Asymmetric and Cubic Nonlinear Energy Sink Inerters for Mitigating Wind-Induced Responses of High-Rise Buildings

Flexible high-rise buildings with low damping are prone to excessive vibration under strong wind loads. To explore a light-weight control device having desirable mitigation efects on responses and sound robustness against deviations in tuning parameters, the performance of two novel inerter-integrated nonlinear energy sinks (NESIs), i.e., asymmetric nonlinear energy sink inerter (Asym NESI) and cubic NESI, on wind-induced vibration control of super high-rise buildings is assessed in the present work. Based on the wind loads obtained from wind tunnel tests, a super high-rise building with a 300m height is taken as the host structure in the numerical case study. Te results show that Asym NESI can achieve reduction ratios of 38.5% and 11.3% on extreme acceleration and displacement, respectively, while the sensitivity indices of Asym NESI on displacement and acceleration control are only 70.5% and 62.5% of those of tuned mass damper inerter (TMDI) having identical mitigation efects. Although the sensitivity indices of cubic NESI are only 5.5% and 29.8% of those of TMDI, the moderate mitigation efects and large nonlinear stifness ratio may prohibit its practical implementation. Overall, Asym NESI could be an alternative to TMDI due to the same mitigation efects but better robustness against possible detuning.


Introduction
Vibration control of fexible high-rise buildings subjected to wind loads has attracted the attention of researchers for decades [1].Especially for high-rise buildings with aspect ratios over 3, the across-wind responses usually exceed along-wind responses due to vortex shedding efects [2], and vibration control strategies are required to satisfy survivability and serviceability criteria.
To mitigate wind-induced vibration, employing passive vibration control devices is one of the widely-adopted strategies attributed to the independence of supplemental energy, low cost, easy implementation, and so on [3,4].Among the passive devices, tuned mass damper (TMD) has drawn extensive attention due to its simplicity and efectiveness and has been implemented in a great number of real-life buildings [5].However, the trend of increasing height of newly constructed high-rise buildings implies a rapidly increasing weight of the structure, in which case a signifcantly heavy attached mass of conventional TMD is demanded to fulfll desirable control performance [6].To alleviate the demand of massive physical masses of control devices, a novel inertia element termed inerter [7] was integrated to conventional control devices, and a series of linear inerter-based vibration absorbers (IVAs) emerged thereafter [3], e.g., tuned inerter damper [8,9], TMDI [10,11], multiple tuned mass damper inerters [12,13], tuned liquid column damper inerter [14,15], tuned tandem mass dampers inerters [16,17], tandem tuned mass damper inerter [18], and inerter-connected double tuned mass damper [19].Beneftting from the considerable mass amplifcation efect, i.e., the apparent mass of an inerter can be thousands of times its physical mass [20], IVAs were commonly reported to outperform their conventional competitors on both response reduction and robustness [14,21,22].Among various IVAs, TMDI is one of the most studied IVAs due to its direct relation with traditional TMD.Giaralis and Petrini [23] numerically investigated the control performance of TMDI on the displacement and acceleration responses of a 76-story building under across-wind loads simulated by an empirical spectrum [2].Te multistory spanning TMDI was found to outperform TMD in terms of better mitigation efects and signifcantly smaller strokes.Based on the wind loads obtained from wind tunnel tests, Wang et al. [24,25] further evaluated the mitigation efects of TMDI on wind-induced vibration of a tall building at 24 wind directions.Considering the practicality of the physical mass of an inerter, a tuned inerter damper (TID) was reported to outperform TMDI and TMD having the same device mass under the assumption that the inertance of an inerter is 200 times its physical mass.Besides investigating the performance, the strategies of parametric design for IVAs were also developed.Based on a simplifed singledegree-of-freedom (SDOF) model, Su et al. [26] derived the closed-form tuning formulae of TMDI considering an arbitrary installation layout of TMDI via a flter-based approach.Kaveh et al. [27] adopted a meta-heuristic optimization algorithm, i.e., colliding bodies optimization (CBO) algorithm, to fnd the optimal parameters of TMDI for controlling a 10-story benchmark building.Further, Kaveh et al. [28] compared the parameters and control performance of TMDI optimized based on the simplifed SDOF model and multi-degree-of-freedom (MDOF) model, pursuing a maximum reduction on the H ∞ norm of the roof displacement transfer function.Te results showed that the MDOF model is recommended to be employed for TMDI design, while the SDOF model should be adopted with caution.Since modal coupling efects were neglected in the process of establishing the simplifed SDOF model, Qiao et al. [29] adopted the Sherman-Morrison matrix inversion operation to derive the explicit forms of entries in the matrix of frequency response functions of the controlled structure based on the MDOF model and, correspondingly, proposed a new parametric design strategy.
Although the robustness of linear IVAs was commonly reported to be better than their conventional counterparts, the linear nature of IVA implies the inherent defciency of detuning; that is, the desirable control performance of IVAs still relies on a proper tuning [26], and thus IVAs are sensitive to the deviations in tuning parameters.It is noted that the additional aerodynamic damping and aerodynamic stifness of high-rise buildings under strong wind loads could alter the dynamic behaviors of the primary structure.Hence, passive IVAs probably encounter detuning issues during their service life.In this regard, it is important to seek control devices having competitive control performance to IVAs with more sound robustness.
In comparison to linear control devices, whose mitigation efects are mainly attributed to the resonant oscillation around the to-be-controlled vibration mode of the primary structure, the nonlinear nature of nonlinear energy sink (NES) endows them with the capability to resonate and irreversibly absorb energy from multiple vibration modes [30,31] and thus generally exhibits attractive vibration mitigation efects but better robustness than linear absorbers like TMD [32].With the ongoing proposals of IVAs, researchers started to introduce inerter to NES to alleviate the requirement of a heavy attached mass and enhance the control performance of traditional NESs.Zhang et al. [33] arranged an inerter to a cubic-type stifness element and a dashpot element of NES in a parallel layout and proposed NESI.Under ground motion excitations, the better control performance of NESI than NES was validated in terms of amplitude-frequency response and energy absorption.Zhang et al. [34] proposed another NESI by replacing the attached mass of NES with a grounded inerter.Since this NESI behaved like an NES having an attached mass of b (inertance of the inerter) and b was usually designed to be much larger than the attached mass of traditional NES, the grounded NESI showed better vibration suppression performance under ground motion excitations than NES.In the same year, the grounded NESI was also independently proposed by Javidialesaadi et al. [35].Besides the better control performance of NESI than NES, the incorporation of a grounded inerter showed a greater enhancement in the control performance of NES than that of TMD.However, all NESIs above were developed by introducing an inerter to cubic-type NES, and its drawback was still inherited in the novel cubic NESI; that is, satisfactory performance was only achieved at a suitable level of input energy.To address this defciency, Wang et al. [36] connected the attached mass of an asymmetric NES and an arbitrary DOF of the structure with an inerter and named the device after Asym NESI.In a numerical case of a three-story steel-frame structure subjected to seismic loads, the robustness of Asym NESI was found to be better than TMDI and cubic NESI against changing frequencies of the host structure and was less sensitive to variations in initial velocity than cubic NESI.Further, Wang et al. [37] updated the equations of motion for Asym NESI by integrating the damping embedded in the inerter part and validating it through experiments.Te updated model was used to investigate the control performance of Asym NESI in the seismic design of a three-story steel-frame structure, wherein Asym NESI exhibited similar control performance as TMDI and strong robustness against variations in structural property and energy level.
Te vibration mitigation efects of both Asym and cubic NESIs were only studied under ground motion excitations, while to the best of the authors' knowledge, no investigations on the parametric optimization and performance assessment of NESIs for wind-induced vibration control have been conducted so far.To address this gap, the present work investigates and compares the mitigation efects and robustness of both Asym NESI and cubic NESI with those of TMDI for wind-induced vibration control.Having formulated the equations of motions of the NESI-controlled MDOF structure based on a planar lumped mass model, the performance indices regarding mitigation efects and robustness are introduced.Based on a real case of a tall building along with wind loads obtained from wind tunnel tests, the optimal parameters of Asym NESI, cubic NESI, and 2 Structural Control and Health Monitoring TMDI are numerically obtained and discussed.Te control performance of the optimum representatives of the three devices is further scrutinized.After evaluating the robustness of control performance against deviations in tuning parameters, practical considerations concerning the realization of nonlinear springs and the initial cost of the devices are discussed.

Mathematical Model of Asym NESI-Controlled High-Rise Buildings Subjected to Wind Loads
Te translational motion of a high-rise building is described by a planar lumped mass model as shown in Figure 1(a), which is broadly adopted in the performance analyses of control devices for the wind-resistant design of high-rise buildings [17,23,38] due to its low computational cost and ease of reproducibility.Since some important issues, e.g., axial deformation and nonlinear geometry, are not considered in the simplifed mode, further analyses based on the fnite element model are still required before the implementation of full-scale control devices.An Asym NESI is installed on the j th foor (j � 1, . .., n, n is the total number of foors) and connected to the i th foor (i � 1,. ..,n) by an ideal inerter having an inertance of b (unit: kg).Te attached mass weighting m Asym is connected to the j th foor by a nonlinear spring having a stifness coefcient of k nl , a linear spring having a stifness coefcient of k l , and a dashpot element having a damping coefcient of c l .In comparison to cubic NESI in Figure 1(b), the nonlinear spring of Asym NESI is prestretched by a set distance r to generate an asymmetric restoring force, and the linear spring is used to keep the mass statically balanced.
Particularly, the realization of nonlinear spring is detailedly introduced to underpin the feasibility of NESIs.One of the most commonly adopted realization approaches of nonlinear spring is arranging a pair of linear springs in serial, in which way the nonlinear restoring force is generated due to the geometric nonlinearity.As illustrated in Figure 2(a), the nonlinear restoring force F Rnl is provided by two linear springs having stifness coefcients of k and unstretched lengths of L.
When the joint moves away from its balance point at displacement x, F Rnl can be explicitly expressed as By Taylor-expanding the fraction term in equation (1) about x � 0, we obtain Tus, the two linear springs behave like a nonlinear spring having a nonlinear stifness of k nl in Figure 1(b) when O(x 5 ) is neglected, as expressed in the following equation: ( Further, when the joint is pulled to a distance of r from its original balance location using a linear spring having a stifness coefcient of k l , the spring system of Asym NESI is obtained, as shown in Figure 2(b).In this system, r can be fexibly adjusted to the design value by changing the deformation length of the linear spring, i.e., x l , as expressed in the following equation: Two photographs of cubic NES and Asym NES (without inerter) are presented in Figure 3 to illustrate the spring systems.
Besides NESIs, TMDI is also presented in Figure 1 as it is set as a competitor to two NESIs in the present work.Particularly, Asym NESI can be regarded as a general form of TMDI and cubic NESI, since TMDI and cubic NESI can be retrieved by setting k nl � r � 0 and k l � r � 0, respectively.In this regard, only the equations of motion of the Asym NESIequipped structure are introduced hereafter.
Te total restoring force of Asym NESI on the j th foor can be expressed as where x Asym is the displacement of the attached mass relative to the j th story.f s � −k nl r 3 is the initial force in the nonlinear spring when x Asym is equal to zero.By substituting f s into equation ( 5), the total restoring force can be rewritten as Tus, the equations of motion of high-rise buildings controlled by Asym NESI under wind loads can be expressed as Structural Control and Health Monitoring x Asym (t) where    Structural Control and Health Monitoring where [M s ], [C s ], and [K s ] are (n + 1)-by-(n + 1) matrices formulated by augmenting a row vector and a column vector of zero elements to the bottom and the rightmost of the mass, damping, and stifness matrices of the uncontrolled structure, respectively.Only the p th (p = 1,. ..,n + 1) element in the (n + 1)-by-1 vector {l} p is equal to one, while the others are zeros.Te superscript T denotes the transposition operation.
Considering the nonlinear terms introduced by Asym NESI, equation ( 7) is numerically solved by using the incremental Newmark-β method with the Newton-Raphson iteration method at each time step.Te initial displacement and velocity of all DOFs are zero.Assuming that the responses of the nonlinear system follow a Gaussian distribution under excitations of weakly stationary wind loads, the extreme value  X of displacement and acceleration responses can be estimated based on the sample of time history as follows [39]: where X and σ x are the mean value and standard deviation of an observation sample, respectively.g is the peak factor evaluated by [40] g � 2 ln ηT w   1/2 + 0.577 where T w is the time duration of the observation sample and η is the efective frequency for structural responses, which can be conservatively taken to be equal to the frst natural frequency of the uncontrolled structure in Hz [40].
Further, a reduction ratio R is defned in equation ( 11) to quantitatively evaluate the mitigation efects of control devices on the wind-induced responses, where  X c and  X uc are the extreme values of responses of the controlled and uncontrolled structures, respectively.
Having in mind that the performance of passive control devices highly relies on tuning parameters, a sensitivity index S is introduced in equation ( 12) to evaluate the robustness of R against deviations in tuning parameters.
where R up is the reduction ratio of the control device whose parameters are not perturbed and R p,j is the reduction ratio of the control device whose parameters are artifcially modifed under the j th set of perturbations (j � 1, . .., J).Te distribution of lumped mass along height is shown in Figure 5(a), wherein the jumps in masses of specifc foors are attributed to the strengthened structural components and emergency devices of refuge foors.As the lateral stifness along the y-axis is smaller than that along the x-axis, preliminary analyses of wind-induced responses show that the responses along the y-axis are larger than those along the x-axis.Tus, only the stifness along the y-axis (cf. Figure 5) is adopted to establish a planar lumped mass model as presented in Figure 1(a).Te frst natural frequency of the planar model of the high-rise building is f 1 = 0.143 Hz, which is close to 0.138 Hz of the FE model.Rayleigh damping is adopted to formulate the damping matrix with an assumed damping ratio of 5% for the frst two modes when calculating displacement but a smaller value of 2% when calculating acceleration.Tis diference is due to the consideration that the wind loads for calculating displacement correspond to a return period of 50 years and are larger than that for calculating acceleration (10-year return period), in which case the structural components are expected to dissipate more energy.

Wind Loads Obtained from Wind Tunnel Tests.
Te synchronous multipoint pressure measurement wind tunnel tests on the scaled high-rise building were carried out in the boundary layer wind tunnel laboratory.Te geometric scale ratio of the model is 1 : 400.According to the location of the building, the C-type wind feld simulating an urban area [41] is generated by roughness elements and spires, as presented in Figure 6(a).Te profles of mean wind velocity and turbulence intensity of the C-type wind feld are presented in Figure 6(b), wherein H and U represent an arbitrary height and the corresponding mean wind speed.H T , U T , and I U are the reference height, wind speed at H T , and turbulence intensity, respectively.A good match between the mean wind speed profle measured in the test and that calculated following the power law with α = 0.22 [41] can be observed.
Te coordinates of wind directions α w are shown in Figure 7. Wind pressures are measured at a total of 36 wind directions at an interval of 10 °.Te specifc parameters of the wind tunnel tests are listed in Table 1.
At the wind direction of 0 °, time histories of the components of aerodynamic forces acting on the 20 th and 60 th foors along the y-axis (across-wind loads) are shown in Figure 8. Te aerodynamic forces on the 60 th foor in Figure 8(b) are mainly contributed by vortex shedding efects and have a mean value of approximately zero.In contrast, Figure 8(a) shows that the mean value of aerodynamic loads on the 20 th foor is nonzero, which may be attributed to the aerodynamic interference of the surrounding buildings as shown in Figures 6(a

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In Figure 9, f and B represent the frequency and alongwind characteristics length of the building, respectively.S p (f ) and σ p 2 are the PSD and variance of aerodynamic forces, respectively.Figure 9 indicates that the energy of aerodynamic forces along the across-wind direction is distributed more narrowly than that along the along-wind direction and mainly distributed around the dimensionless frequency of 0.12, corresponding to the Strouhal number for a square cross-section [42].

Parametric Optimization.
To make a fair comparison among the performance of Asym NESI, cubic NESI, and TMDI, parametric optimizations for the three control devices are carried out.
According to Figure 1(b), a total of eight dimensionless parameters of Asym NESI need to be optimized, i.e., mass ratio μ � m Asym /M tot (M tot is the total mass of the primary building), inertance ratio β � b/M tot , stifness ratio of linear spring κ l � k l /(m Asym + b)ω 2  1 (ω 1 denotes the frst natural circular frequency of the high-rise building), stifness ratio of nonlinear spring κ nl � k nl /(m Asym + b)ω 2  1 , the set distance r, damping ratio ζ � c l /2(m Asym + b)ω 1 , the installation foor of the Asym NESI j, and the downward spanning foors of inerter tp = j − i (cf. Figure 1(a)).
Both mitigation efects and robustness at the most unwanted wind direction (0 °) are pursued in the optimization, which is refected by simultaneously minimizing four By referring to the optimal values reported in the literature regarding IVAs and NESIs [24,25,32,34,36], a constrained multi-objective optimization problem (CMOP) expressed in equation ( 13) is formulated to mathematically describe the optimization problem.Notably, tp is strictly limited below 5, considering the great practical difculties in realizing a control device spanning too many foors.Te upper bound value 4 is determined by referring to the existing pendulum-like TMDs, e.g., the TMD in Taipei 101 tower (spanning 4 foors) [43] and the eddy current TMD in Shanghai Center Tower (spanning more than 20.6 m) [44].
Similarly, CMOPs for cubic NESI and TMDI are defned under the same constraints.All three CMOPs are solved by using the built-in MATLAB ® function, i.e., "gamultiobj," which is a variant of the nondominated sorting genetic algorithm (NSGA-II) [45].It is noted that the solutions of CMOPs are mostly a set of nondominated individuals having preferences on diferent objectives (known as the Pareto set), where there does not exist a single solution that can simultaneously achieve minimum (or maximum) on all objectives.Te four objectives of solutions form a 4dimensional (4D) Pareto front (PF) in the objective space, Structural Control and Health Monitoring and radar plots are used to demonstrate the preferences of all optimal individuals on diferent objectives.To make an intuitive comparison among the objectives of the three devices, the four objectives of the i th individual are transformed into dimensionless form following equations ( 14)-( 17) and shown in Figure 10.
where | x 69 | uncontrolled and € x 69,uncontrolled are the extreme displacement and acceleration on the top foor of the uncontrolled structure, respectively.max( S dis  ) and max( S acc  ) are the maximum values of S dis and S acc among all optimal individuals of the three control devices, respectively.Besides optimal solutions, positive ideal solution (PIS) and negative ideal solution (NIS) are also depicted to demonstrate the limits of each device on the four objectives.PIS and NIS are generated by collecting the minimal and maximum values of all optimal solutions, respectively.Representatives having a good balance among the four objectives are selected and highlighted in red dash-dot lines for the three control devices.
In Figure 10, Asym NESI and TMDI share a similar PIS, indicating that the performance limits of these two devices on each objective are almost identical.PIS of cubic NESI has a similar r 3 and r 4 to those of Asym NESI and TMDI, but its r 1 and r 2 are larger than those of the other two, which demonstrates that the potentials of cubic NESI on decreasing responses are worse than those of Asym NESI and TMDI.However, r 3 and especially r 4 of NISs of Asym NESI and TMDI are larger than those of cubic NESI, which suggests that cubic NESI could be a better choice when robustness on acceleration control performance is the prior design requirement.
Further, the variations of r 3 and r 4 with respect to r 1 and r 2 of the optimal individuals are shown in Figure 11 by projecting the 4D PF on the r 1 -r 3 and r 2 -r 4 planes to demonstrate the relationship between mitigation efects and robustness.
In Figure 11(a), the maximum reductions in displacement and acceleration (corresponding to the minimum r 1 and r 3 ) achieved by Asym NESI and TMDI are close, while cubic NESI is less efective on vibration control than the other two.Between Aysm NESI and TMDI, there mostly exists an optimal individual of Asym NESI that has a similar r 1 or r 2 to that of TMDI with a smaller r 3 or r 4 .Tis indicates that Asym NESI could be an alternative to TMDI for similar mitigation efects but more sound robustness.Although performing poorly on mitigating responses, cubic NESI generally exhibits better robustness on acceleration control, as shown in Figures 10(b) and 11(b).
To provide guidance on the parametric design of NESI for wind-induced vibration control, the optimal parameters and objectives of the three representatives marked in Figure 10 are listed in Tables 2 and 3, respectively.
In Table 2, the optimal β of all three devices is close to or has reached the upper bound of 1, indicating that the incorporation of inerters will enhance the overall performance.Asym NESI and TMDI share similar κ l and ζ, which signifes that Asym NESI is designed to resonate with the frst mode.Besides, κ nl of 67.94 is designed for Asym NESI to enhance its overall performance.In contrast, the optimal κ nl of cubic NESI is signifcantly larger than κ l and κ nl of the others, which may hinder its practical implementation.Unlike NES and TMD which are usually designed to be installed on the top [25,46,47], all three inerterincorporated control devices are optimized to be installed in the middle portion of the building with a multistory spanning layout, which leverages the enhancement efects of inerters [21,48].
Further, the objective values of representatives of the three devices are reported in Table 3.As presented in Figure 11, Asym NESI and TMDI have almost identical mitigation effects and outperform cubic NESI in reducing responses.Regarding robustness, the sensitivity indices of cubic NESI on displacement and acceleration control are only 5.5 % and 29.8 % of those of TMDI, respectively, and the corresponding values of Asym NESI are 70.5 % and 62.5 %'.

Assessment on Mitigation Efects of Asym NESI, Cubic
NESI, and TMDI 3.4.1.Amplitude-Frequency Curves.To investigate the dynamic properties of controlled structures over a broad frequency band, the amplitude-frequency relationship for acceleration and displacement responses on the top foor controlled by two NESIs are numerically calculated through harmonically forced vibrations, while the amplitudefrequency curves of the uncontrolled and TMDIcontrolled structures (i.e., moduli of frequency response functions for linear systems) are analytically calculated [48].
Te results are shown in Figure 12.
Figure 12(a) shows that Asym NESI and TMDI signifcantly decrease the peak values at the frst natural frequency and rationally achieve desirable reductions in displacement responses since the frst mode dominates the wind-induced vibration.Cubic NESI failed to efectively reduce the frst peak and perform worst among the three.Regarding control performance on higher modes, all three control devices have neglectable reductions on the second peak, and minor decreases on the third peak are observed.Similarly, all three absorbers can decrease the peak value of the amplitudefrequency curve at the frst natural frequency in Figure 12(b), and both Asym NESI and TMDI perform better than cubic NESI.Notably, the amplitude-frequency curve of the structure controlled by Asym NESI is almost identical to that of the TMDI-controlled structure in the present case, implying that the whole system behaves almost linearly.In Figure 13, all three absorbers can decrease acceleration responses at both wind directions, wherein Asym NESI has identical performance as TMDI and outperforms Cubic NESI.Between the two wind directions, the control performance of three devices on across-wind vibration is better    than that on along-wind vibration.Tis can be explained by observing Figures 9 and 12; that is, the resonant components contribute more to the across-wind responses and can be mitigated to a larger extent due to the signifcant reduction in amplitude-frequency curves in Figure 12(b).In contrast, the contribution of background components to the alongwind responses is larger than that to across-wind responses.Tus, the minor reduction in amplitude-frequency curves at low frequencies leads to worse control performance than that on across-wind responses.
Further, extreme wind-induced acceleration responses on the top foor at all 36 wind directions are estimated and shown in Figure 14.
In Figure 14, all three control devices can reduce the extreme acceleration responses of the high-rise building at all wind directions.Among the three, Asym NESI and TMDI show desirable mitigation efects at all wind directions, while cubic NESI can only achieve moderate reductions around 0 °.

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x 69 at typical wind directions at 0 °, 90 °, 180 °, and 270 °is further provided in Table 4. Corresponding reduction ratios R are calculated via equation (11) and reported in the parenthesis.As explained before, the mitigation efects of the three devices on across-wind responses are better than those on along-wind responses.
Figures 15(a) and 15(b) show the variation of extreme acceleration along the height at wind directions of 0 °and 90 °, respectively.Asym NESI shares similar control performance as TMDI and efectively decrease  € x from the middle-lower portion to the top of the building at the two wind directions.In contrast, cubic NESI shows moderate mitigation efects at the wind direction of 0 °but exhibits poor mitigation efects at the wind direction of 90 °.

Wind-Induced Displacement
Responses.Figures 16(a) and 16(b) show 10-minute segments of the time histories of wind-induced displacement on the top foor at wind directions of 0 °and 90 °, respectively.
Similar to the results in Figure 13, Asym NESI shares almost identical performance as TMDI, and both control devices outperform cubic NESI.In comparison to control performance on acceleration responses, only moderate reductions at the wind direction of 0 °are achieved by the three devices, and limited mitigation efects at the wind direction of 90 °are observed.Te worse control performance on displacement than that on acceleration can be explained via Figure 12.Specifcally, the amplitude-frequency curves at low frequency in Figure 12(a) are horizontal, which makes the contribution of background components to displacement responses larger than that to acceleration responses, and thus result into worse mitigation efects.
In Figure 17(a), the three passive vibration absorbers fail to reduce mean responses.Te mean displacement around 0 °and 180 °approaches zero, since the wind loads are mainly generated by the vortex shedding efects and have an approximately zero mean.In contrast, reductions in STD value attributed to the fuctuating components are observed at all directions in Figure 17(b).Similar to the mitigation efects on acceleration responses, the three control devices perform better at 0 °and 180 °than 90 °and 270 °(reported in Table 5), as the mean responses contribute more to along-wind responses than across-wind responses.
Further, the variations of extreme displacement along the height of the building at the wind directions of 0 °and 90 °are depicted in Figure 18.Diferences in control performance on across-and along-wind responses are observed again and not discussed for the sake of brevity.

Robustness Analysis.
Considering the great uncertainties embedded in the dynamic characteristics of super high-rise buildings due to construction, aerodynamic stifness, aerodynamic damping, and other factors, the control performance of the vibration absorbers probably sufers deterioration due to detuning.To evaluate the infuences of detuning on the mitigation efects, 21 × 21 − 1 � 440 sets of perturbations are exerted to ζ and κ l for TMDI and ζ and κ nl for cubic NESI.Diferently, Asym NESI has three tuning parameters, i.e., ζ, κ l , and κ nl , and a total of 441 × 21 − 1 � 9260 sets of permutation are considered.Same as the calculation of robustness indices, the 440 sets of perturbations for TMDI and cubic NESI are all permutations of α Z � 0.8 : 0.02 : 1.2 and α K � 0.8 : 0.02 : 1.2 except for α Z � α K � 1.0.An additional vector of permutation coefcient α Knl � 0.8 : 0.02 : 1.2 is considered for κ nl of Asym NESI.
To quantitatively analyze the robustness of these vibration absorbers, a performance index of perturbed control device is defned as where R p and R up are the reduction ratios of the vibration absorber whose optimal parameters (cf.Table 2) are perturbed or not, respectively, at the wind direction of 0 °.
Following this defnition, PI < 1 indicates that the mitigation efect is weakened, and an enhancement efect is achieved when PI > 1.
Figure 19 shows the variation of PI of Asym NESI, cubic NESI, and TMDI for displacement and acceleration control with α Z , α K , and α Knl .
In Figure 19, Asym NESI and TMDI are sensitive to variations in α K but get less infuenced by a changing α Z .Between the two, Asym NESI shows better robustness on both displacement and acceleration control in terms of minor variations in PI in Figures 19(a) and 19(d) than TMDI (cf.Figures 19(c) and 19(f )).Diferently, the robustness of cubic NESI is almost equally afected by both α Z and α Knl as the contour lines share an inclined angle of about 45 °.By comparing the values of PI, cubic NESI shows overall better robustness than Asym NESI and TMDI except for the leftupper zone in Figure 19(e).
Particularly, the infuences of κ nl on PI are shown in Figure 20.In Figure 20 (α Knl � 1.0) and Figures 20(a) and 20(d), the valleys corresponding to low values move downward with an increasing α Knl .Since the stifness ratio is still tuned beside the frst-order frequency of the structure under α K � 0.8 : 1.2, makes the resonance of Asym NESI with the frst mode of the structure the main contributor to the control performance, Asym NESI still exhibits a larger sensitivity to variations in κ l than ζ.

Structural Control and Health Monitoring
In addition, it is noticed in Figure 11(b) that an optimum Asym NESI reaches a reduction ratio of 35.7% on acceleration with a marginal S acc of 1.68 × 10 −5 , which is much smaller than those reported in Table 3.Although this individual can only decrease the extreme displacement to 27.94 cm, the desirable mitigation efects and sound robustness makes it an attractive choice from the perspective of demand-based design when controlling acceleration is of the greatest concern.Te parameters of this individual are listed in Table 6.
From Table 6, this Asym NESI has a much smaller κ l of only 0.02 in comparison to 0.88 of Asym NESI rep .In this way, the nonlinear nature of Asym NESI is expected to be more evident, and its robustness against variations in κ l is rationally enhanced.
Further, the variations of PI of this Asym NESI with α Z , α K , and α Knl for displacement and acceleration control are shown in Figure 21., where E is Young's modulus of the material.Taking steel with E � 210 GPa as an example, the relationship between the diameter and length of cable (i.e., d and L) to realize the nonlinear stifness k nl is shown in Figure 22.
In Figure 22, the required d of Asym NESI is 1/5.87 that of cubic NESI, indicating that the incorporation of an additional linear spring signifcantly alleviates the high requirement of a large k nl and makes it easier to manufacture the nonlinear spring of Asym NESI than cubic NESI.Although an excessive d is still required for both devices when L is large, multiple parallel cables can be adopted to decrease the diameter of individual cables for easy realization.
Subsequently, the initial (upfront) cost of considered three control devices is assessed in terms of two nonmonetary metrics, i.e., device mass and the maximum output forces of mechanical components [49][50][51].Te evaluation on other terms of life-cycle cost, e.g., maintenance cost, falls beyond the scope of the present work and is not analyzed.
From Table 2, Asym NESI owns the lightest auxiliary mass which is 84.0% and 85.7% of cubic NESI and TMDI, respectively.Te inerters of the three control devices share almost identical inertance.Te maximum output forces of the four components, i.e., inertial force of inerter F I , damping force of dashpot element F D , restoring force of nonlinear spring F Rnl , and restoring force of linear spring F Rl , are calculated under wind loads at a return period of 50 years following equations ( 19)- (22).
where € x j,α w , € x Asym,α w , and € x j−tp,α w are the acceleration time histories of j th DOF, DOF of the foor on which Asym NESI is installed, and (j − tp) th DOF at the wind direction of α w , respectively.Te dots over characters indicate time derivative.x l = k nl r 3 /k l is the stretched length of the linear spring at the static balance point.All 36 wind directions from 0 °to 350 °are considered.
Following equations ( 19)-( 22), the maximum output forces of the four mechanical elements in the three devices at all wind directions are calculated and listed in Table 7.
From Table 7, F I and F D of Asym NESI are 102.5% and 111.5% times those of cubic NESI, respectively, but F Rnl of Asym NESI is only 52.8% that of cubic NESI.Although F Rl of Asym NESI is also 74.3% of F Rnl of cubic NESI, the easier realization of a linear spring than a nonlinear one makes Asym NESI more attractive than cubic NESI.In comparison to TMDI, F D and F Rl of Asym NESI are 90.0%and 94.7% of those of corresponding values, while its F I is 123.1% of that of TMDI.Generally speaking, TMDI requires the lowest initial cost among the three, assuming that realization of 1 N of the four forces needs the same cost.Considering that the cost of realizing F Rnl is probably larger than realizing F Rl , Asym Structural Control and Health Monitoring NESI would be superior to cubic NESI due to a much smaller F Rnl .

Conclusions
Tis paper scrutinized the mitigation efects and robustness of Asym NESI and cubic NESI on wind-induced vibration of high-rise buildings in terms of a detailed numerical case study with TMDI being set as a competitor.Te detailed fndings are summarized below.Due to the great difculties in analytically investigating the nonlinear dynamic behaviors of NESI-controlled MDOF structure, numerical integration and numerical search algorithm are employed to solve the CMOPs targeting both mitigation efects and robustness to obtain the optimal parameters of Asym NESI, cubic NESI, and TMDI.Te optimization results show that Asym NESI and TMDI can signifcantly reduce the extreme displacement and acceleration on the top foor at the most unwanted wind direction.Benefting from the nonlinear stifness, Asym NESI exhibits better robustness against perturbations on the design parameters than TMDI having the same mitigation efects.Cubic NESI achieves overall the best robustness on displacement and acceleration control at the cost of the worst mitigation efects on responses and a much larger nonlinear stifness than that of Asym NESI.
Tree representatives of Asym NESI, cubic NESI, and TMDI are selected in terms of a good balance between mitigation efects and robustness.Te representative Asym NESI has an optimal κ l of 0.88, which is close to κ l � 1.14 of TMDI.Te optimal κ nl of Asym NESI is 62.7 and signifcantly smaller than 2.31 × 10 3 of cubic NESI.Te optimal mass and inertance ratios of the three devices are close to or have reached the upper bound, indicating that a larger effective mass contributes to overall better performance.Attributed to the incorporation of inerter, the optimal installation layout of the three inerter-integrated devices is to be installed in the middle portion of the structure and spanning multiple foors.
Regarding mitigating efects, the three vibration absorbers can mitigate the wind-induced responses at all 36 wind directions.Te reduction ratios of the three representatives on extreme displacement are 11.3%, 6.5%, and 11.8% at the wind direction of 0 °, respectively, and corresponding reduction ratios on extreme acceleration are 38.5%,23.4%, and 38.7%.At the wind direction of 90 °(along-wind direction of the structure), the reduction ratios of the three control devices on extreme acceleration decrease to 26.1%, 8.2%, and 27.1%, respectively, and the corresponding values on extreme displacement are only 4.7%, 1.0%, and 4.8%.Tese diferences are mainly attributed to the distinct distribution of energy of across-and along-wind loads and the dynamic behaviors of the controlled structure.
By artifcially changing ζ, κ l , and κ nl , the robustness of the three optimum representatives is evaluated.Te sensitivity indices of Asym NESI on displacement and acceleration responses are 70.5% and 62.5% of those of TMDI having the same reduction ratios.Cubic NESI shows better robustness than Asym NESI but exhibits poor mitigation efects.Considering both control performance and optimal parameters, Asym NESI could be an alternative to TMDI due to the competitive mitigation efects but better robustness against detuning.
Since extreme acceleration is more concerned in windresistant design, the robustness of an optimum Asym NESI having a reduction ratio of 35.7% but extraordinary robustness on acceleration control is particularly assessed.In comparison to the representative Asym NESI having balanced performance on all objectives with a κ l � 0.88, the results suggest that adopting an Asym NESI with a small κ l � 0.02 could result in considerable enhancement in robustness at the cost of minor deterioration on acceleration control performance.
Practicality analyses indicate that the incorporation of a linear spring in Asym NESI efectively reduces its realization difculty in comparison to cubic NESI and probably makes Asym NESI more cost-efective than cubic NESI.TMDI is also attractive due to its desirable control performance and low initial cost evaluated in terms of device mass and maximum output forces of mechanical components.

Figure 1 :
Figure 1: (a) Lumped mass model of a high-rise building controlled by an Asym NESI; (b) mechanical layouts of TMDI, cubic NESI, and Asym NESI.

Figure 4 :
Figure 4: (a) Geometric sizes and (b) FE model of the high-rise buildings.
objectives, i.e., absolute extreme displacement on the top foor | x 69 | � |x 69 | + gσ x69 , extreme acceleration on the top foor  € x 69 � gσ € x69 , sensitivity of control performance on displacement responses S dis , and sensitivity of control performance on acceleration responses S acc .Specifcally, S dis and S acc are calculated by considering eight sets of perturbations {α Z , α K } exerted on ζ and κ nl for Cubic NESI and ζ and κ l for TMDI, i.e., ζ perturbed � α Z ζ, κ nl,perturbed � α K κ nl , and κ l,perturbed � α K κ l .Te eight sets of perturbations are all permutations of α Z � {0.8, 1.0, 1.2} and α K � {0.8, 1.0, 1.2} except α C � α K � 1.0.Particularly, perturbations on only ζ and κ l of Asym NESI are considered in the optimization stage to accelerate the calculation of objectives.Infuences of variations in κ nl of Asym NESI will be discussed in Section 3.4.

Figure 8 :Figure 9 :
Figure 8: Time histories of aerodynamic forces (a) on the 20 th foor and (b) the 60 th foor at the wind direction of 0 °.

10 Structural Control and Health Monitoring 3 . 4 . 2 .
Wind-Induced Acceleration Responses.Figures13(a) and 13(b) present a 10-minute segment of the time histories of the acceleration responses on the top foor of the high-rise building at wind directions of 0 °and 90 °, respectively.

Figure 12 :Figure 13 :
Figure 12: Amplitude-frequency curves of (a) displacement and (b) acceleration responses on the top foor under harmonic excitation (amplitude � 2.5 × 10 5 N) acting on the top foor.

Figure 15 :
Figure 15: Variation of extreme acceleration along the height at wind directions of (a) 0 °and (b) 90 °.

Figure 16 :Figure 17 :
Figure 16: Time histories of displacement on the top foor at wind directions of (a) 0 °and (b) 90 °.

Figure 18 :
Figure 18: Variation of extreme displacement along the height at wind directions of (a) 0 °and (b) 90 °.

Figure 22 :
Figure 22: L and d for realizing nonlinear springs of NESIs.

Table 1 :
Parameters of wind tunnel tests.

Table 3 :
Objective values of the representatives of Asym NESI, cubic NESI, and TMDI.

Table 2 :
Optimal parameters of the representatives of Asym NESI, cubic NESI, and TMDI.

Table 6 :
Parameters of an Asym NESI having desirable mitigation efects and sound robustness on acceleration control.

Table 6 )
18th α Z and α K when (d) α Knl � 0.8, (e) α Knl � 1.0, and (f ) α Knl � 1.2 for acceleration control.18StructuralControlandHealthMonitoringAs expected, this Asym NESI shows much better robustness in controlling both displacement and acceleration responses than Asym NESI rep , which is refected by the maximum and minimum values of contour lines.Diferent from the results in Figures19 and 20, this Asym NESI is more sensitive to variations in ζ and less sensitive to variations in κ l , as contour lines which are almost vertical in most fgures.

Table 2 ,
k nl of Asym NESI and cubic NESI is 1.27 × 10 7 and 4.37 × 10 8 kN/m, respectively, and k l of Asym NESI is 1.65 × 10 5 kN/m.When a metal cable having a circular cross-section with a diameter of d used to provide the stifness of k � Eπd 2 /4 in the nonlinear spring, we

Table 7 :
Maximum output forces of the four mechanical components.