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.     

Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations

A linear oscillator (LO) coupled with two vibro-impact (VI) nonlinear energy sinks (NES) in parallel is studied under periodic and transient excitations, respectively. The objective is to study response regimes and to compare their efficiency of vibration control. Through the analytical study with multiple scales method, two slow invariant manifolds (SIM) are obtained for two VI NES, and different SIM that result from different clearances analytically supports the principle of separate activation. In addition, fixed points are calculated and their positions are applied to judge response regimes. Transient responses and modulated responses can be further explained. By this way, all analysis is around the most efficient response regime. Then, numerical results demonstrate two typical responses and validate the effectiveness of analytical prediction. Finally, basic response regimes are experimentally observed and analyzed, and they can well explain the complicated variation of responses and their corresponding efficiency, not only for periodic excitations with a fixed frequency or a range of frequency, but also for transient excitation. Generally, vibration control is more effective when VI NES is activated with two impacts per cycle, whatever the types of excitation and the combinations of clearances. This observation is also well reflected by the separate activation of two VI NES with two different clearances, but at different levels of displacement amplitude of LO.     

Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation

The nonlinear behaviors and vibration reduction of a linear system with a nonlinear energy sink(NES) are investigated. The linear system is excited by a harmonic and random base excitation, consisting of a mass block, a linear spring, and a linear viscous damper. The NES is composed of a mass block, a linear viscous damper, and a spring with ideal cubic nonlinear stiffness. Based on the generalized harmonic function method,the steady-state Fokker-Planck-Kolmogorov equation is presented to reveal...     

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.     

Integration of a nonlinear energy sink and a piezoelectric energy harvester

A mechanical-piezoelectric system is explored to reduce vibration and to harvest energy. The system consists of a piezoelectric device and a nonlinear energy sink(NES), which is a nonlinear oscillator without linear stiffness. The NES-piezoelectric system is attached to a 2-degree-of-freedom primary system subjected to a shock load. This mechanical-piezoelectric system is investigated based on the concepts of the percentages of energy transition and energy transition measure. The strong target energy transfer occurs for some certain transient excitation amplitude and NES nonlinear stiffness. The plots of wavelet transforms are used to indicate that the nonlinear beats initiate energy transitions between the NES-piezoelectric system and the primary system in the transient vibration, and a 1:1 transient resonance capture occurs between two subsystems.The investigation demonstrates that the integrated NES-piezoelectric mechanism can reduce vibration and harvest some vibration energy.     

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.     

参数和直接激励下非线性压电俘能器性能分析的多尺度法

考虑几何非线性、阻尼非线性和梁的轴向不可伸长条件,利用Hamilton变分原理,建立了参数激励和直接激励下压电俘能器的非线性力电耦合的运动微分方程;利用Galerkin法,将所建立的动力学偏微分方程降阶为力电耦合的Mathieu-Duffing型方程;采用多尺度法获得了梁的位移和输出电压的解析表达式,给出了解的稳定性条件;利用解析表达式研究了单独参数激励以及参数激励和直接激励共同作用下阻尼系数对压电俘能器性能的影响。结果表明,在参数激励情况下,线性阻尼会显著影响超临界分岔点的位置,非线性二次阻尼不会影响超临界分岔点的位置。参数激励和直接激励的结合可以作为提升压电能量俘获器性能的解决方案。     

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。     

Numerical optimisation of a classical stochastic system for targeted energy transfer

The paper studies stochastic dynamics of a two-degree-of-freedom system, where a primary linear system is connected to a nonlinear energy sink with cubic stiffness nonlinearity and viscous damping. While the primary mass is subjected to a zero-mean Gaussian white noise excitation, the main objective of this study is to maximise the efficiency of the targeted energy transfer in the system. A surrogate optimisation algorithm is proposed for this purpose and adopted for the stochastic framework. The optimisations are conducted separately for the nonlinear stiffness coefficient alone as well as for both the nonlinear stiffness and damping coefficients together. Three different optimisation cost functions, based on either energy of the system’s components or the dissipated energy, are considered. The results demonstrate some clear trends in values of the nonlinear energy sink coefficients and show the effect of different cost functions on the optimal values of the nonlinear system’s coefficients.     

Aeroelastic suppression of an airfoil with control surface using nonlinear energy sink

Aeroelasticity exists in airfoil with control surface freeplay, which may induce instability in an incompressible flow. In this paper, a nonlinear energy sink (NES) is used to suppress the aeroelasticity of an airfoil with a control surface. The freeplay and cubic nonlinearity in pitch are taken into account. The harmonic balance method is used to analytically determine the limit cycle oscillations (LCOs) amplitudes of the airfoil–NES system. Linear and nonlinear flutter speeds are detected from the airfoil with control surface freeplay. When NES is attached, both the linear flutter speed of airfoil without freeplay and the nonlinear flutter speed of airfoil with a freeplay are increased. Moreover, the LCO amplitude of airfoil is decreased due to NES. Then, the influences of NES parameters on the increase in flutter boundary of airfoil are carefully studied.     

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.     

具有弹性放大器的双稳态压电俘能器建模及参数影响分析

利用广义Hamilton变分原理,建立了具有弹性放大器的双稳态压电俘能系统BPH+EM的动力学方程。考虑谐波激励,采用调和平衡法获得了BPH+EM系统的位移、输出电压和功率的解析解。利用求得的解析解,讨论了BPH+EM系统扩大能量俘获的频率范围和提高能量俘获效率的机理,研究了弹性放大器的刚度质量比对BPH+EM系统的动力性能影响规律。当弹性放大器的刚度质量比趋于无限大时,具有弹性放大器的双稳态压电俘能系统退化为双稳态压电俘能系统BPH。弹性放大器的刚度质量比趋于0但不等于0时,BPH+EM的俘能效率低于BPH。结果表明,在合适的刚度质量比范围内,BPH+EM的俘能效率显著优于BPH。研究结果为BPH+EM系统的优化设计提供了理论指导。     

Response regimes in forced system with non-linear energy sink: quasi-periodic and random forcing

The system under investigation comprises a linear oscillator coupled to a non-linear energy sink (NES) under quasi-periodic forcing in the regime of 1:1:1 resonance. Interaction of the quasi-periodic excitation with the strongly modulated response (SMR) regime is studied in detail both analytically and numerically. Theoretical study developed in the paper allows establishing the threshold value for the amplitude of modulation beyond which SMR regime is excited. This phenomenon is of great practical use since applying the quasi-periodic excitation beyond the threshold results in elimination of possible undesired regimes causing high-amplitude oscillations of the main structure. Bifurcations of the SMR caused by quasi-periodic excitation were analyzed with the help of semi-analytical procedure based on two-dimensional maps. Numerical evidences for exciting the strongly modulated bursts in the response by a random, quasi-periodic narrow-band excitation are also provided. Fairly good correspondence was observed between analytical model and numerical simulations.     

Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: Optimization of a nonlinear vibration absorber

This paper is the second one in the series of two papers devoted to detailed investigation of the response regimes of a linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. In this paper, we study the performance of a strongly nonlinear, damped vibration absorber with relatively small mass attached to a periodically excited linear oscillator. We present a nonlinear absorber tuning procedure in the vicinity of (1:1) resonance which provides the best total system energy suppression, using analytical and numerical tools. A linear absorber is also tuned according to the same criterion of total system energy suppression as the nonlinear one. Both optimally tuned absorbers are compared under common parameters of damping, external forcing but different absorber stiffness characteristics; certain cases for which nonlinear absorber is preferable over the linear one are revealed and confirmed numerically.     

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.     

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.     

Modeling tangential friction based on contact pressure distribution for predicting dynamic responses of bolted joint structures

Nonlinear Dynamics - The traditional nonlinear energy sink (NES), i.e., a smooth and cubic NES, can cause stable higher branch of response of primary systems with increasing excitation forcing. For...     

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.