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.     

Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.     

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.     

General perturbational solution of a harmonically forced non-linear oscillator equation

An extension of a general perturbational method in the theory of harmonically forced non-linear oscillations has been presented, in which the interplay of two system parameters appears. The method clearly exhibits the entrainment of subharmonic, super-harmonic and other harmonic responses for various interplays of the system parameters.The mathematical insufficiency of the method to predict the behavior of the system in a limiting case of parameter interplay, which is usually attributed to the perturbational method in the Poincaré sense, has been recognized and the method for its removal suggested.     

Forced vibration control of an axially moving beam with an attached nonlinear energy sink

This paper investigates a highly efficient and promising control method for forced vibration control of an axially moving beam with an attached nonlinear energy sink(NES).Because of the axial velocity,external force and external excitation frequency,the beam undergoes a high-amplitude vibration.The Galerkin method is applied to discretize the dynamic equations of the beam–NES system.The steady-state responses of the beams with an attached NES and with nothing attached are acquired by numerical simulation.Furthermore,the fast Fourier transform(FFT)is applied to get the amplitude–frequency responses.From the perspective of frequency domain analysis,it is explained that the NES has little effect on the natural frequency of the beam.Results confirm that NES has a great potential to control the excessive vibration.     

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.     

Chaos control of a nonlinear oscillator with shape memory alloy using an optimal linear control: Part I: Ideal energy source

In this paper, an optimal linear control is applied to control a chaotic oscillator with shape memory alloy (SMA). Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function, which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. This work is presented in two parts. Part I considers the so-called ideal problem. In the ideal problem, the excitation source is assumed to be an ideal harmonic excitation.     

A high-efficient nonlinear energy sink with a one-way energy converter

Nonlinear Dynamics - The nonlinear energy sink (NES) has been presented in the past two decades. Although it has very broad applications, some inherent limitation of the traditional NES is...     

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.     

Effect of 1:3 resonance on the steady-state dynamics of a forced strongly nonlinear oscillator with a linear light attachment

We study the 1:3 resonant dynamics of a two degree-of-freedom (DOF) dissipative forced strongly nonlinear system by first examining the periodic steady-state solutions of the underlying Hamiltonian system and then the forced and damped configuration. Specifically, we analyze the steady periodic responses of the two DOF system consisting of a grounded strongly nonlinear oscillator with harmonic excitation coupled to a light linear attachment under condition of 1:3 resonance. This system is particularly interesting since it possesses two basic linearized eigenfrequencies in the ratio 3:1, which, under condition of resonance, causes the localization of the fundamental and third-harmonic components of the responses of the grounded nonlinear oscillator and the light linear attachment, respectively. We examine in detail the topological structure of the periodic responses in the frequency–energy domain by computing forced frequency–energy plots (FEPs) in order to deduce the effects of the 1:3 resonance. We perform complexification/averaging analysis and develop analytical approximations for strongly nonlinear steady-state responses, which agree well with direct numerical simulations. In addition, we investigate the effect of the forcing on the 1:3 resonance phenomena and conclude our study with the stability analysis of the steady-state solutions around 1:3 internal resonance, and a discussion of the practical applications of our findings in the area of nonlinear targeted energy transfer.     

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%.     

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.     

Analytic evaluation of periodic responses of a forced nonlinear oscillator

The periodic responses of a strongly nonlinear, single-degree-of-freedom forced oscillator with weak excitation and damping are examined. The presented methodology is based on a regular perturbation expansion, whose first term is the solution of the unforced, and undamped nonlinear problem. Higher order approximations are computed by explicitly solving linear differential equations possessing a periodically varying coefficient. The general theory is used for studying the periodic steady state motions of the periodically forced system. Moreover, it is shown that the presented analysis can be used to analytically study the orbital stability of the identified steady state motions. The proposed method can also be used for studying periodic responses due to nonperiodic transient forces, provided that these responses are close to the O(1) periodic generating solution.     

Dynamic analysis of the nonlinear energy sink with local and global potentials: geometrically nonlinear damping

Nonlinear Dynamics - In this paper, the considered two-DOF system consists of a linear oscillator (LO) under external harmonic excitation and an attached lightweight nonlinear energy sink (NES)...     

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...     

Vertical vibration control using nonlinear energy sink with inertial amplifier

To reduce additional mass, this work proposes a nonlinear energy sink(NES)with an inertial amplifier(NES-IA) to control the vertical vibration of the objects under harmonic and shock excitations. Moreover, this paper constructs pure nonlinear stiffness without neglecting the gravity effect of the oscillator. Both analytical and numerical methods are used to evaluate the performance of the NES-IA. The research findings indicate that even if the actual mass is 1% of the main oscillator, the NES-IA...