An electromagnetic vibro-impact nonlinear energy sink for simultaneous vibration suppression and energy harvesting in vortex-induced vibrations

An electromagnetic vibro-impact nonlinear energy sink (EM-VINES) is proposed in the application of vortex-induced vibration (VIV), for both purpose of vibration suppression and energy harvesting. The considered system consists of a cylinder-like bluﬀ body subject to an oncoming ﬂow, coupled to a magnet attachment moving in coil of gap enclosure. The ﬂuid-structure interaction is treated using the classical Van der Pol oscillator model, and the non-smooth dynamics is formulated in a measure diﬀerential complementarity problem adapted with a Moreau-Jean time integration scheme. Comprehensive analyses are then conducted concerning the targeted energy transfer mechanism, as well as the internal competition of the energy ﬂow. A performance indicator is deﬁned over the lock-in region, to obtain the optimal balance between vibration suppression and energy harvesting. It is found that when the system is working in a strongly modulated regime with less than 2 impacts per cycle, a fast-scale targeted energy transfer could be activated over the whole lock-H


Introduction
Vortex-induced vibration (VIV) refers to a special fluidstructure coupling phenomenon, that can be widely observed in various engineering structures such as tall buildings and chimneys, power transmission lines, and off-shore risers.A classical example consists in a flexible bluff body subjected to an oncoming flow: the structure can exhibit near-resonant motions under a certain range of flow velocities, termed as lock-in regime [1].A comprehensive review for understanding the VIV phenomenon of engineering structures and predicting their dynamics within the lock-in regime has been systematically documented in [2,3].
In the vast majority of engineering applications, large amplitude VIVs are unwanted and detrimental.As such, the vibration controlling problem represents a fundamental issue that has been widely discussed in the studies concerning VIVs, where a large number of active and passive methods were proposed [4][5][6][7].In terms of the passive methods, one representative strategy that has gained much attention recently is the so-called Nonlinear Energy Sink (NES) [8][9][10], which is a light-weight nonlinear device that approaches vibration suppression through Targeted Energy Transfer (TET) phenomenon.According to the type of nonlinearity it possesses, NES can be classified into purely cubic NES [11], bistable NES [12,13], piecewise linear NES [14,15], vibro-impact NES [16,17], and inertial NES [18], et.al.Thanks to the TET phenomenon, an NES can resonate with any frequency of vibrations of the primary system, thus showing a broadband property and has also been proved to be effective in various externally excited and selfexcited engineering systems [10,19,20].Being strongly nonlinear, NES systems can produce complex dynamics, including rich bifurcation structures, co-existence of periodic solutions, quasi-periodic regimes, and chaotic responses [21,22].Under certain conditions, A unique response regime known as the strongly modulated response (SMR) can be observed, whose existence is vital in vibration mitigation [23,24].
Prospects of adding an NES to suppress VIV have been extensively studied in [25][26][27][28][29], by considering a classical nonlinear system including a circular cylinder that exposed to uniform flow and coupled to an NES.In [26], an equivalent reduced-order model is developed by combining the finite element method with a data-driven approach.The authors reported there are mainly two mechanisms associated with VIV suppression: namely partial LCO suppression and SMR.Similar findings are also addressed by Mehmood et al in [26] and Blanchard et al in [28].In [26], a linear wake oscillator was considered neglecting all the nonlinear terms.However, the study is limited to only specific values of the flow velocity within the lock-in region, which is insufficient to understand the global behavior of the NES.Dai et al [29] used a separate Van der Pol wake oscillator with acceleration coupling to model the fluid loads acting on the host cylinder, and their numerical investigation obtained SMR as the prominent response in the lock-in region leading to the reduction of cylinder amplitude.Following this line and recently in [30], an analytical treatment based on a complex averaging technique has been successfully employed to reveal the SMR mechanism of the coupled fluid-structure-NES system, confirming the numerical findings in [29].
In addition to active and passive VIV suppression, energy harvesting from VIVs also attracts much attention in recent years.In this realm, the effects of VIV can be beneficial, or even critical, to achieve efficient harvesting of kinetic energy from the fluid motion.For example, Dai et al [31], and Wang et al [32] theoretically investigated the VIV phenomenon for wind energy harvesting.It is seen that in certain low-speed wind region, a lock-in regime occurs to activate resonant VIV in structural dynamics, which is beneficial to wind energy harvesting.Other efforts conducted to enhance the performance of VIV-based energy harvesting can also be found in the literature, see for example the shape optimization in [33,34], the use of multi-stable nonlinearity in [35] and the self-tuning strategy in [36].Despite these successful examples, the development of VIV-based energy harvesting is, however, remains at a very early stage.Current studies mainly focus on structural modification to improve the VIV strength, none of them has considered using the NES as a much more efficient transduction mechanism to improve harvesting performance.
Summarizing the current investigations, one can find that the topics of vibration suppression and energy harvesting concerning VIVs are independently considered, using different devices and methods.Up to now, there is still no such design that can realize simultaneously both vibration suppression and energy harvesting.On the other hand, the applications of NESs in VIVs also concentrate only on vibration suppression, but lack of attention in energy harvesting as it should be.Motivated by the above facts, this paper thus aims at using the NES to achieve both vibration suppression and energy harvesting in VIVs.Such an idea not only brings new dynamical problems such as the strongly nonlinear behaviors of NES in VIVs and the unknown coupling mechanisms between vibration suppression and energy harvesting, but also has instructive meaning for the integrated design of smart systems in engineering fields.
Our key to approaching this idea resides in the concept of EM-VINES [17,37], which is a novel type of NES proposed by the present author that combines a vibroimpact mechanism to obtain fast TET with an electromagnetic transduction method to generate electric energy.The ability of the EM-VINES for both vibration suppression and energy harvesting has been theoretically proved in [37], while the emphasis now will be on examining its application potential in VIVs.As such and without loss of generality, we will consider a coupled system consisting of a host cylinder structure under oncoming airflow and an attached EM-VINES, and the structure of the paper is organized as follows: Section 2 is devoted to the dynamical modeling of the considered system.Then in section 3, a comprehensive numerical investigation will be conducted to uncover the working mechanisms of the EM-VINES in VIVs.Section 4 deals with the optimization problem by highlighting the balance between vibration suppression and energy harvesting.Finally, the concluding remarks are summarized in section 5.

Problem formulation
As shown in Fig. 1(a), the considered system is composed of a single degree of freedom host cylinder structure coupled to an impact magnet.The host cylinder is assumed to have height H and diameter D, with an elastical support defined by stiffness k and damping c, and subjected to an oncoming air flow with constant velocity U along the y direction, so that the vortex induced vibration is activated along the x direction.There is a coil-fixed clearance 2∆ equiped on the host cylinder, inside which the magnet is attached, see Fig. 1(b).With such implementation, the permanent magnet can interact with the coil and undergo twosided impacts when moving inside the clearance, thus acting as an EM-VINES [37] with abilities for both purposes of vibration suppression and energy harvesting.The mass of the cylinder and the EM-VINES are respectively M − m and m, such that the total mass of the system is M .The governing equations for such a coupled system depicted in Fig. 1 can be written as where x 1 and x 2 denote respectively the displacement of the host cylinder and the attached EM-VINES.F V IV represents the vortex-induced force acting on the circular cylinder, F c is the contact force between the magnet and the clearance, and F e stands for the electromagnetic interaction.
The complexity of the system shows multiple dependencies on the geometrical, mechanical, and electrical parameters, a dimensionless model is then introduced to reduce the parameter space and generalize the analysis.It reads where All our analysis hereafter will be focused on this dimensionless model presented in Eq. (2).

Electromagnetic force
The Electromagnetic force F e , which opposes the relative motion between the host cylinder and the EM-VINES, is due to the induced current that flows into the coil.It is by acting against F e , that the mechanical energy is converted into electrical energy.More precisely, by connecting the coil with a load resistance R load , an equivalent harvesting circuit shown in Fig. 1(c) can be obtained.Based on Ohm's law, the induced current i is computed via: where R coil represents the internal resistance of the coil.U s is the induced electromotive force, which is determined according to Faraday's law, with k t the so-called flux linkage gradient.To simplify the problem for our analysis, we assume that k t is a constant, i.e., the time variation of the magnetic field generated by the EM-VINES can be neglected.On the other hand, applying Lenz's law, the F e can be expressed in terms of the induced current i as Combing the three equations Eq. (4)-Eq.( 6) yields F e = k2 t ( ẋ2 − ẋ1 ) / (R load + R coil ), meaning that the electromagnetic force can be equivalently viewed as a linear damping force with an electrical damping coefficient c e = k 2 t / (R load + R coil ).Parallely, the corresponding non-dimensional f e in Eq. (2) reads with ξ e = ce

Vortex-induced force
There are mainly two classes of techniques to predict the behaviors of vortex-induced vibrations, solving the Navier-Stokes equations [25,27,38] or establishing reduced order models [29,30,39,40].The former directly deals with the coupling between the main structure of the cylinder and the fluid, thus can provide very accurate predictions, but also requires huge computation costs.Instead, the latter does not compute the fluidstructure coupling problem but models its principal features using an approximated dynamical system, thereby reducing the order of the original problem and simplifying the analysis.For our situation, in order to better simplify the problem and keep focus on the effect of EM-VINES, the reduced order model method referring to the developments in [30, 39,40] is performed, wherein the vortex-induced force is represented as where C D is the steady mean drag coefficient, C L0 is the steady mean lift coefficient.D is the dimeter of the cylinder structure, ρ s is the air density, U is the speed of the air flow.Finally, q(t) is the wake variable, and can be modeled by the modified Van der Pol oscillator equation as q + λω s (q 2 − 1) q + ω 2 s q = P η1 with ω s = 2πS t U/D is the vortex shedding frequency, and S t is called the Strouhal number, λ and P are constents, and can be determined experimentally.

Contact force
The treatment for the contact dynamics comprises a Signorini condition to determine the occurence of a contact, and the Newton impact law to define the state change after the contact.Contact occurs when the EM-VINES approaches either side of the clearance constraint, given as According to the Signorini contact condition, the corresponding force f c in Eq. ( 2) could be modeled as implying that either The contact dynamics involve velocity jumps, which is described by the Newton impact law, in terms of the relative velocity η2 − η1 , as where η+ i (t) and η− i (t) are respectively the right and left limit for the velocity ηi at the time t for i = 1, 2, and r ∈ [0, 1] is the restitution coefficient that defines the contact loss.It should be emphasized that in this expression, the velocity function ηi (t) is thus assumed to be right continuous with η+ i (t) = ηi (t) and of bounded variation.
A differential measure d ηi , instead of the classical ηi , is then introduced to provide a more rigorous definition for the acceleration due to the discontinuities in velocity: where dt is the Lebesgue measure of time t, and dσ = i a i δ ti is a discrete measure with given sequences a i and δ ti .δ s stands for the Dirac measure supported at t.In this expression, d ηi (t) = ηi (t) dt almost everywhere, while at contact instants t * one has d ηi = η+ i − η− i dδ t * .In this way, the reaction force can also be rewritten in terms of dσ, where f c stands for the continuous contact force and Γ is the impulse corresponding to velocity jumps.Altogether, the complete dynamics of the system are thus given by: dv where the associated matrices are Eq. ( 14) is known as a measure differential complementarity problem.It is then numerically solved using a Moreau-Jean time integration method to handle the non-smooth dynamics and associated with an energy conserving scheme for the linear parts, so as to finally get the time response of the system.The details are not shown here for sake of conciseness, one can see [17] for more information.
3 Fundamental mechanisms for VIV based vibration suppression and energy harvesting using EM-VINES For the nonlinear system depicted in Fig. 1, high amplitude resonant vibrations of the cylinder shall be activated in a certain interval of vortex velocity U which is referred to as lock-in in the literature.Our main purpose is to examine the possible vibration suppression and energy harvesting potential of the attached EM-VINES when the host structure is in the lock-in regime.As it has been proved in [17,37], the fundamental mechanisms for the EM-VINES to approach effective vibration suppression and energy harvesting rely on the vibro-impact induced TET, as well as the internal competition of energy flow.As such, the main focus of this section is thus to provide a comprehensive analysis on the lock-in regime, TET, and the internal competition of energy flow, so as to understand the complex dynamics of the coupled system and evaluate the overall efficiency of the attached EM-VINES.
In order to carry out the analysis, some geometric and material parameters for the system are listed in Tab. 1.These values are obtained thanks to a relevant previous research in [39], where a system of a cylinder under airflow is investigated numerically and experimentally, and observed that when the non-dimensional parameters in the Van der Pol model are selected as λ = 4.2 and P = 28, the predictions by the proposed model can agree very well with the experiments.Thus, the same has been adopted in our study.Besides, for all the numerical simulations later on, the sampling frequency is selected as f s = 10 5 , which is determined following a preliminary convergence study in order to ensure accurancy on the numerical simulation of nonsmooth dynamics.The other parameters not listed in Tab. 1 will be detailed in the later simulations.

lock-in regime
One can imagine that adding an EM-VINES may bring influences on the lock-in regime in two aspects: changing the region for the occurrence of lock-in, or affecting the overall performance within the lock-in region.To see whether this is the case, Fig. 2 and Fig. 3 plot respectively the variation trends of the vibration amplitude η 1 and the electric power P elec , as functions of the flow velocity U .It is seen in Fig. 2 that in the absense of the attached EM-VINES, the lock-in of the host cylinder alone can be identified as U ∈ [1.14, 1.35] m/s, as verified by the fact that the vibration amplitude η 1 within this region is much higher than that in the other regions.When the EM-VINES is introduced to the system, the whole lock-in region undergoes a right shift to U ∈ [1.18, 1.38] m/s.Combining Fig. 2 and Fig. 3 and considering the performance changes within the lock-in region, one can find that the addition of an EM-VINES also changes dramatically the behavior of the system within lock-in, it results in a reduction on the vibration amplitude and produces a considerable electric power output.This indicates that in the lock-in region, the EM-VINES can be capable for both vibration suppression and energy harvesting.
In order to further distinguish how the characteristic parameters of the EM-VINES affect the lock-in, Fig. 4(a-d) displays the dependence of η 1 on variations of mass ratio ϵ, electric damping ξ, clearance ∆, and restitution coefficient r, respectively.One can clearly see that increasing the mass ratio can lead to a right shift of the lock-in, while increasing the electric damping narrows the lock-in region.On the other hand, the vibro-impact parameters ∆ and r, although have significant influences on the performance within the lock-in region, they do not affect the existence interval of lockin.

Targeted energy transfer
The analysis in the last subsection evidenced that the addition of an EM-VINES can realize both the vibration suppression and energy harvesting in the lock-in region, while leaving an open question regarding the mechanisms that govern the effectiveness of the EM-VINES.In fact, as it has been reported in [17,37], For such an EM-VINES depicted in Fig. 1.Its effectiveness for either vibration suppression or energy harvesting is awaited upon the same fundamental nonlinear mechanism termed as targeted energy transfer.The main purpose of this subsection is to determine the conditions for the occurence highly efficient TET, so as to provide possible guidelines for the design of EM-VINES.
To begin with, we introduce the following indicator E dif f to quantify the TET efficiency, where E tot refers to the total energy dissipation by the whole coupled system (i.e., including both the host cylinder and the EM-VINES), while E nes accounts for the part contributed by the EM-VINES.Such an indicator is also widely used in many other NES systems, and has been proven to be successful in evaluating the TET efficiency of many different types of NESs, the interested reader may see the discussions in [41].
Fig. 5 plots the above performance indicator E dif f as a function of ∆, for air velocity selected as U = 1.25 m/s.As ∆ changes, four distinctive response regions can be classified (denoted as I-IV) according to the contact conditions.The first region I is the trivial linear region, in which vibro-impact is not activated, and the TET efficiency E dif f is simply identical to the ratio ξ e /(ξ + ξ e ) = 33%, meaning that no effective TET can be observed in this linear regime.When ∆ decreases and crosses the threshold value ∆ = 0.09, the system undergoes a dramatic change, vibro-impact dymamics is now activated to produce a non-periodic response regime, resulting in a significant increase on the TET efficiency up to almost 80%, such high efficiency can  keep in the whole region II.With further decreasing ∆ into the interval [0.057, 0.067], the system then enters into another region III remaining high efficiency TET at around 70%, but with a periodic regime.Finally, when ∆ < 0.057, the TET efficiency decreases again.
One important conclusion that can be summarized from Fig. 5 is that, the response regimes in region II and III give the most efficient TET.It is thus necessary to take a deeper look into these two response regimes, so that to better understand the TET mechanism induced by the EM-VINES.As an example, the time response of the system in region II with ∆ = 0.08 and in region III with ∆ = 0.06 are then presented in Fig. 6(a) and Fig. 6(b), respectively.It is observed that the nonperiodic response performed in region II is a Strongly Modulated Response (SMR), where the amplitude of η 1 shows a strong modulation and the vibro-impact condition is observed to be less than 2 impacts per cycle.As for the periodic regime in region III, one can find that in each period there are exactly two vibro-impacts.Recalling the notations in reference [37], we call such a pe-riodic regime as VI-2 response.Thus, in order to obtain highly efficient TET within the lock-in region, the EM-VINES should be tuned to make it working in a condition either with less than 2 impacts per cycle to produce SMR response or with equal to 2 impacts per cycle to have VI-2 response.Note that these findings agree very well with the existing documents related to NESs.For example, the evidence of SMR for highly efficient TET could be broadly found in various types of NES systems under forced vibrations, and the importance of SMR and VI-2 responses for a conventional VINES to realize TET is also reported by many other authors, see [42][43][44][45].Especially, in [17,37], it is also proved that SMR and VI-2 are the most important two regimes for TET in EM-VINES under harmonically forced vibrations.
In order to uncover how the observed TET is affecting the vibration suppression and energy harvesting performance of the EM-VINES, let us consider at first the most simple case with r = 1 and R coil =0, for which all the energy absorbed by the EM-VINES can be fully converted to the electric energy.The main results are summarized in Fig. 7, where two performance indicators are introduced, and plotted as functions of the clearance ∆.The vibration suppression performance is described by the amplitude of the host cylinder η 1 , and its variation trend is plotted in the brown line, while the energy harvesting performance, is represented by the output electric power P elec and plotted in the blue line.Furthermore, the Targeted energy transfer efficiency, as well as the vibro-impact condition is also displayed in the upside subfigure.It can be seen that, when vibro-impact is activated to achieving high efficicency TET, both the vibration suppression and energy harvesting performance are improved, as verified by the fact that η 1 reduces and P elec increases when the system enters into the vibro-impact regime.On the contrary, when TET is not effectively realized, regardless in the linear regime without impact or in the regime with inappropriate impact conditions (more than 2 impacts per cycle), the performance is not desirable.This result clearly addresses the importance of TET to the performance of EM-VINES for both vibration suppression and energy harvesting, i.e., the effectiveness of the EM-VINES is not simply dependent on whether there is vibro-impact, but more importantly on whether the vibro-impact condition is able to achieve TET.In addition, once TET is optimized, the two corresponding indicators η 1 and P elec are optimized, meaning that in this simplified situation with r = 1 and R coil =0, the purposes of vibration suppression and energy harvesting are in essential consistent, they exhibit the same variation trend depended directly by TET efficiency.
Extending the current conclusions from U = 1.25 m/s to the whole lock-in region, one can find that the main observations in Figs.5-7 also valid within the whole lock-in region, see Fig. 8. Whatever the velocity is, the performances of vibration suppression and energy harvesting always consistently affected the TET efficiency, implying that the consistency between vibration suppression and energy harvesting is a common fact in all the lock-in region.On the other hand, Fig. 8 also shows that the lock-in is independent of impact condition.

Competition of energy flow
The previous analysis has revealed the importance of TET by focusing on the special simplified case of r = 1 and R coil =0.It was emphasized that in that case, all the energy transferred into the EM-VINES can be fully converted to electric energy and harvested, the vibration suppression and energy harvesting purposes thus show a strong consistent relationship that is both directly dependent on the TET efficiency.However, in the general case of r ̸ = 1 and R coil ̸ = 0, the picture is usually more complicated.The major difference is that the energy transferred to the EM-VINES is now not completely harvested, but flows into three parts: (i) dissipated by the VI damping due to vibro-impact loss, (ii) dissipated by the coil resistance R coil , and harvested by the load resistance R load .While all the three components could contribute to vibration suppression, only the final part flows into R load shall be counted for energy harvesting.As the three components undergo unknown competition, the consistent relationship between vibration suppression and energy harvesting no more hold, and their performances may also no longer be solely dependent on TET efficiency, but also affected significantly by competition of the enegy flow.Therefore, in this subsection, our main purpose is to investigate the competition of the energy flows, so as to clarify the important factors that determine the vibration suppression and energy harvesting performances.
To carry out our analysis step by step, let us first consider the energy flow of the system.One can easily prove that in the permanent regime, the energy of the system satisfies the following balance relationship: P tot = P main + P coil + P load + P V I (17) where P tot refers to the total energy that is dissipated by the system (or equivalently, the input energy ), P main is the contribution of linear damping in the host cylinder, P coil and P load account respectively for the electric power of the load resistance and the coil resistance, and finally P V I is the loss due to vibro-impacts.With such definitions, and by normalizing all the values with respect to the total power P tot , one can easily determine the contribution of the energy flow in each part.In addition, the TET efficiency E dif f in Eq. ( 16) could also be rewritten now as The first parameter to be investigated is ∆, which plays the most important role in depending on the contact conditions and response regimes of the system.Fig. 9 illustrates the competition relationship of the energy flow with varying ∆.With a relatively large ∆, the system works in the linear regime without viboimpact, yielding P V I = 0 and the competition among P coil , P load , and P main is governed by a simple proportional relationship determined by ξ/ξ e and R load /R coil .When ∆ is in the interval [0.05, 0.07], vibro-impact occurs and SMR is activated.As a consequence, P V I shows a dramatical increase and keeps increasing as ∆ decreases in this region, meanwhile the contribution of P load and P coil also increases.Thus, each component that contributes to TET efficiency shows an increasing trend in the SMR regime as compared to that in the linear regime, leading to a sharp shrink on P main and the majority of the energy is hence absorbed by EM-VINES.This provides an explanation to the observation in the last section from a viewpoint of energy competition, that why the TET efficiency improves dramatically when SMR is activated, i.e, the components P coil , P load and P V I gains competitive advantages versus P main .With ∆ decreases to the next interval [0.04, 0.05], the system enters into the VI-2 regime, in this regime, P V I reaches to its highest value, P load and P coil drop slightly as compared to that shown in SMR, but still at a relatively high level.In general, in VI-2, highly efficient TET could still remain thanks to the excellent contribution of P V I .Finally, when ∆ is smaller than 0.04, the VI-n regime (similar the definition of VI-2, but here n > 2, meaning the nonlinear regime with more than 2 impacts per cycle) is performed, one can see that although the contribution of P V I do not change a lot, P coil and P load show significant decreases and lose their advantage to P main , resulting in a decrease on TET efficiency.
Another important parameter is in the electrical domain, the ratio R load /R coil .Its possible effects on the energy competition is comprised of two aspects: first of all, R load /R coil determines the voltage distribution in the energy harvesting circuit, thus affect directly the local power competition between P coil and P load ; secondly, the variation of R load /R coil may also lead to a change of response regimes of the coupled system, thus may have a global effect on the competition of each energy component.To verify this hypothesis, an example is shown in Fig. 10, in which the competition relationship is now plotted as a function of R load /R coil .
Clearly enough, when R load /R coil increases, the system response undergoes a gradual change from linear to SMR, to VI-2, and finally to VI-n.Considering the global effects, one can see that at each response regime region, the competition relationship is similar to that was observed in Fig. 9, i.e, P coil , P load and P V I gains competitive advantages versus P main in SMR and VI-2.
As for the local competition between P coil and P load , it is found that by increasing R load /R coil , the total contribution of electric power P coil +P load decrease, but as on the other hand the part P load gains more advantages in the circuit, the final result makes that P load could keep increasing in the linear and SMR region even though the sum of P coil + P load shows a decreasing trend.Fig. 11 displays the varying trends of vibration and energy harvesting performance due to the competition relationships given in Fig. 9 and Fig. 10.The results confirm again the fact that due to the internal competition of energy flow, the consistency of vibration and energy harvesting can not hold for the general case, and the performances no more simply depend on the TET efficiency.What one can observe from Fig. 11 is that, by taking into account the energy competition, the purpose of vibration suppression and energy harvesting also shows a competition relationship, which means that one can not simply obtain the optimal performance for both vibration suppression and energy harvesting.There should be a balance between the two purposes in the optimization of the system performance, and this will be discussed in the next section.

Balanced performance optimization for vibration suppression and energy harvesting
In order to quantitatively evaluate the performance of the EM-VINES, so to realize an optimal balance between vibration suppression and energy harvesting in the lock-in region, the following performance indicator is introduced The physical meaning of the indicator is to maximize the output power P load harvested by the load resistance while keeping the vibration energy of the host structure at the lowest level.From this sense, the both purpose of vibration suppression and energy harvesting are combined in one indicator, and once the indicator is optimized, one is expected to achieve the optimal balancing between vibration suppression and energy harvesting.
Based on the indicator defined in Eq. ( 18), we mainly consider optimizing the parameter ∆ and R load /R coil .First of all, one optimization example for r = 0.95 is shown in Fig. 12(a).Here, the optimum is reached when ∆ = 0.05 and R load /R coil = 3.5, with optimized indicator I opt = 247.6.Then the comparison of I opt at different r is presented in Fig. 12(b), and it is proved that r = 0.95 gives the best results among other cases.The response of these optimized results is then plotted in Fig. 13, it can be found that in the lock-in region, the vibration of the host cylinder could be largely reduced, and the harvesting power could be kept at a high value.
By checking the response regimes, it is found that the optimum is corresponding to the SMR response.
Finally, the robustness of the optimized results with respect to initial conditions is also considered, and the main results are performed in Fig. 14.For this analysis, the performance indicator I at ∆ = 0.05 and R load /R coil = 3.5 are calculated for different initial conditions [η 10 , η 20 − η 10 ].More precisely, in Fig. 14 noting the average value and the error bar illustrating the standard variation.It is seen that there is a slight fluctuation on the performance indicator I when changing the initial conditions, but the overall performance remains at a high level.Moreover, by selecting the worst case and the best case and comparing their performance in both vibration suppression and energy harvesting, the results are then given in Fig. 14(b-c).It is also concluded that two cases can give very close performances within the lock-in region, meaning that the optimized result behaves robustly under the change of initial condition.

Conclusion
By considering a nonlinear system consisting of a host circular cylinder and an EM-VINES attachment, we examined the potential of an EM-VINES for both vibration suppression and energy harvesting in vortexinduced vibrations.The complex dynamics of the coupled system including lock-in regime, TET mechanisms, as well as competition of energy flow, are comprehensively investigated, in order to optimally design the EM-VINES.
It is found that the lock-in region is dominated by linear parameters like the mass ratio and the linear electric damping but does not depend on the vibro-impact parameters such as clearance size and impact loss.The influence of vibro-impact parameters is mainly reflected in the dynamical behavior within the lock-in region, i.e., on the TET efficiency and competition of energy flow.There are mainly two regimes that can be responsible for activating efficient TET, namely the SMR regime (a non-periodic regime with less than two impacts per cycle) and the VI-2 regime (a periodic regime with 2 impacts per cycle), where the SMR represents a most efficient regime for the electric damping to gain advantages among other terms in the competition of energy flow.As such, when considering simultaneously vibration suppression and energy harvesting, the SMR is more effective than VI-2.When working under the SMR regime, the EM-VINES can reach an optimal balanced performance for both effective vibration suppression and energy harvesting over the whole lock-in region.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.

Fig. 1
Fig. 1 Schematic of a host cylineder structure subjected to an oncoming flow of velocity U , and coupled to an EM-VINES.(a): the coupled system, (b): zoom for the EM-VINES, (c): the harvesting circuit.

Fig. 2 1 .Fig. 3
Fig. 2 Vibration damping performance of an EMVINES over the lock-in region, black is the reference case without VINES, and red is the case with VINES with clearance ∆ = 0.08.(a): average amplitude of the host cylinder η 1 over the lock-in region, (b): time response of the host cylinder at U = 1.25 m/s.System parameters are aligned as ϵ = 0.05, ξ = 0.002, ξ e = 0.001, r = 1.

Fig. 12 Fig. 13
Fig. 12 Balanced performance optimization based on the indicator I defined in Eq. (18).(a): contour plot of I shown for r = 0.95, (b): Comparison of optimized results I opt at different r.System parameters are aligned as ϵ = 0.05, ξ = 0.002, ξ e = 0.001.

Fig. 14
Fig. 14 Robust analysis with respect to initial conditions of optimized results, (a): indicator I varies as function of initial conditions [η 10 , η 20 − η 10 ], where each value of η 20 − η 10 ranging from -0.05 to 0.05, the performance indicator I is calculated at four different initial displacement η 10 = 0.01, 0.04, 0.07, 0.1, and the circle is the average value and errorbar indicating the standard variation.(b-c): comparision of the amplitude η 1 and the harvested power P load between the worst case (green) and best case (blue) shown in subfigure (a).

Table 1
Parameters used in the numerical simulation