Targeted Energy Transfer Research Articles

Non-reciprocal frequency conversion in a two-dimensional waveguide incorporating a local nonlinear gate

We study non-reciprocal frequency conversion in a linear two-dimensional (2D) acoustic waveguide consisting of coupled oscillators incorporating a local asymmetric gate of nonlinearly coupled oscillators. The 2D waveguide supports an acoustic and an optical frequency band. We consider a monochromatic harmonic excitation at the acoustic band that initiates a propagating wave in the linear waveguide. As the wave scatters across the nonlinear gate, its frequency is converted to a higher (triple) harmonic in the optical band. This frequency conversion is due to a local 1:3 internal resonance at the nonlinear gate. The efficacy and robustness of the frequency conversion are validated by direct numerical simulations. A large portion of the energy of the wave is transmitted across the nonlinear gate with converted frequency and this frequency conversion is realized robustly over a range of excitation frequencies and magnitudes. In addition, owing to the nonlinearity, the frequency conversion is tunable to frequency and energy. The transmissibility (i.e., transmitted over reflected energy at the gate) is sensitive to the excitation parameters indicative of bifurcations of the nonlinear acoustics. Lastly, frequency conversion induces strong non-reciprocity in the acoustics since the energy transfer with frequency conversion is only realizable in a preferred direction of wave propagation and prohibited in the reverse direction. The methodology and results of the present nonlinear frequency converter are applicable to various fields including targeted energy transfer in phononic systems and acoustic computing.

Variable-potential bistable nonlinear energy sink for enhanced vibration suppression and energy harvesting

A variable-potential bistable nonlinear energy sink coupled electromagnetic harvesters (VBNES-EH) is proposed, that can simultaneously achieve high efficiency, broadband vibration suppression and energy harvesting under ultra-low and ultra-wide input energy level excitation. The performance is enhanced by introducing two horizontal energy harvesters (HEHs) to dynamically reduce the potential barrier height of the bistable nonlinear energy sink (BNES) or called bistable energy harvester (BEH) to form lower excitation thresholds of the chaotic and strong modulated responses. Firstly, the dimensionless mathematical model of the VBNES-EH is proposed and verified by the experiment. The experimental results are qualitatively consistent with the simulation results. The transient dynamics and the corresponding energy dissipation rates of five target energy transfer (TET) mechanisms for the weakly damped system are summarized numerically. The variable-potential energy effect and its benefits for the dual functions of vibration suppression and energy harvesting are analyzed. Afterward, the stiffness of the VBNES-EH and the fixed-potential BNES-EH (FBNES-EH) are optimized, and their energy dissipation rates under impact excitation are compared. The results indicate that the former has better performance under low- and intermediate-level impact excitations. Furthermore, the influence of system parameters on the energy harvesting rate is discussed to guide the optimal design of the performance. Finally, the dynamical evolution of the VBNES-EH and the parameter effects on the dual performance of the system under harmonic sweep excitation are investigated. Numerical results demonstrate that the VBNES-EH can achieve high performance under the different amplitudes of the harmonic excitation through stiffness optimization.

NES cell

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

Shaking table test study on seismic performance of RC frame structure with NES

A new type of nonlinear vibration absorber NES (Nonlinear Energy Sink) studied in this paper belongs to the structural passive control technology, which consists of a horizontal track and an additional mass block. Acting as a damper absorb and dissipate energy, the NES completes the target energy transfer by generating a horizontal nonlinear restoring force and reduces the response of the main structure. In this paper theoretical analysis of the new NES was analyzed by the expression of the restoring force of the NES and the equation of motion of the additional NES system are deduced. Shaking table test was carried out on the 1:4 scale RC frame structure model installed with the new NES shock absorber, focusing on the research on the energy dissipation and shock absorption effect of the NES shock absorber, and whether it can improve the seismic performance of the RC frame structure damaged in the earthquake. The results show that the NES installed in the RC frame structure has sufficient nonlinear displacement and energy dissipation capacity under the moderate and strong earthquakes, showing a very good energy dissipation and shock absorption effect. The NES can reduce the top story displacement and acceleration response of the main structure under the action of earthquake; the top story displacement and absolute acceleration response are related to the natural frequency of the frame and the spectral characteristics of seismic waves, so that the vibration control effect is affected by the frequency spectrum of seismic waves. In the frequency domain analysis of the frame dynamic response, the nonlinear energy well exhibits the characteristics of broadband control. This type of NES significantly reduces the seismic response of damaged structures, improves the seismic performance of damaged structures under moderate to strong earthquakes, and decreases the risk of potential collapse and construction costs during earthquake aftershocks.

Targeted energy transfer in a vibro-impact cubic NES: Description of regimes and optimal design

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).

Bistable enhanced passive absorber based on integration of nonlinear energy sink with acoustic black hole beam

The nonlinear energy sink (NES) has a broader frequency bandwidth than the linear tuned-mass-damper (TMD), making it more robust. However, the NES is only efficient in a narrow energy range. The mitigation property of NES may deteriorate when input energy is too small or large. To overcome this shortage, a novel NES named ABH-BNES is proposed in this paper, by combining a bistable nonlinear energy sink (BNES, which has negative linear stiffness, cubic nonlinear stiffness, and linear damping) and acoustic black hole (ABH) beam. The negative linear stiffness and cubic nonlinear stiffness elements can generate a large cross-well motion to improve the performance of vibration mitigation for low input energy. By replacing the attached mass in traditional NES as an ABH beam, multiple high-damping modes can be introduced into the vibration absorber. So that the performance of ABH-BNES is also enhanced for large input energy. First, the Gaussian Expansion Method and Modal Approach are employed to establish the dynamic model of a single-dof linear oscillator coupling with an ABH-BNES. Such a modeling approach is verified by comparing it with the models in the existing literature and established by the finite element method (FEM). The slow invariant flow (SIM) theory and time-domain integration are employed to analyze the novel NES's vibration mitigation mechanism. In addition, a multi-solution boundary related to the existence of targeted energy transfer is derived for optimizing the nonlinear absorber. Then, the energy method is used to verify the superiority of the novel NES by comparing it with TMD, NES, and the Linear ABH absorber. The results show that the ABH-BNES can behave with an excellent mitigation effect and an extensive energy bandwidth when the parameters are appropriately selected. Finally, an experiment is carried out to verify the new NES's superiority and the correctness of the theoretical model.

Hamiltonian dynamics and targeted energy transfer of a grounded bistable nonlinear energy sink

A two-degree-of-freedom nonlinear mechanical system describing a damped linear oscillator weakly coupled to a lightweight grounded bistable nonlinear energy sink is proposed to investigate the targeted energy transfer (TET) under asymmetric local nonlinear potentials. The bifurcations of equilibria are discussed to reflect the bistable and asymmetric characteristics of the system. Through projections of isoenergetic manifolds, the topological features of the Hamiltonian dynamics under 1:1 internal resonance have been investigated, which induces conditions for energy localization and exchange. In the general case, the topological features of the Hamiltonian dynamics related to the nonlinear normal modes (NNMs) are classified by the Poincaré map, the relations between the NNMs and the Hamiltonian dynamics have been interpreted through the frequency energy plots. The results show that introducing the asymmetric characteristics may enhance the TET and realize the complete TET.

Influence of Boundary Impedance of 3D Cavity on Targeted Energy Transfer between a Damped Acoustic Mode and a Nonlinear Membrane Absorber

In order to apply the nonlinear energy sink (NES) to reduce the low-frequency noise inside a 3D acoustic cavity with an impedance boundary, a two degrees-of-freedom (DOF) dynamic model of the coupled system of one damped acoustic mode of a regular 3D acoustic cavity and a nonlinear membrane absorber as the NES is established. The damping coefficient of the acoustic mode is obtained based on the finite element analysis method and the relationship of the boundary impedance and the absorption coefficient of the cavity. Based on the set-up of the system, the damping coefficient of the acoustic mode is determined. The frequency responses and the targeted energy transfer (TET) phenomenon of the coupled system are analyzed, and the theoretical and numerical results of the frequency responses of the system are in good agreement with the experimental ones. The effects of the wall impedance of the cavity on the optimal TET are discussed. With the increase in the impedance ratio of the wall, the amplitude of the acoustic displacement plateau decreases, and the frequency bandwidth of the plateau and the two thresholds of the optimal TET interval of the excitation increase. It provides a comprehensive theoretical model and experimental basis for the application of NES in the actual complex structure and provides a reliable design method and control strategy for controlling vehicle interior low-frequency broadband noise.

Research on Dynamic Characteristics of Joint of RC Frame Structure with NES

The NES (nonlinear energy sink) is a new type of nonlinear tuned mass damper that is connected to the shock-absorbing main structure through strong nonlinear stiffness and viscous damping. The vibrational energy in the main structure is transferred to the NES oscillator by means of target energy transfer. A shaking table test of a 1:4 scaled RC (Reinforced Concrete) frame structure model with a new type of NES shock absorber was conducted to study the damping effect of the NES shock absorber, especially for the influence of joint strength and deformation. The NES used in this experiment has a relatively large nonlinear stiffness and a wide vibration absorption frequency band. The variation of reinforcement strains, node failure mode, and structural natural frequency of 1 story and two-layer joints of the model frame structure with NES were studied. The test results showed that NES could effectively reduce the strains of longitudinal reinforcement and stirrup in beams and columns and delay the plastic hinge development at the bottom and the top of the column. The frame model with NES installed has failures at the beam ends and shear failures at the nodes, realizing the seismic mechanism of solid columns and weak beams. Compared with ordinary seismic structures, the NES can effectively reduce the shear stress of concrete at the joints and alleviate the shear failure of joints. The final failure of the NES shock absorbing structure was the yielding of the steel bars at the bottom of the column and the crushing of the concrete at the foot of the column, and the connection between the column foot and the backplane became loose simultaneously. The decreasing rate of the vibration frequency declined due to the NES with varied broadband absorbing capability. It can be seen that the NES shock absorber not only has a good effect on reducing the seismic response of the structure, but more importantly, the damage of the structural nodes is greatly reduced, and therefore, the seismic capacity of the structure improved.

Experimental testing and system identification of the sliding bistable nonlinear energy sink implemented to a four-story structure model subjected to earthquake excitation

Bistable nonlinear energy sink system has been explored as an efficient vibration absorber in many areas. This paper aims to experimentally validate the seismic mitigation performance of sliding bistable nonlinear energy sink (SBNES) for multi-story building structures. An SBNES device was designed and placed on top of a four-story building structure model. Earthquake ground motions were applied through a shake table as the base excitation input into the structure model. By matching the time history envelope curves between the experiment and numerical simulation, the structural parameters of the primary structure, as well as the constitutive model of the SBNES, were experimentally determined. The comparisons between mitigated and unmitigated structural responses, including inter-story drift, acceleration, base-shear force and base-overturning moment of the primary structure, show that substantial reduction of structural seismic responses can be achieved by the SBNES device. The 1:1 transient internal resonance is verified as the dominant scheme for highly efficient targeted energy transfer. With respect to variation of peak ground acceleration, a relatively optimal interval distributed near the borderline of intra-well and inter-well regime is proposed for robust SBNES design. The test validates the potential of the SBNES strategy for both acceleration and displacement control of multi-story building structures.

Low-to-high frequency targeted energy transfer using a nonlinear energy sink with softening-hardening nonlinearity

This paper investigates the performance of a nonlinear energy sink (NES) equipped with a softening–hardening (SH) element for mitigating vibrations that occur below its linear, low-energy frequency. First, an SH elastic struct is constructed, tested quasi-statically, and the resulting restoring force is modeled using a polynomial. Next, the SH polynomial is non-dimensionalized and a theoretical system composed of a linear oscillator (LO) and NES is studied where the NES has a linear, low-energy frequency above that of the LO. The underlying nonlinear normal modes (NNMs) of the system are studied by using harmonic excitation and tracking the changes in the frequency response functions. The transient performance of the NES is investigated and compared with that of a NES with the same linear stiffness but only a cubic nonlinearity. The theoretical performance of the NES is verified using a comparable experimental system that incorporates the SH elastic struct built in the beginning of the paper. The results of this work demonstrate that an SH NES is capable of mitigating vibrations that occur at frequencies below its own linear, low-energy frequency.

Experimental Inter-Modal Targeted Energy Transfer in a cantilever beam undergoing Vibro-impacts

This work experimentally demonstrates inter-modal targeted energy transfer (IMTET) in a cantilever beam system with vibro-impacts. IMTET is known to achieve non-resonant rapid energy transfer from low-to-high frequency structural modes in mechanical systems. While previous works have computationally demonstrated the efficacy of IMTET, the theory and methodology supporting IMTET still lacks experimental verification. To address this task, an experiment is constructed whereby a steel beam is clamped at one end and excited at the other, and steel tips with clearances anchored to a host fixture serve as vibro-impacts (VIs) which induce non-resonant energy redistribution within the modal space of the beam. The system is modeled as an Euler–Bernoulli cantilever beam which is discretized by the finite element method, and the VI forces are simulated by a single-sided dissipative Hertzian contact model. The time-responses, modal dissipation ratios, and characteristic decay times measured in the experiment are compared to the computational models to confirm the validity of the theoretical predictions. It is found for both the experiment and computational model that the VIs transfer energy from low frequency structural modes to high frequency ones which, in turn, results in a substantial reduction in the characteristic decay time and thus confirms the efficacy of IMTET in practice. The diverse potential applications of IMTET in engineering practice are then emphasized.

Passive suppression of vortex-induced vibrations using a nonlinear energy sink—Numerical and analytical perspective

This study investigates the suppression mechanism of instabilities induced by fluid–structure interactions (FSI) using passive vibration absorption devices, such as nonlinear energy sink (NES). The present FSI framework comprises a low-order phenomenological model, wherein the wake effect is modeled using the classical Van der Pol oscillator. The structure is represented as a cylindrical bluff body with degree-of-freedom along the cross-flow direction. The response of the NES-augmented structure exhibits specific relaxation type oscillations, referred to as strongly modulated response (SMR), passively suppressing the high amplitude vortex-induced vibrations (VIV). The underlying mechanism of SMR is studied using an analytical approach based on the Complexification-Averaging (CXA) technique. Using the CXA technique, the slow flow for the coupled FSI system with the NES attachment is modeled effectively, revealing nonlinear beating regimes and initiation of the targeted energy transfer (TET) mechanism. Subsequently, a transient resonance capture, implying a significant energy transfer from the structure to the NES, results in effective VIV suppression. The occurrence of SMRs is explained by analyzing the global dynamics using the slow invariant manifold (SIM) derived from the slow flow of the coupled system. The resultant SIM topology reveals a jump phenomenon between the stable branches, wherein the flow jumps from a lower stable branch to a higher stable branch through SMR. The novelty of this study lies in identifying the occurrence of SMRs as the mechanism of energy transfer and uses the CXA technique to explain the global dynamics by modeling the SIM for the 3-DOF coupled FSI framework with the NES attachment. Furthermore, the optimal operational parameter ranges for an efficient NES design are identified through a parametric study using the slow-flow model.

An improved nonlinear energy sink with electromagnetic damping and energy harvesting

A nonlinear energy sink can achieve vibration reduction without changing the resonant frequency of the system. In this work, an improved nonlinear energy sink is proposed by integrating with an electromagnetic system. The vibration reduction performance is further enhanced and an energy harvesting function is achieved at the same time. The dynamic equations of the primary system with the improved nonlinear energy sink are established, and the electromagnetic analysis is performed. The steady-state response of the system is solved with the harmonic balance method and verified by numerical simulations. The integration of the electromagnetic system provides additional damping to the nonlinear energy sink, and the vibration reduction performances under the harmonic and the impulsive excitations are enhanced. At the same time, the vibrational energy is effectively harvested by the electromagnetic system. The integration of the electromagnetic system does not change the resonant frequency of the system and retains the targeted energy transfer characteristic. The proposed system is applicable to the nonlinear energy sink with originally insufficient damping, and the electromagnetic coupling coefficient is a key parameter affecting both the vibration reduction and the energy harvesting performances. This integrated system is expected to provide a new strategy for multifunctional devices.

A non-traditional variant nonlinear energy sink for vibration suppression and energy harvesting

In this article, an apparatus is proposed for simultaneous vibration suppression and energy harvesting in a broad frequency band. Traditionally, both the spring and the damper of a vibration absorber are directly connected to a primary system. Alternatively, a non-traditional way is to place the damper of the vibration absorber between the absorber mass and the base. A nonlinear energy sink (NES) is a type of nonlinear vibration absorber with an essential nonlinearity. When a strong nonlinear oscillator with a small linear stiffness is weakly coupled to a primary system, it behaves similarly to the NES, which is termed “variant NES”. In the proposed apparatus, a variant NES is arranged in the non-traditional way in which a cantilever beam is placed between a pair of continuous-contact blocks which force the beam to deflect nonlinearly. An electromagnetic energy harvester, which also acts as a damper in the non-traditional variant NES, is formed by attaching a pair of permanent magnets to the free end of the beam and fixing a pair of coils to the base. The transient performances of the developed apparatus are investigated in terms of vibration suppression and energy harvesting using numerical simulation and compared with those of an optimum non-traditional vibration absorber. The simulation results are validated by an experimental study. It is found that the proposed non-traditional variant NES exhibits typical NES characteristics such as 1:1 resonance, targeted energy transfer (TET), and an initial energy threshold. With the prerequisite of TET being established under high-level initial energy, the best trade-off between vibration suppression and energy harvesting is achieved when the NES’s oscillation decays slowly under low electrical damping or quickly under high electrical damping.

Generalization of the Concept of Bandwidth

In the sciences and engineering, there is no generally accepted definition of bandwidth beyond those used to characterize low-loss linear time-invariant systems of low order operating at steady-state. In fact, the concept of bandwidth is often subject to interpretation depending upon context and the requirements of a specific community. The focus of this work is to formulate this concept for a general class of passive oscillatory dynamical systems, including but not limited to mechanical, structural, acoustic, electrical, and optical. Typically, the (linearized) bandwidth of these systems is determined by the half-power (-3 dB) method, and the result is often referred to as “half-power bandwidth.” The fundamental assumption underlying this definition is that the system performance degrades once its power decreases by 50%; moreover, there are restrictive conditions, rarely met, that render a system amenable to the use of this approach, such as linearity, low-dimensionality, low-loss, and stationary output. Here the concept of root mean square (RMS) bandwidth is considered, justified by the Fourier uncertainty principle, to generalize the definition of bandwidth to encompass linear/nonlinear, single/multi-mode, low/high loss and time-varying/invariant oscillating systems. By tying the bandwidth of an oscillatory dynamical system directly to its dissipative capacity, one can formulate a definition based solely on its transient energy evolution, effectively circumventing the previous restrictions. Further, applications are given that highlight the limitations of the traditional half-power bandwidth; these include a Duffing oscillator with hardening nonlinearity, and a bi-stable, geometrically nonlinear oscillator with tunable hardening or softening nonlinearity. The resulting energy-dependent bandwidth computations are compatible with the nonlinear dynamics of these systems, since at low energies they recover the (linearized) half-power bandwidth, whereas at high energies they accurately capture the nonlinear physics. Moreover, the bandwidth computation is directly tied to nonlinear harmonic generation in the transient dynamics, so that the contributions to the bandwidth of the individual harmonics and of inter-harmonic targeted energy transfers can be directly quantified and studied. The new bandwidth definition proposed in this work has broad applicability and can be regarded as a generalization of the traditional linear half-power bandwidth which is used widely in the sciences and engineering.

Estimation of Energy Pumping Time in Bistable Nonlinear Energy Sink and Experimental Validation

Abstract The bistable nonlinear energy sink (NES) shows high efficiency in mitigating vibration through targeted energy transfer (TET). It performs well in low- and high-energy input cases, whereas, for a cubic NES, TET occurs only above a certain energy threshold. In this work, the measure of energy pumping time is extended to a harmonic excitation case by the application of a particular integration assumption. An equivalent point in the slow invariant manifold (SIM) structure can represent the average variation of the amplitudes of linear oscillator (LO) and NES. The marked robustness of this semi-analytical prediction method under parameter perturbation is investigated numerically here. The influence of parameters on the rate at which the amplitude declines is also investigated for both impulsive and harmonic excitation. The pumping time estimation is validated in a low-energy input experimental test.

Periodic Motion and Frequency Energy Plots of Dynamical Systems Coupled With Piecewise Nonlinear Energy Sink

Abstract The frequency-energy plots (FEPs) of two-degree-of-freedom linear structures attached to a piecewise nonlinear energy sink (PNES) are generated here and thoroughly investigated. This study provides the FEP analysis of such systems for further understanding to nonlinear targeted energy transfer (TET) by the PNES. The attached PNES to the considered linear dynamical systems incorporates a symmetrical clearance zone of zero stiffness content where the boundaries of the zone are coupled with the linear structure by linear stiffness elements. In addition, linear viscous damping is selected to be continuous during the PNES mass oscillation. The underlying nonlinear dynamical behavior of the considered structure-PNES systems is investigated by generating the fundamental backbone curves of the FEP and the bifurcated subharmonic resonance branches using numerical continuation methods. Accordingly, interesting dynamical behavior of the nonlinear normal modes (NNMs) of the structure-PNES system on different backbones and subharmonic resonance branches has been observed. In addition, the imposed wavelet transform frequency spectra on the FEPs have revealed that the TET takes place in multiple resonance captures where it is dominated by the nonlinear action of the PNES.

Nonlinear targeted energy transfer: state of the art and new perspectives

Nonlinear targeted energy transfer: state of the art and new perspectives

Hamiltonian Dynamics and Targeted Energy Transfer of a Grounded Bistable Nonlinear Energy Sink

Hamiltonian Dynamics and Targeted Energy Transfer of a Grounded Bistable Nonlinear Energy Sink