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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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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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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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