Passive Targeted Energy Transfer Research Articles

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

The nonlinear energy sink (NES), which has been proven to conduct quick and passive targeted energy transfer (TET), has been implemented in several structures to reduce vibration. For an NES to perform efficiently via targeted energy transfer (TET), the vibration levels must surpass a well-defined threshold; otherwise, the NES will perform inefficiently. The bistable NES (BNES) produces a nonlinear force with a negative linear and a positive nonlinear stiffness component. This investigation examines the energy flow characteristics of a bistable nonlinear energy sink coupled with a nonlinear oscillator (NO). Where frequency–energy graphs (FEPs) demonstrated the nonlinear normal modes (NNMs) and dynamic behaviour of the resultant NO-BNES system. At low energy levels, the emergence of many symmetrical and asymmetrical in-phase and out-of-phase backbone branches is emphasized. The superimposed wavelet frequency spectra of the NO-BNES response on the FEP have validated the robustness of the TET mechanism, where the contribution of the asymmetrical NNM backbones in TET can be viewed with clarity. Comparing the BNES and cubic NES by superimposing the wavelet frequency spectra on FEPs and energy dissipation efficiency reveals that the BNES performs better in TET.

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.

Reliability-based optimization of nonlinear energy sink with negative stiffness and sliding friction

This paper presents implementation of a reliability-based design optimization (RBDO) for Nonlinear Energy Sink (NES) system. The RBDO is implemented with consideration of uncertainties in structural system's parameters and ground acceleration. Negative stiffness based NES and sliding friction are utilized to enhance passive targeted energy transfer from the primary structure to the proposed control device. For a one-story steel moment resistant frame, shake table tests were carried out to show effectiveness of the NES, and the results are used to validate numerical models. Sensitivity analysis is carried out to demonstrate the performance envelops of the proposed control strategy for different stiffness ratios and peak ground acceleration values. Reliable performance of the proposed controller, through tuning parameters of the proposed NES, are determined through the RBDO framework. To reduce computational time, polynomial-chaos-based Kriging surrogate model is used in the RBDO analysis. The numerical results demonstrate efficiency of finding optimal design parameters of the proposed NES system under uncertainties with reasonable computational effort.

Passive nonlinear targeted energy transfer.

Nonlinearity in dynamics and acoustics may be viewed as scattering of energy across frequencies/wavenumbers. This is in contrast with linear systems when no such scattering exists. Motivated by irreversible large-to-small-scale energy transfers in turbulent flows, passive targeted energy transfers (TET) in mechanical and structural systems incorporating intentional strong nonlinearities are considered. Transient or permanent resonance captures are basic mechanisms for inducing TET in such systems, as well as nonlinear energy scattering across scales caused by strongly nonlinear resonance interactions. Certain theoretical concepts are reviewed, and some TET applications are discussed. Specifically, it is shown that the addition of strongly nonlinear local attachments in an otherwise linear dynamical system may induce energy scattering across scales and 'redistribution' of input energy from large to small scales in the linear modal space, in similarity to energy cascades that occur in turbulent flows. Such effects may be intentionally induced in the design stage and may lead to improved performance, e.g. it terms of vibration and shock isolation or energy harvesting. In addition, a simple mechanical analogue in the form of a nonlinear planar chain of particles composed of linear stiffness elements but exhibiting strong nonlinearity due to kinematic and geometric effects is discussed, exhibiting similar energy scattering across scales in its acoustics. These results demonstrate the efficacy of intentional utilization of strong nonlinearity in design to induce predictable and controlled intense multi-scale energy transfers in the dynamics and acoustics of a broad class of systems and structures, thus achieving performance objectives that would be not possible in classical linear settings.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Enhanced Targeted Energy Transfer by Vibro Impact Cubic Nonlinear Energy Sink

Passive targeted energy transfer (TET) that describes a highly efficient manner of energy absorption is considerably enhanced by a new form of absorber proposed in this paper. The absorber is attached to the primary linear oscillator (LO) through cubic stiffness and bilateral barriers that set to induce vibro-impact (VI). Both essential nonlinearity and non-smooth nonlinearity are considered. Energy pumping phenomenon is found, and complexification averaging method is used to give an analytical treatment for the essential stiffness nonlinearity. At a low level of impulse excitation where energy pumping of nonlinear energy sink (NES) does not occur, by introducing VI energy pumping is brought up. At the optimal TET state, the vibro-impact cubic (VIC) absorber improves the efficiency of cubic NES on energy reduction to a certain degree. For a two-degree-of-freedom LO, the new absorber can absorb most energy of the broadband excitation which is a novel improvement compared with normal NES. Broadband excitations like input with sufficient bandwidth and random signals are found to be absorbed extensively by the VIC NES, meaning that the VIC NES as a nonlinear passive vibration absorber can be very efficient on broadband vibration energy absorption.

Nonlinear Energy Sinks With Nontraditional Kinds of Nonlinear Restoring Forces

The nonlinear energy sink (NES) is usually coupled with a linear oscillator (LO) to rapidly transfer and immediately dissipate a significant portion of the initial shock energy induced into the LO. This passive energy transfer and dissipation are usually achieved through strong resonance captures between the NES and the LO responses. Here, a nontraditional set of nonlinear coupling restoring forces is numerically investigated to introduce enhanced versions of the NESs. In this new set of nonlinear coupling restoring forces, one has a varying nonlinear stiffness that includes both of hardening and softening stiffness components during the oscillation, which appear in closed-loops under the effect of the damping. The obtained results by the numerical simulation have shown that employing this kind of the nonlinear restoring forces for passive targeted energy transfer (TET) is promising for shock mitigation.

Experimental investigation and analytical description of a vibro-impact NES coupled to a single-degree-of-freedom linear oscillator harmonically forced

In this paper, the dynamics of a system composed of a harmonically forced single-degree-of-freedom linear oscillator coupled to a vibro-impact nonlinear energy sink (VI-NES) is experimentally investigated. The mass ratio between the VI-NES and the primary system is about $$1\%$$ . Depending on the external force’s amplitude and frequency, either a strongly modulated response (SMR) or a constant amplitude response (CAR) is observed. In both cases, an irreversible transfer of energy occurs from the linear oscillator toward the VI-NES: process known in the literature as passive targeted energy transfer. Furthermore, the problem is analytically studied by using the method of multiple scales. The obtained slow invariant manifold shows the existence of a stable and of an unstable branch of solutions, as well as of an energy threshold (a saddle-node bifurcation) for the solutions to appear. Subsequently, the fixed points of the problem are calculated. When a stable fixed point is reached, the system is naturally drawn to it and a CAR is established, whereas when no stable point is attained, the system exhibits a SMR regime. Finally, a good correlation between the experimental and the analytical results is presented.

Passive targeted energy transfer in the steady state dynamics of a nonlinear plate with nonlinear absorber

In this work, the inducement of passive nonlinear sink for vibration control of nonlinear plate model was studied. The considered system was composed of simply supported thin plate under harmonic excitation with a nonlinear attachment. By varying parameters and locations of the nonlinear energy (NES), an optimized passive targeted energy transfers (TETs) from the plate to the attachments was performed. Moreover, comparative studies of the performance of NESs and linear tuned mass dampers (TMDs) are given. By performing complex averaging and arclength continuations techniques, semi-analytical approximations for nonlinear steady-state response was developed. To justify the averaging method, numerical simulation was also performed. The obtained results show that the NES designed for specific amplitude of excitation has better performance with respect to TMD, but with increase in amplitude of excitation force, NES has lower robustness with respect to TMD. The obtained results show that, if the phase differences of plate and NES with exciting force are ±π2 and ±π respectively, then the best condition for energy transfer from plate to NES will occurs, but if both the plate and NES have π with exciting force, the amplitude of vibration for both of them are considerably high.

Shock Mitigation by Means of Low- to High-Frequency Nonlinear Targeted Energy Transfers in a Large-Scale Structure

In this computational study, the implementation of passive nonlinear vibro-impact attachments (termed nonlinear energy sinks (NESs)) for shock mitigation of an otherwise linear multistory large-scale structure is investigated. This is achieved by inducing passive targeted energy transfer (TET) from the fundamental (lowest-frequency and most energetic) structural mode to high-frequency modes, through a series of vibro-impacts induced by the attachments. The functionality of the passive attachments is based on single-sided vibro-impacts (SSVIs), enabling rapid and one-way scattering of shock energy from low- to high-frequency structural modes. Hence, redistribution of shock energy in the modal space of the structure occurs as energy gets nonlinearly scattered to high frequencies. In turn, this energy scattering rapidly reduces the overall amplitude of the transient structural response, and increases the effective dissipative capacity of the integrated NES-structure assembly. The effective modal dissipation rates of the integrated assembly can be controlled by the inherent damping of the NESs, and can be qualitatively studied in detail by defining appropriate dissipative measures which track the TETs from low- and high-frequency structural modes. Ideally, the optimized NESs can passively scatter up to 80% of the input shock energy from the fundamental structural mode to high-frequency modes in the limit when their inherent damping is zero and the coefficient of restitution during vibro-impacts is unity. When dissipative effects are introduced into the NESs, additional energy exchanges between the NESs and high-frequency modes occur. Our study facilitates the predictive design of vibro-impact NESs for optimal and rapid shock mitigation of large-scale structures.

Asymmetric Magnet-Based Nonlinear Energy Sink

The nonlinear energy sink (NES) is a light-weighted device used for shock mitigation in dynamic structures through its passive targeted energy transfer (TET) mechanism. Here, a new design for the NES is introduced based on using an asymmetric NES force. This force is strongly nonlinear in one side of the NES equilibrium position, whereas it is either weakly nonlinear or weakly linear in the other side. This is achieved by introducing the asymmetric magnet-based NES in which the asymmetric nonlinear magnetic repulsive force is generated by two pairs of aligned permanent magnets. Consequently, this proposed design is found to provide a considerable enhancement in the shock mitigation performance compared with the symmetric stiffness-based NESs for broadband energy inputs.

Targeted Energy Transfer Between a Swept Wing and Winglet-Housed Nonlinear Energy Sink

A prototype nonlinear energy sink, whose design is based upon parameters determined to effectively suppress transonic aeroelastic instabilities of a wind-tunnel wing model (denoted as generic transport wing) via passive targeted energy transfer, is introduced. The lightweight nonlinear energy sink is mounted within a low-profile winglet, which is attached to the tip of the generic transport wing. In addition, safety features and measurement hardware have been built in to the prototype to facilitate a future transonic wind-tunnel experiment. The effects of the nonlinear energy sink on the structural dynamics of the model wing are demonstrated in ground vibration tests, both experimental and computational. Results show that the nonlinear energy sink causes a significant increase in the dissipation rate of energy in the second bending mode of the generic transport wing, even for small wing-tip oscillations. This is a strong indication that the prototype nonlinear energy sink will be effective in wind-tunnel tests because the frequency of the second bending mode is within the range of experimental and computational flutter frequencies of the generic transport wing. Furthermore, accompanying computational analysis is used to show that moderate friction damping in the nonlinear energy sink is unlikely to have a significant effect on the qualitative interaction between the nonlinear energy sink and the generic transport wing.

Vibration reduction in unbalanced hollow rotor systems with nonlinear energy sinks

Reduction of whirling vibration amplitudes in rotordynamic systems resulting from unbalance force during the passage through the critical speeds is very important for preventing propagation of fatigue cracks, excessive noise and instability problems. In this paper, numerical investigations of passive targeted energy transfer (TET) to mitigate the whirling vibration amplitude in rotor systems at the critical speeds are performed by applying nonlinear energy sinks (NESs) to the rotor systems. The three-degree-of-freedom hollow shaft-NES system has been modeled based on Lagrange’s method. Numerical simulations have been performed to optimize the NES parameters in order to obtain the optimum performance for vibration reduction. The influence of critical parameters, including the angle between the unbalance mass and the NES, the NES damping and the unbalance mass on the NES performance in whirling vibration reduction, is also investigated. Additionally, the application of the NES is compared with that of the tuned mass damper (TMD). Numerical results show that the NES is able to efficiently reduce the resonant amplitude of the rotor systems when the stiffness and mass ratios are optimized appropriately. Furthermore, it is found that, unlike the NES, the TMD requires an offset distance from the shaft centerline to be able to reduce the resonant amplitude; otherwise, it fails in mitigating the whirling amplitude. Identifying this offset for the TMD requires pre-knowledge of the exact unbalance value and its eccentricity from the shaft centerline. However, the NES does not require a priori knowledge of the unbalance value which is an advantage of the NES compared to the TMD. The TMD, applied here with an offset distance able to mitigate the resonant vibration, only works as a balancer when positioned opposite to the location of the unbalance mass, which limits its application in vibration reduction of rotor systems.

Highly efficient nonlinear energy sink

The performance of the nonlinear energy sink (NES) that composed of a small mass and essentially nonlinear coupling stiffness with a linear structure is considerably enhanced here by including the negative linear and nonlinear coupling stiffness components. These negative linear and nonlinear stiffness components in the NES are realized here through the geometric nonlinearity of the transverse linear springs. By considering these components in the NES, very intersecting results for passive targeted energy transfer (TET) are obtained. The performance of this modified NES is found here to be much improved than that of all existing NESs studied up to date in the literature. Moreover, nearly 99 % of the input shock energy induced by impulse into the linear structures considered here has been found to be rapidly transferred and locally dissipated by the modified NES. In addition, this modified NES maintains its high performance of shock mitigation in a broadband fashion of the input initial energies where it keeps its high performance even for sever input energies. This is found to be achieved by an immediate cascade of several resonance captures at low- and high- nonlinear normal modes frequencies. The findings obtained here by including the negative linear and nonlinear stiffness components are expected to significantly enrich the application of these stiffness components in the TET field of such nonlinear oscillators.

Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators

Accurate formulas for obtaining the energy, velocity and displacement decay envelopes in purely nonlinear damped oscillators are presented here. These purely nonlinear oscillators have no linear stiffness components, and the new formulas derived in this paper are found to be highly accurate in determining the decay envelopes, regardless of the system’s physical parameters and its initial condition. The mechanism of these new formulas for obtaining the decay envelopes of the nonlinear oscillators is found to be exactly the same as the mechanism of the well-known amplitude decay formulas used for the linear oscillator. Finding such formulas is significant to many scientific applications of these nonlinear oscillators, such as the passive targeted energy transfer through the stiffness-based nonlinear energy sinks (NESs). In these types of NESs, these formulas can be easily applied for system identification and dynamic analysis. In addition, they are expected to have a significant application in different scientific fields for such oscillators.

Enhanced passive targeted energy transfer in strongly nonlinear mechanical oscillators

Single-degree-of-freedom (SDOF) nonlinear energy sinks (NESs) can efficiently mitigate broadband disturbances applied to primary linear systems by means of passive targeted energy transfer (TET), but for a rather limited range of energies. We demonstrate that the TET can be significantly enhanced for broad range of energies by introducing additional internal degrees of freedom to the NES in a highly asymmetric fashion. Numerical simulations demonstrate that the enhanced performance is due to a positive synergistic effect of the internal degrees of freedom of the proposed NES with highly asymmetric stiffnesses.

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

We study targeted energy transfers (TETs) and nonlinear modal interactions attachments occurring in the dynamics of a thin cantilever plate on an elastic foundation with strongly nonlinear lightweight attachments of different configurations in a more complicated system towards industrial applications. We examine two types of shock excitations that excite a subset of plate modes, and systematically study, nonlinear modal interactions and passive broadband targeted energy transfer phenomena occurring between the plate and the attachments. The following attachment configurations are considered: (i) a single ungrounded, strongly (essentially) nonlinear single-degree-of-freedom (SDOF) attachment—termed nonlinear energy sink (NES); (ii) a set of two SDOF NESs attached at different points of the plate; and (iii) a single multi-degree-of-freedom (MDOF) NES with multiple essential stiffness nonlinearities. We perform parametric studies by varying the parameters and locations of the NESs, in order to optimize passive TETs from the plate modes to the attachments, and we showed that the optimal position for the NES attachments are at the antinodes of the linear modes of the plate. The parametric study of the damping coefficient of the SDOF NES showed that TETs decreasing with lower values of the coefficient and moreover we showed that the threshold of maximum energy level of the system with strong TETs occured in discrete models is by far beyond the limits of the engineering design of the continua. We examine in detail the underlying dynamical mechanisms influencing TETs by means of empirical mode decomposition (EMD) in combination with wavelet transforms. This integrated approach enables us to systematically study the strong modal interactions occurring between the essentially nonlinear NESs and different plate modes, and to detect the dominant resonance captures between the plate modes and the NESs that cause the observed TETs. Moreover, we perform comparative studies of the performance of different types of NESs and of the linear tuned mass dampers (TMDs) attached to the plate instead of the NESs. Finally, the efficacy of using this type of essentially nonlinear attachments as passive absorbers of broadband vibration energy is discussed.

Suppressing Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 2: Experiments

This paper presents experimental results corroborating the analysis developed in the companion paper, Part 1 (Lee, Y., Vakakis, A., Bergman, L., McFarland, M., and Kerschen G., Suppression Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 1: Theory, AIAA Journal, Vol. 45, No. 3,2007, pp. 693-711), and demonstrates that a nonlinear energy sink can improve the stability of an aeroelastic system. The nonlinear energy sink was, in this case, attached to the heave (plunge) degree of freedom of a rigid airfoil which was supported in a low-speed wind tunnel by nonlinear springs separately adjustable in heave and pitch. This airfoil was found to exhibit a limit cycle oscillation at flow speeds above the critical (flutter) speed of 9.5 m/s, easily triggered by an initial heave displacement. After attachment of a single degree of freedom, essentially nonlinear energy sink to the wing, the combined system exhibited improved dynamic response as measured by the reduction or elimination of limit cycle oscillation at flow speeds significantly greater than the wing's critical speed. The design, application, and performance of the nonlinear energy sink are described herein, and the results obtained are compared to analytical predictions. The physics of the interaction of the sink with the wing is examined in detail.

Karhunen–Loeve analysis and order reduction of the transient dynamics of linear coupled oscillators with strongly nonlinear end attachments

The Karhunen–Loeve (K–L) decomposition method has become a popular technique to create low-dimensional, reduced-order models of dynamical systems. In this paper this technique is applied to a multi-degree-of-freedom chain of linear coupled oscillators with a strongly nonlinear (nonlinearizable), lightweight end attachment. By performing K–L decomposition we show that the lightweight nonlinear attachment (possessing 0.5% of the total mass of the chain) can affect the global dynamics of the linear chain, exhibiting nonlinear energy-pumping phenomena; that is, irreversible passive targeted energy transfers from the linear chain to the nonlinear end attachment, where this energy is locally confined and dissipated without ‘spreading back’ to the primary system. It is shown that the occurrence of energy pumping can be identified by studying the dominant K–L modes of the dynamics, as well as, the energy distribution among them. Moreover, by comparing the action of the strongly nonlinear attachment to the classical linear vibration absorber, we show robustness of passive nonlinear energy absorption over wide parameter ranges. On the other hand, the case-sensitive nature of K–L-based reduced-order models has always been a constraint for K–L decomposition, since one cannot quantify a priori the error bound of such low-dimensional reduced-order models when different initial conditions are applied to the system. To alleviate this constraint, the paper proposes a multiple correlation coefficient (MCC) as a quantitative measure to effectively assess the applicability of a K–L-based reduced-order model derived for a specific set of initial conditions to a small neighborhood of initial conditions containing that initial state. The derived reduced-order models are validated through reconstruction of the system responses and comparisons to direct numerical integrations.

Using passive nonlinear targeted energy transfer to stabilize drill-string systems

Torsional vibrations of drill strings used in drilling oil and gas wells arise from a complex interaction of the dynamics of the drilling structure with speed-dependent effective rock-cutting forces. These forces are often difficult to model, and contribute substantially to the instability problems of controlling the drilling operation so as to produce steady cutting. In this work we show how nonlinear passive targeted energy transfer to a lightweight attachment can be used to passively control these instabilities. This is performed by means of a nonlinear energy sink (NES), a lightweight attachment which has been shown to be effective in reducing or even completely eliminating self-excited motions in aeroelastic and other systems. The NES is a completely passive, inherently broadband vibration absorber capable of attracting and dissipating vibrational energy from the primary structure to which it is attached, in this case a nonlinear discontinuous model of a drill-string system. In this paper we describe a prototypical drill string-NES system, briefly discuss some of the analytical and computational tools suitable for its analysis, and then concentrate on mathematical results on the efficacy of the NES in this application and their physical interpretation.

Suppression Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 1: Theory

We study passive and nonlinear targeted energy transfers induced by resonant interactions between a single-degree-of-freedom nonlinear energy sink (NES) and a 2-DOF in-flow rigid wing model. We show that it is feasible to partially or even completely suppress aeroelastic instability by passively transferring vibration energy from the wing to the NES in a one-way irreversible fashion. Moreover, this instability suppression is performed by partially or completely eliminating its triggering mechanism. Numerical parametric studies identify three main mechanisms for suppressing aeroelastic instability: recurring burstout and suppression, intermediate suppression, and complete elimination. We investigate these mechanisms both numerically by the Hilbert-Huang transform and analytically by a complexification-averaging technique. Each suppression mechanism involves strong 1:1 resonance capture during which the NES absorbs and dissipates a significant portion of energy fed from the flow to the wing. Failure of suppression is associated with restoring the underlying triggering mechanism of instability, which is a series of superharmonic resonance captures followed by escapes from resonance. Finally, using a numerical continuation technique, we perform a bifurcation analysis to examine sensitive dependence on initial conditions and thus robustness of instability suppression.