Energy transfers in a system of two coupled oscillators with essential nonlinearity: 1 : 1 resonance manifold and transient bridging orbits Kerschen, Gaëtan ; ; et al in Nonlinear Dynamics (2005), 42(3), 283-303 The purpose of this study is to highlight and explain the vigorous energy transfers that may take place in a linear oscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear ... [more ▼] The purpose of this study is to highlight and explain the vigorous energy transfers that may take place in a linear oscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear energy sink. Although these energy exchanges are encountered during the transient dynamics of the damped system, it is shown that the dynamics can be interpreted mainly in terms of the periodic orbits of the underlying Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits of the system is constructed which demonstrates that, thanks to a 1:1 resonance capture, energy can be irreversibly and almost completely transferred from the linear oscillator to the nonlinear attachment. Furthermore, it is observed that this nonlinear energy pumping is triggered by the excitation of transient bridging orbits compatible with the nonlinear attachment being initially at rest, a common feature in most practical applications. A parametric study of the energy exchanges is also performed to understand the influence of the parameters of the nonlinear energy sink. Finally, the results of experimental measurements supporting the theoretical developments are discussed. [less ▲] Detailed reference viewed: 16 (1 ULg)Irreversible passive TEF in the damped response of coupled oscillators with nonlinearity Kerschen, Gaëtan ; ; et al in ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, 2005 (2005, September) We study, numerically and analytically the dynamics of passive energy transfer from a damped linear oscillator to an essentially nonlinear end attachment. This transfer is caused either by fundamental or ... [more ▼] We study, numerically and analytically the dynamics of passive energy transfer from a damped linear oscillator to an essentially nonlinear end attachment. This transfer is caused either by fundamental or subharmonic resonance capture, and in some cases is initiated by nonlinear beat phenomena. It is shown that, due to the essential nonlinearity, the end attachment is capable of passively absorbing broadband energy both at high and low frequencies, acting, in essence, as a passive broadband boundary controller. Complicated transitions in the damped dynamics can be interpreted based on the topological structure and bifurcations of the periodic solutions of the underlying undamped system. Moreover, complex resonance capture cascades are numerically encountered when we increase the number of degrees of freedom of the system. The grounded, essentially nonlinear end attachment discussed in this work can find application in numerous practical settings, including vibration and shock isolation of structures, seismic isolation, flutter suppression and packaging. [less ▲] Detailed reference viewed: 10 (1 ULg)Triggering mechanisms of limit cycle oscillations in a 2DOF wing flutter model ; ; et al in ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, 2005 (2005, September) We show numerically that the triggering mechanisms of limit cycle oscillations (LCOs) due to aeroelastic instability are composed of a series of resonance captures. We consider a two degree-of-freedom ... [more ▼] We show numerically that the triggering mechanisms of limit cycle oscillations (LCOs) due to aeroelastic instability are composed of a series of resonance captures. We consider a two degree-of-freedom (DOF) wing model with cubic nonlinear stiffnesses in the support, assuming quasi-steady aerodynamics and subsonic flow around the wing. Then, we establish the slow flow dynamics model, using the complexification / averaging technique and considering three frequency components; i.e., the two linear natural frequencies corresponding to heave and pitch and the superharmonic component which appears as three times the pitch frequency. It turns out that the LCO triggering mechanisms consist of mainly three stages: (i) transient resonance capture (TRC); (ii) escape; and (iii) permanent resonance capture (PRC). We examine the characteristics of each stage by way of time response, wavelet transform, phase plane, and instantaneous energy. [less ▲] Detailed reference viewed: 115 (2 ULg)Second-order blind identification for operational modal analysis Poncelet, Fabien ; Kerschen, Gaëtan ; Golinval, Jean-Claude et al Conference given outside the academic context (2005) For modal analysis of large structures, it is unpractical and expensive to use artificial excitation (e.g., shakers). However, engineering structures are most often subject to ambient loads(e.g., traffic ... [more ▼] For modal analysis of large structures, it is unpractical and expensive to use artificial excitation (e.g., shakers). However, engineering structures are most often subject to ambient loads(e.g., traffic and wind) that can be exploited for modal parameter estimation. One difficulty is that the actual loading conditions cannot generally be measured, and output-only measurements are available. This paper proposes to explore the utility of blind source separation (BSS) techniques for operational modal analysis. The basic idea of BSS is to recover unobserved source signals from their observed mixtures. The feasibility and practicality of the proposed method are demonstrated using an experimental application. [less ▲] Detailed reference viewed: 34 (6 ULg)Dynamic analysis of the nonlinear behavior of MEMS using the finite element formulation Rochus, Véronique ; Kerschen, Gaëtan ; Golinval, Jean-Claude in ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, 2005 (2005, September) The dynamic response of a mass-spring model of electrostatically actuated MEMS is analyzed when a voltage step is applied. The dynamics of a clamped-clamped beam representing a micro-bridge system are ... [more ▼] The dynamic response of a mass-spring model of electrostatically actuated MEMS is analyzed when a voltage step is applied. The dynamics of a clamped-clamped beam representing a micro-bridge system are also considered. To model and simulate the strong coupling between the electric and mechanical fields, a general finite element program has been developed based on a new monolithic formulation. The methodology proposed in this program does not require any iterative numerical scheme based on separate electrostatic and mechanical solvers, respectively. [less ▲] Detailed reference viewed: 58 (24 ULg)Surprisingly complicated dynamics of a SDOF linear oscillator coupled to a nonlinear attachment ; Kerschen, Gaëtan ; et al We study the dynamics of a two-degree-of-freedom nonlinear system consisting of a linear oscillator with an essentially nonlinear attachment. For the undamped system, we perform a numerical study based on ... [more ▼] We study the dynamics of a two-degree-of-freedom nonlinear system consisting of a linear oscillator with an essentially nonlinear attachment. For the undamped system, we perform a numerical study based on non-smooth temporal transformations to determine its periodic solutions in a frequency–energy plot. It turns out that there is a sequence of periodic solutions bifurcating from the main backbone curve of the plot. We then study analytically the periodic orbits of the undamped system using the complexification / averaging technique in order to determine the frequency contents of the various branches of solutions, and to understand the types of oscillation performed by the system at the different regimes of the motion. The transient responses of the weakly damped system are then examined, and numerical wavelet transforms are used to study the time evolutions of their harmonic components. We show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics, as it causes complicated transitions between modes in the damped transient motion. In addition, there is the possibility of strong passive energy transfer from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions. [less ▲] Detailed reference viewed: 45 (1 ULg)Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators ; Kerschen, Gaëtan ; et al in Journal of the Acoustical Society of America (2005), 118(2), 791-799 Our focus in this study is on experimental investigation of the transient dynamics of an impulsively loaded linear oscillator coupled to a lightweight nonlinear energy sink. It is shown that this ... [more ▼] Our focus in this study is on experimental investigation of the transient dynamics of an impulsively loaded linear oscillator coupled to a lightweight nonlinear energy sink. It is shown that this seemingly simple system exhibits complicated dynamics, including nonlinear beating phenomena and resonance captures. It is also demonstrated that, by facilitating targeted energy transfers to the nonlinear energy sink, a significant portion of the total input energy can be absorbed and dissipated in this oscillator. (C) 2005 Acoustical Society of America. [less ▲] Detailed reference viewed: 16 (2 ULg)Experimental irreversible passive energy transfer in coupled nonlinear oscillators ; Kerschen, Gaëtan ; et al in Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, 2005 (2005, August) Experimental verification of passive nonlinear energy pumping in two configurations of coupled oscillators with essential nonlinearities is carried out. In the experiments presented the nonlinear ... [more ▼] Experimental verification of passive nonlinear energy pumping in two configurations of coupled oscillators with essential nonlinearities is carried out. In the experiments presented the nonlinear attachment interacts with a single linear mode, and passive targeted energy transfer occurs at a single ‘fast’ frequency in the neighborhood of the eigen frequency of a linear mode. Good agreement between simulated and experimental results is observed, in spite of the strongly (essentially) nonlinear and transient nature of the dynamics of the systems considered. The experiments bear out earlier predictions that a significant fraction of the energy introduced directly to a linear structure by an external impulsive (broadband) load can be transferred (pumped) to an essentially nonlinear attachment, and dissipated there locally without spreading back to the system. In addition, the reported experimental results confirm that (a) nonlinear energy pumping in systems of coupled oscillators can occur only above a certain threshold of the input energy, and (b) there is an optimal value of the energy input at which a maximum portion of the energy is absorbed and dissipated at the NES. [less ▲] Detailed reference viewed: 16 (1 ULg)TRANSIENT DYNAMICS OF A DISPERSIVE ELASTIC WAVEGUIDE WITH AN ESSENTIALLY NONLINEAR ATTACHMENT ; ; et al in Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, 2005 (2005, August) We consider the dynamics of a dispersive semiinfinite linear rod weakly connected to an essentially nonlinear end attachment. We examine resonant interactions of the attachment with incident traveling ... [more ▼] We consider the dynamics of a dispersive semiinfinite linear rod weakly connected to an essentially nonlinear end attachment. We examine resonant interactions of the attachment with incident traveling waves propagating in the pass band of the rod, as well as resonance capture phenomena where the attachment engages in transient 1:1 internal resonance with the in-phase mode of the rod, at the bounding frequency of its pass and stop bands. The studied resonant interactions of the nonlinear attachment with traveling waves are similar to resonance capture cascades occurring in finite chain–nonlinear attachment configurations; these are the transient dynamics of the attachment as it engages in a series of instantaneous resonances with a continuous spectrum of frequencies in the pass band of the medium. As the energy of the attachment decreases due to damping and energy radiation back to the rod, the attachment engages in 1:1 resonance capture with the in-phase mode of the rod; this resonance capture occurs due to the dispersion property in the linear medium. [less ▲] Detailed reference viewed: 18 (1 ULg)The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview Kerschen, Gaëtan ; Golinval, Jean-Claude ; et al in Nonlinear Dynamics (2005), 41(1-3), 147-169 Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New ... [more ▼] Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study. [less ▲] Detailed reference viewed: 36 (8 ULg)Complex dynamics of a linear oscillator with a nonlinear attachment ; Kerschen, Gaëtan ; et al in Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, 2005 (2005, August) We study the dynamics of a two-degree-of-freedom (DOF) nonlinear system consisting of a grounded linear oscillator coupled to a light mass by means of an essentially nonlinear (nonlinearizable) stiffness ... [more ▼] We study the dynamics of a two-degree-of-freedom (DOF) nonlinear system consisting of a grounded linear oscillator coupled to a light mass by means of an essentially nonlinear (nonlinearizable) stiffness. We show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics, as it causes complicated transitions between modes in the damped transient motion. In addition, there is the possibility of strong passive energy transfer (energy pumping) from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions.. [less ▲] Detailed reference viewed: 17 (1 ULg)Multi-frequency nonlinear energy transfer from linear oscillators to mdof essentially nonlinear attachments ; Kerschen, Gaëtan ; et al in Journal of Sound & Vibration (2005), 285(1-2), 483-490 We report on multi-frequency energy transfer from a two-mode, initially excited linear system to a multi-degree-of-freedom (mdof) essentially nonlinear attachment. This occurs through simultaneous ... [more ▼] We report on multi-frequency energy transfer from a two-mode, initially excited linear system to a multi-degree-of-freedom (mdof) essentially nonlinear attachment. This occurs through simultaneous resonance interactions of both linear modes with a set of nonlinear normal modes (NNMs) of the attachment, and is studied utilizing numerical wavelet transforms. The multi-frequency nonlinear energy transfer discussed herein differs from multi-frequency energy transfer caused by resonance capture cascading where sequential energy transfer from a set of linear modes to single-dof nonlinear attachments takes place. (c) 2004 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 40 (1 ULg)Structural damage diagnosis under varying environmental conditions - Part I: A linear analysis ; Kerschen, Gaëtan ; et al in Mechanical Systems & Signal Processing (2005), 19(4), 847-864 A damage detection method is proposed for structural health monitoring under varying environmental and operational conditions. The method is based on principal component analysis (PICA) applied to ... [more ▼] A damage detection method is proposed for structural health monitoring under varying environmental and operational conditions. The method is based on principal component analysis (PICA) applied to vibration features identified during the monitoring of the structure. The advantage of the method is that it does not require to measure environmental parameters because they are taken into account as embedded variables. The number of principal components of the vibration features is implicitly assumed to correspond to the number of independent environmental factors. Since the environmental effects may be effectively eliminated by the proposed procedure, the residual error of the PCA prediction model remains small if the structure is healthy, and it increases significantly when structural damage occurs. Novelty analysis on the residual errors provides a statistical indication of damage. In the present paper, the environmental conditions are assumed to have a linear (or weakly non-linear) effect on the vibration features, and the PCA-based damage detection method is illustrated using computer-simulated and laboratory testing data. The extension of the proposed method to non-linear cases is addressed in a companion paper where the efficiency of the method is verified using data obtained from a 1-year in situ monitoring of a bridge. 2005 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 31 (3 ULg)Structural damage diagnosis under varying environmental conditions - Part II: local PCA for non-linear cases ; Kerschen, Gaëtan ; et al in Mechanical Systems & Signal Processing (2005), 19(4), 865-880 It is well known that changes in vibration features of structures due to damage may be masked by the effects of environmental variations. This influence has to be eliminated in the structural health ... [more ▼] It is well known that changes in vibration features of structures due to damage may be masked by the effects of environmental variations. This influence has to be eliminated in the structural health-monitoring process, especially when a long-term in situ monitoring is expected. In the companion paper [1] a linear method based on principal component analysis (PCA) has been proposed and has shown encouraging results for linear or even weakly non-linear cases. The present paper concerns a further extension of the proposed method to handle non-linear cases, which may be encountered in some complex structures. The method involves a two-step procedure, namely a clustering of the data space into several regions and then the application of PCA La each local region. The application of local PCA allows performing a piecewise linearisation of the non-linear problem. A close look at the choice of the distortion function used in data clustering leads to two new clustering strategies. Whereas the first strategy is specifically suitable for the application treated in this paper, the second one is more general. The local PCA-based damage detection method is applied for the structural health monitoring of a real bridge using vibration data measured in situ over a one-year period. 2005 Elsevier Ltd. All rights reserved. [less ▲] Detailed reference viewed: 42 (1 ULg)Multi-frequency passive non-linear targeted energy transfers in systems of couples oscillators ; ; Kerschen, Gaëtan et al in 5th GRACM International Congress on Computational Mechanics, Limassol, 2005 (2005, June) We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment ... [more ▼] We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment with damping can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible, over wide frequency and energy ranges. We numerically demonstrate that the topological structure of the periodic orbits in the frequency – energy plane of the underlying hamiltonian system greatly influences the strength of targeted energy transfer in the damped system, and governs to a great extent the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers. [less ▲] Detailed reference viewed: 36 (1 ULg)Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment ; Kerschen, Gaëtan ; et al in Physica D: Nonlinear Phenomena (2005), 204(1-2), 41-69 We study the dynamics of a two-degree-of-freedom (DOF) nonlinear system consisting of a grounded linear oscillator coupled to a light mass by means of an essentially nonlinear (nonlinearizable) stiffness ... [more ▼] We study the dynamics of a two-degree-of-freedom (DOF) nonlinear system consisting of a grounded linear oscillator coupled to a light mass by means of an essentially nonlinear (nonlinearizable) stiffness. We consider first the undamped system and perform a numerical study based on non-smooth transformations to determine its periodic solutions in a frequency-energy plot. It is found that there is a sequence of periodic solutions bifurcating or emanating from the main backbone curve of the plot. We then study analytically the periodic orbits of the undamped system using a complexification/averaging technique in order to determine the frequency contents of the various branches of solutions, and to understand the types of oscillation performed by the system at the different regimes of the motion. The transient responses of the weakly damped system are then examined, and numerical wavelet transforms are used to study the time evolutions of their harmonic components. We show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics, as it causes complicated transitions between modes in the damped transient motion. In addition, there is the possibility of strong passive energy transfer (energy pumping) from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions. (c) 2005 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 18 (0 ULg)Distortion function and clustering for local linear models Kerschen, Gaëtan ; ; Golinval, Jean-Claude in Journal of Sound & Vibration (2005), 280(1-2), 443-448 Principal component analysis (PCA) is a ubiquitous statistical technique for data analysis. PCA is however limited by its linearity and may sometimes be too simple for dealing with real-world data ... [more ▼] Principal component analysis (PCA) is a ubiquitous statistical technique for data analysis. PCA is however limited by its linearity and may sometimes be too simple for dealing with real-world data especially when the relations among variables are nonlinear. Recent years have witnessed the emergence of nonlinear generalizations of PCA, as for instance nonlinear principal component analysis (NLPCA) [1] or vector quantization principal component analysis (VQPCA) [2]. VQPCA involves a two-step procedure, namely a clustering of the data space into several regions and the application of PCA in each local region. In Ref. [3], VQPCA was applied for the reconstruction of dynamical response and it was shown that it is potentially a more effective tool than conventional PCA. The purpose of this technical note is to further investigate VQPCA and to have a closer look at the choice of the distortion function used for clustering the data space. [less ▲] Detailed reference viewed: 21 (1 ULg)Transient dynamics of a dispersive elastic wave guide weakly coupled to an essentially nonlinear end attachment ; ; et al in Wave Motion (2005), 41(2), 109-132 We consider the dynamics of a dispersive semi-infinite linear rod weakly connected to an essentially nonlinear end attachment. Using a Green's function formulation we reduce the dynamics to an integro ... [more ▼] We consider the dynamics of a dispersive semi-infinite linear rod weakly connected to an essentially nonlinear end attachment. Using a Green's function formulation we reduce the dynamics to an integro-differential equation, which we express in the form of an infinite set of ordinary differential equations using Neumann expansions. We examine resonant interactions of the attachment with incident traveling waves propagating in the pass band of the rod, as well as resonance capture phenomena where the attachment engages in transient 1:1 internal resonance with the in-phase mode of the rod, at the bounding frequency of its pass and stop bands. The studied resonant interactions of the nonlinear attachment with traveling waves are similar to resonance capture cascades occurring in finite-chain nonlinear attachment configurations; these are the transient dynamics of the attachment as it engages in a series of instantaneous resonances with a continuous spectrum of frequencies in the pass band of the medium. As the energy of the attachment decreases due to damping and energy radiation back to the rod, the attachment engages in 1:1 resonance capture with the in-phase mode of the rod; this resonance capture occurs due to the dispersion property in the linear medium. It is shown that the transient dynamics of resonance capture can be reduced to a set of two 'slow flow' amplitude and phase modulation equations. The analytical results are in agreement with direct numerical simulations. (C) 2004 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 28 (7 ULg)Sensor validation using principal component analysis Kerschen, Gaëtan ; ; Golinval, Jean-Claude et al in Smart Materials & Structures (2005), 14(1), 36-42 For a reliable on-line vibration monitoring of structures, it is necessary to have accurate sensor information. However, sensors may sometimes be faulty or may even become unavailable due to failure or ... [more ▼] For a reliable on-line vibration monitoring of structures, it is necessary to have accurate sensor information. However, sensors may sometimes be faulty or may even become unavailable due to failure or maintenance activities. The problem of sensor validation is therefore a critical part of structural health monitoring. The objective of the present study is to present a procedure based on principal component analysis which is able to perform detection, isolation and reconstruction of a faulty sensor. Its efficiency is assessed using an experimental application. [less ▲] Detailed reference viewed: 24 (4 ULg)Generation of accurate finite element models of nonlinear systems - Application to an aeroplane-like structure Kerschen, Gaëtan ; Golinval, Jean-Claude in Nonlinear Dynamics (2005), 39(1-2), 129-142 Model updating and validation is currently a central issue in the fields of computational structural mechanics and dynamics. The vast majority of applications however concerns linear structures. On the ... [more ▼] Model updating and validation is currently a central issue in the fields of computational structural mechanics and dynamics. The vast majority of applications however concerns linear structures. On the other hand, updating nonlinear models is something the structural dynamicist prefers to avoid mainly because tools such as modal analysis are no longer available. The objective of the present study is to propose a two-step methodology for dealing with nonlinear systems. Its most appealing feature is that it decouples the estimation of the linear and nonlinear parameters. A numerical application consisting of an aeroplane-like structure is used to assess the efficiency of the procedure. [less ▲] Detailed reference viewed: 18 (1 ULg) |
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