Martin Arnold

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.

Explicit Runge–Kutta methods are the gold standard of time-integration methods for nonstiff problems in system dynamics since they combine a small numerical effort per time step with high accuracy, error control, and straightforward implementation. For the analysis of beam dynamics, we couple them with a local coordinates approach in a Lie group setting to address large rotations. Stiff shear forces and inextensibility conditions are enforced by internal constraints in a coarse-grid discretization of a geometrically exact beam model. The resulting nonstiff constrained systems are handled by a half-explicit approach that relies on the constraints at velocity level and avoids all kinds of Newton–Raphson iteration. We construct half-explicit Runge–Kutta Lie group methods of order up to five that are equipped with an adaptive step-size strategy using embedded Runge–Kutta pairs for error estimation. The methods are tested successfully for a roll-up maneuver of a flexible beam and for the classical flying-spaghetti benchmark.

The present work aims at describing hysteresis behaviour arising from cyclic bending experiments on cables by means of the Preisach operator. Pure bending experiments conducted in previous work show that slender structures such as electric cables behave inelastically and open hysteresis loops arise, with noticeable difference between the first load cycle and the following ones. The Preisach operator plays an important role in describing the input-output relation in hysteresis behaviours and it can be expressed as a superposition of relay operators. Here, we utilise data collected from pure bending experiments for a first approach. We introduce a mathematical formulation of the problem, and starting from the curvature of the cable specimen, we recursively define the Preisach plane for this specific case. Therefore, we derive a suitable kernel function in a way that the integration of such function over the Preisach plane results in the bending moment of the specimen.

As commonly known, standard time integration of the kinematic equations of rigid bodies modeled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using the $\mathbb{R}^{3}\times SO(3)$ R 3 × S O ( 3 ) Lie group formulation and several typical rigid multibody systems. It is shown that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turns out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the Lie group integration method is computationally the more efficient one. It also turns out that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly.

The simulation of inelastic effects in flexible slender technical devices has become of increasing interest in the past years. Different approaches have been considered depending on the effects relevant for the specific application. Recently, a mixed stress strain driven computational homogenisation has been proposed to model the dissipative nonlinear bending response of spiral strands subjected to axial force. In this study, we propose two different approaches, namely a rheological model and a databased greybox model, to predict the cyclic response of these strands using only their monotonic response. In the first approach, a system of so-called bending springs and sliders is used to model different contributions to the bending stiffness of the strands. The data-based approach makes use of mathematical tools called hysteresis operators. The Prandtl-Ishlinskii operator plays a relevant role in modelling the input-output relation in phenomena showing hysteretic behaviour and can be expressed as a weighted superposition of elementary stop operators. Comparing the two approaches leads to a better understanding and an explicit physical interpretation of the parameters of a specific class of hysteresis operator models.

We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce non-expansive systems on such manifolds and define B-stability of integrators. In this first exposition, we provide concrete results for a geodesic version of the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on Riemannian manifolds with non-positive sectional curvature. We show through numerical examples that the GIE method is expansive when applied to a certain non-expansive vector field on the 2-sphere, and that the GIE method does not necessarily possess a unique solution for large enough step sizes. Finally, we derive a new improved global error estimate for general Lie group integrators.