Analytic conditions for targeted energy transfer between nonlinear oscillators or discrete breathers

Targeted energy transfer by Fermi resonance

We propose a simple, novel mechanism for inducing highly selective and efficient energy transfer and focusing in certain discrete nonlinear systems. Under a precise condition of nonlinear resonance, when a specific amount of energy is injected as a discrete breather at a donor system, it can be transferred as a discrete breather to another weakly coupled acceptor system. This general mechanism could be relevant for energy transfer in bioenergetics and electron transfer in chemical reactions and could be used for engineering functional materials and devices.

The cubic or third-power (TP) nonlinear energy sink (NES) has been proven to be an effective method for vibration suppression, owing to the occurrence of targeted energy transfer (TET). However, TET is unable to be triggered by the low initial energy input, and thus the TP NES would get failed under low-amplitude vibration. To resolve this issue, a new type of NES with fractional nonlinearity, e.g., one-third-power (OTP) nonlinearity, is proposed. The dynamic behaviors of a linear oscillator (LO) with an OTP NES are investigated numerically, and then both the TET feature and the vibration attenuation performance are evaluated. Moreover, an analogy circuit is established, and the circuit simulations are carried out to verify the design concept of the OTP NES. It is found that the threshold for TET of the OTP NES is two orders of magnitude smaller than that of the TP NES. The parametric analysis shows that a heavier mass or a lower stiffness coefficient of the NES is beneficial to the occurrence of TET in the OTP NES system. Additionally, significant energy transfer is usually accompanied with efficient energy dissipation. Consequently, the OTP NES can realize TET under low initial input energy, which should be a promising approach for micro-vibration suppression.

Targeted energy transfer between a Rotor and a Morse oscillator: A model for selective chemical dissociation

Targeted energy transfer analysis of a nonlinear oscillator coupled with bistable nonlinear energy sink based on nonlinear normal modes

The flow-induced vibration of two-dimensional wing coupled with two nonlinear energy sinks (NESs) under freestream is studied by numerical methods, and the relationship between the vibration suppression and targeted energy transfer (TET) of the system is analyzed in detail. First, the model of the coupling system, which includes heave and pitch modes, is presented, and the NESs are located at the leading edge and trailing edge (NES1 and NES2) separately. Then, the vibrations suppressed by NESs are also investigated from the viewpoint of energy transfer, and the resonance captures (RCs) in the nonlinear coupling system are studied by using spectrum analysis. Furthermore, the ensuing TET through the modes of wing (heave and pitch) and the NESs is discussed in detail. The results show that the NESs can absorb the energy from every single mode of the wing, and the TET and RCs between modes can be more significant in the coupling system. Therefore, the TET is more efficient between the wing and NESs. It leads to the increase of the critical velocity of freestream under which the nonlinear vibration of wing can be suppressed by NESs effectively.

Targeted energy transfer (TET) is one of the passive approaches for the attenuation of vibration by facilitating the irreversible energy transfer from the primary oscillator (LO) to an auxiliary system. Recent studies on TET through vibro-impact nonlinear energy sink (VINES) have shown improved performance over a broad frequency spectrum. In VINES, we consider a ball which oscillates inside the linear oscillator (LO), transferring energy through the impacts and mitigating the vibration of the LO. Previous studies have given impetus on energy transfer through VINES under deterministic external excitation. However, in reality, external excitation is often stochastic. While randomness has been primarily considered for conventional TET mechanisms, this study adopts a fully non-smooth system, where external excitation will have random fluctuations which potentially influence the energy transfer mechanism. We investigate the bifurcation structure of the vibro-impact system and the TET phenomenon and the results are directly related to several energy transfer performance measures of VINES within the noisy environment. Also, we derived the analytical expressions to understand the noise-driven shifts in the bifurcation sequences.

Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, analytical study

We present a review on one of the latest developments in the field of dynamical systems, The nonlinear Targeted Energy Transfer (TET). The great significance of the phenomenon lies in the fact that the systems in which Nonlinear TET occurs present a form of self-tuning and can transfer energy over a wide variety of frequencies (resonances). This makes nonlinear TET particularly suitable in practical applications where it is necessary to extract energy from multiple ways of oscillation. Dynamical systems where nonlinear TET occurs are systems with different time scales and are singular. This property allows us to study such systems with the use of singular perturbation theory. It has been shown that Nonlinear TET is related to the bifurcation of the Slow Invariant Manifold of such systems and their slow flow.

The targeted energy transfer (TET) phenomenon has been observed in the field of acoustics, which provides a new approach to passive sound control in low frequency domain. The TET phenomenon has been investigated firstly inside one tube (1D acoustic system) with a membrane nonlinear energy sink (NES) or a loudspeaker nonlinear absorber, then inside an acoustic cavity (3D acoustic system) with a membrane NES. 3D acoustic cavities have been considered as more general geometry for the acoustic medium in view of applications in the acoustic field and the membrane NES is mounted directly on the wall of the acoustic cavity. The placement of a membrane NES on the wall involves a weak coupling between the membrane NES and a considered acoustic mode, which constitute the two degrees-of-freedom (DOF) system. The beginning of TET phenomenon of the two DOFs system has been analyzed and the desired working zone for the membrane NES has also been defined. The two thresholds of the zone have been determined by an analytical formula and semi-analytically, respectively. The parametric analysis of the membrane NES by using the two DOFs system has been investigated to design the membrane NES. In order to enhance the robustness and the effective TET range in acoustic cavities, a three DOFs system with two membranes and one acoustic mode is studied in this paper. We consider two different membranes and two almost identical membranes to analyze the TET phenomenon, respectively. The desired working zone for the membrane NES and the value of the plateau which are obtained by the two DOFs system are applied to analyze the three DOFs system. We observe that two membranes can enlarge the desired working zone of the NES.

Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments

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

Non-linear localization phenomena in biological lattices have attracted a steadily growing interest and their existence has been predicted in a wide range of physical settings. We investigate the non-linear proton dynamics of a hydrogen-bonded chain in a semi-classical limit using the coherent state method combined with a Holstein-Primakoff bosonic representation. We demonstrate that even a weak inherent discreteness in the hydrogen-bonded (HB) chain may drastically modify the dynamics of the non-linear system, leading to instabilities that have no analog in the continuum limit. We suggest a possible localization mechanism of polarization oscillations of protons in a hydrogen-bonded chain through modulational instability analysis. This mechanism arises due to the neighboring proton-proton interaction and coherent tunneling of protons along hydrogen bonds and/or around heavy atoms. We present a detailed analysis of modulational instability, and highlight the role of the interaction strength of neighboring protons in the process of bioenergy localization. We perform molecular dynamics simulations and demonstrate the existence of nanoscale discrete breather (DB) modes in the hydrogen-bonded chain. These highly localized and long-lived non-linear breather modes may play a functional role in targeted energy transfer in biological systems.

Targeted energy transfer (TET) represents the phenomenon where energy in a primary system is irreversibly transferred to a nonlinear energy sink (NES). This only occurs when the initial energy in the primary system is above a critical level. There is a natural asymmetry in the system due to the desire for the NES to be much smaller than the primary structure it is protecting. This asymmetry is also essential from an energy transfer perspective. To explore how the essential asymmetry is related to TET, this work interprets the realization of TET from a symmetry breaking perspective. This is achieved by introducing a symmetrized model with respect to the generically asymmetric original system. Firstly a classic example, which consists of a linear primary system and a nonlinearizable NES, is studied. The backbone curve topology that is necessary to realize TET is explored and it is demonstrated how this topology evolves from the symmetric case. This example is then extended to a more general case, accounting for nonlinearity in the primary system and linear stiffness in the NES. Exploring the symmetry-breaking effect on the backbone curve topologies, enables the regions in the NES parameter space that lead to TET to be identified.

Numerical simulations were conducted to study flow-induced vibration of a two-dimensional airfoil with two nonlinear energy sinks (NES). The relationship between targeted energy transfer (TET) and vibration suppression is analyzed in detail. The main system has two degrees of freedom – the pitch and heave. The two NES are treated as subsystems, in which the first NES is place at the leading edge and the second NES is placed at the trailing edge. The limit cycle oscillation (LCO), which is to be suppressed by the NES, is studied from the viewpoint of the TET. The resonance capture (RC) in the coupled nonlinear system is also discussed by the means of the energy and spectrum analysis. This is followed by a detailed target energy transfer discussion of the heave and pitch modes and the NES. In addition, the empirical mode decomposition (EMD) is utilized to obtain an intrinsic mode function (IMF) to analyze resonance capture in the system. The results show that the NES can absorb vigorous amount of energy from one of the specified vibration modes. As the RC occurs, the TET between the vibration modes in the coupled system becomes more significant. In particular, the TET between the NES and the wing becomes more efficient. This results in an increase in the critical freestream velocity as the NES suppresses the nonlinear vibration of the main system in a very effective way. As the total energy exceeds the suppression range of the subsystem, the NES loses its effectiveness on vibration suppression effect on the main system. The IMF of the EMD exhibits special super-harmonic resonance and frequency competition characteristics.