Browsing by Author "Leine, Remco I."

Now showing 1 - 7 of 7

Open Access
The direct method of Lyapunov for nonlinear dynamical systems with fractional damping
(2020) Hinze, Matthias; Schmidt, André; Leine, Remco I.
In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.
Open Access
Model reduction of a periodically forced slow-fast continuous piecewise linear system
(2023) Karoui, A. Yassine; Leine, Remco I.
In this paper, singular perturbation theory is exploited to obtain a reduced-order model of a slow-fast piecewise linear 2-DOF oscillator subjected to harmonic excitation. The nonsmooth nonlinearity of piecewise linear nature is studied in the case of bilinear damping as well as with bilinear stiffness characteristics. We propose a continuous matching of the locally invariant slow manifolds obtained in each subregion of the state space, which yields a reduced-order model of the same nature as the full dynamics. The frequency-response curves obtained from the full system and the reduced-order models suggest that the proposed reduction method can capture nonlinear behaviors such as super- and subharmonic resonances.
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.
Open Access
A nonsmooth generalized‐alpha method for mechanical systems with frictional contact
(2021) Capobianco, Giuseppe; Harsch, Jonas; Eugster, Simon R.; Leine, Remco I.
In this article, the existing nonsmooth generalized‐α method for the simulation of mechanical systems with frictionless contacts, modeled as unilateral constraints, is extended to systems with frictional contacts. On that account, we complement the unilateral constraints with set‐valued Coulomb‐type friction laws. Moreover, we devise a set of benchmark systems, which can be used to validate numerical schemes for mechanical systems with frictional contacts. Finally, this set of benchmarks is used to numerically assert the properties striven for during the derivation of the presented scheme. Specifically, we show that the presented scheme can reproduce the dynamics of the frictional contact adequately and no numerical penetration of the contacting bodies arises - a big issue for most popular time‐stepping schemes such as the one of Moreau. Moreover, we demonstrate that the presented scheme performs well for multibody systems containing flexible parts and that it allows general parametrizations such as the use of unit quaternions for the rotation of rigid bodies.
Open Access
Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis
(2023) Bayer, Fabia; Leine, Remco I.
In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting , which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n -DOF generalization are used to exemplify these findings.
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, Andrin
The 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.
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.