07 Fakultät Konstruktions-, Produktions- und Fahrzeugtechnik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/8
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Item Open Access A family of total Lagrangian Petrov-Galerkin Cosserat rod finite element formulations(2023) Eugster, Simon R.; Harsch, JonasThe standard in rod finite element formulations is the Bubnov-Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov-Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.Item Open Access Dynamics of finite-dimensional mechanical systems(2019) Winandy, Tom; Leine, Remco I. (Prof. Dr. ir. habil.)This monograph deals with the description of mechanical systems having finitely many degrees of freedom using the language of global differential geometry. The mechanical systems may be explicitly time-dependent and involve nonpotential forces. The focus is on the mathematically rigorous formulation of the physical theory dealing with the aforementioned mechanical systems with the objective to introduce the involved physical quantities as well-defined mathematical objects. The geometric presentation of the physical theory is erected upon a generalized space-time known as Galilean manifold. The state space of a mechanical system is defined as an affine subbundle of the tangent bundle of its associated Galilean manifold. The system's motion is considered to be an integral curve of a second-order vector field on the state space. With the coordinate-free characterization of the motion in terms of second-order vector fields, differential forms appear on stage. A one-to-one correspondence between second-order vector fields and action forms is established. Action forms are differential two-forms with additional properties. The definition of action forms and the derivation of this bijective relation relies on the geometry of double tangent bundles, in which vector bundle homomorphisms and their differential concomitants play an important role. A coordinate-free definition of forces is given and different geometric interpretations are discussed. With the definition of kinetic energy and of potential forces, the equations of motion are postulated in a coordinate-free way using the action form of the mechanical system. Lagrange's, Hamel's, and Hamilton's equations become local representations of this postulate in terms of a respective chart of the state space. Moreover, the connection between action forms and the concept of virtual work is established. This allows to obtain Lagrange's and Hamel's central equation. This variational perspective is pursued by showing that motions characterized by an exact action form satisfy Hamilton's principle. For this purpose, a coordinate-free definition of the action integral is given. Finally, constraints are defined as distributions compatible with the time structure of the Galilean manifold on which they are defined. Consequently, the distinction between holonomic and nonholonomic constraints is made using the Frobenius theorem.Item Open Access State observers for the time discretization of a class of impulsive mechanical systems(2022) Preiswerk, Pascal V.; Leine, Remco I.In this work, we investigate the state observer problem for linear mechanical systems with a single unilateral constraint, for which neither the impact time instants nor the contact distance is explicitly measured. We propose to attack the observer problem by transforming and approximating the original continuous‐time system by a discrete linear complementarity system (LCS) through the use of the Schatzman-Paoli scheme. From there, we derive a deadbeat observer in the form of a linear complementarity problem. Sufficient conditions guaranteeing the uniqueness of its solution then serve as observability conditions. In addition, the discrete adaptation of an existing passivity‐based observer design for LCSs can be applied. A key point in using a time discretization is that the discretization acts as a regularization, that is, the impacts take place over multiple time steps (here two time steps). This makes it possible to render the estimation error dynamics asymptotically stable. Furthermore, the so‐called peaking phenomenon appears as singularity within the time discretization approach, posing a challenge for robust observer design.Item Open Access Mechanical systems with frictional contact : geometric theory and time discretization methods(2021) Capobianco, Giuseppe; Leine, Remco I. (Prof. Dr. ir. habil.)This dissertation deals with the mathematical description and the simulation of mechanical systems with frictional contact. First, a geometric theory for the description of smooth mechanical systems is developed, which is then extended to allow for nonsmooth motions, i.e., motions with discontinuous velocities. The developed nonsmooth theory of mechanics is used to describe mechanical systems with frictional contact. Finally, two numerical schemes for the simulation of such systems are derived by using a time finite element method and the generalized-alpha approach, respectively.Item Open Access Stability of rigid body motion through an extended intermediate axis theorem : application to rockfall simulation(2021) Leine, Remco I.; Capobianco, Giuseppe; Bartelt, Perry; Christen, Marc; Caviezel, AndrinThe stability properties of a freely rotating rigid body are governed by the intermediate axis theorem, i.e., rotation around the major and minor principal axes is stable whereas rotation around the intermediate axis is unstable. The stability of the principal axes is of importance for the prediction of rockfall. Current numerical schemes for 3D rockfall simulation, however, are not able to correctly represent these stability properties. In this paper an extended intermediate axis theorem is presented, which not only involves the angular momentum equations but also the orientation of the body, and we prove the theorem using Lyapunov’s direct method. Based on the stability proof, we present a novel scheme which respects the stability properties of a freely rotating body and which can be incorporated in numerical schemes for the simulation of rigid bodies with frictional unilateral constraints. In particular, we show how this scheme is incorporated in an existing 3D rockfall simulation code. Simulations results reveal that the stability properties of rotating rocks play an essential role in the run-out length and lateral spreading of rocks.Item Open Access Model reduction of the tippedisk : a path to the full analysis(2021) Sailer, Simon; Leine, Remco I.The tippedisk is a mechanical-mathematical archetype for friction-induced instability phenomena that exhibits an interesting inversion phenomenon when spun rapidly. The inversion phenomenon of the tippedisk can be modeled by a rigid eccentric disk in permanent contact with a flat support, and the dynamics of the system can therefore be formulated as a set of ordinary differential equations. The qualitative behavior of the nonlinear system can be analyzed, leading to slow-fast dynamics. Since even a freely rotating rigid body with six degrees of freedom already leads to highly nonlinear system equations, a general analysis for the full system equations is not feasible. In a first step the full system equations are linearized around the inverted spinning solution with the aim to obtain a local stability analysis. However, it turns out that the linear dynamics of the full system cannot properly describe the qualitative behavior of the tippedisk. Therefore, we simplify the equations of motion of the tippedisk in such a way that the qualitative dynamics are preserved in order to obtain a reduced model that will serve as the basis for a following nonlinear stability analysis. The reduced equations are presented here in full detail and are compared numerically with the full model. Furthermore, using the reduced equations we give approximate closed form results for the critical spinning speed of the tippedisk.Item Open Access A total Lagrangian, objective and intrinsically locking‐free Petrov-Galerkin SE(3) Cosserat rod finite element formulation(2023) Harsch, Jonas; Sailer, Simon; Eugster, Simon R.Based on more than three decades of rod finite element theory, this publication combines the successful contributions found in the literature and eradicates the arising drawbacks like loss of objectivity, locking, path-dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jelenić in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SE(3)-structure of the Cosserat rod kinematics. Applying a Petrov-Galerkin projection method, we propose a rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time-derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parameterization in terms of a minimal number of nodal unknowns are demonstrated by representative numerical examples in both statics and dynamics.Item Open Access Nonlinear dynamics of the tippedisk : a holistic analysis(2023) Sailer, Simon; Leine, Remco I. (Prof. Dr. ir. habil.)This dissertation deals with the tippedisk which is a new mechanical-mathematical archetype for friction-induced instabilities and exhibits an energetically counterintuitive inversion phenomenon. In a holistic analysis, the dynamics of the tippedisk is investigated numerically in the field of multibody simulation, theoretically in the field of nonlinear dynamics, and experimentally in the focus of applied physics. Based on different nonsmooth rigid body models with set-valued force laws, the main physical mechanisms inducing the inversion behavior are identified and the governing system equations are derived. Subsequent model reduction results in a reduced system in the form of an ordinary differential equation, which is suited to be studied in the context of nonlinear dynamics. Both the local stability behavior of the non-inverted and inverted stationary spinning motions as well as the global proof of an existing heteroclinic saddle connection allow the dynamic behavior of the tippedisk to be captured analytically. The particular structure of the mathematical model reveals a singularly perturbed dynamics that evolves on multiple time scales and is characterized by slow rolling and fast sliding motions of the tippedisk. Utilizing perturbation expansions and an analysis in dimensionless quantities, the qualitative dynamics is characterized by closed-form expressions, from which a global stability map is deduced. Based on this complete stability map, three different bifurcation scenarios are identified, which correspond to different geometric and inertia properties, defining three qualitatively different types of tippedisks. Finally, the mathematical investigation is complemented by high-speed experiments on a real test specimen. Qualitative comparison of experimental measurements with simulations at different levels of abstraction completes the holistic approach to the dynamic analysis of the tippedisk.