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Dynamic response of a heavy ball rolling inside a spherical dish under external excitation
Náprstek, Jiří ; Fischer, Cyril
The set of a heavy metallic ball which is rolling freely inside a semispherical dish with larger diameter, being fixed to structure, is frequently used as tuned mass damper of vibration. Ratio of both diameters, mass of the rolling ball, quality of contact surfaces and other parameters should correspond with characteristics of the structure. The ball damper is modelled as a non-holonomic system. Hamiltonian functional including an adequate form of the Rayleigh function is formulated in moving coordinates using Euler angles and completed by ancillary constraints via Lagrangian multipliers. Subsequently Lagrangian differential system is carried out. Together with rolling conditions the governing system of seven equations is formulated. Later Lagrangian multipliers character is analysed and redundant motion components are eliminated. First integrals are derived and main energy balances evaluated together with their physical interpretation. Discussion of basic dynamic properties of the system is provided.
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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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Massive parallel implementation of ODE solvers
Fischer, Cyril
The paper maps the possibilities of exploitation of the massive parallel computational hardware (namely GPU) for solution of the initial value problems of ordinary differential equations. Two cases are discussed: parallel solution of a single ODE and parallel execution of scalar ODE solvers. Whereas the advantages of the special architecture in the case of a single ODE are problematic, repeated solution of a single ODE for different data can prot from the parallel architecture. However, special algorithms have to be used even in the latter case to avoid code divergence be- tween individual computational threads. The topic is illustrated on several examples.
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Analytic model of a ball rolling on a spherical surface under harmonic kinematic excitation
Náprstek, Jiří ; Fischer, Cyril
The set of a heavy metallic ball which is rolling freely inside a semi-spherical dish with larger diameter, being fixed to structure, is frequently used as tuned mass damper of vibration. Ratio of both diameters, mass of the rolling ball, quality of contact surfaces and other parameters should correspond with characteristics of the structure. The ball damper is modeled as a non-holonomic system. Hamiltonian functional including an adequate form of the Rayleigh function is formulated in moving coordinates using Euler angles and completed by ancillary constraints via Lagrangian multipliers. Subsequently Lagrangian differential system is carried out. Together with rolling conditions the governing system of seven equations is formulated. Later Lagrangian multipliers character is analyzed and redundant motion components are eliminated. First integrals are derived and main energy balances evaluated together with their physical interpretation.
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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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Dynamics of a vibration damperworking on a principle of a heavy ball rolling inside a spherical dish
Náprstek, Jiří ; Fischer, Cyril ; Pirner, Miroš
Wind excited vibrations of slender structures such as towers, masts or certain types of bridges can be reduced using passive or active vibration absorbers. If there is available only a limited vertical space to install such a device, a ball type of absorber can be recommended. In general, it is a semispherical horizontal dish in which a ball of a smaller diameter is rolling. Ratio of both diameters, mass of the rolling ball, quality of contact surfaces and other parameters should correspond with characteristics of the structure. The ball absorber is modeled as a holonomous system. Using Lagrange equations of the second type, governing non-linear differential system is carried out. The solution procedure combines analytical and numerical processes. As the main tool for dynamic stability investigation the 2nd Lyapunov method is used. The function and effectiveness of the absorber identical with those installed at the existing TV towers was examined in the laboratory.
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GPU computation in engineering practice
Fischer, Cyril
In many research fields the numerical problems demand extremely large computational power. As a consequence, researchers are constantly demanding new and more powerful computers. Recently, the graphics chip producer NVIDIA launched the CUDA parallel computing architecture, which enables scientists to utilize the extreme power available on modern and relatively cheap Graphical Processor Units (GPUs). Necessity to change the common approach together with significant difference in the GPU architecture prevents the wide usage of the graphics hardware.
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Experimental and theoretical stability analysis of damped auto-parametric pendulum
Pospíšil, Stanislav ; Fischer, Cyril ; Náprstek, Jiří
The article describes an experimental and theoretical treatment of the auto-parametric pendulum damper modelled as a double-degree-of-freedom (DDOF) system. It is strongly non-linear, with the harmonic excitation being applied in the suspension point and interpreted for instance as an excitation by wind. The stability of the motion in a vertical plane is analysed in the meaning of semi-trivial vibration. This effect of stability loss and possible stability regain is very important from practical point of view, because the motion of an damper beyond the stability limits may act negatively and thus endanger the structure itself. Special experimental frame was developed.The stability of the system is analysed experimentally and compared with 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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