Targeted Energy Transfer Research Articles

Nonlinear flutter suppression and performance evaluation of periodically embedded nonlinear vibration absorbers in a supersonic FGM plate

To break the limitation of the conventional passive flutter and nonlinear aeroelastic suppressions, a supersonic functionally graded material (FGM) plate featuring periodically embedded nonlinear vibration absorbers (NVAs) is proposed based on the metamaterial concept. Using the von Karman large deformation theory and supersonic piston aerodynamic theory, the motion equations of a supersonic FGM plate coupled with NVAs are derived by using the Hamilton principle. Linear flutter analysis shows that the multiple-NVA design can change the flutter coupling mechanism of the original aeroelastic system and significantly enhance its aerothermoelastic stability. Subsequently, the nonlinear aeroelastic behaviors of the plate and the energy transfer mechanism between it and the NVAs are examined using an energy-based analysis approach. The comparison of bifurcation diagrams indicates that the attachment of periodic NVAs realizes a superior suppression of vibration absorption than a single NVA. Numerical results show that the nonlinear dynamic responses of the plate can be substantially reduced via the targeted energy transfer of NVAs in the post-flutter regime. In particular, the passive control performance of the periodic NVAs does not degrade under an increase in the dynamic pressure and it can also be applicable to different airflow directions. Furthermore, a significant reduction of more than 95% in the response amplitude of the plate can be achieved by properly tuning the NVA nonlinear parameters. The present work demonstrates that the design of periodic array of NVAs is effective at enhancing the flutter stability and mitigating nonlinear aeroelastic responses of the supersonic plate.

Tuned bistable nonlinear energy sink for simultaneously improved vibration suppression and energy harvesting

Despite the combination of nonlinear vibration suppression (VS) and energy harvesting (EH) has received great attention in recent years, the strategy to simultaneously improve the performance in both purposes and extend the operating amplitude range to low-level excitations is still an open issue. To achieve this goal, this paper proposes a tuned bistable nonlinear energy sink (TBNES) to outperform the typical bistable nonlinear energy sink (BNES). The TBNES replaces the fixed magnet that is opposite to the bistable piezoelectric beam (BPB) in the BNES by a tuning piezoelectric beam (TPB) with a movable tip magnet. On one hand, the TPB with a linear viscous damping and a piezoelectric element can improve both VS and EH performances. On the other hand, the tuning effect can dynamically reduce the elastic potential barrier of the BPB, helping it induce the initial snap-through motion and produce the strong targeted energy transfer at low-level excitations, therefore leading to the extension of its operating amplitude range. The electromechanical model and the magnetic force model are derived for both TBNES and BNES based on Hamilton’s principle and the geometrical dipole–dipole method. Parametric studies are conducted to investigate the influences of key system parameters on their VS and EH performances with three defined figure of merits. With the system parameters optimized by the presented dynamic optimization algorithm, the simulation results show that the performance of the TBNES is better than that of the BNES in VS and EH, and its operating range of excitation amplitudes can be successfully extended. The deep mechanisms for these phenomena are discussed from the aspects of 1:1 resonance, induction of nonlinearity, elastic potential energy, etc.

Extreme intermodal energy transfers through vibro-impacts for highly effective and rapid blast mitigation

This work investigates intermodal targeted energy transfers (IMTET) for passive mitigation of a large-scale nine-story structure subjected to blast excitation. This is achieved by inducing extreme, fast time scale energy transfers from lower-frequency structural modes which are mainly excited by the blast to higher-frequency ones. These targeted (directed) energy transfers are governed by a non-resonant nonlinear dynamical mechanism induced by inelastic Hertzian vibro-impacts between the nine-story structure (referred to “primary structure”) and an internal secondary “core structure” assumed to be rigid. The clearance distribution between the primary structure and the core structure is optimized using a multi-objective genetic algorithm by minimizing both the characteristic damping time of the transient response of the primary structure, and the energy redistributed from the lowest-frequency (fundamental) structural mode to higher modes. The results show that the IMTET mechanism enables extremely rapid and nearly irreversible low-to-high frequency scattering of the blast energy in the primary structure. In turn, this nonlinear energy scattering rapidly reduces the overall amplitude of the transient structural response, even in the case of purely elastic Hertzian contacts. The mitigation performance is substantially enhanced when more realistic inelastic vibro-impact nonlinearities are considered. In the studied example with the realistic model of a nine-story structure, the synergy between extremely rapid low-to-high frequency energy redistribution and dissipation due to inelastic vibro-impacts yields a reduction of the characteristic damping time by a factor of 20 compared to the linear case. We envision that the proposed concept of rapid nonlinear IMTET is of broad applicability to general classes of dynamical and acoustical systems.

Frequency–energy plot and targeted energy transfer analysis of coupled bistable nonlinear energy sink with linear oscillator

The nonlinear energy sink (NES), which is proven to perform rapid and passive targeted energy transfer (TET), has been employed for vibration mitigation in many primary small- and large-scale structures. Recently, the feature of bistability, in which two nontrivial stable equilibria and one trivial unstable equilibrium exist, is utilized for passive TET in what is known as bistable NES (BNES). The BNES generates a nonlinear force that incorporates negative linear and multiple positive or negative nonlinear stiffness components. In this paper, the BNES is coupled to a linear oscillator (LO) where the dynamic behavior of the resulting LO-BNES system is studied through frequency–energy plots (FEPs), which are generated by analytical approximation using the complexification-averaging method and by numerical continuation techniques. The effect of the length and stiffness of the transverse coupling springs is found to affect the stability and topology of the branches and indicates the importance of the exact physical realization of the system. The rich nonlinear dynamical behavior of the LO-BNES system is also highlighted through the appearance of multiple symmetrical and unsymmetrical in- and out-of-phase backbone branches, especially at low energy levels. The superimposed wavelet frequency spectrums of the LO-BNES response on the FEP have verified the robustness of the TET mechanism where the role of the unsymmetrical NNM backbones in TET is clearly observed.

Experimental study on influence of wall acoustic materials of 3D cavity for targeted energy transfer of a nonlinear membrane absorber

In view of applying the membrane nonlinear energy sink (NES) to reduce the low-frequency noise inside vehicle interior with acoustic absorption materials, a simplified acoustic cavity is used to model vehicle interior cavity. The system of one acoustic mode of the parallelepiped cavity and one membrane is established and the experimental set-up is built. The forced vibration responses in frequency-domain and time-domain are analyzed by using the set-up with and without the acoustic materials. Under certain excitation amplitude and frequency range, the strongly modulated responses (SMR) occurs and the plateau of suppressed resonance peaks appears. Meanwhile, the energy of the system is studied when the SMR exists. The targeted energy transfer (TET) phenomenon and the theoretical model of the system is verified by the experimental method. The wall acoustic materials of 3D cavity could affect the TET of the system, where the excitation amplitude for the beginning threshold of the desired working zone becomes larger and the excitation amplitude interval of TET becomes larger simultaneously. The ending threshold of the desired working zone of the NES is redefined. The numerical results are in good agreement with the experimental results. The application of membrane NES in 3D acoustic cavity proposed in this paper provides a new technical idea for vehicle interior low-frequency noise reduction in the future.

Importance of gravity and friction on the targeted energy transfer of vibro-impact nonlinear energy sink

The effect of gravity and sliding friction on the dynamics around targeted energy transfer of an inclined vibro-impact nonlinear energy sink (VINES) is addressed. The non-smooth dynamics is first formulated in Signorini’s contact law and treated using a Moreau-Jean scheme adapted with an energy-conserving integration method. Exhaustive discussions are then carried out regarding the vibration damping performance of the VINES, as well as their dependence on the underlying vibro-impact induced response regimes, under both transient and steady-state responses. It is demonstrated that gravity is responsible for reducing the energy barrier required for achieving targeted energy transfer in the transient response, or enlarge the existing bandwidth of the effective strongly modulated response (SMR) regimes, which is an equivalent representation of TET in the steady-state response, to improve considerably the overall damping efficiency. Some adverse effects brought by gravity and friction are also reported, and it is found that, when the incline angle or friction coefficient grows large, the bandwidth of SMR reduces again and leads to a deterioration of efficiency. The results open a door for appropriately taking advantage of the gravity into the VINES design to improve its damping performance.

Discovering nonlinear resonances through physics-informed machine learning

For an ensemble of nonlinear systems that model, for instance, molecules or photonic systems, we propose a method that finds efficiently the configuration that has prescribed transfer properties. Specifically, we use physics-informed machine learning (PIML) techniques to find the parameters for the efficient transfer of an electron (or photon) to a targeted state in a nonlinear dimer. We create a machine learning model containing two variables, χ D and χ A , representing the nonlinear terms in the donor and acceptor target system states. We then introduce a data-free physics-informed loss function as 1.0 − P j , where P j is the probability, the electron being in the targeted state, j . By minimizing the loss function, we maximize the occupation probability to the targeted state. The method recovers known results in the targeted energy transfer (TET) model, and it is then applied to a more complex system with an additional intermediate state. In this trimer configuration, the PIML approach discovers desired resonant paths from the donor to acceptor units. The proposed PIML method is general and may be used in the chemical design of molecular complexes or engineering design of quantum or photonic systems.

Analytical Analysis for Parameter Design of Attached Nonlinear Energy Sink

The dynamic responses of a linear primary structure coupled with a nonlinear energy sink (NES) are investigated under harmonic excitation in the 1 : 1 resonance regime. In civil engineering, initial conditions are usually zero or approximately zero. Therefore, in this study, only these conditions are considered. The strongly modulated response (SMR), whose occurrence is conditional, is the precondition for effective target energy transfer (TET) in this system. Therefore, this study aims to determine the parameter range in which the SMR can occur. The platform phenomenon and other related phenomena are observed while analyzing slow-varying equations. An excitation amplitude interval during which the SMR can occur is obtained, and an approximate analytical solution of the optimal nonlinear stiffness is found. The numerical results show that the NES based on the optimal stiffness performs better in terms of control performance.

Study on target energy transfer of 3D acoustic cavity - plate coupling system with the membrane nonlinear energy sink

The target energy transfer (TET) between a membrane nonlinear energy sink (NES) and the acoustic medium inside a rectangular cavity is studied. The acoustic medium is interacted with a plate and multi-order modes coupling of the 2 structure is considered. Based on the modal expansion approach, with Green's function, Helmholtz equation and the boundary conditions of the acoustic medium and the plate, the coupling coefficient matrix of the mode of 2 structures is derived. The equations of the membrane NES, multi-order modes of the acoustic medium and multi-order modes of the plate are established, and numerical analysis is used to investigate the TET phenomenon. The results show that in condition of a single-point excitation to the plate, under a certain range of excitation levels, the membrane can be seen as a kind of NES, and the energy in the acoustic medium can be unidirectionally transmitted to the membrane NES and attenuated, reducing the sound pressure level in the cavity. At the same time, it is found that the NES can suppress multi-order sound pressure of the acoustic medium at the same time, and realize the control of cascaded resonance noise.

Numerical simulation of impact sound transmission control across a smart hybrid double floor system equipped with a genetically-optimized NES absorber

An idealized detailed 2D formulation is presented for suppression of transient impact sound transmission across a hybrid smart double-leaf sandwich beam (floor-ceiling) structure into a rectangular (receiving) room with ideally flat and rigid walls. The smart double wall structure, which is mechanically inter-connected at an arbitrary point with a lightweight nonlinear energy sink (NES) absorber, incorporates spatially distributed and electrically independent non-collocated semiactive (electro-rheological fluid- or ERF-incorporated) and fully-active (piezoceramic- or PZT-incorporated) actuator layers functioning in a closed loop control framework. Extensive time-domain numerical simulations initially calculate both the uncontrolled and controlled transient acoustic pressure fields in absence of the dynamic vibration absorber for four separate settings of the active (PZT-) and semiactive (ERF-) actuation elements. Subsequently, the remarkable performance of the GA-optimized hybrid smart active/semi-active/passive (PZT/ERF/NES) configuration, which benefits from the multi-mode targeted energy transfer (TET) mechanism of the NES, in significant broadband (low frequency) attenuation of the transmitted shock energy with a much lower actuator energy demand, is demonstrated. Furthermore, some important aspects of the transient fluid-structure interaction (TFSI) control problem like weakening of the acoustic shock focusing effects (focal zones) within the source room are illustrated through selected early-to-late-times 2D images and animations of the cavity pressure fields. Limiting situations are studied and correctness of the derivations is established against accessible data in addition to numerical (FEM) simulations.

Understanding targeted energy transfer from a symmetry breaking perspective

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.

A dual-functional metamaterial for integrated vibration isolation and energy harvesting

Enhancing vibration isolation with locally resonant metamaterials has attracted wide attention due to low-frequency band-gap. Moreover, nonlinear periodic structure could improve the range of targeted energy transfer. In this paper, we propose a dual-functional metamaterial for integrated low-frequency vibration isolation and energy harvesting. A periodic array of nonlinear electrical energy harvesters, realized by implanting a rolling-ball with coils into a spherical magnetic cavity, is explored to isolate mechanical wave and simultaneously harvest electrical energy. The dynamical equation is established for a nonlinear dual-functional metamaterial beam under transverse excitation. The Extended Bloch's theorem is applied to give the dispersion relation. Numerical results obtained by finite element method supported the analytical results. Compared to the narrow band-gaps in metamaterials with spherical magnetic cavity, our numerical analysis demonstrates that a cavity mass arrayed beam with a periodic array of nonlinear energy harvesters has more and wider low-frequency band-gaps. Frequency response functions of output power are derived by using finite element analysis. The harvested power is considerable at the local resonant band-gap. Parameter study demonstrates that increasing the cell size and increasing cavity mass could improve elastic waves isolation performance at low frequencies; Increasing the mass of the rolling-ball in the resonator can significantly decrease the frequency of the local resonance band-gap. The existence of multiple band-gaps could be designed for dual-functional vibration attenuation and energy harvesting. Finally, an experimental rig is designed to validate the theoretical results.

Studies on the damping mechanism of shape memory alloy-spring tuned vibration absorber attached to a multi-degree-of-freedom structure

To mitigate the adverse structural responses, an improved version of the traditional tuned vibration absorber has been proposed based on the shape memory alloy spring, referred as the shape memory alloy-spring tuned vibration absorber. The finite element numerical models of the multi-degree-of-freedom structure (e.g., transmission tower) and shape memory alloy-spring tuned vibration absorber are developed by using the commercial software ANSYS, and the nonlinear behavior of the shape memory alloy spring is validated based on a previous experimental study. The damping mechanism of the shape memory alloy-spring tuned vibration absorber attached to a multi-degree-of-freedom structure under seismic excitations is investigated, and the nonlinear hysteretic behavior of the shape memory alloy spring is also discussed. The results show that the proposed damper has a two-stage damping mechanism, and its control performance is remarkable. Because the coupled system response is sensitive to the amplitude level, the optimal configuration of the shape memory alloy-spring tuned vibration absorber can be obtained by parametric analysis. Particularly, because of the nonlinear target energy transfer and transient resonance capture mechanism, the shape memory alloy-spring tuned vibration absorber exhibits stable control ability under different seismic waves, indicating a good stability in vibration control of a multi-degree-of-freedom system.

Nonlinear energy sink to enhance the landing gear shimmy performance

The shimmy phenomenon is a significant concern in aircraft landing gear dynamics. The prediction of the shimmy instability is an essential issue in landing gear design to develop a passive or active suppression method. This work investigates the application of the nonlinear energy sink (NES) concept to mitigate the effects of shimmy in landing gears. The NES concept has been used in recent research on mechanical vibrations. It comprises a passive target energy transfer method that refers to a one-way energy transfer from a primary to a nonlinear subsystem. The landing gear model is based on torsional displacement coupled with the tyre classical elastic string analogy model. The NES device connects to the wheel shaft, and it comprises a mass, a linear damper, and a pure cubic spring. The numerical integration in time was used to assess the shimmy onset speed and the post-shimmy limit cycle oscillations. A parametric analysis of the landing gear nonlinear dynamics without the NES is presented. The design space of possible NES parameters is given, obeying design constraints, and inspected to assess adequate NES designs. The best NES samples are included in the landing gear dynamics to study their influence. Results have shown that the NES can adequately expand the operational speed range for no shimmy and lead to lower LCO amplitudes in the post-shimmy for a reasonable range of speeds. The NES concept’s successful employment for the landing gear dynamics suggests an enormous potential form of passive shimmy control.

Potential of a vibro-impact nonlinear energy sink for energy harvesting

The energy harvesting potential of a vibro-impact nonlinear energy sink (VINES) is investigated. Applying Signorini’s contact law, the non-smooth dynamics is first formulated in a measure differential complementarity problem and treated using a Moreau-Jean scheme adapted with an energy-conserving integration method. The resulting scheme thus shows a good energy preservation property, allowing one to calculate the energy flow of the system in high precision. Exhaustive discussions are then carried out regarding the energy harvesting performance of the proposed VINES harvester, as well as their dependence on the underlying vibro-impact induced response regimes, under both transient and steady-state responses. It is demonstrated that the vibro-impact is responsible for achieving targeted energy transfer in the transient response or controlling the regimes in the steady-state response, to improve significantly the harvesting efficiency. Moreover, a properly designed VINES, working with 1–2 impacts per cycle, is capable for effective broadband energy harvesting over a wide amplitude and frequency range. Two specific considerations are also performed to evaluate further the generality and efficacy of the proposed harvester, and it is proved the VINES can provide higher energy density than the classical linear or cubic NES harvesters, and its effectiveness could be robust over a certain low level of noise.

Development and validation of a piecewise linear nonlinear energy sink for vibration suppression and energy harvesting

In this paper a piecewise linear nonlinear energy sink (PLNES) is developed for the purpose of simultaneous vibration suppression and energy harvesting. The apparatus consists of a cantilever beam attached by a mass at its free end. The beam is situated between a pair of double-stop blocks to constrain its lateral deflection. The stiffness characterization is conducted experimentally. The two combined systems, referred to as weak coupling and strong coupling, are constructed by attaching the PLNES to a stiff primary system and a flexible primary system, respectively. A computer simulation is conducted to investigate the transient performances of the two systems in terms of the targeted energy transfer and energy harvesting. The simulation results are examined by time responses, wavelet transform spectra, and the frequency energy plots. A multi-objective optimization is carried out to investigate the trade-off between vibration suppression and energy harvesting of the two systems. An experimental study is conducted to validate the simulation results.

Nonlinear normal modes and optimization of a square root nonlinear energy sink

Since the vibration mitigation of nonlinear energy sinks (NESs) needs multi-objective optimization, this paper aims to address this issue with efficient multi-objective particle swarm optimization (MOPSO) methods. A non-polynomial NES model, namely a square root NES (SRNES), is targeted for multi-objective optimization and dynamical analysis on an impulsively excited linear oscillator (LO) and compared with the corresponding polynomial model. Different MOPSO methods are developed to balance the mass and dissipation efficiency of SRNESs by utilizing weighted ranking, external archive, and non-dominated sorting. And nonlinear normal modes (NNMs) are extended for the analysis of SRNESs. The optimization results show the efficient targeted energy transfer (TET) and high robustness of the bistable SRNES in different LO initial and damping conditions. Specifically, more than 95% energy absorption can be easily witnessed, even in a small mass ratio (around 0.003), but the amplitude of the SRNES can be very large. The principal manifestation of the optimal TET is 1:1 resonance capture along with considerable low-frequency components of the SRNES. The optimization results of the SRNESs and the linear vibration absorbers show high similarities, while with a large mass ratio, the SRNESs possess broader range of spring stiffness. In the comparison of the two NES models, the differences are found in potential energy surfaces, NNMs, in-well motion, and vibration attenuation performance. Besides, the SRNESs can be divided into two groups in terms of the conservative dynamics, which can be explained by the polynomial model. The frequency component of in-phase NNMs dominates in the conservative and dissipative dynamics of the SRNESs, which lead to the transition of NNMs in the dissipative dynamics. In summary, the efficient MOPSO methods can be designed for different needs of vibration suppression and structure cost. Furthermore, it is important to directly analyze the non-polynomial nonlinear vibration, because transforming non-polynomial models into polynomial ones may lead to unreasonable or imprecise results. Finally, the NES damping structure and the NES performance in small mass conditions should be enhanced.

On nonlinear energy flows in nonlinearly coupled oscillators with equal mass

Due to the near monopoly held by nonlinear energy sinks in the study of targeted energy transfer, little research has been done on the flow of mechanical energy between oscillators with comparable mass. The goal of the present paper is to investigate the flow of mechanical energy between two nonlinearly coupled oscillators with comparable mass that arise due to the breaking of dynamic reciprocity. The first oscillator represents the prototypical linear oscillator (LO), whereas the second oscillator represents a nonlinear oscillator (NO) that is nonlinearly coupled to the LO only. By breaking dynamic reciprocity, one-way energy propagation is achieved in the system, such that energy can only be irreversibly transferred from the NO to the LO. As such, the NO is isolated from the LO for physically reasonable energies whenever it is not directly excited. Moreover, when the NO is directly excited, there exist regimes where the LO, despite being linear and of comparable mass to the NO, behaves like a nonlinear energy sink and parasitically and irreversibly absorbs energy from the NO. The theoretical portion of this works employs direct numerical simulation of the structure to explore the strongly nonreciprocal dynamics and resulting energy transfers. The theoretical results are then verified through experimental measurements of a comparable structure. The present study promotes a new paradigm for investigating energy transfer in mechanical structures and opens the way for passively controlling the flow of energy in complex mechanical systems.

Nonlinear energy sink applied for low-frequency noise control inside acoustic cavities: A review

In recent years, the research of nonlinear energy sink on low-frequency noise control has become a hotspot. By adding a nonlinear energy sink into one primary system, it is possible to obtain the significant target energy transfer characteristics. The target energy transfer can be defined for which the vibration energy of the primary structure is irreversibly transferred to the nonlinear energy sink, quickly concentrated in the nonlinear energy sink and dissipated by the nonlinear energy sink damping. This method has significant advantages to control the broadband low-frequency noise inside the transportations (such as cars, trains, airplanes, etc.). Compared with traditional noise reduction methods such as adding the damping and acoustical materials, the nonlinear energy sink has a simple and lightweight structure. The paper reviews the nonlinear characteristics of the nonlinear energy sink, the main theoretical research methods and the applications of vibration and noise control, and discusses the application of the nonlinear energy sink for the control of low-frequency noise inside the three-dimensional acoustic cavities, which provides the reference and guidance for the low-frequency noise control inside the acoustic cavities of the mean of transportation.

Active nonlinear energy sink using force feedback under transient regime

In this paper, an active nonlinear energy sink (ANES) based on force feedback is investigated. The proposed device is composed of a pair of collocated actuator and force sensor. The control law is implemented by feeding back the output of the force sensor, through one single integrator and one double integrator of its cube. Its working principle can be understood by an equivalent mechanical network which consists of a linear dashpot, linear spring and a cube root inerter. Although the nonlinear assignment between the spring and mass or inerter quantities is different from that of traditional nonlinear energy sinks (NESs), it is found that ANES and NES behave similarly in terms of their slow-scale dynamics and the vibration mitigation effectiveness. Closed-form expressions for properly tuning the feedback gains are derived. Numerical simulations are performed to validate the analytical analysis. The damping mechanism of ANES through targeted energy transfer and resonance capture cascade is demonstrated.