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Experimental Investigation and Design Optimization of Targeted Energy Transfer Under Periodic Forcing
Abstract
In this paper, the dynamic response of a harmonically forced linear oscillator (LO) strongly coupled to a nonlinear energy sink (NES) is investigated both theoretically and experimentally. The system studied comprises an LO with an embedded, purely cubic NES. The behavior of the system is analyzed in the vicinity of 1 : 1 resonance. The complexification-averaging technique is used to obtain modulation equations and the associated fixed points. These modulation equations are analyzed using asymptotic expansion to study the regimes related to relaxation oscillation of the slow flow, called strongly modulated response (SMR). The zones where SMR occurs are computed using a mapping procedure. The slow invariant manifolds (SIM) are used to derive a proper optimization procedure. It is shown that there is an optimal zone in the forcing amplitude-nonlinear stiffness parameter plane, where SMR occurs without having a high amplitude detached resonance tongue. Two experimental setups are presented. One is not optimized and has a relatively high mass ratio (approximate to 13%) and the other one is optimized and exhibits strong mass asymmetry (mass ratio approximate to 1%). Different frequency response curves and associated zones of SMR are obtained for various forcing amplitudes. The reported experimental results confirm the design procedure and the possible application of NES for vibration mitigation under periodic forcing.
... A nonlinear energy sink is an effective device to reduce mechanical and structural vibration passively [1,2]. Recently, much attention has been paid to suppress forced vibrations of structures subjected to external excitations [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The structures were modeled as single-degree-of-freedom oscillators [3][4][5][6][7][8]11], two-degree-of-freedom linear oscillators [9,10,17,18], linear strings [12,13], linear beams [14,15], and single-degree-of-freedom nonlinear oscillators [16]. ...
... Recently, much attention has been paid to suppress forced vibrations of structures subjected to external excitations [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The structures were modeled as single-degree-of-freedom oscillators [3][4][5][6][7][8]11], two-degree-of-freedom linear oscillators [9,10,17,18], linear strings [12,13], linear beams [14,15], and single-degree-of-freedom nonlinear oscillators [16]. A simplest model of a nonlinear energy sink is an essential nonlinear oscillator consisting of a small mass, a cubic stiffness and a linear damper [3-6, 8,9,11-18]. ...
... Most available investigations focused on periodic steady-state responses [3,5,[7][8][9][11][12][13][14][15][16][17][18]. In addition to experimental works [3,4,11,18] and numerical simulations [3,7,10,14,[16][17][18], approximate analytical methods are a powerful approach to predict the steady-state responses by yielding amplitude-frequency response curves and examining their stabilities [3,[7][8][9][11][12][13][14][15][16]. ...
... Depending on the modulation strength, responses with lower modulation intensity are termed weakly modulated responses (WMR), while those with higher modulation intensity are termed strongly modulated responses (SMR), indicating that nonlinear vibration systems with NES have complex response mechanisms [24][25][26]. Gourc et al. [27] studied the dynamic behavior of coupled NES systems on the 1:1 resonance manifold and obtained modulation equations of the slow invariant manifold (SIM) through analytical methods. Further research by Gendelman et al. [28] found that the vibration reduction effect during SMR is better than during steady periodic responses. ...
... From the previous analysis, it can be seen that when the jump phenomenon occurs in the response of the system, the foldback point is located at the extremum of the SIM, so the system jumps at g (N ) = 0. To avoid singularities, the right ends of Eqs. (27) and (28) are remembered as f 1 (N , θ) and f 2 (N , θ), respectively, and the equations are rescaled and remembered as N = f 1 (N , θ) , θ = f 2 (N , θ) . ...
... To determine the magnitude of the critical external excitation amplitude, Eqs. (27) and (28) are set to zero, and the phase of the pair of fold singularities on the folded line N i is obtained as follows ...
Viscoelastic materials are widely used in vibration isolation and reduction devices due to their simple structure and excellent energy dissipation performance. However, the introduction of viscoelastic Maxwell elements typically adds a half degree of freedom to the system, thereby increasing its complexity. The coupling effects among these complex structures and their impact on system dynamics remain unclear. This paper applies the slow-fast analysis method based on the complexification-averaging method to systems containing a half degree of freedom for the first time to study the complicated behavior and its mechanism. An approximate analytical solution for the two-and-a-half degrees of freedom system is obtained using the complexification-averaging method. By applying the multi-scale method, the slow invariant manifold of the system is derived, and the necessary conditions for a strongly modulated response are obtained. The vibration mechanisms of these responses are explained by using slow-fast analysis method with a combination of the slow invariant manifold, slowly variable response curves, and phase trajectory analysis. The result shows that the fast subsystem jumping back and forth between two branches of the slow invariant manifold is the main cause of strongly modulated response.The evaluation of the energy spectrum reveals that the damping ratio and stiffness ratio of the viscoelastic elements can be adjusted to further enhance the vibration reduction efficiency of the system. Additionally, the control equations for the Fold bifurcation and Hopf bifurcation of the system are derived, and the stability of the system response is analyzed.
... It's easy to solve the condition for latter terms in (27) and obtain the expression of force amplitude F i,c . The δ 2 interval for SMR occurs in VIC NES is the same with the pure cubic condition case, for the reason that all the parameters are the same for (28). During the SMR in VIC NES, the typical SMR motion can also be classified as four stages: (1) The jumping interval is (δ 1,1 , δ 1,2 ), which is the same as cubic case. ...
... Both cases share the same fold point N 2,1 . However, considering the other fold point N 2,2 is inaccessible in the modest clearance case, the extra folding point N 2,e is applied in (28). ...
... In the cubic NES case, the analytical amplitude threshold calculated by (28) that the SMR of the VIC NES ends at a much lower excitation amplitude case. Figure. ...
In this study, we address the response regimes of a novel Nonlinear Energy Sink (NES) that couples both cubic and impact nonlinearities. In a non-smooth condition, the conventional multiple scales method is considered with impact condition. By identifying the occurrence of the collision, the asymptotic analysis of the equivalent cubic NES model and Vibro-Impact (VI) NES model can illustrate the fixed point of the Vibro-Impact Cubic (VIC) NES. Three types of VIC NES are described as a function of clearance length. The role of clearance length on the response regimes is provided, offering solid criteria for optimal design. Combined with the simulation results, our experimental observations prove the restraint effect of impact on the robustness of the Strongly Modulated Response (SMR).
... The SMR was also demonstrated by this method. Gourc et al. [37] used theoretical and experimental methods to study the 1:1 resonance responses of a harmonically forced linear oscillator strongly coupled to a NES. Different frequency response curves and associated zones of SMR are obtained by a mapping procedure. ...
... Case III has been reported in Refs. [30,31,35,37]. Case IV has been reported in Ref. [37]. ...
... [30,31,35,37]. Case IV has been reported in Ref. [37]. Case V has been reported in Refs. ...
Nonlinear energy sink (NES) can be used to reduce the forced vibration of the primary system. Many different kinds of periodic responses of the linear system coupled with NES have been presented in the literature. In order to reveal all kinds of periodic responses of the NES system on given parameter planes, this study intends to achieve the classification of different kinds of periodic responses of linear oscillator coupled with a NES under harmonic excitation, which is modeled as a 2-degree-of-freedom system. The amplitudes of periodic responses of the nonlinear system are obtained by using the harmonic balance method. Then a general method based on the singularity theory is proposed to reveal all kinds of periodic responses under a given set of parameters. The periodic responses include 10 kinds known responses and 4 kinds of newly-revealed responses. The complete classification of the responses ensure the reliability and the efficiency of a NES.
... These stable special orbits are responsible for TET [5]. The efficiency of this mechanism in absorbing vibration has been explored both numerically [6,7] and experimentally [8][9][10][11]. Also, its potential applications in structural seismic control [12] and in mitigating the hypersonic 3-D wing flutter [13] have been studied. ...
... The stability of SMR is transferred into a 1-D mapping problem, where the drop point after 'jumping' is located in a certain interval of phase portraits. The necessary condition of external harmonic force is also investigated experimentally to trigger the SMR mechanism [11]. ...
... The SMR should satisfy some conditions to be trigged. The force threshold interval has been indicated in [11,30]. Since the excitation is at the same frequency as the natural frequency of the primary system and the mass ratio is sufficiently small (ε << 1), the effect of the detuning parameter and the mass ratio is beyond the scope of this work. ...
This work mainly concentrates on the optimization of cubic and bistable NES to find the maximum efficiency point under harmonic excitation. The conservative system is considered to reveal the inner property of the damping system. With the application of the multiple scales method and the complex variables method, the threshold of excitation and different response regimes are distinguished under the assumption of 1:1 resonance. The maximum efficiency point of cubic and bistable NES occurs when SMR disappears. The factors that affect the optimal efficiency limit are explored. The result indicates that the maximum absorption efficiency level is mainly determined by the damping parameters. Compared with the cubic case, the bistable case involves more complex regimes in terms of chaos oscillation. The influence of damping parameters on the chaos threshold is discussed to adopt different energy levels. With the help of analytical predictions, the proper nonlinear stiffness is determined for certain harmonic excitation. This work offers some fundamental insights into the optimal design of cubic and bistable NES.
... Too little energy prevents TET activation, while excessive energy can diminish its effectiveness. To enhance the efficiency of TET, a series of increasingly complex NESs, such as the parallel NES [5], series NES [6], and inerter NES [7] etc., have been proposed, starting from the simplest cubic stiffness NES [8]. Among them, the bistable nonlinear energy sink (BNES) [9] has garnered significant attention. ...
... The equations for |W 1 | 2 are derived separately for both types of motion, as given in Eqs. (8) and (12). By subtracting these two equations, we obtain ...
Under harmonic excitation, a bistable Nonlinear Energy Sink (BNES) manifests diverse attractors, rendering the vibration characteristics highly intricate. This study employs the first-order harmonic balance method to analyze a two-degree-of-freedom system with a BNES, predicting the count of periodic attractors within and across potential wells. Formulas for the Fold bifurcation curves are deduced to anticipate the quantity of cross-well period responses and intra-well period responses under varying frequencies and amplitudes of excitation. The interrelation between the Fold bifurcation curves and the shapes of frequency response curves (FRC) is demonstrated, with a discussion on the shape and quantity of Fold bifurcation curves under diverse system parameters. The connecting points of the inter-well FRCs and the cross-well FRCs are resolved, yielding a Pitchfork bifurcation curve. The influence of system parameters on the shape of the Pitchfork bifurcation curve is scrutinized. Utilizing the acquired bifurcation curves allows for predicting the count of periodic attractors under different harmonic excitations. Due to the impact of various local and global bifurcations, the actual number of stable attractors may deviate from the predicted value, and the system may even lack stable periodic attractors. The stability of periodic attractors is assessed using Lyapunov exponents, unveiling the accurate predictive capability of bifurcation curves within certain parameter ranges. Corresponding to bifurcation curve predictions, under specific excitations, the system may exhibit both cross-well and intra-well periodic attractors, multiple cross-well periodic attractors, or multiple intra-well periodic attractors. While comprehensively predicting all periodic attractors remains challenging, the bifurcation curves serve as a valuable tool for identifying potential multiple steady-state responses during NES optimization. The prediction of cross-well motion contributes to designing a more efficient energy-harvesting NES.
... The NES can generate a oneway, irreversible target energy transfer (TET) from the primary system to the attachment, and the energy is eventually dissipated through the damping [2]. So far, various types of NESs have been proposed and studied, such as cubic stiffness NES [3][4][5][6], vibroimpact NES [7][8][9][10][11], rotary NES [12,13], piecewise stiffness NES [14,15], track NES [16,17], nonlinear damped NES [18], bistable NES (BNES) , multi-DOF NES [46][47][48][49] and other types [50,51]. However, previous studies demonstrated that traditional cubic NES has a significant input energy threshold to activate efficient TET. ...
... The dynamic characteristics of the weakly damped systems under different excitation levels are presented in Figs. 6,7,8,9,10,11 where the damping ratio parameters are set as f 0 =0, f 1 = f 2 = 0.005. Figure 6 depicts the responses, and the corresponding phase portraits and the wavelet spectra at A = 5. Figure 6a shows that the LO oscillates harmonically with an amplitude approximately equal to the excitation. ...
A novel vari-potential energy bistable nonlinear energy sink (VBNES) is proposed in this paper. By introducing a pair of tuned oscillators (TOs) to dynamically adjust the potential barrier height of the BNES, the excitation threshold of the strong modulated response (SMR) is reduced and its vibration suppression ability is enhanced, especially under ultra-low and wide-amplitude excitation. Firstly, the dimensionless theoretical models of the VBNES and the fixed-potential BNES (FBNES) are constructed by the Lagrange equation. The actual response trajectories on the potential energy surface and restoring force surface are numerically tracked to verify the benefit of variable potential energy effect on vibration suppression. The dynamical characteristics of the typical target energy transfer (TET) mechanisms of the VBNES and their contributions to energy dissipation are analyzed. Furthermore, the transient responses and energy dissipation rates of the VBNES and FBNES with optimal stiffness under impact excitation are compared. The results indicate that the VBNES has higher impact vibration absorption efficiency and stronger robustness. The influences of system parameters on energy dissipation rate are analyzed. Finally, the experimental and numerical studies under harmonic excitation are carried out. The experimental results verify the correctness of the theoretical model. The complex dynamics under numerical frequency and amplitude sweeps demonstrate that the VBNES has a lower SMR excitation threshold and broadband vibration suppression ability. This work provides a novel and valuable NES model and numerical evidence for low-frequency and low-amplitude vibration suppression.
... The nonlinearity could be that of the elastic restoring force (Alexander and Schilder 2009) or that of the damping force . To overcome this difficulty, the parameters of the NES must be constrained to a certain range of spatial parameters (Gourc et al. 2014). Although efficient, this procedure limits the ability to maximize the attenuation obtained by the NES. ...
... 3 Parameter optimization of the exact equations 3.1 Optimization using literature rules A first optimization, considering only the linear damping component, was performed using literature rules, such as the one proposed for by Starosvetsky and Gendelman (2008). The chosen parameters, which avoid the problem of detached resonance by the procedure described in Gourc et al. (2014), were set to = 0.01, 0 = 0.1, = 2, = 0.03 and = 28. The FRF covering 1/6 of an octave on either side of the wind turbine resonance is shown in Figure 2. In this figure, ( , ) = rms ( , )/ , where rms ( , ) is the root mean square (RMS) of the steady state FOWT amplitude for a given amplitude and forcing frequency , and rms ( , ) is estimated by averaging ( ) over the last half-hour of motion. ...
Passive vibration mitigation of offshore wind turbines using nonlinear absorbers or nonlinear energy sinks has begun to receive attention in the literature. In most cases, little attention has been paid to the possibility of detached resonances that occur when a nonlinear energy sink is attached to the linear system describing the wind turbine. Sea motions that alter the initial conditions of the floating offshore wind turbine may cause the nonlinear energy sink to operate at one or more detached resonances, completely negating its ability to control turbine vibration. In this paper, we are interested in optimizing the parameters of a nonlinear energy sink with nonlinear stiffness and nonlinear viscous damping for vibration control of a toy model (e.g., a linear mass-spring-damper system) of a floating offshore wind turbine over its entire operating range. The mechanism of detached resonance cancellation is studied analytically under the 1:1 resonance. It is shown that nonlinear energy reduction with properly tuned nonlinear viscous damping allows the complete elimination of undesired regimes and fully restores the absorber's ability to strongly limit the vibration of a floating offshore wind turbine over its entire forcing range. The results obtained over a wide range of parameters suggest that both the optimal nonlinear energy sink parameters (linear and nonlinear stiffness and nonlinear damping) and the damping of the vibration of a floating offshore wind turbine depend on simple power laws of the nonlinear energy sink mass and linear damping.
... It is shown that the multiple solutions occur in a frequency band that is lower than the resonance frequency of the primary structure. The characteristics of the multiple solutions in the two-degreeof-freedom vibration isolation system are similar to those observed in systems with nonlinear energy sinks [36][37][38]. Moreover, it is noted that the secondary oscillator does not introduce a new resonance band in the primary oscillator, even for excitation amplitudes that are extremely large, which is a beneficial characteristic for nonlinear energy sinks. ...
... It is shown that the multiple solutions occur in a frequency band that is lower than the resonance frequency of the primary structure. The characteristics of the multiple solutions in the two-degree-of-freedom vibration isolation system are similar to those observed in systems with nonlinear energy sinks [36][37][38]. Moreover, it is noted that the secondary oscillator does not introduce a new resonance band in the primary oscillator, even for excitation amplitudes that are extremely large, which is a beneficial characteristic for nonlinear energy sinks. ...
The quasi-zero stiffness (QZS) isolator shows excellent characteristics of low-frequency vibration isolation. However, the jump behavior caused by the strong nonlinearity is a primary reason for the failure of QZS isolators. In order to grasp the effective frequency range and failure mechanism of a horizontal QZS isolator comprehensively, the dynamics of the isolator were studied in the following two cases. In the first case, the isolator is subject to a base displacement excitation; in the second case, the isolator is installed on a linear structure that is subject to a harmonic force. The nonlinear algebraic equations describing the steady-state response of the two systems were derived via the complexification-averaging method, and the results obtained using the derived expressions were verified by comparing the results of the complexification-averaging method and the Runge–Kutta method. The effective frequency ranges of the isolator were then obtained, and the jump phenomena in the response amplitude induced by the strong nonlinearity of the isolator were analyzed. The results show that when the excitation amplitude is small, the vibration isolation system does not exhibit jumping behavior and the effective frequency range is relatively wide. With increases in the excitation amplitude, the system can exhibit jumping behavior when an additional impact load is considered, and this phenomenon leads to a narrowing of the effective frequency range. The characteristics of the jump phenomena produced in the two cases were analyzed, and the differences in the jump behaviors were elucidated. Furthermore, the effect of the isolator parameters on the effective frequency range was investigated.
... Experimental and theorical studies have shown that the nonlinearity of the NES allows an irreversible transfer of energy, known as Targered Energy Transfer, from the primary system to the NES [25]. Passive control of resonance using a NES was studied analytically [6,7] and experimentally [8,9]. Investigations on the control of aeroelastic instabilities with a nonlinear absorber was also analyzed [1012]. ...
... In order to study the energy transfer, an asymptotic analysis by the method of multiple scales is carried out because the mass ratio ε with a NES, is very low, ε << 1 [8]. First of all, the time τ is decomposed into several sub-scales of time, increasingly smaller. ...
Vibration absorbers are known for their use in vibration mitigation. In particular, nonlinear vibration absorbers have been of great interest for vibratory level reduction as they do not have to be tuned to the natural frequency of their supporting structure. In order to obtain satisfactory operations of the absorber (energy necessary for its activation and dissipated vibratory level), it is necessary to identify the correct parameters of the absorber which are its nonlinear stiffness and damping. However, when moving from analytics to designing an experimental prototype, it is complicated to have the appropriate parameter values, especially damping, because it is very difficult (or even impossible) to adjust a precise value for a mechanical assembly. As a consequence, this bad adjustment leads to an inefficiency of the absorber and unsatisfactory results. To avoid this lack of robustness, the addition of a multiphysical coupling, to a nonlinear absorber is studied in this paper in order to create an equivalent damping coefficient from another nature : electro-magnetomechanical. This new damping generated is adapted and allows to adjust the equivalent damping coefficient of the absorber to get the best efficiency, analytically and experimentally.
... A NES has usually no inherent natural frequency and is able to absorb energy over a wide range of frequencies. Different types of coupling between the primary structure and the auxiliary mass have been studied theoretically [1,2,13,16,17] and experimentally [6,14,15,21,22] in order to understand the essential changes caused by such a coupling. Many studies (e.g. ...
... where the prime symbol (·) represents the differentiation w.r.t. the dimensionless time τ . The dynamics described by Eq. (7) look similar and result from substituting (6) ...
The aim of the paper is to formulate a complete set of design rules for a vibro-impact nonlinear energy sink (VI NES). Hereto, analytical and numerical methods to extract the backbone curve of vibro-impact systems are presented. The adopted approach exploits the multiple scales method and is applied to a linear oscillator coupled with a VI NES. The dynamics of the system are explored and the system’s response under harmonic forcing in the vicinity of resonance is analyzed and approximated. The presented method allows for the derivation of a closed-form approximation of a nonlinear mode. The relation between the steady state response and the input parameters is established. The resonance frequency of the examined nonlinear system for different excitation levels is estimated and the corresponding backbone curve is identified. The theoretical predictions are confirmed through a comparison with the standard resonant decay method combined with a phase-locked loop. Finally, the closed-form approximations are used to derive a complete set of design rules for a vibro-impact nonlinear energy sink.
... They utilized splitting harmonics and iterative techniques to explore dynamic behaviors of the three-DOF system and to examine the effects of the system parameters. For a two-DOF system of a harmonically excited linear oscillator coupled strongly with a NES, Gourc et al. [203] found theoretically and experimentally an optimal area in the forcing amplitude-nonlinear stiffness plane, where strongly modulated responses occur with a small amplitude resonance threshold. The complexificationaveraging technique was used to determine the existence zone of stable strongly modulated response. ...
... The complexificationaveraging technique was used to determine the existence zone of stable strongly modulated response. The NES was optimized to achieve as small mass ratio as 1% [203]. Lin and Oguamanam [204] investigated the targeted energy transfer efficiency or the energy transfer rate between a single-DOF primary system and a NES. ...
Nonlinear energy sink (NES) is an appropriately designed nonlinear oscillator without positive linear stiffness. NES can suppress vibrations over a wide frequency range due to its targeted energy transfer characteristics. Thus, investigations on NES have attracted a lot of attention since a NES was proposed. Designs, analysis, and applications of NESs are still active since different configurations are needed in various practical circumstances. The present work provides a comprehensive review of state-of-the-art researches on NESs. The work begins with a survey of the generation of a NES and its important vibration control characteristics. The work highlights possible complex dynamics resulting in a NES coupled to a structure. The work also summarizes some significant design on the implements of optimal damping effects and the offsets of NES shortcomings. Then, the work details the applications of NESs in all engineering fields. The concluding remarks suggest further promising directions, such as NESs for multidirectional vibration reduction, NESs with nonlinearities beyond the cubic, potential deterioration caused by a NES, low-cost NESs, NESs for extremely low frequency range, and NESs integrated into active vibration controls. There are 383 references in the review, including some publications of the authors.
... Recently, the focus on nonlinear energy sinks [22][23][24][25] and nonlinear tuned vibration absorbers [26][27][28][29] demonstrates the topical interest in detached resonance curves. Particular attention is drawn to the influence of internal resonances on the emergence of isola, leading to a series of publications [30][31][32][33][34][35][36]. ...
... From a maximum, two initial points toward the eigendirections related to the largest negative eigenvalue of the HessianH(z) are provided to the optimization scheme in Eq. (22). Saddle-nodes are treated in both ways: The smallest inclining eigendirections are forwarded to the uphill tracking in Eq. (24) and the largest declining eigendirections to the downhill optimization in Eq. (22). As new minima, maxima, and saddlenodes are determined, the overall procedure stores these points and repetitively applies the subroutines. ...
Two novel methods to determine detached periodic solution branches of low-dimensional and large-scale friction-damped mechanical systems have been developed. The approach for low-dimensional systems is an extension of the global terrain method and consists of three steps: First, the non-smooth elastic Coulomb slider is temporarily modified to pose a continuous energy surface in the frequency domain. Second, the global terrain method is extended to rescale the search space consecutively along multiple eigendirections of the Hessian at the solution points, facilitating the determination of disconnected solutions. Finally, found solutions are used as the starting points for a homotopy strategy to detect the solutions of the original non-smooth problem. The method for large-scale systems based on an invariant manifold approach consists of three steps: First, the system’s dimension is decreased following a nonlinear modal reduction leading to a two-dimensional surrogate problem. Second, various subspaces for direct extrema, direct bifurcation, constant frequency, and constant amplitude solution detection are defined. Finally, a deterministic line search, a stochastic global solution method, and a newly developed deterministic line deflation are applied to the problem. The proposed methods are applied to low-dimensional and large-scale friction-damped mechanical systems simultaneously subjected to external and self-excitation as well as a low-dimensional mechanical system with shape-memory alloy nonlinearity subjected to external excitation. Their ability to find all solutions is discussed and compared with existing solution procedures such as bifurcation tracking, deflation, homotopy, and the global terrain method. Depending on the chosen subspace for the global search, the proposed methods are capable of providing additional detached solution branches (isola).
... One of the main drawbacks of NES when used to control an harmonically excited system is the possible presence of a high amplitude detached resonance curve. Specific design procedures have been developed to ensure a safe operating region of the system, at the cost of limited performance of the NES [10,4]. A promising way to overcome this difficulty is the introduction of nonlinear damping in addition to linear viscous damping. ...
The paper deals with the passive control of resonant systems using nonlinear energy sink (NES). The objective is to highlight the benefits of adding nonlinear geometrical damping in addition to the cubic stiffness nonlinearity. The behaviour of the system is investigated theoretically by using the mixed harmonic balance multiple scales method. Based on the obtained slow flow equations, a design procedure that maximizes the dynamic range of the NES is presented. Singularity theory is used to express conditions for the birth of detached resonance cure independently of the forcing frequency. It is shown that the presence of a detached resonance curve is not necessarily detrimental to the performance of the NES. Moreover, the detached resonance curve can be completely suppressed by adding nonlinear damping. The results of the design procedure are then compared to numerical simulations.
... This has been disclosed by analytical investigations [6,7,9,10] and experimental works [11][12][13][14]. Additionally, there are many studies focusing on steadystate responses of a primary system with a NES under harmonic excitation, which have been investigated analytically [15][16][17][18][19][20] and experimentally [19][20][21][22]. In addition to the common periodic response, the compound system may present other types of steady-state responses, namely a quasi-periodic response [15,20,23] and a strongly modulated response (SMR) [15,16,24], which are usually analyzed along with Saddle-Node bifurcation and Hopf bifurcation. ...
This paper discloses the unique dynamic behavior and vibration mitigation performance of a polynomial nonlinear vibration absorber (PNVA), a single-degree-of-freedom nonlinear vibration absorber with positive linear and cubic stiffnesses and negative quadratic stiffness, attached to a linear primary system subjected to harmonic excitation. The slow flow equations of the compound system are derived analytically using the harmonic balance method, based on which the analyses of bifurcation, stability and frequency response are carried out. The results show that the primary system coupled with a PNVA has different dynamic behavior compared to that coupled with a traditional nonlinear energy sink (NES), such as the existence of at most five periodic solutions to the slow flow equations, the possible existence of two closed regions enclosed by boundaries for Hopf bifurcations, and the possible existence of two folds in a slow invariant manifold curve. Numerical simulations are performed to validate theoretical analyses, explore comprehensive dynamics, and optimize the design of a PNVA. It is shown that the numerical and analytical results of steady-state periodic response present a good agreement with each other. The mass ratio and damping ratio of a PNVA, initial conditions, and external excitations all have effects on the response regimes of the compound system such that it may undergo a desirable low-amplitude periodic/quasi-periodic response, strongly modulated response (SMR) or an undesirable large-amplitude periodic response. Comparison among the vibration absorbers attached to a primary system with stiffness uncertainty shows that the optimal PNVA has better vibration mitigation efficiency and robustness against frequency variations than the optimal tuned mass damper and Cubic NES, and the optimal PNVA has better robustness against variations in external excitation amplitude than the optimal Cubic NES.
... Although very different types of vibration absorbers exist, most of them exploit internal resonances, making them prone to generation of IRCs. The nonlinear energy sink (NES), consisting of a purely nonlinear resonator, if attached to single-or multi-DoF primary systems, presents IRCs [37,38], whose existence was verified also experimentally [39]. An attempt to eliminate this undesired phenomenon demonstrated that IRCs can be avoided if the absorber has a properly-tuned piecewise-quadratic damping characteristic [40]. ...
We analyze isolated resonance curves (IRCs) in a single-degree-of-freedom system with nonlinear damping. The adopted procedure exploits singularity theory in conjunction with the harmonic balance method. The analysis unveils a geometrical connection between the topology of the damping force and IRCs. Specifically, we demonstrate that extremas and zeros of the damping force correspond to the appearance and merging of IRCs.
... This highlights that no general conclusion can be drawn regarding the influence of quasiperiodic attractors. Detached resonance curves (DRCs), also termed isolas, are generated by the multivaluedness of nonlinear responses and may limit the practical applicability of nonlinear absorbers [24,25]. An important difficulty with DRCs is that they can easily be missed, because they are detached from the main resonance branch [5,26]. ...
The nonlinear tuned vibration absorber (NLTVA) is a recently-developed nonlinear absorber which generalizes Den Hartog's equal peak method to nonlinear systems. If the purposeful introduction of nonlinearity can enhance system performance, it can also give rise to adverse dynamical phenomena, including detached resonance curves and quasiperiodic regimes of motion. Through the combination of numerical continuation of periodic solutions, bifurcation detection and tracking, and global analysis, the present study identifies boundaries in the NLTVA parameter space delimiting safe, unsafe and unacceptable operations. The sensitivity of these boundaries to uncertainty in the NLTVA parameters is also investigated.
... However, VI systems present analytical challenges due to their non-smooth and discontinuous dynamics. Most existing studies on VI-NES utilize the method of multiple scales [18,19], which is applicable only when the mass of the ball is significantly smaller than the mass of the main system [20]. This method helps determine the optimal conditions for TET, such as the cavity length and the number of impacts per period [21]. ...
This study investigates a system comprising a linear oscillator (LO) equipped with a Vibro-Impact Nonlinear Energy Sink (VI-NES) and a coil. The LO's damping and stiffness coefficients are represented by ( C ) and ( k ), respectively, while a ball inside the LO moves within a cavity, colliding with the walls at both ends. The primary focus is on optimizing the cavity length (( L_c )) and the coefficient of restitution (( \kappa )) and coil parameter () using a Genetic Algorithm (GA). The motion equations for the LO incorporating the VI-NES are derived, considering the electromagnetic force (( F_e )) and contact force (( F_c )). The study also explores the energy harvesting circuit, which generates electric power by connecting a load resistance to the coil, transforming mechanical energy into electrical energy. The dynamic behavior of the system is analyzed, highlighting the transition from predictable patterns to chaotic responses. Efficiency metrics are defined to measure the VI-NES's performance in absorbing and dissipating excitation energy. The optimization formulation addresses the influence of initial conditions, proposing a multi-objective approach to handle uncertainties. Validation against existing studies confirms the reliability of the proposed optimization method. The results demonstrate that optimizing both ( L_c ) and ( \kappa ) using GA effectively minimizes the amplitude response of the LO, outperforming approximate methods.
... Congurations featuring more complex NESs consisting of several degrees-of-freedom have also been investigated, [33,34,35]. Finally, the NES can be employed a broadband isolator or enhance existing ring isolators [36,37] A harmonically forced host system with a mechanical NES [38,39] has three characteristic properties. The rst one is that, given sucient energy, the vibration amplitude in the host structure will saturate even as the forcing level increases. ...
The theoretical study and experimental validation of a nonlinear shunt circuit for piezoelectric vibration damping is investigated here. The circuit consists of a resistor, an inductor and a nonlinear cubic voltage source. The shunt acts as an electrical analog to the mechanical nonlinear energy sinks (NESs). These mechanical NESs are passive vibration absorbers that typically have a cubic nonlinear stiness. They have attractive properties such as saturation of the host system's vibration amplitude and strongly modulated response. This increases its operational frequency bandwidth and robustness against variations in the properties of host systems compared to linear vibration absorbers. However, the nonlinear nature may induce isolated responses in the host system that induce high vibration amplitudes. This paper investigates if these attractive properties also occur in the electric nonlinear energy sink shunt. An analytical expression for the frequency response is derived through the complexication-averaging method. Bifurcations in the frequency response reveal the occurrence of a quasi-periodic vibration energy exchange between the host system and the voltage over the electrodes of piezoelectric material. This is the main mechanism behind the amplitude saturation of the host system. Other bifurcations also reveal the existence of isolated responses. The nonlinear shunt is then realized with analog multipliers and a synthetic inductor and its performance in vibration damping is experimentally veried for a cantilever beam.
... Congurations featuring more complex NESs consisting of several degrees-of-freedom have also been investigated, [33,34,35]. Finally, the NES can be employed a broadband isolator or enhance existing ring isolators [36,37] A harmonically forced host system with a mechanical NES [38,39] has three characteristic properties. The rst one is that, given sucient energy, the vibration amplitude in the host structure will saturate even as the forcing level increases. ...
The theoretical study and experimental validation of a nonlinear shunt circuit for piezoelectric vibration damping is investigated here. The circuit consists of a resistor, an inductor and a nonlinear cubic voltage source. The shunt acts as an electrical analog to the mechanical nonlinear energy sinks (NESs). These mechanical NESs are passive vibration absorbers that typically have a cubic nonlinear stiffness. They have attractive properties such as saturation of the host system's vibration amplitude and strongly modulated response. This increases its operational frequency bandwidth and robustness against variations in the properties of host systems compared to linear vibration absorbers. However, the nonlinear nature may induce isolated responses in the host system that induce high vibration amplitudes. This paper investigates if these attractive properties also occur in the electric nonlinear energy sink shunt. An analytical expression for the frequency response is derived through the complexification-averaging method. Bifurcations in the frequency response reveal the occurrence of a quasi-periodic vibration energy exchange between the host system and the voltage over the electrodes of piezoelectric material. This is the main mechanism behind an amplitude saturation of the host system. Other bifurcations also reveal the existence of isolated responses. The nonlinear shunt is then realized with analog multipliers and a synthetic inductor and its performance in vibration damping is experimentally verified for a cantilever beam.
... As seen in previous NES studies, the frequency response of the primary system is optimal for vibration mitigation at a critical excitation amplitude before the appearance of large amplitude detached resonant toughs [36]. In this study, parameters were tuned to maximize at = 0.01 while simultaneously discarding parameter combinations that result in detached resonances. ...
Efficient passive vibration absorption can prevent the failure of systems without requiring sensors or energy sources. Nonlinear energy sinks (NES) have gained popularity as passive vibration absorbers due to their targeted energy transfer (TET) mechanisms that extract and dissipate vibrational energy over broad frequency ranges. In this work, the vibration suppression performance of a novel bistable rotary nonlinear energy sink (BRNES) is studied numerically in the cases of impulse and harmonic excitation. The BRNES consists of a secondary mass that connects to the primary system by a rigid arm and spring that pivot at each connection point. The spring produces an irrational nonlinear restoring force that introduces bistability and favorable oscillatory TET mechanisms. The BRNES outperforms the traditional rotary NES and, in some cases, even the bistable NES. Moreover, unlike most NESs restricted to rectilinear motion, the BRNES is efficient at multiple orientations, thus demonstrating its potential to passively suppress vibrations in any in-plane direction over broad excitation magnitude and frequency ranges.
... To achieve optimal damping efficiency with the NES, both the parameters of the NES and the excitation should be selected within a specific range [39,60]. Gourc et al. [61] provided the range of existence of the SMR in this system. It was realized that NES will have more stable vibration reduction ability when it is nonlinearly coupled with the primary system without linear components [29,62]. ...
Nonlinear energy sinks (NES) have numerous advantages, such as wide vibration bandwidth and excellent vibration reduction performance. However, under high excitation intensity, its high vibration attenuation effect often becomes ineffective. Therefore, exploring methods to address this issue and broaden their application range remains a subject for further research. This paper investigates the dynamic characteristics of systems composed of linear oscillators and multiple NES cells and studies the vibration reduction effect of NES cells using the Complexifiction-Averaging (CxA) method, and the obtained results were numerically verified using the Runge-Kutta (R-K) method. The results show that when NES cells are present in the form of cells, increasing the number of cells can reduce the system's saddle-node (SN) bifurcation region, especially shrinking the frequency island region produced by the system under strong excitation. When the number of cells reaches a certain value, the frequency island of the system disappears. Additionally, regardless of whether the system generates frequency islands or not, increasing the number of cells generally improves the vibration reduction efficiency of NES cells. Thus, the cellular strategy proposed in this paper effectively addresses the ineffectiveness of traditional NES under strong excitation and expands its application range.
... The literature shows that nonlinear attachments can interact with a linear host structure to yield enhanced and broadband reductions around its resonances [39][40][41]. Since the nonlinear oscillator's resonating frequency depends on both its nonlinear coefficient and the excitation level [42,43], its nonlinear frequency response [44][45][46][47][48][49][50][51][52], or Unlike the currently well-established body of research on linear metamaterials, a few examples of nonlinear metamaterials that take advantage of the rich variety of dynamic behavior of nonlinear attachments are found in the literature. Most of the research in nonlinear metamaterials and metastructures has been devoted to the study of their transmissibility properties based on wave propagation approaches. ...
... To achieve optimal damping efficiency with the NES, both the parameters of the NES and the excitation should be selected within a specific range [39,60]. Gourc et al. [61] provided the range of existence of the SMR in this system. It was realized that NES will have more stable vibration reduction ability when it is nonlinearly coupled with the primary system without linear components [29,62]. ...
Although the nonlinear energy sink (NES) has the characteristic of wideband vibration reduction, there is no unified reference standard for the excitation frequency range and amplitude range where NES can achieve damping effect. In this paper, the response characteristics of the dynamic system composed of a NES and a linear oscillator under harmonic excitation are analyzed by using the method of Complexification-Averaging (CxA), multiple scales, and Runge–Kutta (R–K) method. In addition, the conditions that need to be satisfied for the excitation frequency and amplitude to achieve damping effect of NES are obtained. This study presents that the necessary condition for NES to achieve vibration reduction effect is that the excitation frequency and amplitude reach a certain threshold. The minimum frequency at which NES can have a damping effect is provided. Moreover, the effective damping zone of NES in the parameter space of excitation frequency-excitation amplitude is given. In particular, this paper identifies effective damping zones where thresholds do not exist, invalid zones where NES does not work, and deterioration zones where NES worsens the vibration of the linear oscillator. Therefore, the in-depth study of the effective damping zone of NES in this work is helpful to understand the vibration suppression characteristics of NES.
Graphical abstract
... The Nonlinear Energy Sink (NES) [1], a passively nonlinear vibration absorber, has become an active research field in recent decades. The application of NES for the suppression of unwanted vibrations is an important issue in the modern manufacturing industry and the civil fields such as mechanical engineering [2][3][4][5][6], vehicle suspensions [7,8], acoustical engineering [9,10], and aero-structures [11]. Compared with a traditional linear absorber, called the tuned mass damper (TMD) [12], which acts on the natural frequency of structure requiring vibration reduction, NES can passively absorb vibratory energy over a wide range of frequencies. ...
This paper focuses on the investigation of the dynamics of novel 2-DOF coupled oscillators. The system consists of a linear oscillator (main structure) and an attached lightweight nonlinear oscillator, called a nonlinear energy sink (NES), under harmonic forcing in the regime of 1:1:1 resonance. The studied NES has geometrically nonlinear stiffness and damping. Due to the degeneracies that the NES brings to the system, diverse bifurcation structures and rich dynamical phenomena such as nonlinear beating and strongly modulated response occur. The latter two phenomena represent different patterns of energy transfer. To capture the bifurcation structure, the slow flow of the system can be acquired with the use of the complex-averaging method. Furthermore, by applying the bifurcation analysis technique, we get curve boundaries of several bifurcation points in the parameter space. These boundaries will induce different types of folding structures, which can lead to complicated patterns of strongly modulated responses, in which intense energy transfer from the main structure to NES occurs. To study the necessary parameter conditions of strongly modulated responses, we analyzed the dynamics of different time scales of the slow flow in detail and determined the corresponding parameter ranges finally. It is worth noting that the small parameter ε may have a qualitative impact on the dynamics of the system.
... When the nonlinear stiffness reaches a certain threshold, a new FRC called detached resonance curve (DRC), which is generally higher than the main FRC, appears. Therefore, another strategy to determine the optimal parameters is to keep the nonlinear stiffness as close to the threshold as possible while ensuring the existence of the quasiperiodic response [12,13]. Transmissibility, an indicator commonly used to analyze the vibration under random excitations, is also introduced to analyze the quasiperiodic response [14]. ...
Nonlinear energy sinks (NESs) have become a research hotspot due to their frequency robustness, but the optimization problem under harmonic excitation has not been adequately addressed. In this paper, the response of a single-degree-of-freedom system with a cubic stiffness NES attached under harmonic excitation is simplified to 1:1 internal resonance through the harmonic balance method, and H∞ optimization is carried out on this basis. By comparing the frequency response curves (FRCs) of the system and the response of the system under the chirp excitation, the reliability of this approximation method is verified. Through the analysis of the fold bifurcations, the variation trend of the FRCs changing with the external excitation and the conditions for detached resonance curves to appear are obtained. The performance in three special cases is analyzed analytically, while the optimal parameters in general cases are analyzed numerically and fitted to obtain empirical formulas. The results show that the NES will fail applied on the undamped system due to a fixed point tending to infinity. When the external excitation is too large, the NES will amplify vibration of the linear system, and the damping of the NES has a very important influence on the vibration control if the damping of the system is relatively high. Finally, the performance of the NES is compared with the traditional linear vibration absorber (LVA). Although the best performance of the NES is not as good as the LVA, it can achieve multi-mode control under certain conditions.
... Among those, time marching [5], the method of normal forms [6], shooting techniques [7], harmonic balance method [8], and the method of multiple scales [8], are the most commonly used methods that have provided valuable analysis and insights when applied to welldefined systems [9][10][11][12]. In particular, the method of multiple scales (MMS) [8,[13][14][15] has been found to be a useful technique in the field of nonlinear stability analysis of dynamical systems. The MMS provides an analytical expression for the post-bifurcation amplitudes as a function of the nonlinearity and asymptotic analysis of the nonlinear problem [16]. ...
Nonlinear stability analysis plays a key role in the design and evaluation of dynamical systems. Model-based analysis methods require extensive calibration and computational resources. While data-driven methods might offer a solution to this challenge, they are limited to available data and fail to generalize. Incorporating physical information about the system into data-driven methods can extend the applicability of these methods and improve their accuracy. In this paper, we present a physics-informed forecasting approach to predict bifurcation diagrams in nonlinear systems prone to instabilities using measurements of system dynamics before instabilities occur. The proposed method is a hybrid approach that combines an asymptotic analysis provided by the method of multiple scales and a data-driven forecasting technique. In particular, the approach uses information about the nonlinearities exhibited in the system to obtain the normal form using the method of multiple scales. The coefficients of this generic pre-identified form are then approximated using data from time series in the pre-bifurcation regime. We demonstrate the applicability of the proposed method in identifying post-bifurcation dynamics of nonlinear systems prone to instabilities through the application of the approach to several classes of nonlinear systems.
... Generally, the TET process was completed with a very small mass ratio [10]. In recent years, the complex dynamic characteristics of the NES have been widely investigated through theoretical and experimental methods [11][12][13][14][15]. For various excitation environments, such as transient [16], steady-state [17], and random forms [18], the NES presents a high damping performance. ...
Based on theoretical and experimental investigations, this paper proposes a nonlinear energy sink (NES) with piecewise linear springs to enhance vibration suppression effects. A cubic nonlinear oscillator is coupled with a piecewise linear spring to form an enhanced NES (E-NES). Without adding a new resonance region and changing the resonance frequency of the primary system, the enhanced NES can achieve better vibration suppression effects. Based on the free vibration and the forced vibration, the influence of piecewise linear stiffness on the vibration suppression is profoundly examined. The results show that the E-NES has a better suppression effect on the vibration of the primary structure than that of the cubic NES in most cases. Furthermore, the piecewise stiffnesses and gap displacements are optimized through the differential evolution algorithm. The best damping effect of the E-NES on the free vibration is presented. However, it also happens that the vibration suppression effect of the E-NES is weaker than that of the conventional NES. Moreover, the experimental results verify the analytical and numerical results. The vibration elimination efficiency of the E-NES exceeds 90%. Therefore, this research on using piecewise springs to enhance the vibration suppression efficiency will attract attention.
... (a) primary system displacement; (b) primary system velocity 3.2. A periodically forced linear oscillator with impact attachment has been studied in [18,27,28]. ...
The nonlinear energy sink (NES) is defined as a single-degree-of-freedom structural element with relatively small mass and weak dissipation, attached to a primary structure via essentially nonlinear coupling. It is a passive energy dissipation device designed to rapidly absorb vibration energy (due to shock, blast, earthquakes, etc.) from a primary structure and locally dissipate it. The article contains a mini-review of the works on NESs. Design schemes for single-sided and double-sided vibro-impact NESs (SSVI and DSVI NESs) are proposed on the basis of conceptual and design NES schemes that exist in the world scientific literature. The motion equations and the impact rule are given. The quasistatic Hertz contact law is adopted as the impact rule. Various representations of the impulsive loading on the primary structure are discussed. These are excitations by initial velocities only, periodic excitation, a shock in the half-sine form, single-sided periodic impulses of a rectangular shape,wind, seismic and broadband excitation. The Tables of some numerical parameters that can be accepted for VI NES are given. Using the presented data, the authors intend to investigate both the efficiency of SSVI and DSVI NESs under different types of impulsive load, and their dynamical behavior with the changing in their parameters.
... Gourc et al. coupled NES into LO oscillator, analyzed the dynamic behavior of the system from experiment and theory under periodic forcing, studied the strongly modulated response (SMR) of the system, and determined the range of NES parameters and excitation amplitude when high amplitude detached resonant tongue does not occur. The experimental results are basically consistent with the theoretical prediction [6]. Without doubt, NES is promising, and it has seen much practical application, being applied in many engineering fields [7]. ...
As a simplified model of structures of many kinds, the Euler Bernoulli beam has proved useful for studying vibration suppression. In order to meet engineering design requirements, inertial nonlinear energy sinks (I-NESs) can be installed on the boundaries of an elastic beam to suppress its vibration. The geometric nonlinearity of the elastic beam is here considered. Based on Hamilton's principle, the dynamic governing equations of an elastic beam are established. The steady-state response of nonlinear vibration is obtained by the harmonic balance method and verified by numerical calculation. It is found that the geometric nonlinearity of the beam principally affects the first-order main resonance and reduces the response amplitude. An uncoupled system and the coupled I-NES system both show strong nonlinear hardening characteristics. I-NES achieves good vibration suppression. Finally, the optimal range of parameters for different damping is discussed. The results show that the vibration reduction effect of an optimized inertial nonlinear energy sink can reach 90%.
... However, a significant challenge is the practical realization of the sought nonlinearity. Mechanical nonlinearities such as cables [39] and springs [40] have some inherent limitations. As such, the practical relevance of these designs is questionable for real-life applications and the lack of tuning flexibility is also seen as a potential problem. ...
Acoustic Black Hole (ABH) phenomenon features unique wave retarding and energy focusing of flexural waves inside thin-walled structures whose thickness follows a power-law variation. Existing studies, mostly focusing on linear aspects, show the deficiency of the linear ABH structures in coping with low-frequency problems, typically below the so-called cut-on frequency. In this paper, electrical nonlinearities are intentionally imposed via PZT patches over an ABH beam to tactically influence its dynamics through electromechanical coupling. Using a fully coupled electromechanical beam model, typical electromechanical coupling phenomena between the beam and the external nonlinear circuits, as well as the resultant salient nonlinear features of the system, are numerically investigated. Results show the beneficial effects arising from the intentional electrical nonlinearity in terms of generating energy transfer from low to high frequencies inside the beam, before being dissipated by the ABH covered by a small amount of damping materials. As such, the effective frequency range of the ABH is broadened, conducive to low-frequency vibration control problems. Meanwhile, different from existing mechanical means, the introduced intentional electrical nonlinearity allows for flexible tuning to accommodate specific frequency ranges arising from different applications.
... Pham et al. (2012) studied the energy transfer from linear system to NES under two different harmonic excitations, and proved that control of the two one-to-one resonances of the system is possible simultaneously by endowing the idea of the relative mode, splitting harmonics, and iterative techniques. Gourc et al. (2014b) analyzed the dynamic response of a harmonically forced linear oscillator (LO) strongly coupled to NES both theoretically and experimentally, and studied the behavior of the system in the vicinity of 1:1 resonance. Samani et al. (2012) made a comparative study of a series of NES, such as NES with cubic, higher odd-order monomials and piecewise linear stiffness. ...
With the increase of excitation, the nonlinear energy sink (NES) will cause the controlled system to produce high branch response and lead to sudden failure. Increasing the mass of the NES can prevent the generation of high branch within a certain range, but high-precision instruments such as aerospace have strict requirements for additional mass. A parallel NES (PNES) is proposed to improve the robustness without increasing the mass. The slow flow equations of the system are derived by using complexification-averaging (CX-A) method, and the vibration suppression performance of the PNES is analyzed from the frequency domain. Compared with purely cubic stiffness NES (CNES), it is found that PNES has better performance near the main resonance. And it is not easy to produce high branch response under the same excitation intensity. Finally, the performance of PNES and CNES is compared by numerical method from time domain and energy spectrum. The results show that when PNES is attached, the attenuation time of the controlled system is shorter under impact excitation, much more, the controlled system has a smaller energy amplitude near the main resonance under harmonic excitation.
... It also has a better robustness [10,11]. The most obvious characteristic of the NES is that the vibration energy of the primary structure can be irreversibly transferred to the NES [12][13][14][15], namely targeted energy transfer. Furthermore, the NES has a good damping effect for various excitation forms, such as transient [16,17], steady-state [18,19] and random forms [20,21]. ...
For the application and research of the nonlinear energy sink (NES), the vibration of the NES is generally not restricted. However, the vibration amplitude of the NES may be greatly excited, which is unacceptable in engineering. A limited NES consisting of a conventional NES and a piecewise spring is investigated to restrict the vibration of the NES. The effects of the piecewise stiffness and the gap on the dynamic response of the primary system and the NES are considered in forced vibration. The piecewise function is fitted into a continuous function to obtain the nonlinear vibration responses with harmonic balance method. Then the results are verified wtih the Runge-Kutta method. The results show that the piecewise stiffness has a good performance on limiting the NES. Although the introduction of the piecewise spring will weaken the damping effect of the NES on the primary system, the NES obtain a considerable damping effect with proper parameters. In a word, this work provides a simple and reliable method to restrict the NES, which is beneficial to the design of the NES and broadening the application of the NES in engineering.
... Experimental realizations to control vortex-induced vibrations employing NES have been discussed in the literature, most involving the rotational NES [50]. The practical employment of NES with cubic stiffness generally is achieved by inducing a geometric nonlinearity with two or more linear springs in parallel configuration [21] or using conical springs [40]. The practical implementation of MDOF-NES can involve combining a linear spring, whose coupling can be adjusted by its stiffness, attached in the main structure. ...
The application of multi-degree of freedom nonlinear energy sinks (MDOF-NES) to control the vortex-induced vibrations of a sprung cylinder passively is investigated in this paper. The flow-induced loads on the cylinder are modeled by the wake Van der Pol oscillator. The MDOF-NES consists of three oscillators connected in series, where the first mass is linked to the cylinder structure by a linear spring. The second mass is connected with the first and third masses by pure cubic springs and linear viscous dampers. The cylinder-MDOF-NES assemble model is analyzed in two scenarios involving lower and strong linear stiffness couplings. Numerical investigations of the bifurcation and limit cycle oscillation (LCO) behavior associated with parametric changes are performed. An appropriate design of the MDOF-NES parameters allows mitigating the sub-critical behavior and control LCO amplitudes. The MDOF-NES approach based on the lower linear stiffness coupling has shown to be more effective in controlling the LCO amplitudes. Moreover, it is verified that the MDOF-NES masses have a significant effect on the LCO stability, amplitudes, and synchronization frequency range. Comparison between the MDOF-NES and a classical Type I NES was also performed to ensure the superiority of the MDOF-NES in suppressing the vortex-induced oscillations.
... Generally, the TET can be completed with a very small mass ratio [11]. In addition, the complex dynamic characteristics of the NES have been widely investigated through theoretical and experimental methods [12][13][14][15][16]. Moreover, the NES also presents a high damping performance for various excitation environments, such as transient [17], steady-state [18], and random forms [19]. ...
Based on theoretical and experimental investigations, this paper proposes a nonlinear energy sink (NES) with piecewise linear springs to enhance vibration suppression effects. A cubic nonlinear oscillator is coupled with a piecewise linear spring to form an enhanced NES (E-NES). Without adding a new resonance region and changing the resonance frequency of the primary system, the enhanced NES can achieve better vibration suppression effects. Based on the free vibration and the forced vibration, the effect of piecewise linear stiffness and gap displacement on the vibration suppression is profoundly investigated. Moreover, the parameters of the piecewise spring are optimized through the differential evolution algorithm to obtain the best damping effect of the E-NES. Furthermore, the experiments are conducted to verify the theoretical results. The results show that the E-NES has a better suppression effect on the vibration of the primary structure than that of the cubic NES in most cases. However, it also happens that the vibration suppression effect of the E-NES is weaker than that of the conventional NES. The design parameters can be optimized efficiently by the differential evolution algorithm. Experiments show that vibration elimination efficiency of the E-NES can exceed 90%. Therefore, it is believed that research on using piecewise springs to enhance the vibration suppression efficiency will attract attention.
... Because of the nonlinear hysteretic effect of the SMA spring, the SMA-spring TVA has nonconstant natural frequencies; therefore, the damper can achieve transient resonance capture with a series of the primary structure's modes to expand its vibration reduction band (Gourc et al., 2014). Figure 6 illustrates the wavelet transform of acceleration at the top of the transmission tower. ...
To mitigate the adverse structural responses, an improved version of the traditional tuned vibration absorber has been proposed based on the shape memory alloy spring, referred as the shape memory alloy-spring tuned vibration absorber. The finite element numerical models of the multi-degree-of-freedom structure (e.g., transmission tower) and shape memory alloy-spring tuned vibration absorber are developed by using the commercial software ANSYS, and the nonlinear behavior of the shape memory alloy spring is validated based on a previous experimental study. The damping mechanism of the shape memory alloy-spring tuned vibration absorber attached to a multi-degree-of-freedom structure under seismic excitations is investigated, and the nonlinear hysteretic behavior of the shape memory alloy spring is also discussed. The results show that the proposed damper has a two-stage damping mechanism, and its control performance is remarkable. Because the coupled system response is sensitive to the amplitude level, the optimal configuration of the shape memory alloy-spring tuned vibration absorber can be obtained by parametric analysis. Particularly, because of the nonlinear target energy transfer and transient resonance capture mechanism, the shape memory alloy-spring tuned vibration absorber exhibits stable control ability under different seismic waves, indicating a good stability in vibration control of a multi-degree-of-freedom system.
... The higher NES stiffness results in superior performance under the low excitation amplitude, and however, the NES is then more prone to generate higher branches [29,34]. Moreover, when the NES stiffness is low, its performance is reduced for low excitation amplitudes and the higher branches are generated until the excitation amplitude increases to a high value. ...
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 this reason, the traditional NES is only effective in a certain excitation range. A kind of non-smooth NES with descending stiffness is proposed for expanding this effective range. The non-smooth NES has a higher cubic nonlinear stiffness in the initial range, and the stiffness is reduced as its amplitude exceeds the initial range. The governing equation of motion for a linear primary oscillator attached to the non-smooth NES is obtained in the case of harmonic excitation. The complexification-averaging method is used to obtain the steady-state equation of the system. A least square-based program with the help of a Runge–Kutta-based program is used to analyze the dynamic behaviors of the system. The results demonstrate that the non-smooth NES can eliminate the stable higher branch, therefore expanding the effective excitation range, until the excitation amplitude increases to a very high level. The influences of the piecewise boundary and the stiffness of the secondary stage of the non-smooth NES on the vibration absorption performance are investigated, and the drawbacks of this NES design are discussed. Finally, a structural design based on the theoretical results of the non-smooth NES is proposed, which is composed of permanent magnets, a smooth and discontinuous oscillator and linear springs.
Vibration control has been of great interest to scientists and engineers for many years. Although linear vibration absorbers have been shown to be effective in mitigating vibrations at specific frequencies, their vibration reduction effect is usually limited to a narrow frequency bandwidth. Nonlinear energy sinks have attracted attention due to their better vibration reduction effect over a wider frequency bandwidth. In practical applications, the nonlinear energy sink devices can effectively absorb, dissipate, and convert energy from broadband excitation, so as to achieve vibration reduction and energy harvesting. However, research on energy harvesting based on the nonlinear energy sink is less mature than in linear systems. Multiple parameters of device design (e.g., damping size) affect the actual performance of the nonlinear energy sinks, but there is no exact method to simplify the design of the multiparameter nonlinear energy sinks. Since it is more difficult to implement electromagnetic and electrostatic energy harvesters, more research has focused on piezoelectric energy harvesters. This paper summarizes the research on the nonlinear energy sink and energy harvesting technology, including the introduction of the nonlinear energy sink, energy harvesting based on the nonlinear energy sink, and its application in various fields of energy harvesting. The paper also summarizes some important methods for solving the dynamical equations, as well as their advantages and disadvantages. The conclusions provide an outlook on the subsequent research of the nonlinear energy sink technology, such as the introduction of piezoelectric materials with high energy density, the benefits of balanced vibration suppression and harvesting of vibration energy, and the self-tuning of parameters in complex environments. It provides a powerful reference for the popularization and application of energy harvesting technology based on nonlinear energy sinks.
In recent times, the vibro-impact nonlinear energy sink (VINES) has emerged as a promising passive mechanism for vibration mitigation in engineering systems. The VINES system consists of a ball traveling within a cavity of an externally excited linear oscillator (LO). The ball impacts either end of the cavity, transferring energy from the LO to the ball and mitigating excess oscillations of the LO. Earlier studies of VINES analyzed scenarios with the mass of the ball to be small relative to the LO, with low forcing amplitude near the resonant frequency of the LO. Improvements in targeted energy transfer (TET), observed for an increased mass of the ball, motivate an investigation of VINES for larger mass ratios, using a recently developed semi-analytical map-based approach that provides the exact solution without the limitations of previous analyses. Complementary analytical and numerical approaches treat larger mass ratios and higher amplitudes of the external harmonic excitation for forcing frequencies away from the natural frequency of the LO, identifying parameter regimes for efficient and inefficient performance based on standard measures of energy transfer. The analysis identifies multiple regions for the desired behavior with two alternating impacts per forcing period and provides relevant stability conditions. Numerical results indicate chattering behavior in regimes where energy transfer is minimal, yielding performance that appears similar to resonance. This phenomenon can be directly related to the passive nature of the VINES design, where the natural frequency of the VINES system decreases as the mass of the ball, and thus that of the system, increases. Then the peak response of the LO is shifted away from its resonant frequency, allowing excellent energy transfer to be realized there.
The nonreciprocity of energy transfer is constructed in a nonlinear asymmetric oscillator system that comprises two nonlinear oscillators with different parameters placed between two identical linear oscillators. The slow-flow equation of the system is derived by the complexification-averaging method. The semi-analytical solutions to this equation are obtained by the least squares method, which are compared with the numerical solutions obtained by the Runge-Kutta method. The distribution of the average energy in the system is studied under periodic and chaotic vibration states, and the energy transfer along two opposite directions is compared. The effect of the excitation amplitude on the nonreciprocity of the system producing the periodic responses is analyzed, where a three-stage energy transfer phenomenon is observed. In the first stage, the energy transfer along the two opposite directions is approximately equal, whereas in the second stage, the asymmetric energy transfer is observed. The energy transfer is also asymmetric in the third stage, but the direction is reversed compared with the second stage. Moreover, the excitation amplitude for exciting the bifurcation also shows an asymmetric characteristic. Chaotic vibrations are generated around the resonant frequency, irrespective of which linear oscillator is excited. The excitation threshold of these chaotic vibrations is dependent on the linear oscillator that is being excited. In addition, the difference between the energy transfer in the two opposite directions is used to further analyze the nonreciprocity in the system. The results show that the nonreciprocity significantly depends on the excitation frequency and the excitation amplitude.
Over the last decade, it has been shown that non-linear absorbers can bring significant benefits in terms of vibration damping and shock resistance, all for a very low added mass. Unlike absorbers already existing in industry, these devices involve a highly non-linear dynamics, which involves developing original approaches to modeling and design. Various studies were conducted on this topic at ONERA in collaboration with ISAE and showed the interest of this concept during a thesis. A continuation envisaged consists in designing a NES which would combine mechanics with a physics of another nature (magnetic, piezoelectric, ...) in order to increase the performances in terms of vibratory attenuation and low mass addition. During this thesis, the candidate will study the various possible multiphysics technologies and how they could integrate into the concepts of NES already studied. The aim of the thesis will be to answer these different issues by relying on a joint theoretical and experimental approach, which will include the realization of a demonstrator.
Background
The effective vibration suppression frequency band of linear vibration absorber is narrow. The nonlinear energy sink (NES) can effectively broaden the vibration suppression frequency band. However, with the increase of excitation intensity, the controlled system is easy to produce high branch response, resulting in the failure of the NES. It is of great significance to study the mechanism of high branch response of the controlled system and improve the stability of the working performance of the NES under different excitation intensity.PurposeThe purpose of this study is to analyze the boundary conditions for the controlled system to produce high branch response. While ensuring the performance of the NES under low disturbance, the generation of high branch response of the controlled system is restrained to a certain extent. That is to improve the vibration suppression performance of the NES under a wider excitation intensity.
Methods
Based on the existing NES, the negative stiffness is introduced to improve the working performance under low disturbance. At the same time, the slow variable flow equation of the system is analyzed by using the complex variable averaging (CX-A) method, and the boundary conditions of saddle node bifurcation are obtained. According to the influence of the structure parameters on the boundary conditions, the structure parameters of the NES are optimized.ResultsThe diagrams of saddle node bifurcation and Hopf bifurcation show that the excitation frequency near the main resonance has an important influence on the bifurcation boundary. When the nonlinear stiffness of the NES is 0.8, 1.0, 1.2 and 1.4 respectively, the performance of the NES will be gradually improved, and then the nonlinear stiffness continues to increase, resulting in high branch response. Similarly, when the damping ratio is 0.8, 0.6 and 0.4 respectively, the performance of the NES is gradually improved, and then when the damping ratio continues to decrease, high branch response will occur.Conclusion
The results show that the excitation frequency and amplitude have important effects on the truncation damping and bifurcation boundary. Increasing the nonlinear stiffness or reducing the damping of the PNES can improve the vibration suppression performance near the main resonance, but when it exceeds a certain threshold, it will cause the high branch response.
View Video Presentation: https://doi.org/10.2514/6.2022-0654.vid This paper proposes a methodology to design a Nonlinear Energy Sink (NES) to control passively the vibration of an unstable dynamic system. The passive device is composed of purely nonlinear stiffness (cubic stiffness), a linear damper and a mass attached directly to a simplified dynamic model representation of the main system (i.e., unstable mode of vibration). Asymptotic methods (Method of Multiple Scale mixed with Harmonic Balanced Method, MMS-HBM) is used to treat nonlinear equations which results to a singular perturbed system. Such a system is studied with Geometric Singular Perturbation Theory (GSPT), also with analytical developments. Response of the dynamical system is related to information obtained through GSPT applications: slow invariant manifold, slow-flow fixed points and their stability. This information is correlated with steady-state response regime attained with the NES and, therefore, leading to analytical expressions about necessary conditions to attain different responses. Parametric investigations are then performed and design maps are created in order to predict the response regime for large combination of NES's parameters (i.e., mass ratio, damping factor and nonlinear stiffness). Results highlight that mainly mass ratio and damping factor parameters affects the response regime while nonlinear stiffness parameters affect the amplitude of the motion. Moreover, effects of initial condition and level of the instability severity on robustness of the NES are investigated.
The steady-state dynamic characteristics of non-smooth vibration absorbers are investigated. The complexification-averaging method is used to obtain the steady-state response equation of a harmonic excited primary system attached to the non-smooth absorbers, with the equation solved using a Matlab program based on the least square method. Research results indicate that the traditional purely nonlinear absorber loses its efficacy after the excitation amplitude reaches a certain value. The non-smooth absorber with piecewise linear damping, by contrast, can suppress vibration of the primary system within a larger range of excitation amplitude than the purely nonlinear absorber. Then, the cubic stiffness component is substituted by a piecewise stiffness component to further enhance the performance of the above non-smooth absorber and good results are obtained. The non-smooth absorber with both piecewise damping and stiffness shows the stronger vibration absorption performance. In addition, the differences of higher branches of response which induced by the three absorbers are analyzed and discussed.
Les travaux effectués durant cette thèse portent sur l’étude du comportement mécanique dynamique de solutions amortissantes passives utilisées pour la réduction des niveaux vibratoires et la stabilisation des systèmes optroniques embarqués au sein de l’entreprise Thales LAS France. Ces solutions intègrent des matériaux élastomères au fort pouvoir dissipatif dont le comportement doit être parfaitement maîtrisé pour un bon dimensionnement de l’isolation vibratoire, et ce malgré leur dépendance à la température et à la fréquence. L’objectif général est d’améliorer la connaissance du comportement de ces matériaux, leur caractérisation, leur prise en compte dans les simulations numériques afin d’améliorer les pratiques employées dans les bureaux d’études qui conçoivent les structures accueillant ces systèmes.Dans ce cadre, les travaux présentés portent tout d’abord sur la caractérisation, la modélisation et l’identification du comportement viscoélastique des élastomères employés dans des amortisseurs de Thales LAS France. Un modèle de type Maxwell généralisé (GMM) est utilisé pour décrire ce comportement, et est introduit dans un modèle éléments finis de l’amortisseur afin d’obtenir une représentation physique satisfaisante de son comportement mécanique dynamique. Le problème est réécrit sous la forme d’une représentation d’état originale qui est associée à une stratégie de réduction de modèle pour réduire les temps de calcul. Différentes simulations sont alors réalisées pour illustrer le potentiel de l’approche proposée, analyse modale complexe, réponse fréquentielle et réponse temporelle. La température ayant une influence primordiale sur le comportement mécanique des élastomères, un modèle matériau thermomécanique spécifique est proposé en identifiant l’évolution en température de paramètres du GMM, et une analyse de robustesse portant sur la capacité de dissipation de l’amortisseur témoin en présence de méconnaissances sur cette variable est réalisée en se basant sur la théorie Info-Gap.L’analyse d’une campagne d’essais a permis de constater l’apparition d’un assouplissement de la structure sous de fortes sollicitations, laissant augurer la présence de non-linéarités. Un autre aspect abordé durant cette thèse porte ainsi sur la caractérisation, la modélisation et l’identification des phénomènes non-linéaires pouvant impacter le comportement dynamique de l’amortisseur. Deux sources ont été mises en évidence : une non-linéarité matérielle liée à la dépendance des caractéristiques mécaniques des élastomères au taux de déformation (effet Payne), et une non-linéarité de type contact liée à la présence de butées. Ces comportements ont été implémentés dans une représentation réduite de l’amortisseur afin d’expliquer les phénomènes non-linéaires observés expérimentalement au cours des campagnes de qualification du produit.Enfin, la dernière partie de ces travaux de thèse porte sur la conception d’un réseau d’absorbeurs à masses accordées (MTMD) afin de réduire le niveau vibratoire d’une pièce structurale supportant les systèmes optiques. Après une formulation du problème éléments finis, une procédure d’optimisation des paramètres du MTMD est mise en œuvre et une analyse de robustesse de la solution optimale en présence d’incertitudes sur la fréquence propre à contrôler est effectuée. Cette étude est menée pour différents jeux de paramètres et une méthode d’optimisation robuste est proposée en combinant la procédure d’optimisation et la théorie Info-Gap. Pour finir, une maquette du système étudié est réalisée ainsi qu’une version simplifiée de son MTMD associé afin de mettre à l’épreuve les règles d’accordage issues des études numériques grâce à une série d’essais vibratoires.
Most aeroelastic systems suffer from nonlinear behavior. The characterization of nonlinear response in aeroelastic systems is, therefore, a relevant issue. When designing for best performance, for instance, to expand the flutter boundaries, the nonlinear aeroelastic behavior is also a formidable challenge. Towards optimal aeroelastic tailoring, an alternative approach may be admitting acceptable levels when nonlinearities are present to the problem. To do this post-flutter behavior can be added to the optimization reasoning, along with the traditional flutter boundaries expansion. The current work proposes an investigation on the aeroelastic tailoring of a typical section with hardening nonlinearity in pitching stiffness seeking to expand the flutter onset boundary and minimum stable LCO amplitudes in post-flutter. This investigation uses the traditional typical section model to develop a reduced order model based on the multiple scales method viewing fast evaluations of the flutter onset and values of LCO amplitudes at some post-flutter airspeeds. Aeroelastic tailoring is based on the differential evolution algorithm to yield Pareto frontiers for the two selected objectives. An analysis of the design variables is presented, and the optimization results reveal that adequate compromise solutions can be assessed. Therefore, possible optimal post-flutter conditions for minimum LCO amplitudes can also be achieved.
One of the goals of this thesis is to enhance the comprehension of nonlinear dynamics, especially MEMS nonlinear dynamics, by proposing new methods for parametric analysis and for nonlinear normal modes computation. In a first part, methods for the detection, the localization and the tracking of bifurcation points with respect to a single parameter are recalled. Then, a new method for parametric analysis, based on recursive continuation of extremum, is presented. This method is then applied to a Nonlinear Tuned Vibration Absorber in order to push isolated solutions at higher amplitude of forcing. Secondly, a method is presented for the computation of nonlinear normal modes. An optimal phase condition and a relaxation of the equation of motion are proposed to obtain a continuation method able to handle modal interactions. Then, a quadratic eigenvalue problem is shifted to compute the stability and bifurcation points. Finally, nonlinear normal modes are extended to non-conservatives systems permitting the continuation of phase and energy resonances. Thirdly, the nonlinear dynamics of MEMS array, based on multiple resonant micro-beams, is analyzed with the help of the proposed methods. A frequency synchronization of bifurcation points due to the electrostatic coupling is discovered. Then, the nonlinear dynamics of a MEMS array after symmetry breaking event induced by the addition of a small mass onto one of the beam of the array is analyzed. Finally, mass detection mechanisms exploiting the discovered phenomena are presented.
This paper deals with the application of the concept of targeted energy transfer to the field of acoustics, providing a new approach to passive sound control in the low frequency domain, where no efficient dissipative mechanism exists. The targeted energy transfer, also called energy pumping, is a phenomenon that we observe by combining a pure nonlinear oscillator with a linear primary system. It corresponds to an almost irreversible transfer of vibration energy from the linear system to the auxiliary nonlinear one, where the energy is finally dissipated. In this study, an experimental set-up has been developed using the air inside a tube as the acoustic linear system, a thin circular visco-elastic membrane as an essentially cubic oscillator and the air inside a box as a weak coupling between those two elements. In this paper, which mainly deals with experimental results, it is shown that several regimes exist under sinusoidal forcing, corresponding to the different nonlinear normal modes of the system. One of these regimes is the quasi-periodic energy pumping regime. The targeted energy transfer phenomenon is also visible on the free oscillations of the system. Indeed, above an initial excitation threshold, the sound extinction in the tube follows a quasi-linear decrease that is much faster than the usual exponential one. During this linear decrease, the energy of the acoustic medium is irreversibly transferred to the membrane and then damped into this element called nonlinear energy sink. We present also the frequency responses of the system which shows a clipping of the original resonance peak of the acoustic medium and we finally demonstrate the ability of the nonlinear absorber to operate in a large frequency band, tuning itself to any linear system.
This paper aims to experimentally verify the theoretical effects of energy pumping especially with external excitation. Energy pumping is irreversible transfer of energy from a linear or linearized structure to a nonlinear energy sink (NES) with relatively small mass. This NES can be used as a nonlinear absorber. This phenomenon is analyzed for different kinds of excitation. In suitable range of amplitudes of the external forcing, the damped system exhibits quasiperiodic vibrational regime rather than periodic responses reported in earlier publications. This regime can be explained by using nonlinear normal mode theory. Mechanical experiments confirm the theoretical results by using a small building model. In particular, the case of earthquake excitations is investigated.
We study the stiffening and damping effects that local essentially nonlinear attachments can have on the dynamics of a primary linear structure. These local attachments can be designed to act as nonlinear energy sinks (NESs) of shock-induced energy by engaging in isolated resonance captures or resonance capture cascades with structural modes. After the introduction of the NESs, the effective stiffness and damping properties of the structure are characterized through appropriate measures, developed within this work, which are based on the energy contained within the modes of the primary structure. Three types of NESs are introduced in this work, and their effects on the stiffness and damping properties of the linear structure are studied via (local) instantaneous and (global) weighted-averaged effective stiffness and damping measures. Three different applications are considered and show that these attachments can drastically increase the effective damping properties of a two-degrees-of- freedom system and, to a lesser degree, the stiffening properties as well. An interesting finding reported herein is that the essentially nonlinear attachments can introduce significant nonlinear coupling between distinct structural modes, thus paving the way for nonlinear energy redistribution between structural modes. This feature, coupled with the well-established capacity of NESs to passively absorb and locally dissipate shock energy, can be used to create effective passive mitigation designs of structures under impulsive loads.
In the present works, we examine experimentally and theoretically the dynamic behavior of linear oscillator strongly coupled to a nonlinear energy sink under external periodic forcing. The nonlinear oscillator has a nonlinear restoring force realized geometrically with two linear springs that extend axially and are free to rotate. Hence, the force-displacement relationship is cubic. The linear oscillator is directly excited via an electrodynamic shaker. Experiments realized on the test bench consist of measuring the displacement of the oscillators while increasing and decreasing frequencies around the fundamental resonance of the linear oscillator. Many nonlinear dynamical phenomena are observed on the experimental setup such as jumps, bifurcation, and quasiperiodic regimes. The retained nonlinear model is a two degree of freedom system. The behavior of the system is then explained analytically and numerically. The complexification averaging technique is used to derive a set of modulation equation governing the evolution of the complex amplitude at the frequency of excitation, and a stability analysis is performed.
The concept of the vibratory energy transfer between a linear master DOF and a nonsmooth nonlinear energy sink (NES) in the presence of gravity forces is studied. Different invariant manifolds of the system at different time scales are revealed and necessary conditions for leading the behavior of the system to strongly modulated response (SMR) are enlightened.
A study of the targeted energy transfer (TET) phenomenon between an acoustic resonator and a thin viscoelastic membrane has recently been presented in the paper [R. Bellet et al., Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber, Journal of Sound and Vibration 329 (2010) 2768–2791], providing a new path to passive sound control in the low frequency domain where no efficient dissipative device exists. This paper presents experimental results showing that a loudspeaker used as a suspended piston working outside its range of linearity can also be used as a nonlinear acoustic absorber. The main advantage of this technology of absorber is the perspective to adjust independently the device parameters (mass, nonlinear stiffness and damping) according to the operational conditions. To achieve this purpose, quasi-static and dynamic tests have been performed on three types of commercial devices (one with structural modifications), in order to define the constructive characteristics that it should present. An experimental setup has been developed using a one-dimensional acoustic linear system coupled through a box (acting as a weak spring) to a loudspeaker used as a suspended piston acting as an essentially nonlinear oscillator. The tests carried out on the whole vibro-acoustic system have showed the occurrence of the acoustic TET from the acoustic media to the suspended piston and demonstrated the efficiency of this new kind of absorber at low frequencies over a wide frequency range. Moreover, the experimental analyses conducted with different NES masses have confirmed that it is possible to optimize the noise absorption with respect to the excitation level of the acoustic resonator.
We study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques. First, we transform the equations of motion using the action-angle variables of the underlying Hamiltonian system and bring them into the form where two-frequency averaging cart be applied. We then show that energy pumping is due to resonance capture in the 1:1 resonance manifold of the system, and perform a perturbation analysis in an O (root epsilon) neighborhood of this manifold in order to study the attracting region responsible for the resonance capture. The second method is based on the assumption of 1:1 internal resonance in the fast dynamics of the system, and utilizes complexification and averaging to develop analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The results compare favorably to numerical simulations. The practical implications of the energy pumping phenomenon are discussed.
The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) ''channeling'' of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.
We study targeted energy transfers and nonlinear transitions in the damped dynamics of a two degree-of-freedom system of coupled oscillators (a linear oscillator with a lightweight, essentially nonlinear, ungrounded attachment), caused by 1:1 resonance captures of the dynamics. Part I of this work deals with the underlying structure of the Hamiltonian dynamics of the system, and demonstrates that, for sufficiently small values of viscous damping, the damped transitions are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the corresponding Hamiltonian system. Focusing exclusively on 1:1 resonance captures in the system, it is shown that the topology of these damped transitions affect drastically the efficiency of passive energy transfer from the linear system to the nonlinear attachment. Then, a detailed computational study of the different types of nonlinear transitions that occur in the weakly damped system is presented, together with an analytical treatment of the nonlinear stability of certain families of periodic solutions of the underlying Hamiltonian system that strongly influence the said transitions. As a result of these studies, conditions on the system and forcing parameters that lead to effective or even optimal energy transfer from the linear system to the nonlinear attachment are determined. In Part II of this work, direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed, in order to analytically model the dynamics in the region of optimal energy transfer, and to determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator.
We examine passive energy pumping in a system of damped coupled oscillators. This is a one-way, passive and irreversible energy flow from a linear main system to a nonlinear attachment that acts, in essence, as a nonlinear energy sink (NES). Energy pumping is caused by 1:1 resonance captures on resonant manifolds of the damped systems. We show that the NES is capable of absorbing significant portions of the energies generated by transient, broadband external excitations. By performing a series of numerical simula-tions we confirm that the energy dependence of the nonlinear normal modes (NNMs) of the underlying undamped, unforced system determines, in essence, the resonance capture and energy pumping dynamics in the corresponding damped system. We present numeri-cal simulations of single-and multi-mode energy pumping, that involve isolated reso-nance captures or resonance capture cascades, respectively. In addition, we discuss meth-odologies for enhancing the nonlinear energy pumping phenomenon by properly selecting the system parameters. The described technique of passively localizing and locally elimi-nating externally induced energy provides a new paradigm for vibration and shock iso-lation of mechanical oscillators.
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.
System under investigation comprises a harmonically forced linear oscillator and a nonlinear energy sink (NES). The NES is
a small mass (relative to that of the linear oscillator) which is attached to the primary system via a linear damper and strongly
nonlinear spring (pure cubic nonlinearity). Among possible responses there exists one characterized by extremely deep modulation
of the oscillations and referred to as a strongly modulated response regime (SMR). Numeric simulations demonstrate that the
SMR can exist only for sufficiently small values of the NES mass. Known analytical approximations for description of the SMR
deal with the lowest order of the asymptotic approximation and, consequently, work fairly well only for very small values
of the NES mass and do not take into account its actual value. In the present study, we develop the analytical tools to investigate
the higher-order asymptotic approximation. This enables us to depict the qualitative changes in the regime for the growing
values of a NES mass and also to provide a crude estimation for a NES mass threshold. It is also demonstrated that in some
cases the mechanisms of loss of stability by SMR (due to the growing values of NES mass) can be illustrated and explained
via one-dimensional mapping diagrams. The described novel analytical approach is verified numerically and a fairly good agreement
between the numerical and analytical models is observed.
Linear oscillator coupled to damped strongly nonlinear attachment with small mass is considered as a model design for nonlinear energy sink (NES). Damped nonlinear normal modes of the system are considered for the case of 1:1 resonance by combining the invariant manifold approach and multiple scales expansion. Special asymptotical structure of the model allows a clear distinction between three time scales. These time scales correspond to fast vibrations, evolution of the system toward the nonlinear normal mode and time evolution of the invariant manifold, respectively. Time evolution of the invariant manifold may be accompanied by bifurcations, depending on the exact potential of the nonlinear spring and value of the damping coefficient. Passage of the invariant manifold through bifurcations may bring about destruction of the resonance regime and essential gain in the energy dissipation rate.
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.
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.
We present an asymptotic approach to the analysis of coupled nonlinearoscillators with asymmetric nonlinearity based on the complexrepresentation of the dynamic equations. The ideas of the approach arefirst developed for the case of the system with two degrees of freedom.Special attention is paid to the study of localized normal modes in achain of weakly coupled nonlinear oscillators. We discuss also certainpeculiarities of the localization of excitations in the case of strongcoupling between the oscillators.
Targeted energy transfer (TET) in a compound nonlinear 2 degrees-of-freedom system during free and forced excitations is studied
analytically and numerically. The nonlinearity of the system is represented intentionally by a non-smooth piece-wise linear
function for the sake of practical investigations. Further stability analysis of the system is demonstrated and commented
upon and the behavior of the system during relaxation and its strongly modulated response is studied and pinpointed.
Dynamic responses of a linear oscillator coupled to a nonlinear energy sink (NES) under harmonic forcing in the regime of 1:1:1 resonance are investigated. Primary attention is paid to the detailed investigation of the so-called strongly modulated response (SMR), which is not related to the fixed points of average modulation equations of the system. Essential mass asymmetry allows a global analysis of the responses despite strong nonlinearity. It is demonstrated that the strongly modulated response is related to a relaxation-type motion and its description in the limit of small mass ratio maybe reduced to the 1D return map of a subset at a fold line of slow invariant manifold. The SMR exists in the O(ε)-vicinity of the exact resonance, where ε≪1 characterizes the mass asymmetry. It is also shown that the SMR appears in the system as a result of global fold bifurcation of limit cycles and exhibits some properties pertinent to generic 1D nonlinear maps, such as period doubling. Transient responses with finite number of relaxation cycles and subsequent attraction to stable periodic attractor are revealed. Analytic results are compared to numeric simulations and a good agreement is observed.
Dynamical system under investigation in the current work is comprised of harmonically forced linear oscillator with attached nonlinear energy sink. External forcing frequency detuning near the main resonance (1:1) is included in the system investigation. The detailed study of the periodic and quasiperiodic regimes is done in the work via (adaptive) averaging method. Local bifurcations of the periodic regimes are revealed and fully described in the space of system parameters (amplitude of excitation, damping, and frequency detuning). Novel analytical approach for predictions of strongly modulated response (SMR) is presented. This approach provides a sufficient condition for the SMR existence contrary to the previous studies. Various possibilities of coexistence of the response regimes are predicted analytically and demonstrated numerically. Among those is a coexistence of two distinct periodic regimes together with the SMR. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.
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.
The purpose of this paper is to report an experimental study of transient resonance capture that may occur in a system of two coupled oscillators with essential (i.e., nonlinearizable) nonlinearity. It is shown that during transient resonance capture the two oscillators are in a state of resonance, the frequency of which varies with time, which leads to targeted nonlinear energy transfer. Further evidence of resonance capture is a non-time-like behavior of the phase difference between the oscillators; this quantity is monitored using the Hilbert transform or the Huang–Hilbert transform in the case of multifrequency response signals.
We study targeted energy transfers and nonlinear transitions in the damped dynamics of a two degree-of-freedom system of coupled oscillators (a linear oscillator with a lightweight, essentially nonlinear, ungrounded attachment), caused by 1:1 resonance captures of the dynamics. Part I of this work deals with the underlying structure of the Hamiltonian dynamics of the system, and demonstrates that, for sufficiently small values of viscous damping, the damped transitions are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the corresponding Hamiltonian system. Focusing exclusively on 1:1 resonance captures in the system, it is shown that the topology of these damped transitions affect drastically the efficiency of passive energy transfer from the linear system to the nonlinear attachment. Then, a detailed computational study of the different types of nonlinear transitions that occur in the weakly damped system is presented, together with an analytical treatment of the nonlinear stability of certain families of periodic solutions of the underlying Hamiltonian system that strongly influence the said transitions. As a result of these studies, conditions on the system and forcing parameters that lead to effective or even optimal energy transfer from the linear system to the nonlinear attachment are determined. In Part II of this work, direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed, in order to analytically model the dynamics in the region of optimal energy transfer, and to determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator.