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 the response of the system.The path integral method based on the Gauss-Legendre polynomial is used to achieve the numerical solutions.The performance of vibration reduction is evaluated by the displacement and velocity transition probability densities,the transmissibility transition probability density,and the percentage of the energy absorption transition probability density of the linear oscillator.The sensitivity of the parameters is analyzed for varying the nonlinear stiffness coefficient and the damper ratio.The investigation illustrates that a linear system with NES can also realize great vibration reduction under harmonic and random base excitations and random bifurcation may appear under different parameters,which will affect the stability of the system.     

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.     

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。     

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.     

DYNAMICAL CHARACTERISTICS ANALYSIS FOR UNILATERAL CONSTRAINED SIMPLY SUPPORTED BEAM WITH HARMONIC WAVE EXCITATION1)

为研究间隙碰撞对系统动力学响应的影响,以理想简支梁的振型函数为Rayleigh--Ritz基函数建立了单侧约束简支梁系统的非线性离散动力学方程组,应用数值方法研究了系统在基础谐波激励下的动力学响应特征及其对共振频率线性等效方法适用性的影响。研究表明：非线性动力学系统间隙产生的局部碰撞,使得系统振动能量在系统各阶模态之间转移,使得线性等效方法失效;即使进行非线性分析,也需要考虑系统固有频率远大于激励频段上限的模态。     

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.     

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.     

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.     

Supercritical forced response of coupled motion of a nonlinear transporting beam

Steady-state periodic responses of nonlinear coupled planar motions are investigated for transporting beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates into multiple equilibrium positions in the supercritical regime. The finite-difference schemes are developed to calculate the non-trivial static equilibrium and the steady-state response under simply supported or clamped boundary conditions. The forced vibration is assumed to be spatially uniform and temporally simple harmonic. Based on the long time series, the steady-state transversal amplitudes of nonlinear planar motions are recorded with changing load frequencies. A?resonance exists if the external load frequency approaches the fundamental frequency. The effects of material parameters and vibration amplitude on the resonance responses are investigated. The coupled planar model can be reduced to two nonlinear models on transversal vibrations, an integro-partial?Cdifferential equation and a partial?Cdifferential one. Numerical examples are displayed for the pros and cons between the two transversal models. It is also revealed that the increased axial speed converts the hardening-type behavior into the softening-type one.     

Reducing thermal shock-induced vibration of an axially moving beam via a nonlinear energy sink

Nonlinear Dynamics - Thermal shock-induced vibration suppression of an axially moving beam with a nonlinear energy sink (NES) is investigated. Owing to thermal shock on the beam, the beam is...     

Targeted energy transfer of a parallel nonlinear energy sink

A parallel nonlinear energy sink(NES) is proposed and analyzed. The parallel NES is composed of a vibro-impact(VI) NES and a cubic NES. The dynamical equation is given, and the essential analytical investigation is carried out to deal with the cubic nonlinearity and impact nonlinearity. Multiple time-scale expansion is introduced, and the zeroth order is derived to give a rough outline of the system. The underlying Hamilton dynamic equation is given, and then the optimal stiffness is expressed. The clearance is regarded as a critical factor for the VI. Based on the periodical impact treatment by analytical investigation, the relationships of the cubic stiffness, the clearance, and the zeroth-order attenuation amplitude of the linear primary oscillator(LPO) are obtained.A cubic NES under the optimal condition is compared with the parallel NES. Harmonic signals, harmonic signals with noises, and the excitation generated by a second-order?lter are considered as the potential excitation forces on the system. The targeted energy transfer(TET) in the designed parallel NES is shown to be more e?cient.     

Resonance response of fluid-conveying pipe with asymmetric elastic supports coupled to lever-type nonlinear energy sink

This paper studies the vibration absorber for a fluid-conveying pipe, where the lever-type nonlinear energy sink (LNES) and spring supports are coupled to the asymmetric ends of the system. The pseudo-arc-length method integrated with the harmonic balance method is used to investigate the steady-state responses analytically. Meanwhile, the numerical solution of the fluid-conveying pipe is calculated with the Runge-Kutta method. Moreover, a special response, called the collapsible closed detached response (CCDR), is first observed when the vibration response of mechanical structures is studied. Then, the relationship between the CCDR and the main structure primary response (PR) is obtained. In addition, the closed detached response (CDR) is also observed to research the resonance response of the fluid-conveying pipe. The appearance of either the CCDR or the CDR does affect the resonance attenuation. Furthermore, the mentioned two phenomena underline that the trend of vibration responses under external excitation goes continuous and gradual. Besides, the main advantage of the LNES is presented by contrasting the LNES with the nonlinear energy sink (NES) coupled to the same pipe system. It is found that the LNES can reduce the resonance response amplitude by 91.33%.     

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.     

Numerical study of flow-induced vibrations of cylinders under the action of nonlinear energy sinks (NESs)

An isolated two-dimensional circular cylinder with two linear degrees of freedom, parallel and perpendicular to the free-stream direction, and owning a nonlinear energy sink (NES) is investigated by fluid–structure interaction (FSI) simulations to assess vortex-induced vibrations (VIV) at moderate Reynolds numbers. Subsequently, the wake-induced vibration (WIV) of a pair of identical cylinders under the action of two NES in a tandem arrangement and in a proximity–wake interference regime is explored using the same approach. The NES parameters (mass, nonlinear stiffness and damping) are investigated to determine their effects on the dynamic response of a single degree of freedom (in transverse flow direction) coupled system by a reduced-order model based on an experimentally validated van der Pol oscillator. The CFD model coupled with FSI method is also validated against VIV experimental data for an isolated cylinder in a uniform flow. The study is aimed to investigate the effect of the passive suppression NES device on VIV and WIV. The amplitude response, trajectories of cylinder motion and temporal evolutions of vortex shedding are obtained by conducting a series of numerical simulations. It is found that placing a tuned NES in the cylinders can provide good suppression effect; however, the effectiveness is function of the reduced velocity.     

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自由度非线性能量阱系统。     

具任意脱层复合材料梁的非线性谐波响应

研究了具任意脱层复合材料梁的非线性谐波响应问题。基于弹性理论，建立了考虑剪切变形时的复合材料梁脱层的基本方程式。在空间上采用B样条函数和Galerkin积分法，在时间上采用增量谐波平衡法进行计算。通过实例计算，得出了简谐力作用下的非线性动力响应曲线。认为基谐波振动仍是非线性振动的主要部分。     

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.     

Design criteria of bistable nonlinear energy sink in steady-state dynamics of beams and plates

Nonlinear Dynamics - A bistable nonlinear energy sink (BNES) conceived for the passive vibration control of beam and plate structures under harmonic excitation is investigated. By applying an...     

均质简支梁简化为单自由度系统条件研究

简支梁是土木工程中的常见结构形式，工程计算中常将其简化为单自由度系统进行初步分析和计算．本文系统地讨论了简谐荷载作用下将简支梁简化为单自由度系统进行竖向位移计算的限定条件．对限定条件分析发现将梁上质量的一半集中到梁跨中处所得近似结果精度最好．给出公式可以通过单自由度系统的位移得到梁上任意点处的竖向位移．本文对结构力学教材中的集中质量法进行了进一步的分析和讨论，给出的近似计算公式对土木工程中简支梁在简谐力作用下竖向位移的快速估算具有一定的参考价值．