Keywords: non-linear vibration, spherical pendulum, auto-parametric systems, dynamic stability, bifurcation points, asymptotic methods.

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 non-linear system with two degrees of freedom of the auto-parametric 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 non-linear terms. The differential system admits the semi-trivial solution. It means that one response component is non-trivial while the second one remains trivial. The semi-trivial solution can lose its stability under certain conditions producing various types of the post-critical 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 semi-trivial solution.

The first type of the resonance domain is encountered when both stability limits are intersected by the resonance curve. Four post-critical response regimes of a deterministic, chaotic, stationary and non-stationary 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 semi-trivial solution continues throughout the whole resonance domain linking smoothly with those in the sub and super-resonance 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 along-wind as well as in cross-wind 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.

1

J. Náprstek, M. Pirner, "Non-linear behaviour and dynamic stability of a vibration spherical absorber", in: "Proc. 15th ASCE Engineering Mechanics Division Conference", A. Smyth et al. (editors), Columbia Univ., New York, 2002, 10 pp. CD ROM.

2

A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, "Autoparametric Resonance in Mechanical Systems", Cambridge University Press, Cambridge, 2000.

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