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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.

We perform a detailed phenomenological study of high-energy neutrino deep inelastic scattering (DIS) focused on LHC far-forward experiments such as FASER$\nu$ and SND@LHC. To this aim, we parametrise the neutrino fluxes reaching these LHC far-forward experiments in terms of `neutrino PDFs' encoding their energy and rapidity dependence by means of the LHAPDF framework. We integrate these neutrino PDFs in the recently developed POWHEG-BOX-RES implementation of neutrino-induced DIS to produce predictions accurate at next-to-leading order (NLO) in the QCD coupling matched to parton showers (PS) with Pythia8. We present NLO+PS predictions for final-state distributions within the acceptance of FASER$\nu$ and SND@LHC as well as for two experiments of the proposed Forward Physics Facility (FPF), FASER$\nu$2 and FLArE. We quantify the impact of NLO QCD corrections, of the parton showering and hadronisation settings in Pythia8, of the QED shower, and of the incoming neutrino flavour for the description of these observables, and compare our predictions with the GENIE neutrino event generator. Our work demonstrates the relevance of modern higher-order event generators to achieve the key scientific targets of the LHC neutrino experiments., Comment: 25 pages, 12 figures

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.

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.

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.

We analyze the signatures of new physics scenarios featuring third-family quark-lepton unification at the TeV scale in lepton-quark fusion at hadron colliders. Working with complete UV dynamics based on the $SU(4)$ gauge symmetry in the third-family fermions, we simulate the resonant production of a vector leptoquark at the next-to-leading order, including its decay and matching to the parton showers. The precise theoretical control over this production channel allows us to set robust bounds on the vector leptoquark parameter space which are complementary to the other production channels at colliders. We emphasize the importance of the resonant channel in future searches and discuss the impact of variations in the model space depending on the flavor structure of the vector leptoquark couplings., Comment: 11 pages, 7 figures, Monte Carlo generator can be downloaded from https://github.com/peterkrack/3rd-Lepton-Quark-Fusion v2: version accepted for publication in EPJC

Proton-proton collisions at the LHC generate a high-intensity collimated beam of neutrinos in the forward (beam) direction, characterised by energies of up to several TeV. The recent observation of LHC neutrinos by FASER$\nu$ and SND@LHC signals that this hitherto ignored particle beam is now available for scientific inquiry. Here we quantify the impact that neutrino deep-inelastic scattering (DIS) measurements at the LHC would have on the parton distributions (PDFs) of protons and heavy nuclei. We generate projections for DIS structure functions for FASER$\nu$ and SND@LHC at Run III, as well as for the FASER$\nu$2, AdvSND, and FLArE experiments to be hosted at the proposed Forward Physics Facility (FPF) operating concurrently with the High-Luminosity LHC (HL-LHC). We determine that up to one million electron- and muon-neutrino DIS interactions within detector acceptance can be expected by the end of the HL-LHC, covering a kinematic region in $x$ and $Q^2$ overlapping with that of the Electron-Ion Collider. Including these DIS projections into global (n)PDF analyses, specifically PDF4LHC21, NNPDF4.0, and EPPS21, reveals a significant reduction of PDF uncertainties, in particular for strangeness and the up and down valence PDFs. We show that LHC neutrino data enables improved theoretical predictions for core processes at the HL-LHC, such as Higgs and weak gauge boson production. Our analysis demonstrates that exploiting the LHC neutrino beam effectively provides CERN with a "Neutrino-Ion Collider" without requiring modifications in its accelerator infrastructure., Comment: 42 pages, 22 figures. Added detailed studies on fluxes uncertainties

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)., Comment: 24 pages, 14 figures, preprint submitted to Journal of Computers and Structures

Understanding interlayer and intralayer coupling in two-dimensional layered materials (2DLMs) has fundamental and technological importance for their large-scale production, engineering heterostructures, and development of flexible and transparent electronics. At the same time, the quantification of weak interlayer interactions in 2DMLs is a challenging task, especially, from the experimental point of view. Herein, we demonstrate that the use of X-ray absorption spectroscopy in combination with reverse Monte Carlo (RMC) and ab initio molecular dynamics (AIMD) simulations can provide useful information on both interlayer and intralayer coupling in 2DLM 2H$_c$-MoS$_2$. The analysis of the low-temperature (10-300 K) Mo K-edge extended X-ray absorption fine structure (EXAFS) using RMC simulations allows for obtaining information on the means-squared relative displacements $\sigma^2$ for nearest and distant Mo-S and Mo-Mo atom pairs. This information allowed us further to determine the strength of the interlayer and intralayer interactions in terms of the characteristic Einstein frequencies $\omega_E$ and the effective force constants $\kappa$ for the nearest ten coordination shells around molybdenum. The studied temperature range was extended up to 1200 K employing AIMD simulations which were validated at 300 K using the EXAFS data. Both RMC and AIMD results provide evidence of the reduction of correlation in thermal motion between distant atoms and suggest strong anisotropy of atom thermal vibrations within the plane of the layers and in the orthogonal direction.

In the past decades many density-functional theory methods and codes adopting periodic boundary conditions have been developed and are now extensively used in condensed matter physics and materials science research. Only in 2016, however, their precision (i.e., to which extent properties computed with different codes agree among each other) was systematically assessed on elemental crystals: a first crucial step to evaluate the reliability of such computations. We discuss here general recommendations for verification studies aiming at further testing precision and transferability of density-functional-theory computational approaches and codes. We illustrate such recommendations using a greatly expanded protocol covering the whole periodic table from Z=1 to 96 and characterizing 10 prototypical cubic compounds for each element: 4 unaries and 6 oxides, spanning a wide range of coordination numbers and oxidation states. The primary outcome is a reference dataset of 960 equations of state cross-checked between two all-electron codes, then used to verify and improve nine pseudopotential-based approaches. Such effort is facilitated by deploying AiiDA common workflows that perform automatic input parameter selection, provide identical input/output interfaces across codes, and ensure full reproducibility. Finally, we discuss the extent to which the current results for total energies can be reused for different goals (e.g., obtaining formation energies)., Comment: Main text: 23 pages, 4 figures. Supplementary: 68 pages. Nature Review Physics 2023

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., Comment: The final version of this article is available online at https://doi.org/10.1016/j.ymssp.2023.110265

Recent progress in calculating lepton density functions inside the proton and simulating lepton showers laid the foundations for precision studies of resonant leptoquark production at hadron colliders. Direct quark-lepton fusion into a leptoquark is a novel production channel at the LHC that has the potential to probe a unique parameter space for large masses and couplings. In this work, we build the first Monte Carlo event generator for a full-fledged simulation of this process at NLO for production, followed by a subsequent decay using the POWHEG method and matching to the parton showers utilizing HERWIG. The code can handle all scalar leptoquark models with renormalisable quark-lepton interactions. We then comprehensively study the differential distributions, including higher-order effects, and asses the corresponding theoretical uncertainties. We also quantify the impact of the improved predictions on the projected (HL-)LHC sensitivities and initiate the first exploration of the potential at the FCC-hh. Our work paves the way toward performing LHC searches using this channel., Comment: 33 pages, 7 figures, 3 tables

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., Comment: The final version of this article is available online at https://doi.org/10.1016/j.ymssp.2022.109551

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.

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.

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.

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., Comment: The final version of this article is available online at https://doi.org/10.1016/j.compstruc.2021.106698

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., Comment: The final version of this article is available online at https://doi.org/10.1016/j.jsv.2021.116527

The High Energy Stereoscopic System (H.E.S.S.) is an array of five Imaging Atmospheric Cherenkov Telescopes located in the Khomas Highland of Namibia. H.E.S.S. observes gamma rays above tens of GeV by detecting the Cherenkov light that is produced when Very High Energy gamma rays interact with the Earth's atmosphere. The H.E.S.S. Data Acquisition System (DAQ) coordinates the nightly telescope operations, ensuring that the various components communicate properly and behave as intended. It also provides the interface between the telescopes and the people on shift who guide the operations. The DAQ comprises both the hardware and software, and since the beginning of H.E.S.S., both elements have been continuously adapted to improve the data-taking capabilities of the array and push the limits of what H.E.S.S. is capable of. Most recently, this includes the upgrade of the entire computing cluster hosting the DAQ software, and the accommodation of a new camera on the large 28m H.E.S.S. telescope. We discuss the performance of the upgraded DAQ and the lessons learned from these activities., Comment: Proceedings of the 37th International Cosmic Ray Conference (ICRC2021)

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.

Flutter stability is a dominant design constraint of modern gas and steam turbines. To further increase the feasible design space, flutter-tolerant designs are currently explored, which may undergo Limit Cycle Oscillations (LCOs) of acceptable, yet not vanishing, level. Bounded self-excited oscillations are a priori a nonlinear phenomenon, and can thus only be explained by nonlinear interactions such as dry stick-slip friction in mechanical joints. The currently available simulation methods for blade flutter account for nonlinear interactions, at most, in only one domain, the structure or the fluid, and assume the behavior in the other domain as linear. In this work, we develop a fully-coupled nonlinear frequency domain method which is capable of resolving nonlinear flow and structural effects. We demonstrate the computational performance of this method for a state-of-the-art aeroelastic model of a shrouded turbine blade row. Besides simulating limit cycles, we predict, for the first time, the phenomenon of nonlinear instability, i.e., a situation where the equilibrium point is locally stable, but for sufficiently strong perturbation (caused e.g. by an impact), the dry frictional dissipation cannot bound the flutter vibrations. This implies that linearized theory does not necessary lead to a conservative design of turbine blades. We show that this phenomenon is due to the nonlinear contact interactions at the tip shrouds, which cause a change of the vibrational deflection shape and frequency, which in turn leads to a loss of aeroelastic stability. Finally, we extend the well-known energy method to capture these effects, and conclude that it provides a good approximation and is useful for initializing the fully-coupled solver., Comment: Journal of Fluids and Structures, Status: accepted for publication, 22 pages, 19 figures

The recently developed generalized Fourier-Galerkin method is complemented by a numerical continuation with respect to the kinetic energy, which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes. In order to enhance the practicability regarding the investigation of complex large-scale systems, it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations. A novel reduced order model (ROM) is developed for those regimes where internal resonances are absent. The approach allows for an accurate approximation of the multi-harmonic content of the resonant mode and accounts for the contributions of the off-resonant modes in their linearized forms. The ROM facilitates the efficient analysis of self-excited limit cycle oscillations, frequency response functions and the direct tracing of forced resonances. The ROM is equipped with a large parameter space including parameters associated with linear damping and near-resonant harmonic forcing terms. An important objective of this paper is to demonstrate the broad applicability of the proposed overall methodology. This is achieved by selected numerical examples including finite element models of structures with strongly nonlinear, non-conservative contact constraints.

A pure frequency domain method for the computation of periodic solutions of nonlinear ordinary differential equations (ODEs) is proposed in this study. The method is particularly suitable for the analysis of systems that feature distinct states, i.e. where the ODEs involve piecewise defined functions. An event-driven scheme is used which is based on the direct calculation of the state transition time instants between these states. An analytical formulation of the governing nonlinear algebraic system of equations is developed for the case of piecewise polynomial systems. Moreover, it is shown that derivatives of the solution of up to second order can be calculated analytically, making the method especially attractive for design studies. The methodology is applied to several structural dynamical systems with conservative and dissipative nonlinearities in externally excited and autonomous configurations. Great performance and robustness of the proposed procedure was ascertained.

As the motions of nonconservative autonomous systems are typically not periodic, the definition of nonlinear modes as periodic motions cannot be applied in the classical sense. In this paper, it is proposed 'make the motions periodic' by introducing an additional damping term of appropriate sign and magnitude. It is shown that this generalized definition is particularly suited to reflect the periodic vibration behavior induced by harmonic external forcing or negative linear damping. In a large range, the energy dependence of modal frequency, damping ratio and stability is reproduced well. The limitation to isolated or weakly-damped modes is discussed.

A novel method for the numerical prediction of the slowly varying dynamics of nonlinear mechanical systems has been developed. The method is restricted to the regime of an isolated nonlinear mode and consists of a two-step procedure: In the first step, a multiharmonic analysis of the autonomous system is performed to directly compute the amplitude-dependent characteristics of the considered nonlinear mode. In the second step, these modal properties are used to construct a two-dimensional reduced order model (ROM) that facilitates the efficient computation of steady-state and unsteady dynamics provided that nonlinear modal interactions are absent. The proposed methodology is applied to several nonlinear mechanical systems ranging form single degree-of-freedom to Finite Element models. Unsteady vibration phenomena such as approaching behavior towards an equilibrium point or limit cycles, and resonance passages are studied regarding the effect of various nonlinearities such as cubic springs, unilateral contact and friction. It is found that the proposed ROM facilitates very fast and accurate analysis of the slow dynamics of nonlinear systems. Moreover, the ROM concept offers a huge parameter space including additional linear damping, stiffness and near-resonant forcing.

A novel probabilistic approach for the design of mechanical structures with friction interfaces is proposed. The objective function is defined as the probability that a specified performance measure of the forced vibration response is achieved subject to parameter uncertainties. The practicability of the approach regarding the extensive amount of required design evaluations is strictly related to the computational efficiency of the nonlinear dynamic analysis. Therefore, it is proposed to employ a recently developed parametric reduced order model (ROM) based on nonlinear modes of vibration, which can facilitate a decrease of the computational burden by several orders of magnitude. The approach was applied to a rotationally periodic assembly of a bladed disk with underplatform friction dampers. The robustness of the optimum damper design was significantly improved compared to the deterministic approach, taking into account uncertainties in the friction coefficient, the excitation level and the linear damping. Moreover, a scale invariance for piecewise linear contact constraints is proven, which can be very useful for the reduction of the numerical effort for the analysis of such systems.

A method is presented for tracing the locus of a specific peak in the frequency response under variation of a parameter. It is applicable to periodic, steady-state vibrations of harmonically forced nonlinear mechanical systems. It operates in the frequency domain and its central idea is to assume a constant phase lag between forcing and response. The method is validated for a two-degree-of-freedom oscillator with cubic spring and a bladed disk with shroud contact. The method provides superior computational efficiency, but is limited to weakly-damped systems. Finally, the capability to reveal isolated solution branches is highlighted.

A mechanical system consisting of an elastic beam under harmonic excitation and an attached sliding body is investigated. Recent experimental observations suggest that the system passively (self-)adapts the axial location of the slider to achieve and maintain a condition of self-resonance, which could be useful in applications such as energy harvesting. The purpose of this work is to provide a theoretical explanation of this phenomenon based on an appropriate model. A key feature of the proposed model is a small clearance between the slider and the beam. This clearance gives rise to backlash and frictional contact interactions, both of which are found to be essential for the self-adaptive behavior. Contact is modeled in terms of the Coulomb and Signorini laws, together with the Newton impact law. The set-valued character of the contact laws is accounted for in a measure differential inclusion formulation. Numerical integration is carried out using Moreau's time-stepping scheme. The proposed model reproduces qualitatively most experimental observations. However, although the system showed a distinct self-adaptive character, the behavior was found to be non-resonant for the considered set of parameters. Beside estimating the relationship between resonance frequency and slider location, the model permits predicting the operating limits with regard to excitation level and frequency. Finally, some specific dynamical phenomena such as hysteresis effects and transient resonance captures underline the rich dynamical behavior of the system., Comment: The final version of this article is available online at http://link.springer.com/article/10.1007/s00419-016-1218-5