Abstract
AbstractWe propose a method for the description and simulation of the nonlinear dynamics of slender structures modeled as Cosserat rods. It is based on interpreting the strains and the generalized velocities of the cross sections as basic variables and elements of the special Euclidean algebra. This perspective emerges naturally from the evolution equations for strands, that are one‐dimensional submanifolds, of the special Euclidean group. The discretization of the corresponding equations for the three‐dimensional motion of a Cosserat rod is performed, in space, by using a staggered grid. The time evolution is then approximated with a semi‐implicit method. Within this approach, we can easily include dissipative effects due to both the action of external forces and the presence of internal mechanical dissipation. The comparison with results obtained with different schemes shows the effectiveness of the proposed method, which is able to provide very good predictions of nonlinear dynamical effects and shows competitive computation times also as an energy‐minimizing method to treat static problems.
Highlights
The modeling and simulation of beams is of great importance in the engineering practice to analyze the configurations and stress distributions of a wide variety of mechanical structures, with sizes ranging from those of pipelines and cables to those of microactuators
It has become clear that the theory of special Cosserat rods provides the optimal mathematical framework to deal with slender structures, as it comprises all of the classical beam models as special cases
We propose a method for the description of the nonlinear dynamics of slender beams that is based on extensions of the SE (3)-strand equations described by Holm and Ivanov [15], with a suitable mechanical interpretation of stress and momenta as dual to strains and velocities
Summary
The modeling and simulation of beams is of great importance in the engineering practice to analyze the configurations and stress distributions of a wide variety of mechanical structures, with sizes ranging from those of pipelines and cables to those of microactuators. To simplify the derivation and the structure of the rod equations, a fundamental step is to view the strains and the generalized velocities of the rigid cross sections as elements of the Lie algebra associated with the special Euclidean group of rigid body motions. This perspective led Simo, Marsden & Krishnaprasad [26] and Hodges [13, 14] to derive the intrinsic rod equations from the variations of a Hamiltonian functional expressed solely in terms of Lie algebraic quantities. In spite of the simplicity of our formulation and of the discretization schemes that we have adopted, the method achieves very good results in solving both static and nonlinear dynamical problems, with competitive computational times
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