits a wide variety of response types. Depending on the amplitude and frequency of the excitation the response can vary from stationary to chaotic regime. It has been already shown that the response can be described using time variable parameters of an inscribed ellipse. Such a description has a potential to close the gap between description of stationary, quasi-periodic and chaotic types of response. This study presents methodology and results of an analysis of the experimentally measured data providing the time dependent parameters of a twisting ellipse. The obtained results are discussed and some open problems are indicated.
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Experimental verification of resonance behaviour of a damped spherical pendulum
Fischer, Cyril ; Pospíšil, Stanislav ; Náprstek, Jiří
Theoretical, experimental and numerical analysis of a spherical pendulum is carried out. The stability of the response in a vertical plane is analysed in the theoretically predicted resonance region. Mathematical model respects the non-linear character of the pendulum and allows to introduce asymmetrical damping. Experimental pendulum is hanging from the Cardan joint and placed to carriage. Uni-directional harmonic excitation is applied to the system. The pendulum is damped by two magnetic units. These units are able to reproduce linear viscous damping independently in both principal response components. Response in in-plane and out-of-plane directions is measured and analyzed.
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Resonance behaviour of spherical pendulum – influence of damping
Fischer, Cyril ; Náprstek, Jiří ; Pospíšil, Stanislav
Experimental and numerical model of a uni-directionally driven pendulum-based tuned mass damper is presented in the paper. Stability of the motion in a vertical plane is analysed in the theoretically predicted resonance region. For the experimental part, special experimental frame is used, allowing independent change of linear viscous damping in the both perpendicular directions. Mathematical model respects the non-linear character of the pendulum and allows to introduce asymmetrical damping. Sensitivity of the resonance behaviour on the change of damping in both directions is studied and commented in the paper. The stability of the system is analysed experimentally and compared with numerical and theoretical results.
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Dynamic stability and post-critical processes of pendulum related auto-parametric systems
Náprstek, Jiří ; Fischer, Cyril
The existence and stability of the semi-trivial solution of the auto-parametric system is analyzed. Individual types of post-critical states are discussed (limit cycles, quasi-periodic response types, chaotic processes, transition processes). General considerations are demonstrated on particular DDOF and MDOF cases in both classes mentioned above. Sensitivity of several systems to stability loss with respect to their parameters, excitation amplitudes and other factors are evaluated. Bifurcation mechanism and diagrams are developed and analyzed. Important transition effects together with physical interpretation are investigated. Some open problems and possible future research strategy are outlined.
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Some properties of non-linear resonance of the pendulum damper
Fischer, Cyril ; Náprstek, Jiří
The pendulum damper modelled as a two degree of freedom strongly non-linear auto-parametric system is investigated using an approximate differential system. Uni-directional harmonic external excitation at the suspension point is considered. The resonance phenomenon cannot be described using the semi-trivial solution. In this contribution, nature of the numerical solution in the state of resonance is thoroughly investigated as a preliminary work for the detailed analytical study. Stationary and non-stationary (quasiperiodic) character of the resonance solution is determined and characterised. General form of a solution is proposed
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