An energy-momentum co-rotational formulation for nonlinear dynamics of planar beams. (15th July 2017) - Chhang, Sophy; Sansour, Carlo; Hjiaj, Mohammed; Battini, Jean-Marc
Highlights: A new time integration scheme that converse the mechanical energy has been proposed for co-rotating planar beams. In absence of external loads, the linear and angular momenta remain constant with the proposed scheme. Formal proofs of conservation properties are given. Stability and accuracy are achieved in long term dynamics. Four numerical examples highlight the merits of the new integration scheme. Abstract: This article presents an energy-momentum integration scheme for the nonlinear dynamic analysis of planar Euler-Bernoulli beams. The co-rotational approach is adopted to describe the kinematics of the beam and Hermitian functions are used to interpolate the local transverse displacements. In this paper, the same kinematic description is used to derive both the elastic and the inertia terms. The classical midpoint rule is used to integrate the dynamic equations. The central idea, to ensure energy and momenta conservation, is to apply the classical midpoint rule to both the kinematic and the strain quantities. This idea, developed by one of the authors in previous work, is applied here in the context of the co-rotational formulation to the first time. By doing so, we circumvent the nonlinear geometric equations relating the displacement to the strain which is the origin of many numerical difficulties. It is rigorously shown that the proposed method conserves the total energy of the system and, in absence of external loads, the linear and angular momenta remain constant. The accuracy and stability of the proposed algorithm, especially in long term dynamics with a very large number of time steps, is assessed through four numerical examples.