A New Methodology for Optimal Design of Hybrid Vibration Control Systems (MR + TMD) for Buildings under Seismic Excitation

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Introduction
Excessive vibrations can generate critical damage to the building structure and even structural collapse, which causes economic losses and, the most serious, life losses. Regarding seismic loads, when the excitation frequencies align with the building's frst natural frequencies, resonance phenomena can occur, leading to potentially catastrophic damage. In this setting, vibration control emerges as a viable alternative for mitigating these vibrations to levels deemed acceptable, according to the normative criterion adopted, as in specifcations of NBR 15421 [1] used in Brazil, in NSR-10 regulation [2] adopted in Colombia, and in ANSI/AISC 360− 16 [3] in the United States.
Control systems can be categorized into passive, active, semiactive, and hybrid confgurations [4][5][6]. Concerning the semiactive system, it gathers characteristics from passive and active devices, necessitating minimal energy input to generate control forces. In hybrid systems, a blend of passive and active, or passive and semiactive devices, can be employed to regulate structural responses, thereby achieving optimal performance and yielding a highly efcient structural control [6].
One of the semiactive devices frequently utilized, which has been extensively investigated by numerous researchers over the past three decades, is the magneto-rheological damper (MR), which can generate controllable damping forces through the application of electrical current in magneto-rheological fuids (MRF). According to [7], the application of a magnetic feld to MRF leads to a modifcation in their mechanical properties. MR devices have found diverse applications in a range of structures and contexts. For instance, these devices have been integrated into the Dongting Lake Bridge located in China [8], incorporated within the Nihon-Kagaku-Miraikan Building situated in Japan [9], and utilized in a residential building in Japan where they were combined with a baseisolation system, resulting in a hybrid control mechanism [10]. Additionally, MR devices have been employed on the Eiland Bridge in the Netherlands [11], serving as a shock isolation system to supplant a conventional passive shock isolation system for commercial-of-the-shelf (COTS) equipment operating within demanding military tactical environments [12]. Furthermore, they have been implemented as semiactive primary suspensions for heavy trucks [13], among other varied applications.
Concerning passive devices, the tuned mass damper (TMD) stands out as one of the extensively adopted supplementary damping mechanisms. Te TMD confguration comprises a mass interconnected to the system through a spring and a viscous damper [14,15]. Within the literature, instances of TMD applications can be identifed across a spectrum of structural contexts. Notable examples include its implementation in prominent structures such as the John Hancock Tower in Boston, USA; Sydney Tower in Sydney, Australia; Millennium Bridge in London, UK; and the Rio-Niterói Bridge in Brazil [16,17].
TMD is the goal of interest of many designers and researchers such as the author of [18] who has authored a comprehensive book providing an in-depth analysis of the optimal tuning frequency of a TMD with internal damping; the authors of [19] calculated the parameters of multiple tuned mass dampers (MTMDs) arranged horizontally at the uppermost level of a ten-story building subjected to seismic excitations; the authors of [20] used a single TMD (STMD) to mitigate the dynamic response of two buildings under seismic excitations; the authors of [21] introduced a hybrid formulation featuring two distinct algorithms, namely, the frefy algorithm and the Nelder-Mead algorithm, for global optimizing of MTMDs in structures exposed to seismic excitations; the authors of [22] proposed a comprehensive study focusing on the robust optimal design of a TMD system to be installed in a tall building subjected to vibrations induced by wind forces; the authors of [23] proposed a robust optimization of MTMDs for vibration control of footbridges under human-induced vibrations. With the objective of enhancing the damping efectiveness of TMD, diferent confgurations of the device were being developed such as vibration absorbers with linear plus cubic spring support [24], impact vibration absorbers [25], and particletuned mass damper [15,26,27].
Combining MR damper with passive devices, such as TMD, a hybrid control is generated, which can obtain excellent results as demonstrated by many researchers, for example, the work by [28] who extensively explored the semiactive control of a building complex utilizing an MR damper and TMD under earthquake excitation through numerical simulations. Te authors of [28] used as a case study a building complex that includes a 14-story main building and an 8-story podium structure. Te assessment of performance encompassed three distinct categories: frst, the analysis of semiactive control strategies; second, the examination of hybrid semiactive control involving both the TMD and MR damper; and third, the evaluation of passive control using solely the TMD. To facilitate efective control forces, fuzzy logic was adopted to design a controller capable of determining and applying the appropriate voltage to the MR damper. Te numerical fndings demonstrated substantial mitigation of seismic responses in both buildings through the implementation of semiactive control and hybrid semiactive control.
In [29], the performance of a semiactive TMD with an adaptive MR damper was investigated using type-1 and type-2 fuzzy controllers for seismic vibration mitigation of an 11-degree of freedom building model. Te authors of [29] explore the efectiveness of a semiactive TMD equipped with an adaptive MR damper. Te study delves into the application of both type-1 and type-2 fuzzy controllers to mitigate seismic vibrations in an 11-degree-of-freedom building model. Te location of the TMD was on the roof, while the MR damper was situated on the 11th story. Notably, the MR damper possessed the capability to generate a control force of 1000 kN. Te design of the fuzzy system was based on the acceleration and velocity measurements of the top foor. Tis system determined the requisite input voltage for producing the control force based on the acceleration or deceleration movements of the building. Te outcomes of the study showed that the implementation of the type-2 fuzzy controller yielded additional reductions in the maximum displacement, acceleration, and base shear of the structure, amounting to 11.7%, 14%, and 11.2%, respectively, in comparison to the results achieved with the type-1 fuzzy controller.
In [30], an investigation is presented regarding the formulation of hybrid control strategies employing three distinct combinations of MR dampers and TMDs for the purpose of regulating seismic responses in building frames. Te controlled responses were derived from the analysis of four earthquakes (El Centro, Uttarkashi, Spitak, and Kobe), employing a total of four distinct control algorithms. Te fndings indicated that by incorporating a combination of TMD and a reduced number of MR dampers, a noteworthy enhancement in control response of up to 40%-45% could be achieved.
When optimization is used to design the vibration control systems, its efciency can be further improved. Regarding the optimized design of passive systems based on the TMD, in [31], the application of the diferential evolution (DE) algorithm is discussed with regard to designing optimal parameters for a tuned impact damper (TID), which draws inspiration from the TMD. According to [31], this design process is rooted in the utilization of an equivalent reducedorder model, and it uses the TID attached to a 20-story nonlinear benchmark building. As main conclusions, the authors highlight the possibility of using the reduced-order model for large and complex engineering structures; signifcantly mitigating the dynamic response of the principal structure-notably, in terms of peak displacement, RMS displacement, and interstory drift ratio-through the integration of an optimized TID system; evidencing a notable reduction in the occurrence of plastic hinges by virtue of the optimized TID system, and the optimized TID system exhibits remarkable robustness, and a validation corroborated by its profcient performance in dynamically responding to a large number of earthquake records afecting the main structure.
In [32], the application of an MR damper is explored to enhance the operational efciency of a benchmark baseisolated building. Te MR damper is strategically positioned between the structure's base and the foundation. Te optimization of MR damper parameters is executed through the utilization of the particle swarm optimization (PSO) algorithm, employing a suite of benchmark earthquake records. Te outcomes of the study demonstrated a notable improvement in the dynamic response under all soil conditions, as contrasted with scenarios involving nonoptimized confgurations and base isolation. Furthermore, the efcacy of the optimization process in enhancing the response of base-isolated structures was well-established.
In [33], an exploration is conducted on a hybrid control system that combines an MR damper and a TMD. Te investigation is carried out using a 15-story shear building as the experimental context. Specifcally, the MR damper is afxed to the TMD structure, enabling it to generate an active control force for the TMD mechanism. Te control voltage for the MR damper is generated through the integration of two distinct control algorithms. Tese algorithms are optimized using the observer-teacher-learner-based optimization (OTLBO) algorithm and aimed to minimize the maximum displacement of the building's rooftop. Tis optimization process is conducted under the infuence of both far-feld and near-feld earthquake excitations. Te fndings revealed a substantial average reduction of 35.06% in building rooftop displacement achieved through the implementation of the fractional-order proportional-integral-derivative (FOPID) control system, coupled with the interval type-2 fuzzy logic controller (IT2FLC). Tis reduction was observed across sixteen distinct far-feld and near-feld earthquake records. Furthermore, it was established that the hybrid system (MR + TMD), outperformed conventional controllers, underscoring its superior efcacy in seismic response mitigation.
In this context, this study presents a novel methodology for designing a hybrid control system. Tis system involves the utilization of a magnetorheological (MR) damper and a tuned mass damper (TMD) positioned at distinct locations within the structure. Te ultimate goal is to mitigate the structural response of a 10-story shear building when subjected to seismic loads, specifcally a nonstationary artifcial earthquake (NSAE) generated by the Kanai-Tajimi spectrum [34,35]. To execute the optimization process, the study harnesses the power of the whale optimization algorithm (WOA). Developed by the authors of [36], this metaheuristic algorithm mimics the hunting behavior of humpback whales.

Mathematical Model of the Dynamic Problem
To describe the dynamic behavior of multiple degrees of freedom (n-DOF) system, with linear behavior, subjected to seismic excitation and control forces generated by a semiactive or hybrid vibration control system, which in this study is MR + TMD, the following diferential equation of motion is employed: in which M s , C s , and K s represent the structure global matrices of mass, damping, and stifness, respectively. € x(t), _ x(t), and x(t) are, respectively, the acceleration, velocity, and displacement vectors. Te seismic and control force vectors are given by the term on the right of equation (1), where Λ is the location vector of the seismic forces, which is associated with the vector of seismic accelerations, € x g (t). Finally, Γ is the control force location matrix of the control force vector, f mr (t). And using the state-space (SS) formulation to solve this problem, and considering an n-DOF system with m MR actuators, equation (1) can be rewritten as follows: in which A is a square matrix that represents the state or characteristics of the system; B c describes the position of the vector control forces (f mr (t)) in the system; E locates the seismic acceleration vector (€ x g (t)); C is a square matrix and represents the output matrix; and D c and F describe the positions of the vectors of control forces and seismic accelerations, respectively. Considering the equation of the vector of states, _ x (2nx1) , the equations that govern each term of its equation are as follows:

Shock and Vibration 3
And, to the output vector, y(t) (3nx1) , the equations are as follows: Rewriting equations (2) and (3) and considering that: Finally, it arrives at the SS equation of the n-DOF system subjected to control forces and seismic excitation, as represented in equation (1): in which B ce and D ce are the location matrices of the control and excitation forces, respectively. It should be noted that, when a passive control system through TMD or multiple TMDs to generate a hybrid control (MR + TMD) is added to the system, the global matrices of the structure (M s , C s , and K s ) are modifed to consider the infuence of these passive devices. In equations (7) and (8), n is the number DOF, and m is the number of MR actuators.

Seismic Excitation.
In conducting the dynamic analysis of the examined building within this study, a nonstationary artifcial earthquake (NSAE) was employed. Tis earthquake record was specifcally generated using the Kanai-Tajimi spectrum [34,35]. Te model's equation is defned through a power spectral density (PSD) function denoted as S(ω), as presented in equation (9). Within this equation, S 0 represents the constant spectral density, while ξ g and ω g denote the soil damping and frequency, respectively. Te soil damping was specifed as ξ g � 0.6 a type of rocky soil, obtained from [37], and to the soil frequency, a value of ω g � 4π, which is an intermediate value between the frst and second mode of vibration of the building under analysis, was employed.
Equation (9) delineates a function within the frequency domain. To facilitate its transition into the time domain, the utilization of equation (10), developed by [38], was adopted. Within this equation, Δω denotes the frequency increment, N ω signifes the interval number of band frequencies, and φ j corresponds to the random phase angle. Te values of φ j are uniformly distributed over the range from 0 to 2π.
Finally, aiming to simulate the nonstationarity of the earthquakes, an envelope function (equation (11)) adapted from [39] was applied to multiply the stationary accelerogram, resulting in the generation of a new record exhibiting analogous behavior to real earthquakes. Te initial parameters that governed this process were adjusted to a 1 � 1.35 s − 1 and a 2 � 1/2 s − 1 . Figure 1 shows the generated NSAE and its PSD.
Te peak ground acceleration (PGA) of the NSAE was specifed as 0.3 g, and its duration was set to 10 s with a time step of 0.002 s.

Analyzed Building and MR Damper Modeling.
Te analyzed structure in this study is a 10-story shear building adapted from [19], which has one DOF per story and linear behavior. Te mass, stifness, and damping are uniform for each story whose adapted values are m i � 3.5 × 10 4 kg, k i � 6.5 × 10 7 N/m, and c i � 6.0 × 10 5 N·s/m, respectively. Te height of each story was defned as 3.0 m. Te ten natural frequencies, obtained using eigenvalues and eigenvectors formulation, are 1.0251 Hz; 3.0524 Hz; 5.0115 Hz; 6.8587 Hz; 8.5527 Hz; 10.0556 Hz; 11.3339 Hz; 12.3590 Hz; 13.1080 Hz; 13.5642 Hz.
In the literature, many numerical models able to describe the behavior of the MR damper can be found, and among them, the modifed Bouc-Wen model (MBW), proposed by [40] and illustrated in Figure 2, was employed in this study.
According to [40], the Bouc-Wen model efectively forecasts the force-displacement characteristics of the damper, displaying a force-velocity behavior that aligns more closely with experimental data. Nevertheless, akin to the Bingham model, the Bouc-Wen model does not exhibit a roll-of in its nonlinear force-velocity response within the realm where acceleration and velocity exhibit opposing signs, and velocity magnitudes remain modest. To enhance the predictive capabilities of the damper's response within this specifc context, a modifed variant of the model was introduced, denoted as MBW. Te damping force (F � f mr ) of this model is as follows: where solving for derivative internal displacement, _ y, results in and, where the evolutionary variable, z, is governed by Te terms that remain to be defned in equations (12)- (14), according to [40], are k 1 which signifes the accumulator stifness; c 0 which represents the viscous damping observed at higher velocities; c 1 which incorporates a dashpot component aimed at introducing the roll-of phenomenon evident in experimental data at lower velocities; k 0 which is utilized to govern stifness at elevated velocities; x 0 which designates the initial displacement associated with spring k 1 a contribution to the nominal damper force linked with the accumulator; x and _ x which denote the displacement and velocity, respectively, of the controlled structure; and the parameters α, β, c, A bw , and n bw which describe the hysteresis of the system that depends on the physical characteristics of each MR damper, such as MR fuid, numerical model, and piston rest position. In the context of semiactive control, the building under study is subject to control through a single MR damper. Tis MR damper is located between the ground and the frst story of the building, as shown in Figure 3(a). Te MR damper was placed in this position due to the possible high value of the story drift for this story (usually, the frst story has higher values) and easy installation in real life. Furthermore, once the MR damper is placed in this location, it puts a large amount of damping force on this story, which has efects on the other stories of the building. Te MR used in the analyses was proposed by the authors of [41], and it is a large-scale device with a maximum damping force of approximately 200,000 N (20 tons) and a maximum operating current of 2.0 A.
According to [41], the MR damper has an internal diameter of 20.3 cm with electromagnetic coils connected in three sections of the piston, which results in four efective regions of the valve. Te full device has approximately 1 m in length, mass of 250 kg, and contains approximately 6 liters of MR fuid. Te MBW model parameters were obtained experimentally by the authors of [41], and some of these (α, c 0 , and c 1 ) depend on the input current. Te obtained values, as well as their equations, are shown in Table 1.

Control Law.
To calculate the damping force generated by the MR damper which is applied to the structure under analysis, the clipped optimal control technique developed by the authors of [42][43][44] is employed. According to [8], the clipped optimal control is an optimal control strategy that has already been implemented in many civil engineering applications. Te control law of this technique is based on the linear state feedback controllers, designed for active or semiactive systems, whose desired control signal is calculated using a linear controller, such as linear quadratic Gaussian (LQG) or linear quadratic regulator (LQR), combined with a clipped algorithm to limit the actuation signal to the achievable working range of the control device. In this study, the LQR combined with clipped optimal control, named here as CO-LQR, is employed to obtain the in which I max , I c , f o , f mr , and H(− ) are the maximum current, the MR damper command current, the optimal control force generated by the LQR, the MR force, and the Heaviside function, respectively. To better understand this algorithm, equation (16) shows the Heaviside function, where the values of 0 or 1 depend on the comparison between the force generated by the MR and the desired optimal force.
As per equation (16), when the device is exerting the desired optimal force (f o � f mr ), the command current is maintained unchanged. If the magnitude of the force generated by the MR damper is smaller than the magnitude of the desired optimal force and the two forces share the same sign, the applied current is heightened to its maximum level. Tis adjustment aims to amplify the force produced by the MR damper to align with the intended control force. Conversely, when the two forces exhibit difering signs, the Heaviside function yields a value of zero. In such instances, the command current is set to zero, refecting this condition where forces are dissimilar. Tus, to calculate the vector of control forces, f o (t), generated by the LQR controller, the following equation is used: in which x(t) represents the state vector of the system with n displacements, x(t), and n velocities, _ x(t). n is the number DOF, and G indicates the gain matrix, calculated using the following equation: in which P is the positive-defnite symmetric matrix, which is the solution of the Riccati equation defned as follows: where A represents the system state matrix, B c is the input matrix characterizing the positions of the control forces and for the LQR, and Q and R are its weighting matrices.    [41]).

Current independent parameters
Shock and Vibration

Proposed Hybrid Control.
Te hybrid control (MR + TMD) of the studied building was generated by the MR damper associated with the TMD which, according to [14,45], consists of a mass, an elastic spring, and a viscous (or hysteretic) damper. For this system, many control scenarios are proposed, and the TMD parameters, stifness (k TMD ), damping coefcients (c TMD ), and the weighting matrices of the LQR controller are optimized. Te mass of the TMD (m TMD ) was chosen as 1% of the building's total mass and is not optimized. An illustration of the hybrid control is shown in Figure 3(b) where the MR damper is positioned between the ground and the frst story, and the TMD is positioned at the top foor. In Figure 3(a), only the MR is used as semiactive control to control the response for comparison purposes with the hybrid control.
Te installation of the TMD at the top foor of the building requires modifcations in the global matrices of mass (M s ), stifness (K s ), and damping (C s ) of the structure to incorporate the infuence of the properties of the device. Terefore, equations (20)- (22) show these modifcations: Te control criterion h i /400 of each story height, in which h i is the ith height of the ith story, was employed in the analyses. Tis criterion is presented in the ANSI/AISC 360− 16 code of the American Institute of Steel Construction [3] and was also employed by many authors such as the authors of [46][47][48][49][50][51][52].

Optimization Algorithm.
Te whale optimization algorithm (WOA), a metaheuristic algorithm that mimics the hunting behavior of humpback whales, proposed by the authors of [36], was employed to conduct the optimization processes. In the hybrid vibration control system, both the TMD parameters and the weighting matrices of the LQR controller were simultaneously optimized. Te pseudocode of WOA is shown in Figure 4 in which the parameters A and C are coefcient vectors of the leader position vector and updated positions vector. Te vector a is used to calculate A, and it is linearly decreased from 2 to 0 over the course of iterations (in both exploration and exploitation phases). Te parameters A, C, and a belong to the mechanism encircling prey. Te parameter l is a random number in [− 1, 1] and belongs to the spiral updating position equation. Finally, according to [36], the behavior of humpback whales involves a dual approach: they circle their prey within a diminishing circle while concurrently following a spiral-shaped trajectory. To mimic this simultaneous behavior, the authors adopt a 50% probability of choosing between the shrinking encircling mechanism and the spiral model to update the whales' positions. Tis probability, denoted as p, is governed by a random number in the range of [0, 1]. Te parameters l and p are intricately linked to the bubble-net attacking method, indicative of the exploitation phase within the algorithm. Further information about the mathematical formulation and details about the three operations of WOA (encircling prey, bubble-net attacking, and search for prey) can be found by the authors of [36].
Recent applications of optimization utilizing the whale optimization algorithm (WOA) are documented in [49][50][51], where investigations encompassed the optimization of design parameters and placements of single TMD and multiple TMDs. Tese studies sought to enhance vibration control in buildings subjected to seismic excitations, with the primary objective of diminishing story drift. Furthermore, the authors of [52] conducted optimization of TMD parameters for a building under seismic loading. Additionally, the authors of [53] focused on optimizing the parameters of MTMDs to minimize the maximum vertical displacement of road bridges subjected to vehicle trafc.

Uncontrolled Response.
Once the structure has been modeled and the seismic excitation generated, the uncontrolled response scenario (URS) of the building is evaluated. Considering that all stories of the building have the same height, defned as 3.0 m, and the story drift evaluation criterion presented in the ANSI/AISC 360− 16 [3] indicates, h i /400, the results obtained are presented in Table 2.
By analyzing Table 2, it can be seen that the frst fve stories have story drift higher than the maximum allowed according to the adopted control criterion. Terefore, based on the results, it was verifed that a vibration control system Shock and Vibration is necessary for this building in order to adapt it to the consulted code.

Semiactive Control Response.
Te frst proposed scenario for the building is the semiactive with one MR damper installed between the ground and the frst story (as shown in Figure 3(a)). Te device is modeled based on the MBW, and it can operate in diferent modes: passive OFF/ON (MR-OFF and MR-ON) and semiactive with current control through the CO-LQR algorithm (CO-LQR). Te MR damper used was proposed by the authors of [41] and has a maximum operating current of 2.0 A. Te MBW model parameters were obtained experimentally by the authors of [41] and are shown in Table 1. For the CO-LQR scenario, Q was defned based on equation (24), and R was adjusted to R � 10 − 14 .

Optimized Hybrid Control
Response. Te response of the analyzed structure is evaluated considering an optimized hybrid control system (MR + TMD) through the WOA. For comparison between the proposed hybrid control and other solutions (as semiactive control in the previous section), another control scenario, based on the passive control, is proposed. In this way, for the optimization procedure, two control scenarios are proposed, namely, (1) STMD: single optimized TMD installed at the top foor of the building (2) CO-LQR (MR + TMD): optimized TMD at the top of the building combined with the MR damper installed between the ground and the frst story (as shown in Figure 3(b)), controlled by the optimized CO-LQR algorithm For the STMD scenario, the TMD has mass (m TMD ) corresponding to 1% of the total mass of the structure, and its design parameters are optimized, namely, stifness coefcient (k TMD ) and damping coefcient (c TMD ). Te    (26), where ∆ h, max represents the maximum story drift, which is the objective function of the optimization problem. Te design variables are k TMD and c TMD , and the lower bound and the upper bound of the stifness and damping constants of the TMD are 0-4000 kN/m and 0-1000 kNs/m, respectively.
Subject to: Te results obtained for the STMD were m TMD � 3500 kg, k TMD � 138734 N/m, and c � 469 Ns/m, where it can be seen that the TMD tunes close to the frst vibration mode of the building, with a frequency equal to f TMD � 1.0020 Hz.
For the second scenario, CO-LQR (MR + TMD), the TMD again has a mass of 1% of the structure mass, and it is installed at the top foor of the building, and its design parameters are also optimized. Furthermore, the LQR parameters (q LQR and r LQR ) of the weighting matrices are also optimized. Te objective function is the same as in equation (26), which is the minimization of the maximum story drift. Te lower bound and upper bound of the stifness and damping constants of the TMD are 0-4000 kN/m and 0-1000 kNs/m, respectively, and for the LQR are 10 − 6 -10 18 for q LQR and 10 − 20 -10 10 for r LQR . Te weighting matrices were defned as Q � q LQR diag(K s ; M s ) and R � r LQR Id (mxm) , where m is the number of MR. Te maximum current of the MR is 2.0 A. Equation (29) shows the optimization problem of this control scenario.
Subject to: Te results obtained for this control scenario were m TMD � 3500 kg, k TMD � 1.6906 × 10 5 N/m, c � 5.1553 × 10 3 Ns/m, q LQR � 7.3135 × 10 15 , and r LQR � 1.3133 × 10 5 . For the MR force, a value of f mr,MR+TMD � 199159.2176 N was obtained, and for the TMD, it was verifed that it tunes to f TMD � 1.1061 Hz, again close to the frst vibration mode of the building.

Assessment of the Responses.
In the semiactive control, three approaches were initially evaluated, namely, MR damper in passive mode OFF (MR-OFF); MR damper in passive mode ON (MR-ON); and clipped optimal control with LQR (CO-LQR). In MR-OFF, the current has a constant value of 0 A, while in MR-ON, its value is 2.0 A. In CO-LQR, there is control of the applied value of the current which varies from 0 A to 2.0 A. Finally, regarding the optimized hybrid control, frst, the response reduction was evaluated using an optimized STMD and, after, the CO-LQR (MR + TMD) scenario.
Te response, in terms of displacements on the 10th story, is shown in Figure 5, where it can be seen that the MR-ON and CO-LQR (MR + TMD) scenarios show good reductions, especially between the 6 s and 10 s of the seismic excitation. Regarding the MR-OFF scenario, it is possible to see lower reductions, especially at the end seconds of the earthquake. Figure 5 also shows that the STMD scenario generates an increase in the responses in terms of story drift on the last two stories, as also shown in Figure 6, which shows the story drift for each story, and in Table 3, which shows the story drift reduction percentages for all scenarios. Te negative reductions of the last two stories for the STMD scenario indicate increases in the response, and it is related to the presence of the TMD in the building, because it tunes a diferent mode than the most excited by the earthquake, or it is related to the low mass in relation to the mass of the structure. Figure 6, as previously mentioned, shows the maximum story drift for each story, considering all control scenarios and the limit of the consulted normative. As can be seen in Figure 6, for the URS, the frst fve stories have story drift higher than the maximum allowed. It is also verifed that the semiactive control in modes MR-OFF, MR-ON, and CO-LQR, failed to adapt all the story drifts to the established criterion. Te STMD scenario presents reasonable reductions, ftting the 4th and 5th stories to the normative criterion; however, the last two stories show increases in the response. For CO-LQR (MR + TMD), it can be seen that it was the only one able to efectively control the structural response and adapt all stories to the control criterion.
Considering the percentages of reductions for all scenarios as shown in Table 3, it is verifed that the highest reduction (44.8276%) happens with the MR-ON mode at the 1st story, which has constant current applied by the MR, and the damping force is directly applied to this story. Te CO-LQR (MR + TMD) also presents good reductions, and it was the only one that had efective control of the structural response. Finally, Figure 7 shows the percentages of reductions of the story drifts, where it is observed that the lowest reductions happen in the MR-OFF scenario.
According to Figure 7, the MR-ON and CO-LQR scenarios showed good reductions for all stories and the CO-LQR (MR + TMD). It is observed that the optimized hybrid control CO-LQR (MR + MTD) resulted in the highest reductions, at the lowest stories (except on the 1st foor), precisely those that should be controlled. Furthermore, this control scenario was the only one that had efective control over the structure response; therefore, this is the best control scenario for the considered building.
Overall, the proposed methodology, which uses an optimization procedure to determine the optimal TMD parameters and LQR weighting matrices, improved the efciency of this system and proved to be an excellent tool for designing vibration control systems. Unlike other design methods, such as the authors of [18,54,55] which are Shock and Vibration classical methods used for the design of single TMD, in which the parameters of the device are calculated based on the frst modal shape of the structure, the design based on the optimization algorithms, such as meta-heuristics like WOA, calculates the optimal TMD parameters based on an objective function which in the case of this work was the minimization of story drift and associated with a semiactive device generates a hybrid control which is designed for maximum efciency.
In addition, the optimization procedure by metaheuristics algorithms has advantages over classical methods, such as they do not require function gradient   information; they do not get stuck in local minima, if adjusted correctly; they can be used to solve problems with mixed variables; they can be applied to nonconvex or discontinuous functions; and they provide a set of optimal solutions, so the designer can choose the one that best fts his project. Terefore, the proposed methodology can be a useful tool to assist designers of these types of vibration control systems and contribute to improving the efciency of the project.

Conclusions
Te focus of this work was the reduction of the response of buildings subjected to seismic excitation through a hybrid control (MR + TMD) designed using a new methodology that involves the use of metaheuristic optimization. Diferent control scenarios were proposed and evaluated to determine the best control scenario, in relation to the adopted control criterion. A 10-story shear building was evaluated with a single MR installed between the ground and the frst story, operating in passive OFF and ON modes and in semiactive mode with current control by CO-LQR. Furthermore, the response was also evaluated through traditional passive control, with a single TMD installed on the top foor. Te WOA was employed to design the parameters of the TMD (k TMD and c TMD ) and LQR for the weight matrices (q LQR and r LQR ). Regarding the analyzed scenarios, MR-OFF presented the worst reductions, while in the MR-ON mode, it was verifed the highest reduction (44.8276%) at the 1st story. Te CO-LQR mode showed good reductions, however, similar to the MR-ON mode. Te STMD scenario showed reasonable reductions; however, it generated an increase in the structural response in the last two stories. Nevertheless, it should be noted that the mass of the TMD was taken as just 1% of the total mass of the structure. On the other hand, the MR has a mass of 250 kg, which represents only 7.14% of the mass of the TMD.
Overall, the optimized hybrid control scenario (MR + TMD) shows to be the best alternative to control the response of the building and adapt all story drift to the control criterion. Tus, the proposed methodology can be a useful tool to assist designers of these types of vibration control systems.

Λ:
Location vector of the seismic forces A: System state matrix B c : Vector of the position of the vector control forces, input matrix B ce : Location matrix of the control forces C: Output matrix C s : Global damping matrix of the structure D c : Vector of the position of the vector control forces D ce : Location matrix of the excitation forces E: Vector of the location of the seismic acceleration vector F: Vector of the position of the seismic accelerations G: Gain matrix K s : Global stifness matrix of the structure M s : Global mass matrix of the structure P: Symmetric matrix, which is the solution of the Riccati equation Q: Weighting matrices of the LQR controller R: Weighting matrices of the LQR controller Γ: Control force location matrix of the control force vector f o (t): Vector of control forces f mr (t): Control force vector € u → g (t): Stationary accelerogram x(t): Displacement vector _ x(t): Velocity vector _ x (2nx1) : Vector of states € x(t): Acceleration vector € x g (t): Seismic acceleration vector A bw : Parameter of the MBW that describes the hysteresis of the system a 1 : Parameter 1 of the envelope function a 2 : Parameter 2 of the envelope function α: Parameter of the MBW that describes the hysteresis of the system β: Parameter of the MBW that describes the hysteresis of the system c 0 : Viscous damping of the MBW observed at larger velocities c i : Damping of each story of the building c 1 : Dashpot of the MBW that was included in the model to produce the roll-of that was observed in the experimental data at low velocities c TMD : Damping of the TMD d i− 1 : Maximum displacement of the lower story d i : Maximum displacement of the upper story ∆ h,max : Maximum calculated value for story drift Δω: Frequency increment ξ g : Soil damping F � f mr : Damping force of the MR damper for MBW f o : Optimal control force generated by the LQR f(g): Objective function g(t): Envelope function H: Heaviside function