A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

Abstract

In this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

Evaluation and Implementation of Lie Group Integration Methods for Rigid Multibody Systems

“…In our numerical experiments, we observed effects of round-off errors in the evaluation of the tangent operator T exp respectively its inverse near ∥∆q∥ = 0. The effects of round-off errors when evaluating tangent operators have also been reported in [42] and are termed there as a loss of significance. To obtain accurate and precise results, we use Taylor approximations for those terms that cause the loss of significance.…”

“…In Eq. (A1), the matrices T exp SO(3) ∈ R 3×3 and T −1 exp SO(3) ∈ R 3×3 are the tangent operators on SO(3) and read [2,42]…”

“…There is a loss of significance when Eqs. (A4-A6) are evaluated near ∥∆ψ∥ = 0 on a [42]. In order to obtain accurate and precise results from Eqs.…”

“…We choose the intervals where Eqs. (A4-A6) are evaluated by its Taylor approximation as proposed in [42]. The improved versions of Eqs.…”

Relative-Kinematic Formulation of Geometrically Exact Beam Dynamics based on Lie-Group Variational Integrators

“…as in [56], which can be obtained by differentiating the continuous-time equivalent of Eq. ( 13) and then substituting ġa = g a ηa | t=t k ≈ g k a ηk a .…”