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CivilComp Proceedings
ISSN 17593433 CCP: 86
PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 11
AutoParametric PostCritical Behaviour of a Spherical Pendulum Damper J. Náprstek and C. Fischer
Institute of Theoretical and Applied Mechanics ASCR, v.v.i., Prague, Czech Republic , "AutoParametric PostCritical Behaviour of a Spherical Pendulum Damper", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 11, 2007. doi:10.4203/ccp.86.11
Keywords: nonlinear vibration, spherical pendulum, autoparametric systems, dynamic stability, bifurcation points, asymptotic methods.
Summary
Many structures encountered in civil and mechanical engineering are equipped with various
devices for reducing the dynamic response component. The examples are slender structures,
such as towers, masts, chimneys, or technological and special means of transport. Many
monographs and papers dealing with this topic have been published, for instance [1].
Dynamic behaviour of the pendulum, however, is significantly more complex than it is
supposed by the conventional linear SDOF model. The linear model is satisfactory only if
the amplitude of the excitation is small and if its frequency remains outside a resonance
domain.
The spherical pendulum is a strongly nonlinear system with two degrees of freedom of the autoparametric type, e.g. [2]. The response is described by two simultaneous differential equations of the second order which are independent on the linear level. Their interaction follows from nonlinear terms. The differential system admits the semitrivial solution. It means that one response component is nontrivial while the second one remains trivial. The semitrivial solution can lose its stability under certain conditions producing various types of the postcritical response. Three types of the resonance domains have been identified with respect to the relation of a preliminary resonance curve and two stability limits (in or out of the vertical plane) of the semitrivial solution. The first type of the resonance domain is encountered when both stability limits are intersected by the resonance curve. Four postcritical response regimes of a deterministic, chaotic, stationary and nonstationary character have been observed, when passing through this resonance domain. Above the upper limit of the resonance domain the existence of a stable stationary solution in the vertical plane resumes. The second type follows from an intersection of the resonance curve with out of plane stability limit only. In such a case two different response regimes arise. The third type is represented by the resonance curve below the stability limits without any intersection points. No particular regime is generated and the semitrivial solution continues throughout the whole resonance domain linking smoothly with those in the sub and superresonance domains. From the practical point of view, it is recommended that the damping pendulum be designed in such a way that any intersections of the resonance curve with the stability limits are avoided. Otherwise the negative influence of the pendulum in the resonance domain is to be expected in both alongwind as well as in crosswind directions. Taking into account that excitation in the open air has rather a broad band random character, details of such a device should be thoroughly thought out. Results of this study can help avoid a misapplication of the pendulum damper. References
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