Vibration of a Timoshenko beam supporting arbitrary large pre-deformation

Abstract

International audienceWe present an induced geometrically exact theory for the three-dimensional vibration of a beam undergoing finite transformation. The beam model coincides with a curvilinear Cosserat body and may be seen as an extension of the Timoshenko beam model. No particular hypothesis is used for the constitutive laws (in the framework of hyperelasticity), the geometry at rest or boundary conditions. The method leads to a weak formulation of the equations of vibration. The obtained internal energy is symmetric and leads to a self-adjoint operator that casts into a geometrical and material parts. Both may be written explicitly in terms of the finite transformation. The results are applied on the vibration of a beam supporting a finite longitudinal strain. The nonlinear effects according to the pre-stress are explicitly detailed for this example: instability, buckling

Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes

“…An appropriate beam model should be selected in order to properly capture the geometrical nonlinearities at very large amplitudes of vibration. In this paper, a finite element discretization of the geometrically exact beam model (also known as the Reissner-Simo beam model [21,22,23,24,25]), based on Timoshenko kinematics of the cross-section, is used. The advantage of this model is that the geometrical nonlinearities are kept exact without any truncation or linearization, a strong advantage over other nonlinear models that break down at high amplitudes when the rotation of the cross-section becomes large.…”

Explicit analysis of large transformation of a Timoshenko beam: post-buckling solution, bifurcation, and catastrophes

“…Cosserat's formulation was used by modeling the beam as a curvilinear line with moving directors frame. Inspired by [14] and by adapting dimensionless procedures, equilibrium relations in nondimensional form were found. Even if the physical problem is defined by a mixture of kinematical (position and orientation of the crosssection) and dynamical (force and moments) boundary conditions, the formulation followed in the present paper chooses to emphasize first a Cauchy problem formalism where all strains are prescribed at one end (in contrast to most studies in this field).…”

“…where the angle θ(S) = � e z , d 3 = � e x , d 1 measures in a trigonometric way the rotation of the section around d 2 . Spatial derivation of directors is obtained through (e.g [14]):…”

“…Later on, these results where expanded by Rakotomanana who regarded waves and vibrations of strings, beams and shells [13]. Henceforth, Le Marrec et al gave an exact theory of Timoshenko beam undergoing three-dimensional finite transformation and subjected to dynamical perturbations [14] . Stability of the equilibrium solutions is of crucial importance under such large transformations (e.g.…”