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。     

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.     

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.     

Dynamic design of a nonlinear energy sink with NiTiNOL-steel wire ropes based on nonlinear output frequency response functions

A novel vibration isolation device called the nonlinear energy sink(NES)with Ni Ti NOL-steel wire ropes(Ni Ti-ST) is applied to a whole-spacecraft system. The Ni Ti-ST is used to describe the damping of the NES, which is coupled with the modified Bouc-Wen model of hysteresis. The NES with Ni Ti-ST vibration reduction principle uses the irreversibility of targeted energy transfer(TET) to concentrate the energy locally on the nonlinear oscillator, and then dissipates it through damping in the NES with Ni Ti-ST.The generalized vibration transmissibility, obtained by the root mean square treatment of the harmonic response of the nonlinear output frequency response functions(NOFRFs),is first used as the evaluation index to analyze the whole-spacecraft system in the future.An optimization analysis of the impact of system responses is performed using different parameters of NES with Ni Ti-ST based on the transmissibility of NOFRFs. Finally, the effects of vibration suppression by varying the parameters of Ni Ti-ST are analyzed from the perspective of energy absorption. The results indicate that NES with Ni Ti-ST can reduce excessive vibration of the whole-spacecraft system, without changing its natural frequency. Moreover, the NES with Ni Ti-ST can be directly used in practical engineering applications.     

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     

一类二自由度非线性能量阱系统的减振性能分析

研究了包含一类新型二自由度非线性能量阱的机械振动系统.建立非线性系统的运动微分方程,用泰勒级数展开近似表达二级振子所受到的非线性力.运用数值方法对该非线性能量阱的减振性能进行分析.从其结构参数以及初始能量的角度出发,研究其发生靶向能量传递的条件.结果表明:该非线性能量阱可以有效降低主结构的能量,其振动抑制效果明显优于单...     

Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator

We present a theoretical study of the dynamics of the coupled system of Jiang, McFarland, Bergman, and Vakakis. It comprises a harmonically excited linear subsystem weakly coupled to an essentially nonlinear oscillator. We explored the rich dynamics exhibited by this coupled system by determining its periodic responses and their bifurcations. Not surprisingly, we found a lot of interesting dynamics over a broad frequency range: cyclic-fold, Hopf, symmetry-breaking, and period-doubling bifurcations; phase-locked motions; regions with multiple coexisting solutions; hysteresis; and chaos. We did not find any occurrence of energy transfer via modulation (also known as zero-to-one internal resonance); theoretically, the possibility of its occurrence was ruled out for systems with weak nonlinearity and damping. Finally, we investigated the ef fectiveness of the so-called nonlinear energy sink (NES) in vibration attenuation of forced linear structures. We found that the NES results in an increase in the vibration amplitude of the linear subsystem, especially when the damping is low, contrary to the claim made by Jiang et al. Also, we did not find any indication of nonlinear energy pumping or localization of energy in the NES, away from the directly forced linear subsystem, indicating that the NES is not ef fective for controlling the vibrations of forced linear structures.     

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.     

一类分段光滑隔振系统的非线性动力学设计方法

分段光滑隔振系统是一类具备分段刚度或阻尼的非线性动力学系统,在振动控制领域中具有广泛代表性,诸如限位隔振系统、分级汽车悬挂等. 分段光滑的刚度或阻尼特性能够实现隔振系统的特定动力学性能及提升隔振性能,如抑制共振响应、提升共振区隔振性能等,但是亦会给隔振系统的动力学行为带来诸多不利影响. 以分段双线性分段光滑隔振系统为理论模型,系统研究了摒除不利于隔振的非线性动力学现象设计方法,包括幅值跳跃、周期运动的倍周期分岔等. 首先,利用平均法与奇异性理论给出了主共振频响曲线拓扑特征的完整拼图. 研究结果表明,参数空间分为4 个区域,其中2 个区域存在幅值跳跃,而其产生跳跃原因分别由鞍结分岔与擦边分岔所导致;基于此提出避免主共振跳跃的设计方法. 其次,建立了隔振有效区内周期运动的庞加莱映射,通过特征值分析给出了避免倍周期分岔发生的条件,证实增大阻尼可以抑制倍周期分岔的发生. 最后通过数值仿真分析了噪声对多稳态运动的影响. 研究结果发现在噪声影响下,分段光滑隔振系统的响应会在不同稳态间跃迁,非常不利于隔振. 因此,在完成跳跃与倍周期分岔的防治设计后,应采用数值仿真校验系统是否存在多稳态运动.      

NES cell

A broadband adaptive vibration control strategy with high reliability and flexible versatility is proposed. The high vibration damping performance of nonlinear energy sink (NES) has attracted attention. However, targeted energy transfer may cause severe vibration of NES. Besides, it is difficult to realize pure nonlinear stiffness without the linear part. As a result, the reliability of NES is not high. The low reliability of NES has hindered its application in engineering. In addition, the performance of NES depends on its mass ratio of the primary system, and NES lacks versatility for different vibration systems. Therefore, this paper proposes the concept of NES cell. The advantages of the adaptive vibration control of NES are applied to cellular NES. By applying a large number of NES cells in parallel, the reliability of NES and its versatility to complex vibration structures are improved. An elastic beam is used as the primary vibration structure, and a limited NES is used as the cell. The relationship between the vibration suppression effect of NES cells and the number of NES cell is studied. In addition, the effect of the distribution of NES cells on the multi-mode resonance suppression of the beam is also studied. In summary, the mode of the primary structure can be efficiently controlled by a large number of lightweight NES cell. Therefore, the lightweight NES cell is flexible for vibration control of complex structures. In addition, it improves the reliability of NES applications. Therefore, the distributed application of NES cells proposed in this paper is a valuable vibration suppression strategy.     

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.     

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.     

Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink

We explore the conservative and dissipative dynamics of a two-degree-of-freedom (2-DoF) system consisting of a linear oscillator and a lightweight nonlinear rotator inertially coupled to it. When the total energy of the system is large enough, the motion of the rotator is, generically, chaotic. Moreover, we show that if the damping of the rotator is sufficiently small and the damping of the linear oscillator is even smaller, then the system passes through a cascade of resonance captures (transient internal resonances) as the total energy gradually decreases. Rather unexpectedly, all these captures have the same principal frequency but correspond to different nonlinear normal modes (NNMs). In each NNM, the rotator is phase-locked into periodic motion with two frequencies. The NNMs differ by the ratio of these frequencies, which is approximately an integer for each NNM. Essentially non-integer ratios lead to incommensurate periods of ??slow?? and ??fast?? motions of the rotator and, thus, to its chaotic behavior between successive resonance captures. Furthermore, we show that these cascades of resonance captures lead to targeted energy transfer (TET) from the linear oscillator to the rotator, with the latter serving, in essence, as a nonlinear energy sink (NES). Since the inertially-coupled NES that we consider has no linearized natural frequency, it is capable of engaging in resonance with the linear oscillator over broad frequency and energy ranges. The results presented herein indicate that the proposed rotational NES appears to be a promising design for broadband shock mitigation and vibration energy harvesting.     

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.     

Dynamic analysis of 1-dof and 2-dof nonlinear energy sink with geometrically nonlinear damping and combined stiffness

Nonlinear Dynamics - Nonlinear energy sink (NES) refers to a typical passive vibration device connected to linear or weakly nonlinear structures for vibration absorption and mitigation. This study...     

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.     

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.     

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.     

Bifurcation Control of Parametrically Excited Duffing System by a Combined Linear-Plus-Nonlinear Feedback Control

For a parametrically excited Duffing system we propose a bifurcation control method in order to stabilize the trivial steady state in the frequency response and in order to eliminate jump in the force response, by employing a combined linear-plus-nonlinear feedback control. Because the bifurcation of the system is characterized by its modulation equations, we first determine the order of the feedback gain so that the feedback modifies the modulation equations. By theoretically analyzing the modified modulation equations, we show that the unstable region of the trivial steady state can be shifted and the nonlinear character can be changed, by means of the bifurcation control with the above feedback. The shift of the unstable region permits the stabilization of the trivial steady state in the frequency response, and the suppression of the discontinuous bifurcation due to the change of the nonlinear character allows the elimination of the jump in the quasistationary force response. Furthermore, by performing numerical simulations, and by comparing the responses of the uncontrolled system and the controlled one, we clarify that the proposed bifurcation control is available for the stabilization of the trivial steady state in the frequency response and for the reduction of the jump in the nonstationary force response.     

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally, but the lack of studies of real environmental conditions on these absorbers is felt. The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES). In this paper, the Burgers model is assumed for the viscoelasticity in an NES, and a linear oscillator system is considered for inve...