Vibro-impact Nonlinear Energy Sink Research Articles

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.

Vibration suppression and modal energy transfers in a linear beam with attached vibro-impact nonlinear energy sinks

We present a comprehensive study of nonlinear resonant modal energy scattering and passive vibration suppression in a linear cantilever beam with vibro-impact nonlinear energy sinks (VI NESs) attached to it. It is well-known that vibro-impacts are a strong source of non-smooth nonlinearity, resulting in rapid and intense multi-scale energy scattering from low-to-high frequencies in the modal space of the beam. We present a direct correlation between such low-to-high frequency nonlinear energy scattering induced by vibro-impacts and vibration suppression of the beam vibrations under both sweep- and constant-frequency harmonic excitations. In particular, we study the intensity of the collisions between the tip of the beam and the particles of the VI NESs by means of an event-driven method based on an explicit variable time step integration scheme, and relate the dynamics of the integrated beam-VI-NES system to the induced resonant energy scattering from low beam modes to higher ones. On this basis, the effects of the mass ratio and clearance between the absorber and the beam on the resonant energy scattering are presented. Our aim is to perform predictive design of the VI NESs for effective and robust vibration mitigation of the beam response. To this end we perform optimization studies on the mass ratio and clearance between the absorber and beam for the case of single and multiple VI NESs and identify regions of optimal suppression. Besides, since the single VI NES is only effective in a limited frequency and amplitude range – as its dynamics is energy-dependent – using multiple VI NESs with optimized parameters can extend the frequency range of effective vibration suppression, rendering the resulting vibration mitigation more robust to variations of the applied excitations.

Comparison of a modified vibro-impact nonlinear energy sink with other kinds of NESs

Vibration mitigation is essential to many dynamical and engineering structures that are subjected to destructive vibration amplitudes induced by impulsive loading, seismic excitation, blasts, flutter, collisions, fluid–structure interaction and so on. Unprotected structures by vibration absorbers could be exposed to failure which lead to enormous losses in human lives, major equipment and economy. Employing the nonlinear targeted energy transfer (TET) concept in nonlinear vibration absorbers which are later called as nonlinear energy sinks has ignited a very rapid research interest since 2001. Up-to-date, considerable growth in the NESs field has taken place. Accordingly, various types of NESs have been introduced for vibration mitigation in variety of dynamical and structural engineering systems. The types of introduced NESs included, but not limited to, stiffness-based, rotating and the vibro-impact NESs. Among these common types of NESs, the most effective and efficient one is the single-sided vibro-impact (SSVI) nonlinear energy sink (NES). However, most of investigations has implemented a coefficient of restitution of 0.7 which closely corresponds to a steel-to-steel impact. Therefore, this paper is aimed to further improve the SSVI NES by including the coefficient of restitution in the performance optimization. Accordingly, significant improvement in the SSVI NES performance is obtained when the coefficient of restitution is found to be near 0.45. In addition, performance comparison between the enhanced SSVIe NES with several existing types of NESs is performed here where a nine-story large-scale structure is employed for this numerical comparison. Accordingly, the performance of the enhanced SSVIe NES of nearly 0.45 coefficient of restitution is found to be more robust to the initial impulsive energy levels and to its physical parameters variation than other kinds of existing NESs.

Numerical study of a symmetric single-sided vibro-impact nonlinear energy sink for rapid response reduction of a cantilever beam

This paper concerns an investigation into the control of the transient vibration of an Euler–Bernoulli beam using a symmetric single-sided vibro-impact nonlinear energy sink (SSSVI NES). The non-dimensional system of equations is derived by using the Galerkin method. Consideration and theoretical analysis of the impact dynamics of the device is carried out by introducing the impact modes. This analysis shows that the proposed SSSVI NES has increased complexity in its modes of energy dissipation compared with the single-sided vibro-impact NES (SSVI NES). Simulations are conducted with a wide variety of impulse loads to determine the optimum parameters of the SSSVI NES. The beam vibration suppression performance of the optimized SSSVI NES is then compared with both the SSVI NES and the case in which the NES is locked. When these devices have the same total mass, the SSSVI NES has superior vibration suppression performance, especially when the damping in the control device is light. The vibration suppression performance of the SSSVI NES is investigated when its location along the beam is varied. The effect of the clearance between the NES masses and the impact surface on the vibration suppression performance of the SSSVI NES is also investigated, as well as the device’s damping and the coefficient of restitution. Finally, the efficacy of the SSSVI NES device for seismic loads is investigated. The numerical results of this analysis show that the optimized SSSVI NES can effectively reduce the energy in the system and suppress the maximum bending moment and shear stress of the host cantilever beam.

Rotary-impact nonlinear energy sink for shock mitigation: analytical and numerical investigations

The nonlinear energy sink (NES) is a lightweight, strongly nonlinear dynamical attachment coupled to a (typically linear) large-scale primary structure for passive vibration mitigation. There are two nonlinear mechanisms governing the dynamics of the coupled system: irreversible targeted energy transfer (TET) from the primary structure to the NES, where energy is confined and locally dissipated, and NES-induced nonlinear energy scattering between the structural modes of the primary structure. In the literature, different NES designs have been investigated to optimize their nonlinear effects on the primary structures. One such design is the rotary NES consisting of a small mass inertially coupled to the primary structure through a rigid arm; another is the vibro-impact NES with non-smooth nonlinearities and inelastic collisions with the primary structure. These types have been found to achieve strong and rapid TET and are less sensitive to energy fluctuations. In this work, a hybrid NES design is proposed based on the synergetic synthesis of the rotary and impact-based NESs in a single rotary-impact NES (RINES). The RINES incorporates a fixed rigid barrier attached (typically) to the top floor of the primary structure to inflict impacts between its rotating mass and the top floor. An analytical study to evaluate its capacity to engage in resonance capture with a primary structure is presented first, followed by numerical investigations of cases when the RINES is attached to the top floors of small- and large-scale linear primary structures under impulsive excitation. The non-smooth nonlinearities induced through the consecutive impacts resulted in effective broadband shock mitigation at highly energetic response regimes, whereas the nonlinear inertial coupling enables similar beneficial mitigation capacity at lower-energetic response regimes. Hence, the combined effect of non-smooth and inertial nonlinearities enables effective passive mitigation capacity for a broad range of applied impulsive energies.

Design criteria for optimally tuned vibro-impact nonlinear energy sink

This paper proposes the design criteria for optimally tuned Vibro-Impact (VI) Nonlinear Energy Sink (NES) to control vibration under periodic and transient excitation. Firstly, a generalized dimensionless model of a two degrees of freedom (DOF) system comprising a harmonically excited linear oscillator (LO) strongly coupled to a VI NES is investigated. Bifurcation analysis and efficiency of Targeted Energy Transfer (TET) around the Slow Invariant Manifold (SIM) are studied with the variation of clearance. As a result, the optimal clearances for periodic and transient excitation are calculated from two transition points of the SIM, respectively. Then the procedure is extended to the case of multiple VI NESs in parallel with the LO. Two principles of additivity and separate activities of VI NESs are verified theoretically. Finally, experiments involving the whole system embedded on an electrodynamic shaker are performed. The results show that the design criteria can not only predict the efficient TET at resonance frequency, but can also achieve an optimal performance in a range of frequencies. Furthermore, the criteria can be straightforward for the application of multiple VI NESs, so as to make VI NESs work robustly under different types of excitation.

Effects of modal energy scattering and friction on the resonance mitigation with an impact absorber

A linear vibration absorber can be tuned to effectively suppress the resonance of a particular vibration mode. It relies on the targeted energy transfer into the absorber within a narrow and fixed frequency band. Nonlinear energy sinks (NES) have a similar working principle. They are effective in a much wider frequency band but generally only in a limited range of excitation levels. To design NES, their working principle must be thoroughly understood. We consider a particular type of NES, a small mass undergoing impacts and dry friction within a cavity of a base structure (vibro-impact NES or impact absorber). The nonlinear dynamic regimes under near-resonant forcing and resulting operating ranges are first revisited. We then investigate how off-resonant vibration modes and dissipation via impacts and dry friction contribute to the vibration suppression. Moreover, we assess the effectiveness of the impact absorber for suppressing multiple resonances in comparison to a linear tuned vibration absorber (LTVA) and a pure friction damper with the same mass.

On the energy transfer mechanism of the single-sided vibro-impact nonlinear energy sink

In this paper, the vibration energy transfer mechanism of the single-sided vibro-impact (SSVI) nonlinear energy sink (NES) is studied. The concept of impact mode is proposed where the device velocities are decomposed into an energy free impact mode (EFIM) and an energy dissipation impact mode (EDIM). The impact modes are applied to the analysis of a single-degree-of-freedom (SDOF) vibration system with a SSVI NES. This analysis shows that the energy in the system can be redistributed between the different impact modes and that energy transferred from the EFIM to the EDIM is beneficial, because portions of the energy in the EDIM will be dissipated by impact, while the energy in the EFIM will not. More importantly, the mechanism of the SSVI NES to realize the local dissipation of energy is revealed. Furthermore, in order to better evaluate the energy dissipation performance of the SSVI NES, a new evaluation criterion called the vibro-impact vibration reduction (VVR) factor is proposed. Then, the relationship between the VVR factor and the energy free impact mode coefficient is investigated using the Hilbert transform. Finally, the effect of the SSVI NES parameters on the energy dissipation performance of the SSVI NES with various initial conditions is discussed, and a satisfactory region for the SSVI NES design, which is identified via numerical simulation, is proposed.

Wavelet-bounded empirical mode decomposition for vibro-impact analysis

We consider the response of a single-degree-of-freedom, linear oscillator (LO) coupled to a vibro-impact (VI) nonlinear energy sink (NES), and subject to an impulsive load. A previous study of this system used the empirical mode decomposition (EMD) technique to decompose the LO’s and NES’s responses into smooth components (the first harmonic) and nonsmooth components (all other harmonics). The smooth components were used to demonstrate the existence of a 1:1 transient resonance capture (TRC) between the first harmonics of the LO and the NES. When faced with the possibility of a 3:1 TRC between the first and third harmonic, previous studies were unable to extract the third harmonic from the nonsmooth component and, thus, did not provide an analysis of its nonlinear interaction with the first harmonic. Consequently, it was conjectured that the higher-order TRCs that appeared in the response were nonphysical and the result of numerical artifacts produced by the VIs. In this work, we argue that this conjecture is incorrect, in the sense that higher-order TRCs do populate the dynamics and, moreover, that the higher harmonics and higher-order TRCs contribute significantly to the VI response. To show this, we employ a new and more powerful version of EMD, namely the recently developed wavelet-bounded EMD method, to decompose the responses into multiple smooth components, each physically representative of a single harmonic contained in the original response. The components reveal the relative contributions of each harmonic and are used to demonstrate the existence of higher-order TRCs, including the 3:1 TRC between the first and third harmonics. Hence, the present work highlights the efficacy of wavelet-bounded EMD to extract the nonlinear interactions from VI responses, a result which, to the authors’ knowledge, appears for the first time in the literature.

Chaotic characteristic of a linear oscillator coupled with vibro-impact nonlinear energy sink

The chaotic characteristic of a system with vibro-impact nonlinear energy sink is studied here. An analytical method is developed to calculate Lyapunov exponent. The mechanism by which impact results in chaos is further clarified rather than only by the calculation of Lyapunov exponent. In addition, an approach to identifying Lyapunov exponents from experimental data is proposed, and the estimated results are consistent with numerical results.

Activation characteristic of a vibro-impact energy sink and its application to chatter control in turning

The ultimate goal of this paper is to propose a procedure for the optimal design of a Vibro-Impact (VI) Nonlinear Energy Sink (NES) to control the vibration of any possible linear or nonlinear main systems. To this end, the activation characteristic of VI NES at a range of displacement amplitude of a main system is generalized from linear systems to nonlinear systems. It is theoretically proved and experimentally observed that this activation characteristic is almost independent of frequency, which provides direct proof for the effectiveness of VI NES in a broad frequency bandwidth. In terms of vibration control, this feature is very attractive and builds a bridge between linear and nonlinear systems. Then it is applied for the design of VI NES attached to nonlinear systems. In this way, the design of VI NES for a nonlinear system is simplified to the optimal design for a linear system, which is designed to be similar to this target nonlinear system. Because the latter can be analytically calculated, the proposed method is feasible from a quantitative perspective. Finally, this activation characteristic and a proposed design method are applied to control chatter in a turning process, and results prove its feasibility.

Response regimes in equivalent mechanical model of strongly nonlinear liquid sloshing

Equivalent mechanical model of liquid sloshing in partially-filled cylindrical vessel is treated in the cases of free oscillations and of horizontal base excitation. The model is designed to cover both regimes of linear and essentially nonlinear sloshing. The latter regime involves hydraulic impacts applied to the walls of the vessel. These hydraulic impacts are commonly simulated with the help of high-power potential and dissipation functions. For analytical treatment, we substitute this traditional approach by consideration of the impacts with velocity-dependent restitution coefficient. The resulting model is similar to recently explored vibro-impact nonlinear energy sink (VI NES) attached to externally forced linear oscillator. This similarity allowed exploration of possible response regimes. Steady-state and chaotic strongly modulated responses are encountered. Besides, we simulated the responses to horizontal excitation with addition of Gaussian white noise, and related them to reduced dynamics of the system on a slow invariant manifold (SIM). All analytical predictions are in good agreement with direct numerical simulations of the initial reduced-order model.

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.

Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations

A linear oscillator (LO) coupled with two vibro-impact (VI) nonlinear energy sinks (NES) in parallel is studied under periodic and transient excitations, respectively. The objective is to study response regimes and to compare their efficiency of vibration control. Through the analytical study with multiple scales method, two slow invariant manifolds (SIM) are obtained for two VI NES, and different SIM that result from different clearances analytically supports the principle of separate activation. In addition, fixed points are calculated and their positions are applied to judge response regimes. Transient responses and modulated responses can be further explained. By this way, all analysis is around the most efficient response regime. Then, numerical results demonstrate two typical responses and validate the effectiveness of analytical prediction. Finally, basic response regimes are experimentally observed and analyzed, and they can well explain the complicated variation of responses and their corresponding efficiency, not only for periodic excitations with a fixed frequency or a range of frequency, but also for transient excitation. Generally, vibration control is more effective when VI NES is activated with two impacts per cycle, whatever the types of excitation and the combinations of clearances. This observation is also well reflected by the separate activation of two VI NES with two different clearances, but at different levels of displacement amplitude of LO.

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.

On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink

A periodically forced linear oscillator with impact attachment has been studied. An asymptotical analytical method has been developed to obtain the fixed points and to analyze the transient 1:1 resonance (two impacts per cycle) of the modulated response. The influence of parameters on dynamics has been analyzed around the slow invariant manifold (SIM). Five different response regimes have been observed from theoretical and numerical results. It is demonstrated that they are closely related to the topological structure and relative position of fixed points. The bifurcation, route to chaos and the efficiency of targeted energy transfer (TET) with the variation of different parameters (i.e., amplitude and frequency of excitation, clearance, damping, mass ratio and restitution coefficient) have been investigated and well explained around SIM. Experimental results validate the existence of different regimes and different routes to chaos by the variation of the return map of time difference between consecutive impact moments. TET phenomenon has been analyzed for a strongly modulated response, and different cases of TET have been observed and analyzed. It is clearly observed that TET depends not only on whether there exists 1:1 resonance, but also on impulse strength during the transient resonance capture.

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.

Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink

This paper investigates the passive control of chatter instability in turning processes using a vibro-impact nonlinear energy sink (NES). The workpiece is assumed to be rigid and the tool is flexible. A dynamical model including a nonlinear cutting law is presented and the stability lobes diagram is obtained. The behavior of the system with the vibro-impact NES is investigated using an asymptotic analysis. A control mechanism by successive beating is revealed, similarly to the strongly modulated response in the case of NES with cubic stiffness. It is shown that such a response regime may be beneficial for chatter mitigation. An original experimental procedure is proposed to verify the sizing of the vibro-impact NES. An experimental setup is developed with a vibro-impact NES embedded on the lathe tool and the results are analyzed and validated.

Targeted Energy Transfer Under Harmonic Forcing With a Vibro-Impact Nonlinear Energy Sink: Analytical and Experimental Developments

Recently, it has been demonstrated that a vibro-impact type nonlinear energy sink (VI-NES) can be used efficiently to mitigate vibration of a linear oscillator (LO) under transient loading. The objective of this paper is to investigate theoretically and experimentally the potential of a VI-NES to mitigate vibrations of an LO subjected to a harmonic excitation (nevertheless, the presentation of an optimal VI-NES is beyond the scope of this paper). Due to the small mass ratio between the LO and the flying mass of the NES, the obtained equations of motion are analyzed using the method of multiple scales in the case of 1:1 resonance. It is shown that in addition to periodic response, system with VI-NES can exhibit strongly modulated response (SMR). Experimentally, the whole system is embedded on an electrodynamic shaker. The VI-NES is realized with a ball which is free to move in a cavity with a predesigned gap. The mass of the ball is less than 1% of the mass of the LO. The experiment confirms the existence of periodic and SMR regimes. A good agreement between theoretical and experimental results is observed.

Structural Seismic Response Mitigation using Optimized Vibro-Impact Nonlinear Energy Sinks

A nonlinear targeted energy transfer mechanism called Vibro-Impact Nonlinear Energy Sink (VI NES) is studied in the present work for seismic mitigation of building frames. In order to best tune VI NES parameters including clearance and stiffness ratio, the problem is formulated via an optimization framework. To form the objective function and trace the structural behavior, several seismic performance indices developed in previous works are employed. Carrying out nonlinear dynamic time history analyses on a number of low and medium-rise steel moment frames with an attached VI NES and performing Fourier analyses on the obtained structural responses, reveal a short reaction time and broadband energy dissipation in the passively controlled structure through the attached VI NES. VI NES parameters are also determined via the optimization process under the scaled earthquake time history record that has the closest spectrum to the target spectrum. The seismic performance indices are evaluated to test the trained (optimized) case under other earthquake records in the set and yield appropriate results.