Vibration absorption of parallel-coupled nonlinear energy sink under shock and harmonic excitations

Nonlinear energy sink(NES) can passively absorb broadband energy from primary oscillators. Proper multiple NESs connected in parallel exhibit superior performance to single-degree-of-freedom(SDOF) NESs. In this work, a linear coupling spring is installed between two parallel NESs so as to expand the application scope of such vibration absorbers. The vibration absorption of the parallel and parallel-coupled NESs and the system response induced by the coupling spring are studied. The results show that the responses of the system exhibit a significant difference when the heavier cubic oscillators in the NESs have lower stiffness and the lighter cubic oscillators have higher stiffness. Moreover, the efficiency of the parallel-coupled NES is higher for medium shocks but lower for small and large shocks than that of the parallel NESs. The parallel-coupled NES also shows superior performance for medium harmonic excitations until higher response branches are induced. The performance of the parallel-coupled NES and the SDOF NES is compared. It is found that, regardless of the chosen SDOF NES parameters, the performance of the parallel-coupled NES is similar or superior to that of the SDOF NES in the entire force range.     

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially non-linear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption.     

Takagi–Sugeno fuzzy generalized predictive control for a class of nonlinear systems

The new generations of compact high output power-to-weight ratio internal combustion engines generate broadband torsional oscillations, transmitted to lightly damped drivetrain systems. A novel approach to mitigate these untoward vibrations can be the use of nonlinear absorbers. These act as Nonlinear Energy Sinks (NESs). The NES is coupled to the primary (drivetrain) structure, inducing passive irreversible targeted energy transfer (TET) from the drivetrain system to the NES. During this process, the vibration energy is directed from the lower-frequency modes of the structure to the higher ones. Thereafter, vibrations can be either dissipated through structural damping or consumed by the NES. This paper uses a lumped parameter model of an automotive driveline to simulate the effect of TET and the assumed modal energy redistribution. Significant redistribution of vibratory energy is observed through TET. Furthermore, the integrated optimization process highlights the most effective configuration and parametric evaluation for use of NES.     

Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments

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 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. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency–energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs 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. PACS numbers: 05.45.Xt, 02.30.Hq     

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.     

Numerical and experimental investigations of a rotating nonlinear energy sink

In this work, passive nonlinear targeted energy transfer (TET) is addressed by numerically and experimentally investigating a lightweight rotating nonlinear energy sink (NES) which is coupled to a primary two-degree-of-freedom linear oscillator through an essentially nonlinear (i.e., non-linearizable) inertial nonlinearity. It is found that the rotating NES passively absorbs and rapidly dissipates a considerable portion of impulse energy initially induced in the primary oscillator. The parameters of the rotating NES are optimized numerically for optimal performance under intermediate and strong loads. The fundamental mechanism for effective TET to the NES is the excitation of its rotational nonlinear mode, since its oscillatory mode dissipates far less energy. This involves a highly energetic and intense resonance capture of the transient nonlinear dynamics at the lowest modal frequency of the primary system; this is studied in detail by constructing an appropriate frequency–energy plot. A series of experimental tests is then performed to validate the theoretical predictions. Based on the obtained numerical and experimental results, the performance of the rotating NES is found to be comparable to other current translational NES designs; however, the proposed rotating device is less complicated and more compact than current types of NESs.     

Broadband energy exchanges between a dissipative elastic rod and a multi-degree-of-freedom dissipative essentially non-linear attachment

We analyze complex, multi-frequency, non-linear modal interactions in the damped dynamics of a viscously damped dispersive finite rod coupled to a multi-degree-of-freedom essentially non-linear attachment. We perform a parametric study to show that the attachment can be an effective broadband energy absorber and dissipater of shock energy from the rod. It is shown that strong targeted energy transfer from the rod to the attachment occurs when there is strong stiffness asymmetry in the attachment. For weak viscous dissipation, a clear understanding of dynamical transitions in the integrated rod-non-linear attachment system can be gained by wavelet transforming the time series and superimposing the resulting wavelet spectra in the frequency-energy plot (FEP) of the periodic orbits of the underlying Hamiltonian system. Two distinct NES configurations are analyzed in detail, and their damped responses are analyzed by the Hilbert-Huang transform (HHT). We show that the HHT is capable of analyzing even complex non-linear damped transitions, by providing the dominant frequency components (or equivalently, time scales) at which the non-linear phenomena take place, and clarifying the series of non-linear resonance captures between the rod and attachment dynamics that are responsible for the broadband energy exchanges in this system.     

Targeted energy transfers in vibro-impact oscillators for seismic mitigation

In the field of seismic protection of structures, it is crucial to be able to diminish ‘as much as possible’ and dissipate ‘as fast as possible’ the load induced by seismic (vibration-shock) energy imparted to a structure by an earthquake. In this context, the concept of passive nonlinear energy pumping appears to be natural for application to seismic mitigation. Hence, the overall problem discussed in this paper can be formulated as follows: Design a set of nonlinear energy sinks (NESs) that are locally attached to a main structure, with the purpose of passively absorbing a significant part of the applied seismic energy, locally confining it and then dissipating it in the smallest possible time. Alternatively, the overall goal will be to demonstrate that it is feasible to passively divert the applied seismic energy from the main structure (to be protected) to a set of preferential nonlinear substructures (the set of NESs), where this energy is locally dissipated at a time scale fast enough to be of practical use for seismic mitigation. It is the aim of this work to show that the concept of nonlinear energy pumping is feasible for seismic mitigation. We consider a two degree-of-freedom (DOF) primary linear system (the structure to be protected) and study seismic-induced vibration control through the use of Vibro-Impact NESs (VI NESs). Also, we account for the possibility of attaching to the primary structure additional alternative NES configurations possessing essential but smooth nonlinearities (e.g., with no discontinuities). We study the performance of the NESs through a set of evaluation criteria. The damped nonlinear transitions that occur during the operation of the VI NESs are then studied by superimposing wavelet spectra of the nonlinear responses to appropriately defined frequency – energy plots (FEPs) of branches of periodic orbits of underlying Conservative systems.     

Suppression of vibrations in strongly nonhomogeneous 2DOF systems

In this work we analyze the effects of an optimal linear control for nonlinear systems under the vibrations of a strongly nonhomogeneous nonlinear two-degree-of-freedom system with single anchor spring under condition of initial impact. We compare the results with those obtained for (a) when using only a passive broadband boundary controller which, in essence, corresponds to a one-way passive and almost irreversible energy flow from a heavy or main system to a nonlinear energy sink (NES), and (b) when using the nonlinear energy sink in combination with the optimal linear control for nonlinear system.     

Vibration suppression and higher branch responses of beam with parallel nonlinear energy sinks

The effects of nonlinear energy sink (NES) on vibration suppression of a simply supported beam are investigated in this work. The slow flow equations of the system are derived by using complexification–averaging method, and the validity of the derivation is verified. By comparing the vibration absorption of single and parallel NESs of equal mass, it is found that the latter exhibits superior vibration absorption performance. In addition, the parallel NES can eliminate higher branch responses of the system under the harmonic load. Furthermore, it is found that parallel NES can eliminate the higher branches of the system more effectively by tuning nonlinear stiffness and damping. Moreover, the thermal effect on natural frequencies of the simply supported beam is considered, and the influences of the parallel NES’s parameters on the energy dissipation rate under shock load are investigated. The nonlinear responses of the simply supported beam with parallel NES under harmonic load and with the increase of temperature are described.     

Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results

We study theoretically and experimentally the effect that anonlinear energy sink (NES) has on the steady state dynamics of a weaklycoupled system. The NES possesses essentially nonlinear(nonlinearizable) stiffness nonlinearity of the third degree. We findthat, in contrast to the classical linear vibration absorber, the NES iscapable of absorbing steady state vibration energy from the linearoscillator over a relatively broad frequency range. This results inlocalization of the steady state vibration in the NES, away from thedirectly forced subsystem. For a forward frequency sweep the localizedbranch of steady state motions is suddenly eliminated by a jump to alinearized low-amplitude motion, whereas, for a backward frequency sweepa reverse jump occurs. The difference in the frequencies of the twojumps introduces a nonlinear hysteresis loop. This work extends to thesteady state case of earlier transient passive energy pumping results.The notion of passively transferring vibration energy to an a prioridetermined NES, weakly attached to a main structure, is novel. The useof nonlinear energy sinks for passively absorbing energy from a linearmain structure can form the basis of relatively simple and modularvibration and shock isolation designs of mechanical systems.     

Design criteria for optimally tuned nonlinear energy sinks—part 1: transient regime

The context of present work is related to the study of strongly nonlinear absorbers (NESs) aimed to attenuate vibrations induced in a single degree-of-freedom oscillator and working under the principle of targeted energy transfer (TET). The purpose motivated by practical considerations is here to establish a design criterion permitting to first ensure whether NES absorber is active or not and second to provide a nonlinear stiffness lower bound for optimal energy absorption during pumping phases. An asymptotic expansion of dynamic equations of motion under transient regime enables to emphasize a new definition of activation energy and to investigate the influence of damping upon the efficiency of one-way channeled energy transfer. Methodology is straightforwardly extended to the case of multiple NES attached in parallel to the primary oscillator. Numerical benchmark simulations corroborate the reliability and robustness of proposed design procedure.     

Interactions of propagating waves in a one-dimensional chain of linear oscillators with a strongly nonlinear local attachment

We study the interaction of propagating wavetrains in a one-dimensional chain of coupled linear damped oscillators with a strongly nonlinear, lightweight, dissipative local attachment which acts, in essence, as nonlinear energy sink—NES. Both symmetric and highly un-symmetric NES configurations are considered, labelled S-NES and U-NES, respectively, with strong (in fact, non-linearizable or nearly non-linearizable) stiffness nonlinearity. Especially for the case of U-NES we show that it is capable of effectively arresting incoming slowly modulated pulses with a single fast frequency by scattering the energy of the pulse to a range of frequencies, by locally dissipating a major portion of the incoming energy, and then by backscattering residual waves upstream. As a result, the wave transmission past the location of the NES is minimized, and the NES acts, in effect, as passive wave arrestor and reflector. Analytical reduced-order modeling of the dynamics is performed through complexification/averaging. In addition, governing nonlinear dynamics is studied computationally and compared to the analytical predictions. Results from the reduced order model recover the exact computational simulations.     

Response regimes in linear oscillator with 2DOF nonlinear energy sink under periodic forcing

The system under investigation comprises a linear oscillator coupled to a strongly asymmetric 2 degree-of-freedom (2DOF) purely cubic nonlinear energy sink (NES) under harmonic forcing. We study periodic, quasiperiodic, and chaotic response regimes of the system in the vicinity of 1:1 resonance and evaluate the abilities of the 2DOF NES to mitigate the vibrations of the primary system. Earlier research showed that single degree-of-freedom (SDOF) NES can efficiently mitigate the undesired oscillations, if limited to relatively low forcing amplitudes. In this paper, we demonstrate that the additional degree-of-freedom of the NES considerably broadens the range of amplitudes where efficient mitigation is possible. Efficiency limits of the system with the 2DOF NES are evaluated numerically. Analytic approximations for simple response regimes are also developed.     

分布振子复合板的模态阻尼及多点激励下阻尼减振相关性

分布阻尼振子可拓宽结构减振频带,因此可将振子分布于板中以形成复合板（简称“分布振子复合板”）,进而实现较宽的减振频带．对于多点支撑处受到宽频非一致激励（例如在不同激励点处的激励频率、幅值与相位有差异）的分布振子复合板,目前还缺乏有效简便的优化控制指标．在作者之前的研究中,针对含分布振子的梁推导了基于模态应变能的模态阻尼计算理论,讨论了模态阻尼与单点激励下梁的减振效果的相关性,并应用于宽频减振设计优化．本文进一步将模态阻尼计算理论推广到分布振子复合板,并将研究从梁的单点激励扩展到板的多点非一致激励下的阻尼减振相关性．首先,在利用模态应变能法推导得到分布振子复合板的模态阻尼计算公式后,从理论上讨论了不同边界条件与模态阶次对计算结果的影响,以及计算理论的适用性．而后,进一步通过有限元参数分析了边界条件、频率比、模态阶次与质量比的影响．最后,通过算例分析了无振子板或分布振子复合板在四个激励点具有多种幅值与相位组合情况下的稳态响应．结果表明,推导的模态阻尼计算公式可正确预测不同边界条件下的模态阻尼,且理论预测的模态阻尼与基板的稳态平均加速度减小率、稳态峰值应变能减小率均有较高的相关性．     

Performance measures for targeted energy transfer and resonance capture cascading in nonlinear energy sinks

In vibrating mechanical systems, the targeted energy transfer mechanism (TET) of nonlinear energy sinks (NES) is employed as an alternative to linear tuned mass dampers (TMD) as passive vibrations absorbers for transient vibrations. The major advantages a NES has over a linear TMD are (1) an increased robustness to detuning and (2) the ability to dissipate multiple frequencies with only a single NES through so-called resonance capture cascading (RCC). The performance, especially the speed, of TET and RCC has rarely been a topic of research. In this research, algebraic performance measures for the speed of both TET and RCC are derived, called the pumping time and the cascading time, respectively. It shows that cascading time can be seen as a sum of single-mode pumping times, by introducing a novel modal decomposition. The strength of both measures is that they do not require numerical simulations, allowing easy optimization of the NES. The influence of different nonlinearities on the TET and RCC performance is investigated. Actual numerical simulations presented in the study validate the merit of both the pumping time and cascading time.     

谐波平衡法与复平均法计算强非线性系统稳态响应的区别

近些年，很多学者致力于利用非线性增强振动响应减少的效果或者能量采集器的效率。因而非线性系统的响应值需要从理论计算方面更准确地预测。另外，根据学者已取得的研究成就，非线性能量汇(NES)中存在的立方刚度非线性可以将结构中宽频域的振动能量传递至非线性振子部分。文章将一种由NES和压电能量采集器组成的NES-piezo装置与两自由度主结构耦合连接，系统受谐和激励作用。文章采用谐波平衡法和复平均法分别推导了系统稳态响应，参照数值结果，对比两种近似解析方法在求解强非线性系统稳态响应时的异同。计算结果表明，系统体现较弱非线性时，二者计算结果差异很小；当系统体现强非线性时，复平均法不能准确地呈现系统高阶响应，提高阶数的谐波平衡法能更准确地表示系统响应值。基于谐波平衡法和数值算法，讨论NES-piezo装置对于系统宽频域减振的影响。与仅加入非线性能量汇情况对比，结果表明NES-piezo装置不会恶化宽频域减振效果，并且在第一阶共振频率附近，可以稍微提高结构减振效率。另外，计算结果也表明，采用恰当的NES-piezo装置可实现宽频域范围的结构减振和压电能量采集一体化。此项研究工作为研究不同情形强非线性系统的响应提供了理论方法的指导。另外，研究结果也为宽频域范围的结构减振和压电能量采集一体化提供了理论依据。     

Experimental study of non-linear energy pumping occurring at a single fast frequency

Experimental verification of passive non-linear energy pumping in a two-degree-of-freedom system comprising a damped linear oscillator coupled to an essentially non-linear attachment is carried out. In the experiments presented the non-linear attachment interacts with a single linear mode and, hence, energy pumping occurs at a single ‘fast’ frequency in the neighborhood of the eigenfrequency of the linear mode. Good agreement between simulated and experimental results was observed, in spite of the strongly (essentially) non-linear and transient nature of the dynamics of the system 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 non-linear attachment, and dissipated there locally without spreading back to the system. In addition, the reported experimental results confirm that (a) non-linear 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.     

Clearance-type nonlinear energy sinks for enhancing performance in electroacoustic wave energy harvesting

This paper explores a clearance-type nonlinear energy sink (NES) for increasing electrical energy harvested from non-stationary mechanical waves, such as those encountered during impact and intermittent events. The key idea is to trap energy in the NES such that it can be harvested over a time period longer than that afforded by the passing disturbance itself. Analytical, computational, and experimental techniques are employed to optimize the energy sink, explore qualitative behavior (to include bifurcations), and verify enhanced performance. Unlike traditionally studied single-DOF NESs, both subdomains of the NES (i.e., on either side of the clearance) contain displaceable degrees of freedom, increasing the complexity of the analytical solution approach. However, closed-form solutions are found which quantify the relationship between the impact amplitude and the energy produced, parameterized by system properties such as the harvester effective resistance, the clearance gap, and the domain mass and stiffness. Bifurcation diagrams and trends therein provide insight into the number and state of impact events at the NES as excitation amplitude increases. Moreover, a closed-form Poincaré map is derived which maps one NES impact location to the next, greatly simplifying the analysis while providing an important tool for follow-on bifurcation studies. Finally, a series of representative experiments are carried out to realize the benefits of using clearance-type nonlinearities to trap wave energy and increase the net harvested energy.     

The tuned bistable nonlinear energy sink

A bistable nonlinear energy sink conceived to mitigate the vibrations of host structural systems is considered in this paper. The hosting structure consists of two coupled symmetric linear oscillators (LOs), and the nonlinear energy sink (NES) is connected to one of them. The peculiar nonlinear dynamics of the resulting three-degree-of-freedom system is analytically described by means of its slow invariant manifold derived from a suitable rescaling, coupled with a harmonic balance procedure, applied to the governing equations transformed in modal coordinates. On the basis of the first-order reduced model, the absorber is tuned and optimized to mitigate both modes for a broad range of impulsive load magnitudes applied to the LOs. On the one hand, for low-amplitude, in-well, oscillations, the parameters governing the bistable NES are tuned in order to make it functioning as a linear tuned mass damper (TMD); on the other, for high-amplitude, cross-well, oscillations, the absorber is optimized on the basis of the invariant manifolds features. The analytically predicted performance of the resulting tuned bistable nonlinear energy sink (TBNES) is numerically validated in terms of dissipation time; the absorption capabilities are eventually compared with either a TMD and a purely cubic NES. It is shown that, for a wide range of impulse amplitudes, the TBNES allows the most efficient absorption even for the detuned mode, where a single TMD cannot be effective.