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Following the notation of the Eq. The other quantities appearing in Eq.
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It is important to remark that the continuous model from which the discretized equations of motion have been derived, has been formulated by modeling the Nicolai beam as a one-dimensional polar continuum, geometrically nonlinear and internally constrained. In particular, the constraints are the unshearability, the inextensibility and the untwistability. The first two are commonly used in the modeling of beams while the third one is based on an analysis of the orders of magnitude of the energy contribution of the underlying elastic model, according to Luongo and Zulli By using the discretized system introduced above, we want to highlight the bifurcation mechanisms which guides the Nicolai paradox.
This latter equation represents the key point to understand the Nicolai paradox since, if the system is symmetric i. The stability domain expressed by Eq.
It is worth noticing that, since in the present paper a reduced two degrees of freedom system is considered, the effects of higher modes is ignored. However, the effects of this reduction needs to be deeply investigated since, e. In that sense the present paper is a first although quite rough approach to the matter. Stable zone S in gray. Our post-critical analysis is aimed to numerically investigate the dynamics of the nonlinear system close to the bifurcation point. To this end, a direct integration of the nonlinear equations of motion and a parametric analysis has been performed, in the case of elliptical cross section of the previous paragraph.
Results of this integration are displayed in Fig. Once the large circular motion has been reached, the system increases its velocity unboundedly see Fig. Therefore, the beam whirls with an increasing velocity, experiencing a conical motion. Finally, it is important to remark that, in the first approach to the nonlinear problem carried out in the present paper, the effects of higher modes are ignored since, as we said, a reduced model of two degrees of freedom has been considered.
Moreover, damping has been ignored. The investigation of these aspects will be object of our studies in future works.