Stefan Hante

When structures are subjected to dynamic loading, such as that caused by an earthquake or working machinery, the rocking behavior of objects located on parts of the loaded structure plays an important role in addressing the protection and stability of non-structural components. In this work, the free rocking of a rigid block on a flexible beam and rigid base was investigated using numerical simulations. To this end, a numerical code based on the non-smooth contact dynamics method was developed for this particular problem, and numerical simulations are compared to experimental tests when a rigid base is considered. The purpose of the study was to investigate the predictive capabilities and limitations of the numerical model and address the effect of introducing beam flexibility on the rocking response. The investigated flexibilities were such that the beam deflection under the static weight of the block remains within the common limit of 1/250 of the beam span. For a rigid base, qualitatively good correlation with the experiments was obtained, and good convergence in terms of the time-step is displayed. With the increase in beam base flexibility, it was observed that the simulation results tend to become more sensitive to mesh density and time-step size. Furthermore, we identify a limited flexibility with respect to which unreliable predictions of the overall free rocking are obtained, which corresponds to the stiffness resulting in the beam deflection under the block weight of beam-span/2500. For stiffnesses higher than that, no significant effect of beam flexibility in comparison to the rigid base was noticed in terms of tilt angle and rocking duration, which indicates the adequacy of a rigid base approximation for beams with low flexibility.

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.

We will discuss staggered grid space discretizations for micropolar Cosserat models with nonlinear configuration spaces, that have Lie group structure. The discretization scheme is applied at the level of the variational principle, has second order truncation error and uses a staggered grid, where the spatial derivatives are discretized inbetween the discretization points of the material configuration.

Configuration spaces with Lie group structure display kinematical nonlinearities of mechanical systems. In Lie group time integration, this nonlinear structure is also considered at the time-discrete level using nonlinear updates of the configuration variables. For practical implementation purposes, these update formulae have to be adapted to each specific Lie group setting that may be characterized from the algorithmic viewpoint by group operation, exponential map, tilde, and tangent operator. In this paper, we discuss these practical aspects for the time integration of a geometrically exact Cosserat rod model with rotational degrees-of-freedom being represented by unit quaternions. Shearing and longitudinal extension of the Cosserat rod may be neglected using suitable constraints that result in a differential-algebraic equation (DAE) formulation of the beam structure. The specific structure of unconstrained systems and constrained systems is exploited by tailored algorithms for the corrector iteration of the generalized-α Lie group integrator.