NUMERICAL SOLUTIONS OF FORCED VIBRATION AND WHIRLING OF A NON-LINEAR STRING USING THE THEORY OF A COSSERAT POINT

Abstract

O'Reilly and Holmes [1] performed experiments and analysis of non-linear motion of a string subjected to vertical oscillation of one end. Their analysis suggested that differences between theoretical and experimental results were mainly due to uncertainty in the assumed form for the forcing function that was used to satisfy the forced boundary condition. Here, a numerical solution of the non-linear string problem is considered which incorporates the forced boundary condition directly and thus avoids the need to assume an arbitrary forcing function. Specifically, the numerical formulation of string problems based on the theory of a Cosserat point is used. It is shown that the director inertia coefficient in the Cosserat theory can be chosen so that the theory predicts the first modal frequency of small deformation free vibration exactly for each level of discretizational. Also, examples of forced in-plane motion and free non-linear whirling motion are considered to validate the numerical model. The resulting simulations indicate that the forcing amplitude for the onset of persistent whirling and aperiodic response is about five times smaller than that observed in the experiments even when the uncertainty in the forcing function is removed from the analysis. Additional attempts to bridge the gap between theory and experiment suggest that there is still some mechanism in the experiment that is not properly modelled.