arXiv subjects

Remco I. Leine

Publications and source records attributed to Remco I. Leine.

12 recordsLinked to original sources

arXiv2026

A Mixed Discrete Cosserat Rod Formulation

In this communication we propose a discrete Cosserat rod formulation in which a slender elastic rod is represented as a chain of rigid bodies (nodes) coupled by compliant elastic forces and moments acting between adjacent node pairs. Discrete dilatation, shear, torsion and curvature strain measures are evaluated from the relative kinematics of each node pair, while the constitutive behavior is expressed in compliance form through independent stress degrees of freedom. We show that the resulting model arises rigorously from a mixed Petrov--Galerkin Cosserat rod finite element formulation (FEM) at linear kinematic interpolation order when the internal virtual work is integrated by the midpoint rule and the external and inertial contributions by the trapezoidal rule. The proposed formulation inherits the robustness and the absence of locking from the underlying mixed FEM while simultaneously exposing a two-node coupling structure that mirrors discrete rod models from the computer graphics community. This is in sharp contrast to the dense coupling of strain-parameterized reduced-order models often used in soft robotic applications. Three numerical examples involving piecewise-varying cross sections, tendon-driven actuation under different spatial discretizations, and coupled longitudinal-torsional dynamics confirm the accuracy, robustness, and convergence behavior of the presented approach.

math.NA↗

arXiv2026

Koopman-based stability analysis of differential-algebraic equations with applications to frictional multibody systems

Periodic solutions of differential-algebraic equations (DAEs) and ordinary differential equations (ODEs) can be determined using the harmonic balance method (HBM), which is a frequency-domain approach that approximates the solution by its truncated Fourier series. The Koopman-Hill method, a method to determine the stability of periodic solutions found by HBM, and originally developed for ODEs, is generalized to DAEs in this work. Analogously to the ODE case, the core idea of the proposed Koopman-Hill method for DAEs is to establish a linear time-invariant but high-dimensional DAE which approximately governs the dynamics of small admissible perturbations around the periodic solution. The crucial difference to the ODE case is the fact that the evolution of this linear time-invariant DAE is not simply given by a matrix exponential, but by a more complicated expression involving a Drazin inverse, rendering the resulting monodromy matrix singular. Still, even in the DAE case, this novel relationship between the monodromy matrix and the Hill matrix is essentially given by one single formula, which is the main result of this work. Two academic mechanical systems, a mathematical pendulum formulated as an index-3 DAE and a nonsmooth frictional two-mass oscillator with switching index, demonstrate the applicability of the proposed method and its blindness to the DAE's index.

math.DS↗

arXiv2026

Hill-Type Stability Analysis of Periodic Solutions of Fractional-Order Differential Equations

This paper explores stability properties of periodic solutions of (nonlinear) fractional-order differential equations (FODEs). As classical Caputo-type FODEs do not admit exactly periodic solutions, we propose a framework of Liouville-Weyl-type FODEs, which do admit exactly periodic solutions and are an extension of Caputo-type FODEs. Local linearization around a periodic solution results in perturbation dynamics governed by a linear time-periodic differential equation. In the classical integer-order case, the perturbation dynamics is therefore described by Floquet theory, i.e. the exponential growth or decay of perturbations is expressed by Floquet exponents which can be assessed using the Hill matrix approach. For fractional-order systems, however, a rigorous Floquet theory is lacking. Here, we explore the limitations when trying to extend Floquet theory and the Hill matrix method to linear time-periodic fractional-order differential equations (LTP-FODEs) as local linearization of nonlinear fractional-order systems. A key result of the paper is that such an extended Floquet theory can only assess exponentially growing solutions of LTP-FODEs. Moreover, we provide an analysis of linear time-invariant fractional-order systems (LTI-FODEs) with algebraically decaying solutions and show that the inaccessibility of decaying solutions through Floquet theory is already present in the time-invariant case.

eess.SY↗

arXiv2026

Model order reduction of piecewise linear mechanical systems using invariant cones

We present a methodology that extends invariant manifold theory to a class of autonomous piecewise linear systems with nonsmoothness at the equilibrium, providing a framework for model order reduction in mechanical structures with compliant contact laws. The key idea is to make the absence of a local linearization around the equilibrium tractable by leveraging the positive homogeneity property. This property simplifies the invariance equations defining the geometry of the invariant cones, from a set of partial differential equations to a system of ordinary differential equations, enabling their effective solution. We introduce two techniques to compute these invariant cones. First, an intuitive graph-style parametrization is proposed that utilizes Fourier expansions and Chebyshev polynomials to derive explicit reduced-order models in closed form. Second, an arc-length parametrization is introduced to robustly compute invariant cones with complex folding geometries, which are intractable with a standard graph-style technique. The approach is demonstrated on mechanical oscillators with unilateral visco-elastic supports, showcasing its applicability for systems with both continuous (unilateral elastic) and discontinuous (unilateral visco-elastic) unilateral force laws.

math.DS↗

arXiv2025

The effect of friction on the dynamics of targeted energy transfer by symmetric vibro-impact dampers

This study investigates the nonlinear dynamics of a symmetric vibro-impact nonlinear energy sink (VI-NES) subjected to dry friction, a crucial factor that remains insufficiently explored in previous research. The combined effect of impact and friction leads to intricate behaviors that require further investigation. To address this, the multiple scales method is extended to incorporate frictional effects and is complemented with a generalized impact map approach. This allows for a systematic exploration of periodic solutions, stability, and bifurcations, revealing critical transitions between impact-dominated and sliding-dominated regimes. The activation thresholds and amplitude levels for different response regimes, including stick-slip dynamics, are identified, offering new insights into friction-induced nonlinearities. The results bridge the gap between theoretical modeling and practical implementation, offering a more accurate predictive framework for VINES behavior. This improves design strategies for enhanced energy dissipation and robustness in real-world applications.

nlin.CD↗

arXiv2025

Explicit error bounds and guaranteed convergence of the Koopman-Hill projection stability method for linear time-periodic dynamics

The Koopman-Hill projection method offers an efficient approach for stability analysis of linear time-periodic systems, and thereby also for the Floquet stability analysis of periodic solutions of nonlinear systems. However, its accuracy has previously been supported only by numerical evidence, lacking rigorous theoretical guarantees. This paper presents the first explicit error bound for the truncation error of the Koopman-Hill projection method, establishing a solid theoretical foundation for its application. The bound applies to linear time-periodic systems whose Fourier coefficients decay exponentially with a sufficient rate, and is derived using constructive series expansions. The bound quantifies the difference between the true and approximated fundamental solution matrices, clarifies conditions for guaranteed convergence, and enables conservative but reliable inference of Floquet multipliers and stability properties. Additionally, the same methodology applied to a subharmonic formulation demonstrates improved convergence rates of the latter. Numerical examples, including the Mathieu equation and the Duffing oscillator, illustrate the practical relevance of the bound and underscore its importance as the first rigorous theoretical justification for the Koopman-Hill projection method.

math.NA↗

arXiv2025

Asymmetric VI-NES with dry friction: An impact map approach

This paper examines the dynamics of a vibro-impact nonlinear energy sink (VI-NES) using a generalized impact map approach. The study incorporates asymmetry and dry friction, reflecting realistic conditions. The proposed method identifies all periodic solutions and determines their stability, and is applicable to various VI-NES configurations, including horizontal and vertical orientations. Numerical results validate prior findings for symmetric frictionless cases and extend them to include frictional and asymmetric dynamics, providing a powerful tool for optimizing the performance of VI-NES in vibration mitigation.

nlin.CD↗

arXiv2024

Extended invariant cones as Nonlinear Normal Modes of inhomogeneous piecewise linear systems

The aim of this paper is to explore the relationship between invariant cones and nonlinear normal modes in piecewise linear mechanical systems. As a key result, we extend the invariant cone concept, originally established for homogeneous piecewise linear systems, to a class of inhomogeneous continuous piecewise linear systems. The inhomogeneous terms can be constant and/or time-dependent, modeling nonsmooth mechanical systems with a clearance gap and external harmonic forcing, respectively. Using an augmented state vector, a modified invariant cone problem is formulated and solved to compute the nonlinear normal modes, understood as periodic solutions of the underlying conservative dynamics. An important contribution is that invariant cones of the underlying homogeneous system can be regarded as a singularity in the theory of nonlinear normal modes of continuous piecewise linear systems. In addition, we use a similar methodology to take external harmonic forcing into account. We illustrate our approach using numerical examples of mechanical oscillators with a unilateral elastic contact. The resulting backbone curves and frequency response diagrams are compared to the results obtained using the shooting method and brute force time integration.

math.DS↗

arXiv2021

Singularly perturbed dynamics of the tippedisk

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disk, being reminiscent to (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow-fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.

physics.class-ph↗

arXiv2020

A Phase Resonance Approach for Modal Testing of Structures with Nonlinear Dissipation

The concept of nonlinear modes is useful for the dynamical characterization of nonlinear mechanical systems. While efficient and broadly applicable methods are now available for the computation of nonlinear modes, nonlinear modal testing is still in its infancy. The purpose of this work is to overcome its present limitation to conservative nonlinearities. Our approach relies on the recently extended periodic motion concept, according to which nonlinear modes of damped systems are defined as family of periodic motions induced by an appropriate artificial excitation that compensates the natural dissipation. The particularly simple experimental implementation with only a single-point, single-frequency, phase resonant forcing is analyzed in detail. The method permits the experimental extraction of natural frequencies, modal damping ratios and deflection shapes (including harmonics), for each mode of interest, as function of the vibration level. The accuracy, robustness and current limitations of the method are first demonstrated numerically. The method is then verified experimentally for a friction-damped system. Moreover, a self-contained measure for estimating the quality of the extracted modal properties is investigated. The primary advantages over alternative vibration testing methods are noise robustness, broad applicability and short measurement duration. The central limitation of the identified modal quantities is that they only characterize the system in the regime near isolated resonances.

eess.SY↗

arXiv2017

Continuum Model for Pressure Actuated Cellular Structures

Previous work introduced a lower-dimensional numerical model for the geometric nonlinear simulation and optimization of compliant pressure actuated cellular structures. This model takes into account hinge eccentricities as well as rotational and axial cell side springs. The aim of this article is twofold. First, previous work is extended by introducing an associated continuum model. This model is an exact geometric representation of a cellular structure and the basis for the spring stiffnesses and eccentricities of the numerical model. Second, the state variables of the continuum and numerical model are linked via discontinuous stress constraints on the one hand and spring stiffness, hinge eccentricities on the other hand. An efficient optimization algorithm that fully couples both sets of variables is presented. The performance of the proposed approach is demonstrated with the help of an examples.

q-bio.QM↗

arXiv2017

Shape Optimization of Compliant Pressure Actuated Cellular Structures

Biologically inspired pressure actuated cellular structures can alter their shape through pressure variations. Previous work introduced a computational framework for pressure actuated cellular structures which was limited to two cell rows and central cell corner hinges. This article rigorously extends these results by taking into account an arbitrary number of cell rows, a more complicated cell kinematics that includes hinge eccentricities and varying side lengths as well as rotational and axial cell side springs. The nonlinear effects of arbitrary cell deformations are fully considered. Furthermore, the optimization is considerably improved by using a second-order approach. The presented framework enables the design of compliant pressure actuated cellular structures that can change their form from one shape to another within a set of one-dimensional C1 continuous functions.

q-bio.QM↗