Oleg Gendelman

Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture

We study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques. First, we transform the equations of motion using the action-angle variables of the underlying Hamiltonian system and bring them into the form where two-frequency averaging can be applied. We then show that energy pumping is due to resonance capture in the 1:1 resonance manifold of the system, and perform a perturbation analysis in an O (√e) neighborhood of this manifold in order to study the attracting region responsible for the resonance capture. The second method is based on the assumption of 1:1 internal resonance in the fast dynamics of the system, and utilizes complexification and averaging to develop analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The results compare favorably to numerical simulations. The practical implications of the energy pumping phenomenon are discussed.

Energy Pumping in Nonlinear Mechanical Oscillators: Part I—Dynamics of the Underlying Hamiltonian Systems

The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) channeling of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems I

Volume I: Preface Abbreviations 1 Introduction 2 Preliminary Concepts, Methodologies and Techniques 2.1 Nonlinear Normal Modes (NNMs) 2.2 Energy Localization in Nonlinear Systems 2.3 Internal Resonances, Transient and Sustained Resonance Captures 2.4 Averaging, Multiple Scales and Complexification 2.5 Methods of Advanced Signal Processing 2.5.1 NumericalWavelet Transforms 2.5.2 Empirical Mode Decompositions and Hilbert Transforms 2.6 Perspectives on Hardware Development and Experiments 3 Nonlinear Targeted Energy Transfer in Discrete Linear Oscillators with Single-DOF Nonlinear Energy Sinks 3.1 Configurations of Single-DOF NESs 3.2 Numerical Evidence of TET in a SDOF Linear Oscillator with a SDOF NES 3.3 SDOF Linear Oscillators with SDOF NESs: Dynamics of the Underlying Hamiltonian Systems 3.3.1 Numerical Study of Periodic Orbits (NNMs) 3.3.2 Analytic Study of Periodic Orbits (NNMs) 3.3.3 Numerical Study of Periodic Impulsive Orbits (IOs) 3.3.4 Analytic Study of Periodic and Quasi-Periodic IOs 3.3.5 Topological Features of the Hamiltonian Dynamics 3.4 SDOF Linear Oscillators with SDOF NESs: Transient Dynamics of the Damped Systems 3.4.1 Nonlinear Damped Transitions Represented in the FEP 3.4.2 Dynamics of TET in the Damped System 3.5 Multi-DOF (MDOF) Linear Oscillators with SDOF NESs: Resonance Capture Cascades and Multi-frequency TET 3.5.1 Two-DOF Linear Oscillator with a SDOF NES 3.5.2 Semi-Infinite Chain of Linear Oscillators with an End SDOF NES 4 Targeted Energy Transfer in Discrete Linear Oscillators with Multi-DOF NESs 4.1 Multi-Degree-of-Freedom(MDOF) NESs 4.1.1 An AlternativeWay for Passive Multi-frequency Nonlinear Energy Transfers 4.1.2 Numerical Evidence of TET in MDOF NESs 4.2 The Dynamics of the Underlying Hamiltonian System 4.2.1 System I: NES with O(1) Mass 4.2.2 System II: NES with O(e) Mass 4.2.3 Asymptotic Analysis of Nonlinear Resonant Orbits 4.2.4 Analysis of Resonant Periodic Orbits 4.3 TRCs and TET in the Damped and Forced System 4.3.1 Numerical Wavelet Transforms 4.3.2 Damped Transitions on the Hamiltonian FEP 4.4 Concluding Remarksl Index. Volume 2: 5 Targeted Energy Transfer in Linear Continuous Systems with Singlean Multi-DOF NESs 5.1 Beam of Finite Length with SDOF NES 5.1.1 Formulation of the Problem and Computational Procedure 5.1.2 Parametric Study of TET 5.2 Rod of Finite Length with SDOF NES 5.2.1 Formulation of the Problem, Computational Procedure and Post-Processing Algorithms 5.2.2 Computational Study of TET 5.2.3 Damped Transitions on the Hamiltonian FEP 5.3 Rod of Semi-Infinite Length with SDOF NES 5.3.1 Reduction to Integro-differential Form 5.3.2 Numerical Study of Damped Transitions 5.3.3 Analytical Study 5.4 Rod of Finite Length with MDOF NES 5.4.1 Formulation of the Problem and FEPs 5.4.2 Computational Study of TET 5.4.3 Multi-Modal Damped Transitions and Multi-Scale Analysis 5.5 Plate with SDOF and MDOF NESs 5.5.1 Case of a SDOF NES 5.5.2 Case of Multiple SDOF NESs 5.5.3 Case of a MDOF NES 5.5.4 Comparative Study with Linear Tuned Mass Damper 6 Targeted Energy Transfer in Systems with Periodic Excitations 6.1 Steady State Responses and Generic Bifurcations 6.1.1 Analysis of Steady State Motions 6.1.2 Numerical Verification of the Analytical Results 6.2 Strongly Modulated Responses (SMRs) 6.2.1 General Formulation and Invariant Manifold Approach 6.2.2 Reduction to One-DimensionalMaps and Existence Conditions for SMRs 6.2.3 Numerical Simulations 6.3 NESs as Strongly Nonlinear Absorbers for Vibration Isolation 6.3.1 Co-existent Response Regimes 6.3.2 Efficiency and Broadband Features of the Vibration Isolation 6.3.3 Passive Self-tuning Capacity of the NES 7 NESs with Non-Smooth Stiffness Characteristics 7.1 Systems with Multiple NESs Possessing Clear

Effect of supramolecular structure on polymer nanofibre elasticity

Polymer materials of reduced size and dimensionality, such as thin films, polymer nanofibres and nanotubes, exhibit exceptional mechanical properties compared with those of their macroscopic counterparts. We discuss here the abrupt increase in Youngs modulus in polymer nanofibres. Using scaling estimation we show that this effect occurs when, in the amorphous (non-crystalline) part of the nanofibres, the transversal size of regions consisting of orientation-correlated macromolecules is comparable to the nanofibre diameter, thereby resulting in confinement of the supramolecular structure. We suggest that in polymer nanofibres the resulting supramolecular microstructure plays a more dominant role in the deformation process than previously thought, challenging the commonly held view that surface effects are most significant. The concept we develop also provides a way to interpret the observed--but not yet understood--temperature dependence of Youngs modulus in nanofibres of different diameters.

Dynamics of linear discrete systems connected to local, essentially non-linear attachments

The dynamics of a linear periodic substructure, weakly coupled to an essentially non-linear attachment are studied. The essential (non-linearizable) non-linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena, e.g., passive, one-way, irreversible transfer of energy from the substructure to the attachment. As a specific application the dynamics of a finite linear chain of coupled oscillators with a non-linear end attachment is examined. In the absence of damping, it is found that the dynamical effect of the non-linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain. When damping exists energy pumping phenomena are realized in the system. It is shown that energy pumping strongly depends on the topological structure of the non-linear normal modes (NNMs) of the underlying undamped system. This is due to the fact that energy pumping is caused by the excitation of certain damped invariant NNM manifolds that are analytic continuations for weak damping of NNMs of the underlying undamped system. The bifurcations of the NNMs of the undamped system help explain resonance capture cascades in the damped system. This is a series of energy pumping phenomena occurring at different frequencies, with sudden lower frequency transitions between sequential events. The observed multi-frequency energy pumping cascades are particularly interesting from a practical point of view, since they indicate that non-linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure, in compatibility with the design objectives.

Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators

Redistribution of energy in a highly asymmetric system consisting of coupled linear and highly nonlinear damped oscillators is investigated. Special attention is paid to the excitation of a nonlinear normal mode while the energy is initially stored in other modes of the system. The transition proceeds via the mechanism of subharmonic resonance which is possible because of the strong nonlinearity of the system. The conditions of the energy transition to NNM being effective are revealed and guidelines to design such a systems are formulated in detail.

Bifurcations of Nonlinear Normal Modes of Linear Oscillator with Strongly Nonlinear Damped Attachment

Linear oscillator coupled to damped strongly nonlinear attachment with small mass is considered as a model design for nonlinear energy sink (NES). Damped nonlinear normal modes of the system are considered for the case of 1:1 resonance by combining the invariant manifold approach and multiple scales expansion. Special asymptotical structure of the model allows a clear distinction between three time scales. These time scales correspond to fast vibrations, evolution of the system toward the nonlinear normal mode and time evolution of the invariant manifold, respectively. Time evolution of the invariant manifold may be accompanied by bifurcations, depending on the exact potential of the nonlinear spring and value of the damping coefficient. Passage of the invariant manifold through bifurcations may bring about destruction of the resonance regime and essential gain in the energy dissipation rate.

Janus droplets: liquid marbles coated with dielectric/semiconductor particles.

The manufacturing of water droplets wrapped with two different powders, carbon black (semiconductor) and polytetrafluoroethylene (dielectric), is presented. Droplets composed of two hemispheres (Janus droplets) characterized by various physical and chemical properties are reported first. Watermelon-like striped liquid marbles are reported. Janus droplets remained stable on solid and liquid supports and could be activated with an electric field.

Isolated Resonance Captures and Resonance Capture Cascades Leading to Single- or Multi-Mode Passive Energy Pumping in Damped Coupled Oscillators

We examine passive energy pumping in a system of damped coupled oscillators. This is a one-way, passive and irreversible energy flow from a linear main system to a nonlinear attachment that acts, in essence, as a nonlinear energy sink (NES). Energy pumping is caused by 1:1 resonance captures on resonant manifolds of the damped systems. We show that the NES is capable of absorbing significant portions of the energies generated by transient, broadband external excitations. By performing a series of numerical simulations we confirm that the energy dependence of the nonlinear normal modes (NNMs) of the underlying undamped, unforced system determines, in essence, the resonance capture and energy pumping dynamics in the corresponding damped system. We present numerical simulations of single- and multi-mode energy pumping, that involve isolated resonance captures or resonance capture cascades, respectively. In addition, we discuss methodologies for enhancing the nonlinear energy pumping phenomenon by properly selecting the system parameters. The described technique of passively localizing and locally eliminating externally induced energy provides a new paradigm for vibration and shock isolation of mechanical oscillators.

Asymptotic Analysis of Passive Nonlinear Suppression of Aeroelastic Instabilities of a Rigid Wing in Subsonic Flow

We study theoretically passive suppression of aeroelastic instabilities of a rigid wing in subsonic flow with an essentially nonlinear attachment. The analysis is performed by constructing a reduced-order model, applying complexification/averaging and slow-fast partition of the dynamics. The resulting slow-flow dynamics is then analyzed by singular perturbation. We fully recover computational and experimental results for this system reported in previous works and prove the existence of trivial/nontrivial stable attractors as well as relaxation oscillations in the slow-flow dynamics. These, in turn, correspond to complete/partial instability suppression, and (periodic or quasi-periodic) strongly modulated responses in the full-order dynamics. Moreover, we demonstrate the existence of Shilnikov bifurcations in the dynamics. The analysis of the slow-flow dynamics is confirmed by numerical simulations of the full-order system. The methodology developed in this work can be used in a predictive capacity when ap...