arXiv subjects

Malte Krack

Publications and source records attributed to Malte Krack.

At least 19 recordsLinked to original sources

arXiv2026

AFT Neural Function Approximators for 1D Nonlinear Force Laws

Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.

cs.CE↗

arXiv2026

Enabling topography-resolving structural dynamic contact simulation

Damping of structures and systems is often dominated by frictional dissipation in connections, the prediction of which remains a longstanding scientific challenge. Previous studies have shown that the actual topography of contact interfaces may have a strong effect, especially in the partial slip/liftoff regime. We recently proposed a multi-scale method, which couples finite element and boundary element modeling. The primary benefit of this approach is to analyze the effect of actual contact topography on the dynamics of jointed structures. While this multi-scale modeling method was initially developed for quasi-static analysis, we demonstrate herein how it can be used for time step integration and Harmonic Balance analysis. We cross-verify those fully dynamic analysis methods against each other and quasi-static results, for the S4 Beam benchmark. We compare the multi-scale method against state-of-the-art full-FE analysis, in terms of numerical damping and computational performance. Some discrepancy is found to be of physical origin. Depending on the load history, it is shown that the system settles to a slightly different equilibrium. Transient multi-scale simulations enable the prediction of this interesting phenomenon, for the first time, for a structure with bolted joints.

cs.CE↗

arXiv2025

A sub-structuring approach for model reduction of frictionally clamped thin-walled structures

Thin-walled structures clamped by friction joints, such as aircraft skin panels are exposed to bending-stretching coupling and frictional contact. We propose an original sub-structuring approach, where the system is divided into thin-walled and support regions, so that geometrically nonlinear behavior is relevant only in the former, and nonlinear contact behavior only in the latter. This permits to derive reduced component models, in principle, with available techniques. The Hurty-/Craig-Bampton method, combined with an interface reduction relying on an orthogonal polynomial series, is used to construct the reduction basis for each component. To model geometrically nonlinear behavior, implicit condensation is used, where an original, engineering-oriented proposition is made for the delicate scaling of the static load cases required to estimate the coefficients of the nonlinear terms. The proposed method is validated and its computational performance is assessed for the example of a plate with frictional clamping, using finite element analysis as reference. The numerical results shed light into an interesting mutual interaction: The extent of geometric hardening is limited by the reduced boundary stiffness when more sliding occurs in the clamping. On the other hand, the frictional dissipation is increased by the tangential loading induced by membrane stretching.

eess.SY↗

arXiv2025

A coupled FE-BE multi-scale method for the dynamics of jointed structures

The damping of built-up structures stems largely from the microscopic dry frictional interactions in the contact interfaces. The accurate prediction of friction damping has been an important scientific aim of the past several decades. Recent research indicates that very good agreement with vibration measurements is to be expected if the actual contact surface topography is sufficiently well known and finely resolved, and frictional-unilateral interactions are modeled in terms of the Coulomb-Signorini conditions. Resolving all relevant length scales in one finite element model leads to enormous or even prohibitive computation effort and regularization of the set-valued contact laws might be needed to ensure numerical stability. In this work, we propose a multi-scale approach: The stress and deformation field in the contact region is modeled using elastic half-space theory, implemented on a regular and fine grid of boundary elements (BE), so that the compliance matrix can be expressed in closed form. The vibration behavior of the remaining region is described using a relatively coarse finite element (FE) model, which is further reduced via component mode synthesis. The two models are coupled by enforcing compatibility and equilibrium conditions in the far field. The set-valued Coulomb-Signorini conditions are enforced robustly and efficiently using a projected over-relaxation scheme in conjunction with an appropriate active-set strategy. For the S4 beam benchmark, very good agreement with regard to the amplitude-dependent frequency and damping ratio of the first few modes is achieved, while the computation effort is reduced by several orders of magnitude compared to the full-FE reference. The proposed multi-scale method permits a very fine resolution of the contact surface topography without suffering from numerical instability.

eess.SY↗

arXiv2025

The Black Metal Tribometer: high-resolution measurement of normal load-indentation curves and partial slip hysteresis cycles

A new instrument has been designed for measuring frictional hysteresis cycles and normal load-indentation curves. The primary purpose of the BMT is the validation of more predictive modeling approaches for damping in friction joints. An important original feature of the BMT is its ability to measure both normal and tangential contact behavior without having to separate the samples. Special attention was paid to alignment and smooth motion in order to permit testing nominally flat-on-flat contacts in the microslip regime. Examples are shown for finely resolved, undistorted hysteresis cycles with relative tangential displacement amplitudes in the sub-micrometer range at about 50 MPa nominal normal pressure, which is maintained well over the tangential load cycle.

physics.ins-det↗

arXiv2024

An iteration-free approach to excitation harmonization

Sinusoidal excitation is particularly popular for testing structures in the nonlinear regime. Due to the nonlinear behavior and the inevitable feedback of the structure on the exciter, higher harmonics in the applied excitation are generated. This is undesired, because the acquired response may deviate substantially from that of the structure under purely sinusoidal excitation, in particular if one of the higher harmonics engages into resonance. We present a new approach to suppress those higher excitation harmonics and thus the unwanted exciter-structure interaction: Higher harmonics are added to the voltage input to the shaker whose Fourier coefficients are adjusted via feedback control until the excitation is purely sinusoidal. The stability of this method is analyzed for a simplified model; the resulting closed-form expressions are useful, among others, to select an appropriate exciter configuration, including the drive point. A practical procedure for the control design is suggested. The proposed method is validated in virtual and real experiments of internally resonant structures, in the two common configurations of force excitation via a stinger and base excitation. Excellent performance is achieved already when using the same control gains for all harmonics, throughout the tested range of amplitudes and frequencies, even in the strongly nonlinear regime. Compared to the iterative state of the art, it is found that the proposed method is simpler to implement, enables faster testing and it is easy to achieve a lower harmonic distortion.

eess.SY↗

arXiv2024

Robust and fast backbone tracking via phase-locked loops

Phase-locked loops are commonly used for shaker-based backbone tracking of nonlinear structures. The state of the art is to tune the control parameters by trial and error. In the present work, an approach is proposed to make backbone tracking much more robust and faster. A simple PI controller is proposed, and closed-form expressions for the gains are provided that lead to an optimal settling of the phase transient. The required input parameters are obtained from a conventional shaker-based linear modal test, and an open-loop sine test at a single frequency and level. For phase detection, an adaptive filter based on the LMS algorithm is used, which is shown to be superior to the synchronous demodulation commonly used. Once the phase has locked, one can directly take the next step along the backbone, eliminating the hold times. The latter are currently used for recording the steady state, and to estimate Fourier coefficients in the post-process, which becomes unnecessary since the adaptive filter yields a highly accurate estimation at runtime.The excellent performance of the proposed approach is demonstrated for a doubly clamped beam undergoing bending-stretching coupling leading to a 20 percent shift of the lowest modal frequency. Even for fixed control parameters, designed for the linear regime, only about 100 periods are needed per backbone point, also in the nonlinear regime. This is much faster than what has been reported in the literature so far.

eess.SY↗

arXiv2024

On the locomotion of the slider within a self-adaptive beam-slider system

A beam-slider system is considered whose passive self-adaption relies on an intricate locomotion process involving both frictional and unilateral contact. The system also exploits geometric nonlinearity to achieve broadband efficacy. The dynamics of the system take place on three distinct time scales: On the fast time scale of the harmonic base excitation are the vibrations and the locomotion cycle. On the slow time scale, the slider changes its position along the beam, and the overall vibration level varies. Finally, on an intermediate time scale, strong modulations of the vibration amplitude may take place. In the present work, first, an analytical approximation of the beam's response on the slow time scale is derived as function of the slider position, which is a crucial prerequisite for identifying the main drivers of the slider's locomotion. Then, the most important forms of locomotion are described and approximations of their individual contribution to the overall slider transport are estimated. Finally, the theoretical results are compared against numerical results obtained from an experimentally validated model.

eess.SY↗

arXiv2024

Experimental analysis of the TRC benchmark system

The Tribomechadynamics Research Challenge (TRC) was a blind prediction of the vibration behavior of a thin plate clamped on two sides using bolted joints. The first bending mode's natural frequency and damping ratio were requested as function of the amplitude, starting from the linear regime until high levels, where both frictional contact and nonlinear bending-stretching coupling become relevant. The predictions were confronted with experimental results in a companion paper; the present article addresses the experimental analysis of this benchmark system. Amplitude-dependent modal data was obtained from phase resonance and response controlled tests. An original variant of response controlled testing is proposed: Instead of a fixed frequency interval, a fixed phase interval is analyzed. This way, the high excitation levels required outside resonance, which could activate unwanted exciter nonlinearity, are avoided. Consistency of testing methods is carefully analyzed. Overall, these measures have permitted to gain high confidence in the acquired modal data. The different sources of the remaining uncertainty were further analyzed. A low reassembly-variability but a moderate time-variability were identified, where the latter is attributed to some thermal sensitivity of the system. Two nominally identical plates were analyzed, which both have an appreciable initial curvature, and a significant effect on the vibration behavior was found depending on whether the plate is aligned/misaligned with the support structure. Further, a 1:2 nonlinear modal interaction with the first torsion mode was observed, which only occurs in the aligned configurations.

eess.SP↗

arXiv2023

Fully Coupled Forced Response Analysis of Nonlinear Turbine Blade Vibrations in the Frequency Domain

For the first time, a fully-coupled Harmonic Balance method is developed for the forced response of turbomachinery blades. The method is applied to a state-of-the-art model of a turbine bladed disk with interlocked shrouds subjected to wake-induced loading. The recurrent opening and closing of the pre-loaded shroud contact causes a softening effect, leading to turning points in the amplitude-frequency curve near resonance. Therefore, the coupled solver is embedded into a numerical path continuation framework. Two variants are developed: the coupled continuation of the solution path, and the coupled re-iteration of selected solution points. While the re-iteration variant is slightly more costly per solution point, it has the important advantage that it can be run completely in parallel, which substantially reduces the wall clock time. It is shown that wake- and vibration-induced flow fields do not linearly superimpose, leading to a severe underestimation of the resonant vibration level by the influence-coefficient-based state-of-the-art methods (which rely on this linearity assumption).

cs.CE↗

arXiv2023

Are Chebyshev-based stability analysis and Urabe's error bound useful features for Harmonic Balance?

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors' knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe's error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.

math.DS↗

arXiv2022

Experimental validation of a model for a self-adaptive beam-slider system

A system consisting of a doubly clamped beam with an attached body (slider) free to move along the beam has been studied recently by multiple research groups. Under harmonic base excitation, the system has the capacity to passively adapt itself (by slowly changing the slider position) to yield either high or low vibrations. The central contributions of this work are the refinement of the recently developed system model with regard to the finite stiffness of the beam's clamping, followed by a thorough validation of this model against experimental results. With the intent to achieve repeatable and robust self-adaption, a new prototype of the system was designed, featuring, in particular, a continuously adjustable gap size and a concave inner contact geometry. The initial beam model is updated based on the results of an Experimental Nonlinear Modal Analysis of the system (without slider). By varying the excitation level and frequency in a wide range, all known types of behavior were reproduced in the experiment. The simulation results of the updated model with slider are in excellent agreement with the measurements, both qualitatively (type of behavior) and quantitatively. Minor deviations are attributed to the system's sensitivity to inevitable uncertainties, in particular with regard to the friction coefficient and the linear natural frequency. It is thus concluded that the proposed model is well-suited for further analysis of its intriguing dynamics and for model-based optimization for technical applications such as energy harvesting.

nlin.AO↗

arXiv2022

Computational and experimental analysis of the impact of a sphere on a beam and the resulting modal energy distribution

We consider the common problem setting of an elastic sphere impacting on a flexible beam. In contrast to previous studies, we analyze the modal energy distribution induced by the impact, having in mind the particular application of impact vibration absorbers. Also, the beam is analyzed in the clamped-clamped configuration, in addition to the free-free configuration usually considered. We demonstrate that the designed test rig permits to obtain well-repeatable measurements. The measurements are confronted with predictions obtained using two different approaches, state-of-the-art Finite Element Analysis and a recently developed computational approach involving a reduced-order model. The innovative aspect of the latter approach is to achieve a massless contact boundary using component mode synthesis, which reduces the mathematical model order and numerical oscillations. We show that the novel computational approach reduces the numerical effort by 3-4 orders of magnitude compared to state-of-the-art Finite Element Analysis, without compromising the excellent agreement with the measurements.

eess.SY↗

arXiv2022

Prediction and validation of the strongly modulated forced response of two beams undergoing frictional impacts

We consider two cantilevered beams undergoing frictional impacts at the free end. The beams are designed to be of similar geometry so that they have distinct but close natural frequencies. Under harmonic base excitation near the primary resonance with the higher-frequency fundamental bending mode, the system shows a strongly modulated non-periodic response. The purpose of this work is to analyze to what extent the non-periodic vibro-impact dynamics can be predicted. To this end, we use a recently developed modeling and simulation approach. The approach relies on component mode synthesis, the massless boundary concept and an appropriate time stepping scheme. Unilateral contact and dry friction are modeled as set-valued laws and imposed locally within the spatially resolved contact area. A linear model updating is carried out based on the natural frequencies and damping ratios identified in the regime without impacts. The nonlinear simulation of the steady-state response to forward and backward stepped sine excitation is compared against measurements. The results are in very good agreement, especially in the light of the uncertainty associated with the observed material loss in the contact region and the nonlinear behavior of the clamping.

eess.SY↗

arXiv2022

Nonlinear damping quantification from phase-resonant tests under base excitation

The present work addresses the experimental identification of amplitude-dependent modal parameters (modal frequency, damping ratio, Fourier coefficients of periodic modal oscillation). Phase-resonant testing has emerged as an important method for this task, as it substantially reduces the amount of data required for the identification compared to conventional frequency-response testing at different excitation/response levels. In the case of shaker-stinger excitation, the applied excitation force is commonly measured in order to quantify the amplitude-dependent modal damping ratio from the phase-resonant test data. In the case of base excitation, however, the applied excitation force is challenging or impossible to measure. In this work we develop an original method for damping quantification from phase-resonant tests. It relies solely on response measurement; it avoids the need to resort to force measurement. The key idea is to estimate the power provided by the distributed inertia force imposed by the base motion. We develop both a model-free and a model-based variant of the method. We validate the developed method first in virtual experiments of a friction-damped and a geometrically nonlinear system, and then in a physical experiment involving a thin beam clamped at both ends via bolted joints. We conclude that the method is highly robust and provides high accuracy already for a reasonable number of sensors.

eess.SP↗

arXiv2021

A massless boundary component mode synthesis method for elastodynamic contact problems

We propose to combine the ideas of mass redistribution and component mode synthesis. More specifically, we employ the MacNeal method, which readily leads to a singular mass matrix, and an accordingly modified version of the Craig-Bampton method. Besides obtaining a massless boundary, we achieve a drastic reduction of the mathematical model order in this way compared to the parent finite element model. Contact is modeled using set-valued laws and time stepping is carried out with a semi-explicit scheme. We assess the method's computational performance by a series of benchmarks, including both frictionless and frictional contact. The results indicate that the proposed method achieves excellent energy conservation properties and superior convergence behavior. It reduces the spurious oscillations and decreases the computational effort by about 1-2 orders of magnitude compared to the current state of the art (mass-carrying component mode synthesis method). We believe that the computational performance and favorable energy conservation properties will be valuable for the prediction of vibro-impact processes and physical damping.

cs.CE↗

arXiv2021

Predictive design of impact absorbers for mitigating resonances of flexible structures using a semi-analytical approach

Analytical conditions are available for the optimum design of impact absorbers for the case where the host structure is well described as rigid body. Accordingly, the analysis relies on the assumption that the impacts cause immediate dissipation in the contact region, which is modeled in terms of a known coefficient of restitution. When a flexible host structure is considered instead, the impact absorber not only dissipates energy at the time instances of impact, but it inflicts nonlinear energy scattering between structural modes. Hence, it is crucial to account for such nonlinear energy transfers yielding energy redistribution within the modal space of the structure. In the present work, we develop a design approach for reonantly-driven, flexible host structures. We demonstrate decoupling of the time scales of the impact and the resonant vibration. On the long time scale, the dynamics can be properly reduced to the fundamental harmonic of the resonant mode. A light impact absorber responds to this enforced motion, and we recover the Slow Invariant Manifold of the dynamics for the regime of two impacts per period. On the short time scale, the contact mechanics and elasto-dynamics must be finely resolved. We show that it is sufficient to run a numerical simulation of a single impact event with adequate pre-impact velocity. From this simulation, we derive a modal coefficient of restitution and the properties of the contact force pulse, needed to approximate the behavior on the long time scale. We establish that the design problem can be reduced to four dimensionless parameters and demonstrate the approach for the numerical example of a cantilevered beam with an impact absorber. We conclude that the proposed semi-analytical procedure enables deep qualitative understanding of the problem and, at the same time, yields a quantitatively accurate prediction of the optimum design.

cs.CE↗

arXiv2021

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

Under certain conditions, the dynamics of a nonlinear mechanical system can be represented by a single nonlinear modal oscillator. The properties of the modal oscillator can be determined by computational or experimental nonlinear modal analysis. The simplification to a single-nonlinear-mode model facilitates qualitative and global analysis, and substantially reduces the computational effort required for probabilistic methods and design optimization. Important limitations of this theory are that only purely mechanical systems can be analyzed and that the respective nonlinear mode has to be recomputed when the system's structural properties are varied. With the theoretical extension proposed in this work, it becomes feasible to attach linear subsystems to the primary mechanical system, and to approximate the dynamics of this coupled system using only the nonlinear mode of the primary mechanical system. The attachments must be described by linear ordinary or differential-algebraic equations with time-invariant coefficient matrices. The attachments do not need to be of purely mechanical nature, but may contain, for instance, electric, magnetic, acoustic, thermal or aerodynamic models. This considerably extends the range of utility of nonlinear modes to applications as diverse as model updating or vibration energy harvesting. As long as the attachments do not significantly deteriorate the host system's modal deflection shape, it is shown that their effect can be reduced to a complex-valued modal impedance and an imposed modal forcing term. In the present work, the proposed approach is computationally assessed for the analysis of exciter-structure interaction. More specifically, the force drop typically encountered in frequency response testing is revisited.

physics.class-ph↗