Strongly Modulated Response Research Articles

Research on Dynamics Characteristics of Grounded Damping Nonlinear Energy Sink

The nonlinear energy sink (NES) system mainly dissipates energy through damping elements, and changing the position of the damping element will also change the performance of the NES. In this paper, a grounded damping NES is proposed by grounding the damping element of the traditional cubic stiffness NES. The complex dynamics of this two-degree-of-freedom system are investigated. The slow-varying equations of the system under 1:1 internal resonance are derived by using the complexification-averaging (C×A) method, based on which the influence of the primary structure’s damping on a bifurcation is analyzed. The conditions for the existence of strongly modulated response (SMR) are studied, and the accuracy of the results is verified using the slow invariant manifold (SIM), Poincare mapping, and time-history diagrams. This provides a means of verifying the analytical findings. The vibration suppression effects of the grounded damping NES, as compared to the cubic stiffness NES, are thoroughly studied under both pulse and harmonic excitations. The results indicate that the main structure damping affects the stability of the system and the occurrence of the SMR. Most previous studies of the NESs have overlooked the effect of main structure damping, which may influence the selection of structural parameters. Moreover, under relatively large pulse or harmonic excitations, the vibration suppression effectiveness of the grounded damping NES surpasses that of traditional cubic stiffness NES. This finding has important practical significance for improving the vibration suppression effect and robustness of the NES, and provides a reference for the structural design of the NES.

A comparative study on the torsional vibration suppression of rotor system using variable steady state nonlinear energy sink

Current studies have shown that the introduction of bistable nonlinear energy sink (B-NES) and tristable nonlinear energy sink (T-NES) can reduce the effective threshold and alleviate the separation resonance curve compared with traditional NES under the low and higher external excitation respectively. However, there is a lack of comparative studies on the performance of B-NES, T-NES and monostable nonlinear energy sink (M-NES) in the field of torsional vibration engineering application. Aimed at this limitation, a variable steady state nonlinear energy sink (V-NES) is proposed to suppress the torsional vibration of shafting. Negative stiffness is generated by the pre-compression of the spring. The proposed V-NES can be easily switched between monostable, bistable and tristable cases by the moving position of the spring bases. Based on this, the governing equation of torsional vibration coupled with V-NES after proper orthogonal decomposition (POD) reduction is derived. The approximate analytical solution of the coupled system is obtained by using complex averaging method and verified numerically. Then, the vibration reduction effects of M-NES, B-NES and T-NES on the torsional vibration of shafting are compared. Throughout the study, the motion states of strongly modulated response (SMR), interwell oscillation, and chaotic strongly modulated response (CSMR) are observed in the V-NES. Excessively high external excitation amplitudes can make V-NES exhibit CSMR, diminishing the effectiveness of vibration reduction in comparison to inter-well oscillations.

Study of low-frequency noise suppression effect inside acoustic cavities by applying piezoelectric nonlinear energy sink

In this paper, the piezoelectric nonlinear energy sink (NES) is used to suppress the radiated noise from a thin plate, and the targeted energy transfer (TET) characteristics and mechanism of the coupled system of two piezoelectric NESs and a thin plate are analyzed, and the coupling model of the system with the acoustic cavity, rectangular thin plate and piezoelectric NES is established. The parameters of the piezoelectric NESs are adjusted according to the excitation amplitude, and then applied to the rectangular thin plate. The numerical results show that the piezoelectric NES causes the system to undergo strongly modulated response (SMR) and the resonance peak of radiation noise is effectively suppressed. It provides a new technique for passive control of low-frequency noise.

Study on excitation threshold of strong modulation response and vibration suppression performance of bistable nonlinear energy sink

The dynamics and vibration reduction performance of bistable nonlinear energy sink (BNES) are studied in this paper. First, the negative stiffness of BNES is realized by geometric nonlinearity, and the dynamics model of the system is established. The slow flow equation of the system under 1:1 main resonance is analyzed based on the complexification-averaging (CX-A) method, and the boundary conditions of saddle node bifurcation and Hopf bifurcation are analyzed. Second, the slow invariant manifold (SIM) of the BNES is studied based on multiscale analysis, and the excitation threshold of strongly modulated response (SMR) is analyzed; the analysis results are verified by the numerical method. The results show that the analytical solution is highly consistent with the numerical solution, and the error is less than 1%. Finally, the influence of structural parameters on the vibration reduction performance is analyzed and optimized. The vibration reduction performance of BNES and CNES is compared, and the results show that the BNES has better vibration reduction performance in the full frequency band.

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).

Vibration analysis of a new nonlinear energy sink under impulsive load and harmonic excitation

Nonlinear energy sink (NES) is a passive energy absorption device, which plays an important role in suppressing structural vibrations. In this study, the vibration reduction effects of different NESs under different impulsive load amplitudes are revealed, and the dynamics and vibration reduction effects of the combined damping NES under harmonic excitation are analyzed. Firstly, by analyzing the vibration reduction effects of different NESs under impulsive excitation, it is concluded that the vibration suppression effect of the linear damping NES is better than that of the nonlinear damping NES when the excitation amplitude is small, and nonlinear damping NES has better vibration reduction effect than linear damping NES when the excitation amplitude is large. Meanwhile, a model of combined damping NES is proposed. By comparing the vibration reduction effects of different NESs, it is shown that the combined damping NES has a good vibration suppression effect under different impulsive load amplitudes. Secondly, by studying the bifurcation phenomenon and strongly modulated response (SMR) of the system with combined damping NES under harmonic excitation, the influence of system parameters and excitation amplitude on the dynamic characteristics is revealed. Lastly, by applying the energy spectrum, time response and Poincare mapping, it is verified that the combined damping NES has better vibration reduction effect under harmonic excitation.

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.

Investigations on magnetic bistable PZT-based absorber for concurrent energy harvesting and vibration mitigation: Numerical and analytical approaches

Bistable systems possess broadband characteristics and can have large-amplitude oscillations under low amplitude excitations. This work is aimed to investigate the potential of a bistable piezoelectric-based absorber (BPA) for the purpose of simultaneous energy harvesting and vibration suppression. The BPA composed of a cantilever beam with PZT layers and two permanent magnets. The interactions due to magnets generate bistability in the BPA. Furthermore, the absorber is connected to a main vibrating beam. The governing nonlinear equations are obtained using the Hamilton's principle and a recently developed magnetic force model. Next, numerical analyses in time and frequency domains are performed to characterize the BPA performance in various dynamical regimes. It is found that, the best performance occurs when the BPA exhibits chaotic inter-well oscillations, which lead to strongly modulated response (SMR). Furthermore, approximate-analytical investigations based on the complexification-averaging and multiple scales methods are carried out. Using these, the excitation threshold corresponding to SMR is obtained. The performance of the proposed BPA in terms of harvested power and dissipated energy are examined comprehensively. The results reveal that, the BPA efficiently suppresses the vibration and harvests power over a wide frequency band, and it outperforms a linear absorber and nonlinear energy sink.

Effects of bi-stable grounded nonlinear energy sink on energy dissipation of the system

In this paper, one of the most efficient passive absorbers, called nonlinear energy sink (NES), is analytically studied. A two-degree-of-freedom system is considered which consists of a linear oscillator (LO) with a base excitation and an NES, called grounded NES (GNES), which is connected to the ground with a nonlinear spring. In this study, we proposed a new arrangement of potential elements in GNES and studied invariant manifolds of the system, as well as the energy absorption performance of the NES. The system is considered in the vicinity of 1:1 resonance to investigate the strongly modulated response (SMR). To this end, after obtaining the equations of motion, the Manevitch complex variable and multiple scale method are applied to solve the equations, analytically. Then, the slow invariant manifold (SIM) is obtained. Also, the energy dissipation ratio of the NES and the percentage of the instantaneous total energy stored in the NES are calculated via the time-amplitude diagram. The results show that when the nonlinear effect decreases, the occurrence of energy pumping is less probable. Also, when the excitation amplitude decreases, the percentage of the instantaneous total energy stored in the NES increases as well as the amount of energy dissipation.

Importance of gravity and friction on the targeted energy transfer of vibro-impact nonlinear energy sink

The effect of gravity and sliding friction on the dynamics around targeted energy transfer of an inclined vibro-impact nonlinear energy sink (VINES) is addressed. The non-smooth dynamics is first formulated in Signorini’s contact law and treated using a Moreau-Jean scheme adapted with an energy-conserving integration method. Exhaustive discussions are then carried out regarding the vibration damping performance of the VINES, as well as their dependence on the underlying vibro-impact induced response regimes, under both transient and steady-state responses. It is demonstrated that gravity is responsible for reducing the energy barrier required for achieving targeted energy transfer in the transient response, or enlarge the existing bandwidth of the effective strongly modulated response (SMR) regimes, which is an equivalent representation of TET in the steady-state response, to improve considerably the overall damping efficiency. Some adverse effects brought by gravity and friction are also reported, and it is found that, when the incline angle or friction coefficient grows large, the bandwidth of SMR reduces again and leads to a deterioration of efficiency. The results open a door for appropriately taking advantage of the gravity into the VINES design to improve its damping performance.

Analytical Analysis for Parameter Design of Attached Nonlinear Energy Sink

The dynamic responses of a linear primary structure coupled with a nonlinear energy sink (NES) are investigated under harmonic excitation in the 1 : 1 resonance regime. In civil engineering, initial conditions are usually zero or approximately zero. Therefore, in this study, only these conditions are considered. The strongly modulated response (SMR), whose occurrence is conditional, is the precondition for effective target energy transfer (TET) in this system. Therefore, this study aims to determine the parameter range in which the SMR can occur. The platform phenomenon and other related phenomena are observed while analyzing slow-varying equations. An excitation amplitude interval during which the SMR can occur is obtained, and an approximate analytical solution of the optimal nonlinear stiffness is found. The numerical results show that the NES based on the optimal stiffness performs better in terms of control performance.

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

Nonlinear energy sink (NES) refers to a typical passive vibration device connected to linear or weakly nonlinear structures for vibration absorption and mitigation. This study investigates the dynamics of 1-dof and 2-dof NES with nonlinear damping and combined stiffness connected to a linear oscillator. For the system of 1-dof NES, a truncation damping and failure frequency are revealed through bifurcation analysis using the complex variable averaging method. The frequency detuning interval for the existence of the strongly modulated response (SMR) is also reported. For the system of 2-dof NES, it is reported in a similar bifurcation analysis that the mass distribution between NES affects the maximum value of saddle-node bifurcation. To obtain the periodic solution of the 2-dof NES system with the consideration of frequency detuning, the incremental harmonic balance method (IHB) and Floquet theory are employed. The corresponding response regime is obtained by Poincare mapping, it shows that the responses of the linear oscillator and 2-dof NES are not always consistent, and 2-dof NES can generate extra SMR than 1-dof NES. Finally, the vibration suppression effect of the proposed NES with nonlinear damping, and combined stiffness is analyzed and verified by the energy spectrum, and it also shows that the 2-dof NES system demonstrates better performance.

Nonlinear vibration and dynamic stability analysis of an axially moving beam with a nonlinear energy sink

This paper investigates the forced vibration and dynamic stability of a simply supported axially moving beam coupled to a nonlinear energy sink with the aim of passive vibration control. The equations of motion of the coupled system are solved using harmonic balance and pseudo-arc-length continuation methods. The impacts of the beam velocity, excitation frequency, and magnitude of the external force on the behavior of the system are studied. It is observed that the absorber should be designed in a way that the strongly modulated response (SMR) and weakly modulated response occur in the response of the system when the moving beam is excited near its primary resonances. The results show that the saddle node and Hopf bifurcations boundaries, as well as the absorber performance, would be reduced by increasing the beam velocity. It is also realized that the excitation frequency in which the response of the system enters the SMR region is decreased by enhancing the beam velocity, as well as the magnitude of the external force. Using the results of this paper, one can readily simulate the vibration control of simply supported moving beams utilizing a nonlinear energy sink.

On the analysis of a passive vibration absorber for submerged beams under hydrodynamic forces: An optimal design

This paper investigates the passive vibration control of immersed beams under sinusoidal hydrodynamic forces using a vibro-impact nonlinear energy sink (VI-NES). This type of absorber has a lower activation threshold compared to other nonlinear isolators. It is comprised of a clearance containing an impact element. Neither spring nor damper is required, and the restitution coefficient is the only source of damping. The equation of motion of the coupled system is analytically solved utilizing the multiple scales method and is verified via a numerical solution. The strongly modulated response (SMR), chaotic response, regimes with two impacts per cycle (symmetric and asymmetric) and more than two impacts per cycle have been observed in the nonlinear dynamic behavior of the coupled system which is associated with the location of the fixed points on the slow invariant manifold (SIM). The optimal design of the absorber is also investigated for the impulsive excitation case. Here, it is aimed to decrease the amount of time that takes the absorber to mitigate 95 percent of the primary system’s initial energy. The absorber location, restitution coefficient, clearance length, and mass ratio between the impact element and the beam are treated as design parameters. Furthermore, the influence of the design parameters on the SIM and the transient dynamic behavior of the system is also studied. The results show that the optimally designed absorber can mitigate the initial energy of the structure in a short time. It is observed that the strongly modulated response and the regime with two impacts per cycle (symmetric) are the most optimal responses. Also, comparing the performance of the mentioned absorber to the cubic NES yields satisfying results.

Flow-induced vibration attenuation of a viscoelastic pipe conveying fluid under sinusoidal flow using a nonlinear absorber

In this paper, the dynamics of a viscoelastic pipe conveying fluid attached to a nonlinear energy sink (NES) subjected to a sinusoidal flow is studied, aiming at performance improvement of such fluid-interaction systems. The NES has an essentially nonlinear cubic stiffness and a nonlinear cubic damping. Complexification-averaging and fourth-order Runge-Kutta methods are applied to solve the equations of the motion. The conditions of flow-induced instability, the weak modulated response (WMR), and the strongly modulated response (SMR) are investigated. The influence of the internal fluid velocity, the external flow velocity, the viscoelastic damping coefficient, the NES stiffness, location, and damping on the system efficiency is elucidated. The Poincare-Bendixson theorem implies that there are no closed trajectories in the phase diagram of the system response. The results revealed that the occurrence probability of the SMR and jump phenomenon in the system response increases by ascending the internal fluid velocity. Besides, it is shown that at high external fluid velocities or low viscoelastic damping coefficients, the detached resonance curve (DRC) emerges in the frequency-response curves, and simultaneously the unstable regions expand. Furthermore, by approaching the NES installation location to the pipe supports, the occurrence probability of the SMR, DRC, and WMR descends, the divergence regions are enlarged and the transient response lasts longer.

Design and experimental study of a Nonlinear Energy Sink coupled to an electromagnetic energy harvester

Nonlinear vibration absorbers, commonly referred to as Nonlinear Energy Sinks (NESs), have been the object of several theoretical and experimental studies over the past decade. This work illustrates the theoretical design and experimental realization of a Nonlinear Energy Sink coupled to an energy harvester. The mass of the Magnetic-Strung NES is a magnet which is linked to the primary system by means of two strings with adjustable pretension that work transversally. The restoring elastic force of the strings is modulated by the magnetic force applied by two magnets suitably located on the primary mass. Either a cubic or a bistable configuration may be obtained, depending on the distance of the additional magnets, NES's efficiency as an absorber is studied on a harmonically forced single degree-of-freedom primary system. The Target Energy Transfer (TET) from the primary system to the NES, as well as different response regimes like the Strongly Modulated Response (SMR), are experimentally observed. Furthermore, the harvesting of energy from the NES vibrations is also investigated by coupling the mechanical system with a coil for electromagnetic energy conversion. Consequently, the vibration energy of the primary mass is absorbed by the NES and finally converted into electric energy.

Dynamics of 1-dof and 2-dof energy sink with geometrically nonlinear damping: application to vibration suppression

Nonlinear energy sink (NES) refers to a lightweight nonlinear device that is attached to a primary linear or weakly nonlinear system for passive energy localization into itself. In this paper, the dynamics of 1-dof and 2-dof NES with geometrically nonlinear damping is investigated. For 1-dof NES, an analytical treatment for the bifurcations is developed by presenting a slow/fast decomposition leading to slow flows, where a truncation damping and failure frequency are reported. Existence of strongly modulated response (SMR) is also determined. The procedures are then partly paralleled to the investigation of 2-dof NES for the bifurcation analysis, with particular attention paid to the effect of mass distribution between the NES. To study the frequency response for 2-dof NES, the periodic solutions and their stability are obtained by incremental harmonic balance method and Floquet theory, respectively. Poincare map and energy spectrum are specially introduced for numerical analysis of the systems in the neighborhood of resonance frequency, which in turn are used to compare the efficiency of the NESs to the application of vibration suppression. It is demonstrated that a 2-dof NES can generate extra SMR by adjusting its mass distribution and hence to a great extent reduces the undesired periodic responses and provides with a more effective vibration absorber.

Experimental investigation and analytical description of a vibro-impact NES coupled to a single-degree-of-freedom linear oscillator harmonically forced

In this paper, the dynamics of a system composed of a harmonically forced single-degree-of-freedom linear oscillator coupled to a vibro-impact nonlinear energy sink (VI-NES) is experimentally investigated. The mass ratio between the VI-NES and the primary system is about $$1\%$$ . Depending on the external force’s amplitude and frequency, either a strongly modulated response (SMR) or a constant amplitude response (CAR) is observed. In both cases, an irreversible transfer of energy occurs from the linear oscillator toward the VI-NES: process known in the literature as passive targeted energy transfer. Furthermore, the problem is analytically studied by using the method of multiple scales. The obtained slow invariant manifold shows the existence of a stable and of an unstable branch of solutions, as well as of an energy threshold (a saddle-node bifurcation) for the solutions to appear. Subsequently, the fixed points of the problem are calculated. When a stable fixed point is reached, the system is naturally drawn to it and a CAR is established, whereas when no stable point is attained, the system exhibits a SMR regime. Finally, a good correlation between the experimental and the analytical results is presented.

Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation

This paper is dedicated to exploit the same optimization mechanism of targeted energy transfer under different types of excitation. Specifically, a linear oscillator (LO) coupled with a vibro-impact nonlinear energy sink is analytically studied with an asymptotical method. The optimization mechanism under periodic excitation with a single frequency and under transient excitation is numerically obtained and experimentally validated for the first time. For periodic excitation, the boundary between the regime with two impacts per cycle and that of strongly modulated response (SMR) is proved to be optimal rather than SMR. The chaotic SMR is experimentally observed from the viewpoint of displacement of LO. The above-observed mechanism is further applied to explain the optimization mechanism under transient excitation and that under periodic excitation with a range of frequency. It is experimentally verified that the optimization of the latter can be simplified to the optimization under an excitation with a single resonance frequency. For transient excitation, the efficiency of different transient response regimes is experimentally compared, which agrees with the periodic results. Moreover, the efficiency comparison of different lengths of cavity is also experimentally validated. In short, the close relation of optimization under different excitations is clearly demonstrated.

Dynamics of Cubic and Vibro-Impact Nonlinear Energy Sink: Analytical, Numerical, and Experimental Analysis

This paper is devoted to study and compare dynamics of primary linear oscillator (LO) coupled to cubic and vibro-impact (VI) nonlinear energy sink (NES) under transient and periodic forcing. The classic analytical procedure combining the approach of invariant manifold and multiple scales is extended from the analysis of steady-state resonance to other regimes, especially strongly modulated response (SMR). A general equation governing the variation of motion along the slow invariant manifold (SIM) is obtained. Numerical results show its convenience to explain the transition from steady-state response to SMR and the characteristics of SMR for periodic forcing. Targeted energy transfer (TET) under transient forcing can also be well understood. Experimental results from LO coupled to VI NES under periodic forcing confirm the existence of SMR and its properties (e.g., chaotic). They also verify the feasibility of the general equation to explain complicated case like SMR in experiments.