Quantifying chaos of curvilinear beams via exponents

We propose a procedure for predicting the stability loss and transition into chaos of a network of oscillators lying on a curve, where each of the oscillators can move in two perpendicular directions. Dynamics of the coupled oscillators are governed by the sixth-order PDE, which is directly derived using the classical hypotheses of a curvilinear flexible beam movement theory. We apply FDM (Finite Difference Method) to reduce PDEs into ODEs, and the used number of spatial coordinate positions defines the number of involved oscillators approximating the dynamics of our continuous structural member (beam). Our procedure has a few advantages over the classical approaches, which has been illustrated and discussed. The proposed method has been validated for non-linear dynamical regimes by using the classical vibrational analysis (time histories, frequency power spectra and Poincaré maps).