A finite element approach to the chaotic motion of geometrically exact rods undergoing in-plane deformations

Abstract

The paper is concerned with a hybrid finite element formulation for the geometrically exact dynamics of rods with applications to chaotic motion. The rod theory is developed for in-plane motions using the direct approach where the rod is treated as a one-dimensional Cosserat line. Shear deformation is included in the formulation. Within the elements, a linear distribution of the kinematical fields is combined with a constant distribution of the normal and shear forces. For time integration, the mid-point rule is employed. Various numerical examples of chaotic motion of straight and initially curved rods are presented proving the powerfulness and applicability of the finite element formulation.

Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams

“…The chosen strain measure corresponds to the engineering strainsẽ and κ = θ of the present study and the stress measure is a consequence of the constitutive laws N = E Aẽ and M = E I θ , which are postulated and not a result of a proper derivation from the 3D continuum mechanics laws. This family also groups the studies based on Cosserat rods theories [1,15,40,41,61,73], in which the same strain measure is used, but derived from the derivatives of the crosssection position and orientation in the deformed configuration. The same stress measure and constitutive laws are postulated and used.…”

On the design of energy–momentum integration schemes for arbitrary continuum formulations. Applications to classical and chaotic motion of shells

“…Chaotic vibrations of geometrically exact arches undergoing in-plane deformation are given in detail in Sansour et al [16]. There, the one-dimensional Cosserat line using a rotation tensor was used to model the geometrically exact arch and the mid-point rule was applied for time discretization.…”

Ανάλυση Μη Γραμμικών Συστημάτων Σε Πεπερασμένους Χώρους Φάσεων

“…• περιέχει απόσβεση Για την αντιμετώπιση -επίλυση του συστήματος των εξισώσεων κίνησης μια αναλυτική επεξεργασία χωρίς την απλοποίηση της μη γραμμικότητας δεν είναι εφικτή με μη γραμμικές μεθόδους. Ωστόσο, τέτοιας μορφής συστήματα μπορεί να λυθούν με πολύ μεγάλη ακρίβεια μέσω αποτελεσματικών αλγορίθμων που βασίζονται στη μέθοδο των πεπερασμένων στοιχείων [Sansour et. al 1996;Georgiou and Sansour, 1998;Georgiou and Sansour 2000].…”