Seismic control of multidegree‐of‐freedom structures using a concentratedly arranged tuned viscous mass damper

Connecting a flexible supporting element to an inerter arranged in parallel with a viscous element yields a tuned‐mass damper‐like system, designated as a tuned viscous mass damper (TVMD). The advantage of a TVMD is that it exploits the flexibility of the supporting member, which is usually considered to compromise the energy‐dissipating performance, and the inerter and soft spring form a supplemental oscillator to enhance the damping performance with resonance to the primary structure. The fixed‐point method for the optimal design of a single‐degree‐of‐freedom structure containing a TVMD is further expanded to a multidegree‐of‐freedom structure, in which a TVMD is arranged concentratedly. Furthermore, approximated closed‐form formulae for a concentratedly arranged TVMD are derived under the assumption that the primary structure is undamped, remains in an elastic range, and the target mode dominates the response of the structure. An analytical example illustrates that installing a TVMD spanning three stories in a concentrated manner based on the proposed design methods ensures the efficiency of the damping system, which demonstrates a damping effect similar to that of stiffness‐proportionally distributed TVMDs with less demand for inertance and total control force, thus resulting in a lower cost.


INTRODUCTION
Unlike springs and dampers with two independently movable terminals, a physical mass has a center of mass as a movable terminal, and the other terminal is a fixed point in an inertial frame, which renders the analogy between mechanical and electrical networks incomplete. 1 An inerter, 2 an apparent mass device that has two independently movable terminals created by mass amplifying mechanisms, such as ball-screws and rack pinions, can fill the blank in the analogy between mechanical and electrical networks.Kawamata 3,4 and Nakamura et al. 5 developed a type of fluid inerter device designated as an accelerated liquid damper that utilizes fluid mass and hydraulics.Swift et al., 6 Liu et al., 7 Domenico et al., 8 and Zhang et al. 9 developed fluid inerter-damper devices using external and internal helical channels.While a fluid inerter damper uses the fluid mass, the hydraulic inerter developed by Wang et al. 10 uses a flywheel attached to a hydraulic motor in the fluid passage of the bypass tube to obtain an amplified apparent mass based on Pascal's principle.A rack and pinion system, 2,11,12 a leveraged pendulum system, 13 and crank mechanisms, 14 as well as ball screw mechanisms, [15][16][17][18][19][20][21][22][23] can be used to implement an inerter.
Inerters have been considered a viable option for the vibration control of vehicle and train suspensions. 24,25Several types of inerter-like devices have been investigated for the response control of civil structures.Okumura 26 proposed incorporating a rack-and-pinion-based inerter into a seismically isolated building to achieve the insulation of a specific frequency component of vibration induced by an earthquake.Furuhashi and Ishimaru 27 proposed an inerter placement strategy to eliminate the high modes of a multidegree-of-freedom (MDOF) shear building.Saitoh 11 proposed incorporating an inerter device into a seismically isolated building to reduce isolator displacement.Various configurations of building suspensions containing an inerter were investigated by Wang et al. 28 Nakamura et al. 19 developed an inerter damper, whose damping force was controlled semi-actively using an electric generator by changing its terminal resistance.Lazar, Neild, and Wagg 29 proposed a tuned inerter damper (TID), in which the configurations of the mass, damper, and spring elements are the same as those of a conventional tuned mass damper (TMD).Garrido, Curadelli, and Ambrosini, 30 De Domenico and Ricciardi, 31 Giaralis and Taflanidis, 32 and Pietrosanti et al. 33 improved traditional TMDs using isolators and inerters.Considering the quasidynamic background effect, Krenk and Høgsberg 34 proposed a unified calibration procedure for inerter-based absorbers.Zhang et al. 35 examined the optimal inerter-spring-damper configuration incorporated into a multistory building.Wen et al. 36 determined the optimum locations of tuned inerter-based dampers to minimize a placement index defined based on the  2 norm.Taking the inherent damping of the primary structure into consideration, Taflanidis et al. 37 proposed a multi-objective numerical method for designing inerter-based absorbers and further explored the impact of the inerter vibration absorbers spanning one or more floors and the case of hysteretic structures. 38Zhang et al. 39,40 elucidated the damping enhancement effect of inerter systems and proposed a semi-analytical method for the design of inerter devices to control MDOF structures optimally.Qiao et al. 41 proposed an efficient numerical optimization method for inerter-based vibration absorbers by considering modal interaction effects using the Sherman-Morrison inversion method.However, few of these systems have been successfully adopted in real-life civil structures.
One of the most successful inerter-based devices accepted as building dampers is the tuned viscous mass damper (TVMD). 42,43A TVMD is a viable alternative to TMD, allowing a large mass ratio without excessive weight penalty by utilizing the mass amplification effect of the inerter.A significant feature of the TVMD is that it uses a rotary inerterdamper device that simultaneously generates inertial and viscous resistive forces.The mass amplifying mechanism used in TVMD is an efficient ball screw manufactured with high precision, producing an inertance several thousand times larger than its physical mass.Ikago et al. 43 derived a closed-form expression of an  ∞ optimal control design for a singledegree-of-freedom (SDOF) structure containing a TVMD using fixed points on the dynamic amplification factor (DAF) curves that exist regardless of the damping coefficient of the device.The design method was subsequently extended to stiffness-proportionally distributed TVMDs incorporated into an MDOF shear building. 44Sugimura et al. 45 and Ogino and Sumiyama 46 reported the structural design of high-rise buildings incorporating TVMDs built in Sendai and Tokyo, respectively.Ji et al. 47 proposed to install TVMDs in a zigzag configuration between coupled walls to exploit the shear deformation between them.Li, Ikago, and Yin 23 proposed to use the inertance produced in a rotary eddy current damper to configure a TVMD.
Under the assumption that the primary structure is undamped, remains in a linear elastic range, and the target mode dominates the response, this study discusses the optimum design of MDOF buildings containing a concentratedly arranged TVMD.First, the  ∞ optimal control design method for a TVMD-controlled SDOF structure is reviewed briefly.We then prove the existence of fixed points on the DAF curves of an MDOF structure containing a concentratedly arranged TVMD.Next, we propose a numerical  ∞ optimal design method for the concentratedly arranged TVMD and derive closed-form approximation formulae.The effectiveness of the concentratedly arranged TVMD is demonstrated by comparing them with stiffness-proportionally distributed TVMDs.

Inerter-based TMD-like devices
A TMD is a viable option for controlling wind-induced vibrations in civil structures. 48However, as Kaynia, Veneziano, and Biggs 49 noted, a mass ratio smaller than 2% was insufficient for seismic control; thus, a sufficient supplemental mass required for seismic control might result in an excessive weight penalty.Thus, an inerter is a promising alternative to the mass element in a TMD, providing a large apparent mass with a small physical mass.Two types of inerter-based TMD-like devices are proposed for civil structures: one is proposed by Saito et al., 42 designated as a TVMD, in which, unlike conventional TMDs, the inerter and damper are arranged in parallel, whereas the other is proposed by Lazar, Neild, and Wagg, 29 designated the TID, which has the same configuration as a conventional TMD.
Figure 1 shows a photograph of a real-life rotary inerter damper installed in a building in Sendai, Japan, 45 which is also referred to as the inertial rotary damping tube (iRDT).
Figure 2 shows the mechanism of the rotary inerter damper.The translational motion input to the damper was converted into a rotational motion by the ball screw.The ball-screw mechanism produces an amplified apparent mass activated by the relative acceleration between the two terminals of the device, implementing the concept of the inerter.Silicone oil was enclosed between the fixed internal cylinder and the rotating outer cylinder to generate a viscous damping force.The torque of the viscous damping force between the two cylinders was also amplified when it was converted back to the translational force.Thus, the rotary inerter damper generates inertial and viscous resistive forces, which is shown by the parallel configuration of the inerter and damper.
A force restriction mechanism was installed in the rotary inerter damper as a failsafe.The cylindrical flywheel slips to restrict the resultant force of the inertial and viscous resistive forces when it reaches the prescribed maximum frictional torque, which is adjusted by the normal stress introduced between the frictional material and the part that transmits the torque to the ball nut.Watanabe et al. 18 reported the results of full-scale dynamic tests conducted using a force-restricted rotary inerter damper.
Figure 3 shows the installation of TVMDs in a steel frame. 45Natural rubber plates were inserted between the steel plates; one plate was connected to the steel brace, and the others were connected to rotary inerter dampers.The stiffnesses of the steel brace and rubber plates were designed such that their combined stiffness ensures that the TVMD is tuned to the primary structure.The equations of motion for the TVMD element are as follows:

Equations of motion for a TVMD element
If only external nodes A and B are connected to a structure, and we ignore the inertial force induced by the physical mass of the device, then  c = 0. Thus, Equation (1) can be rewritten as where  d denotes the deformation of the viscous element.

 ∞ optimal design of TVMD-controlled SDOF system
Ormondroyd and Den Hartog 50,51 presented the concept of a TMD and its  ∞ optimal design method using fixed points on resonance curves.Saito et al. 42 and Ikago, Saito, and Inoue 43 derived closed-form  ∞ optimal designs of SDOF structures containing a TVMD based on the same method.
F I G U R E 5 SDOF structure containing a TVMD.SDOF, single degree of freedom; TVMD, tuned viscous mass damper.
Here, as shown in Figure 5, we consider an SDOF structure with mass m and stiffness k, which is incorporated with a TVMD whose inertance, supporting spring stiffness, and damping coefficient are  d ,  b , and  d , respectively.
In Figure 5, ,  0 , and  d are the displacement of the primary mass relative to the ground, ground displacement, and the deformation of the viscous element, respectively.The deformation of the damper-supporting spring is  −  d .The equation of motion for this system is where and ẍ0 denotes the ground acceleration.
Let the mass ratio μ, frequency ratio β, nondimensional relaxation time λ, nominal damping ratio ĥd , and stiffness ratio η for the SDOF structure be defined as follows: where Figure 6 shows the DAF |()∕ g ()|, where  is the angular excitation frequency, and () and  g () are the Fourier transformations of () and ẍ0 (), respectively.γ is the ratio of the angular excitation frequency  to the fundamental natural angular frequency of the undamped primary system  0 .
There are fixed points on the DAF curves regardless of the damping coefficient  d .In a manner similar to that proposed by Ormondroyd and Den Hartog, the optimal design that minimizes the maximum value of the DAF, that is, the  ∞ norm of the DAF, can be determined under the following conditions: The supporting spring stiffness  b is determined such that the ordinates of the fixed points are the same, and the damping coefficient  d is determined such that the DAF curve reaches the maximum values at the fixed points (Figure 6).
Ikago et al. 43 derived closed-form expressions for the optimum designs of the frequency ratio βo , nondimensional relaxation time λo , nominal damping ratio ĥo  , and stiffness ratio ηo for a given mass ratio μ as follows: where the superscript "o" denotes the optimum design.

Extension of the fixed-point method to MDOF systems
For an uncontrolled -degree-of-freedom structure, the equation of motion can be expressed as where  P and  P are the mass and stiffness matrices of the primary MDOF structure, respectively. u denotes the displacement vector of the uncontrolled MDOF structure.() denotes the external dynamic load.It is worth noting that the pure shear deformation should be decoupled from the total deformation if the primary structure is not a shear building because a TVMD is activated by shear deformation when connected via a shear-link.Thus, for simplicity's sake, we confine ourselves to shear buildings in this study.
Figure 7 shows an -degree-of-freedom structure in which the two terminals of the TVMD are connected to two independent floors.The inertance, supporting spring stiffness, and damping coefficient of the TVMD were  d ,  b , and  d , respectively.The installation location of the TVMD is expressed by vector .For example, when the two terminals of the TVMD are connected across the th and th floors ( < ), the th and th components of  are −1 and 1, respectively, and the remaining components are 0. If the th floor is the ground, only the th component is 1, and the remaining components are 0. Thus, the equations of motion of the controlled -degree-of-freedom structure can be expressed as follows: where  and  d are the displacement vector of the controlled MDOF structure and the deformation of the viscous element, respectively.The equations of motion of the uncontrolled MDOF system in the frequency domain are as follows: where  u (), (), and   () are the Fourier transforms of  u (), (), and ẍ0 , respectively, and  is the angular excitation frequency.Thus, where  main () denotes the receptance matrix of the uncontrolled structure.
Similarly, the equations of motion of the MDOF structure controlled by the concentratedly arranged TVMDs in the frequency domain can be expressed as where () and  d () are the Fourier transforms of () and  d (), respectively.Solving Equation ( 14) with respect to  d () yields By substituting Equation (15) into Equation ( 13), the displacement vector of the primary MDOF structure () can be obtained as follows: Consider an invertible matrix  slightly modified by adding a product of two column vectors   .Using the Sherman-Morrison matrix inversion theorem, 52 we obtain its inverse as follows: Substituting  = − Therefore, the modal displacement of the th layer can be expressed as where  u,k () is the modal displacement of the th layer of the uncontrolled structure.
u,w () =    main ()() ℎ main,dw () =    main () ℎ main,ww () =    main () () = − P   () Equation ( 19) is further reduced to where 2 () = ℎ main,dw () ( ℎ main,ww () Equation (22) implies that  1 () denotes the relative displacement between the TVMD installation layers of the uncontrolled structure caused by a unit force applied at the TVMD installation layer.It is worth noting that  2 () = 0 when a TVMD is installed across the observed th layer and the ground, that is,  = .Nevertheless, the TVMD location vector  and observation vector  are independent to each other in this study.
As we can assume that the first mode component is dominant in the external dynamic load (), the external load () is approximated as follows: where  1 is the first mode vector of the uncontrolled system, and   is its participation factor.
Thus, the DAF of the th layer Based on the definition given in Equation ( 12),  main () can be expressed using the eigenvectors of the uncontrolled system   and their corresponding eigenvalues   as follows: where   is the generalized stiffness of the th mode, F I G U R E 8 DAF of an MDOF structure containing a concentratedly arranged TVMD.DAF, dynamic amplification factor; MDOF, multi degree of freedom; TVMD, tuned viscous mass damper.
Substituting Equation (27) into Equation ( 26) yields Furthermore, we choose floor  such that the th component of the th participation mode vector is closest to unity, namely    , ≈ 1.Then, the DAF on the th floor is reduced to We define the mass ratio , frequency ratio , nondimensional relaxation time , and the ratio of excitation frequency to primary fundamental frequency  to simplify the equation further.
where  1 is the generalized mass of the first mode.
Equations ( 30) and (33) indicate that there are fixed points whose ordinates are independent of  d (see Figure 8) if the following relation holds: TA B L E 1 Specifications of the analytical model.Herein, we assume that two fixed points P and Q that satisfy Equation (35) exist in the vicinity of the fundamental natural frequency of the undamped primary structure.The ratios of the frequencies of points P, Q, and  1 are denoted as   and   (  <   ), respectively, as shown in Figure 8.Therefore, by combining the equal peak condition,  P ,  Q , and the corresponding optimum damper frequency  o satisfy the following equations: 2 1  o 2 ᾱ2 ( P ) 1 +  M1  2 1  o 2 ᾱ1 ( P )

Story
Then, an optimization problem can be established to solve the above three equations with respect to  P ,  Q , and  o numerically: 36) and (37), where  L and  U in the side constraints are determined such that  is observed near  = 1. L and  U are set as 0 and 2 in this study, respectively.Furthermore, substituting  and  o into Equation (33) yields a DAF as a function of  and : The optimum nondimensional relaxation time  o can be obtained from the following optimization problem:

Derivation of closed-form formulae for approximating the optimal design
1 () and  2 () in Equations ( 22) and ( 23) are further simplified as follows.
When  = 0, the receptance matrix is Herein, we assume that the contribution of the high modes to the receptance matrix is approximated by a static response, as reported by Hansteen and Bell. 55Then, where  is referred to as a residual receptance matrix.
Hence,  1 () and  2 () can be expressed as follows.The detailed derivations can be found in the Appendix. where Note that  2 () = 1∕ ′′ is constant regardless of the excitation frequency  because of the assumption that the highermode components of the external dynamic force can be omitted, as shown in Equation (24).
Thus, squaring both sides of Equation ( 50) yields where By substituting  = 0( d = 0) and  = ∞( d = ∞) into Equation ( 51), the square of the amplitude ratios of the transfer function for the two extreme damping coefficients of the TVMD can be obtained as follows:

Optimum tuning frequency of TVMD
According to the fixed-point method, the optimum tuning frequency  o can be obtained under the condition that the two fixed points have the same ordinate.
The existence condition for fixed points Participation mode vector of the first mode.

F I G U R E 1 1
Relationship between the optimum parameters.

F I G U R E 1 2
Relationship between the mass and damping ratios.
F I G U R E 1 3 Analytical models of concentratedly arranged and distributed TVMD systems.TVMD, tuned viscous mass damper.
Considering the signs of the square roots on both sides of the above equation, 51 ) Thus, where Let Γ P and Γ Q be the squares of the abscissas of the fixed points; then, the following equation holds according to the factor theorem.
By comparing the coefficients in Equations ( 57) and (59), we obtain Provided that the ordinates of the fixed points are equal, Thus, Substituting Equation (60) into Equation (63), we obtain where Solving Equation ( 64) with respect to  yields where the mass ratio  must satisfy the following condition such that  is a positive real value: Thus, the optimum damping frequency  o is Note that Equation (68) reduces to the following simple formula for an SDOF structure equipped with a TVMD (2-DOF system) when 0 <  < 1 and Δ 1 and Δ 2 diverge to positive infinity: which is identical to βo in Equation ( 6).This is because Δ 1 → ∞ and Δ 2 → ∞ indicate that all the contributions of uncontrolled modes except for the first mode vanish and the system approaches the equivalent 2-DOF system discussed in Ikago et al. 44

Optimum damping coefficient
The optimum nondimensional relaxation time  o can be obtained from the following conditions according to the fixedpoint method: Rewriting Equation ( 51) yields Differentiating both sides of Equation ( 72) with respect to Γ and setting Thus,  2 can be derived as follows: By substituting the optimum damping frequency  o derived from Equation (64) into Equation (58), the corresponding  o 0 ,  o 1 , and  o 2 can be obtained.
Then, Γ o P and Γ o Q as functions of  o can be derived from Equation (57) as follows: Substituting  o and Equation (76) into Equation (54) yields the optimum ordinates of the fixed points as follows: Substituting Equations ( 76) and (78) and  o into Equation (74) yields the optimum nondimensional relaxation time for the fixed point P, which is denoted as  o P .Similarly, the optimum nondimensional relaxation time for the fixed point Q, denoted as  o Q , can be obtained by substituting Equations ( 77) and ( 78) and  o into Equation (74).As there is no  that simultaneously satisfies the stationary condition at points P and Q, we adopt Brock's root mean square 56 to obtain the optimum nondimensional relaxation time  o , and then the optimum nominal damping ratio ℎ o d .
Note that the difference between the values of ℎ o d and ĥd o is negligibly small, which is discussed the following section.
In particular, the equation ℎ o d = ĥd o holds when Δ 1 → ∞ and Δ 2 → ∞.Hence, the lengthy closed-form expression for the optimal damping ratio ℎ o d is omitted here, and it can be practically replaced by ĥd o .

ANALYTICAL EXAMPLE
Table 1 lists the specifications of a 10-story benchmark steel building developed in a joint US-Japan research project. 57able 2 presents the first three fundamental natural periods and their corresponding angular frequencies.The number of stories in this example was  = 10.
Installing an inerter spanning multiple stories ensures a large relative displacement between the two nodes, resulting in an enhanced performance of the TVMD.The concentrated TVMD installed spanning enough floors should be preferred. 37,38However, considering practical implementation issues such as floor openings and designing peripheral members, Ogino and Sumiyama 46 reported that spanning three consecutive stories as shown in Figure 9 is the best feasible option.In this study, the number of stories spanned was selected as three as well following the suggestion by Ogino and Sumiyama.Therefore, the installation location vector  is  = [0, 0, 1, 0, 0, 0, 0, 0, 0, 0]  (81) According to the fundamental participation mode vector shown in Figure 10, the representative height is 27.12m.The seventh floor is chosen as the representative floor because its height is closest to 27.12 m.Therefore, the observation vector  is  = [0, 0, 0, 0, 0, 0, 1, 0, 0, 0]  (82) Based on the modal analysis results, Equations ( 47 Substituting Equation (84) into Equation (64), the condition for ensuring positive real solutions to Equation ( 64) is observed to be  < 0.44.The nominal stiffness ratio  of the TVMD were obtained as follows: By substituting ,  o , and  o into Equations ( 79) and (85), we obtain the optimum damping ratio ℎ o d and stiffness ratio  o .Figure 11 shows the two optimum damper parameters obtained using different methods for specified mass ratios  within the range of [0,0.20].The red line represents the closed-form formula proposed in Section 3.2 for approximating the optimal design.Hereafter, the method using closed-form formulae is referred to as the proposed method.The black line represents the optimum damper parameters under the assumption that Δ 1 = ∞ and Δ 2 = ∞, which is identical to the direct use of the design formula proposed for the SDOF system (see Equation 6) for the MDOF system, as discussed in  Ikago et al. 44 .Hereafter, we will refer to this method as the SDOF method.The blue circles and diamonds represent the numerical method proposed in Section 3.1.Hereafter, we will refer to this method as the numerical method.As shown in Figure 11, these three methods yield similar optimum damping ratios, which validates the assumption that ĥd o provides a practically good approximation for ℎ o d .The SDOF method yields a significantly low stiffness ratio compared with the proposed and numerical methods, which suggests that the effect of how TVMD is placed should be considered when designing the spring element of the TVMD.

Method
Figure 12 shows the first two damping ratios of the structure controlled by the concentratedly arranged TVMDs designed using the aforementioned methods.In addition, as demonstrated in Ikago et al. 44 , stiffness-proportionally distributed TVMDs (see Figure 13) can be designed using an equivalent 2-DOF system to obtain the exact optimal control effect.Herein, the damping effect achieved by optimum stiffness-proportionally distributed TVMDs is set as the baseline for the comparison with the concentratedly arranged TVMD.The solid green line represents the first two damping ratios controlled by the optimum stiffness-proportionally distributed TVMDs.The proposed method makes the damping ratios of the first two modes close to the baselines, which leads to equal peak values of DAF of the first two modes to satisfy the optimality condition, thus demonstrating a control effect similar to the baseline, as shown in Figure 14.
For example, Table 3 lists the optimum TVMD parameters and damping ratios of the entire structure containing the TVMD for  = 0.1.Figure 15 shows the DAF defined in Equation (30) for  = 0.1.The proposed method obtains the lowest peak value around the first natural frequency of the uncontrolled structure ( = 1).The peak value of the DAF produced by the proposed method near  = 1 was 4.30, which is 66.46% of that obtained using the SDOF method.The

Method
Total inertance (t) peak DAF of the stiffness-proportionally distributed TVMD system near  = 1 was 4.22, which is similar to that of the optimum concentratedly arranged TVMD system.Table 4 lists the total optimum inertance and damping coefficient of the TVMD designed using the proposed and SDOF methods, as well as those of the stiffness-proportionally distributed TVMDs.
The optimum inertance required to achieve the mass ratio  = 0.1 is identical for the proposed and SDOF methods because the mass ratio is derived from the same equation using the uncontrolled mode vector by definition, as shown in Equation (31).Note that the concentratedly arranged TVMD system significantly reduces the demand for the amount of inertance compared with the distributed TVMD system.The total required inertance for the distributed system is 10 times larger than that of the concentrated system, which demonstrates the efficiency of the concentrated system.
Stiffness-proportional damping is adopted for the inherent damping; the inherent damping ratio for the first mode is 0.02.
A synthetic ground motion provided by the Building Center of Japan (BCJ) (hereafter referred to as BCJ L2), the North-South component of the 1940 Imperial Valley Earthquake recorded at El Centro (hereafter referred to as El Centro 1940 NS), the North-South component of the 1968 Tokachi-oki Earthquake recoded at Hachinohe Hourbor (hereafter referred to as Hachinohe 1968 NS), and the North-South component of the 2011 Great East Japan Earthquake recorded at Tohoku University (hereafter referred to as Tohoku 2011 NS) are adopted as the input ground motions.Figure 16 shows the time histories of their ground accelerations and their velocity response spectra.
Figure 17 shows the maximum shear coefficient, drift angle, and floor response accelerations on each floor of the uncontrolled and controlled structures.The concentratedly arranged TVMD system designed through the proposed method can obtain a similar control effect as the baseline (optimum stiffness-proportionally distributed TVMDs).The performance of the concentratedly arranged TVMD system was better in terms of controlling the drift angles of the stories lower than the TVMD connected floor (third floor), whereas it is slightly poor for the upper stories.Although the floor response accelerations on some floors in the cases of the El Centro 1940 NS and Hachinohe 1968 NS suffered poorer response than the bare frame, most of the floor response accelerations were better than the bare frame otherwise.The time-history responses of the viscous element and TVMD in Figure 18 show that the concentratedly arranged TVMD system exploits the flexibility of the supporting members in amplifying the deformation in the viscous element, thereby resulting in enhanced energy dissipation.Further, we examine the costs of adding TVMD system.The cost of a TVMD depends on its maximum resistive force, inertance, damping coefficient, supporting spring component, and so forth.The most crucial factor is the maximum resistive force.Herein, we assume that the device cost is roughly proportional to the maximum resistive force.
Table 5 lists the resistive forces of both concentratedly arranged and distributed TVMD systems. M,baseline denotes the maximum device-wise resistive force generated in the distributed system, while  Σ,baseline and  Σ,proposed represent the total sum resistive forces generated by distributed and concentrated TVMD systems, respectively.As the number of TVMDs decreases from 10 to one, the device-wise resistive force of the concentrated system is roughly 2.08 times greater than that of the distributed system, which may result in a higher single device cost for the concentrated system.Nevertheless, the total sum resistive force of the concentrated system is approximately 0.290 times as large as that of the distributed system.This indicates that a concentrated TVMD system has advantage in the total cost, although a single device for a concentrated system can be more expensive than that of a distributed one.

CONCLUSION
In Japan, TVMDs have been put to practical use and applied to high-rise buildings, some of which are concentratedly located as discussed in this paper.In practical design, a simple formula derived from the fixed-point method for an SDOF structure equipped with a TVMD is used.For MDOF systems, the closed-form solution can be obtained only when the TVMD distribution is stiffness proportional.In this study, an approximate closed-form solution of the optimal design for a concentratedly arranged TVMD system is presented.As the change in the structural properties of seismic control systems including TVMDs is usually slight, the Sherman-Morrison theorem is effectively used to approximate its transfer function.A closed-form approximation of the optimal spring stiffness for a concentratedly arranged TVMD was presented by omitting the participation of the high-mode components of the ground motion excitation.As for the optimal damping coefficient in a TVMD, the simple formula for an equivalent 2-DOF TVMD system is observed to be an excellent alternative because it provides a value similar to that derived from the excessively complicated closed-form solution for an MDOF structure equipped with a TVMD.The contributions of this study are summarized as follows: • Using the Sherman-Morrison matrix inversion method, a closed-form expression of the response of an MDOF system controlled by a concentratedly arranged TVMD has been derived, and thus, the existence of fixed points on the DAF curves has been proven.• Two methods to derive optimal  ∞ control designs of a concentrated TVMD based on the fixed-point method are presented.One is a numerical method, and the other derives closed-form formulae.To obtain the numerical solution, an optimum design problem is formulated to determine the optimal spring stiffness of the TVMD and then, using its solution, another optimum design problem is formulated to determine the optimal damping coefficient of the TVMD.The closed-form solution is derived by approximating the contributions of higher modes in the dynamic response with static response.Although the closed-form solution of the optimal damping coefficient is too complicated and thus is not presented in this paper, it is shown that the optimal damping coefficient for a concentratedly arranged TVMD system can be replaced with the simple formula for a 2-DOF system containing a TVMD.
• Analytical examples using a shear building model demonstrated that the proposed approximated closed-form optimum design formulae exhibited good agreement with the numerical optimization results.• A TVMD installed spanning multiple stories achieved the same inertance ratio as that of a distributed TVMD system with a small inertance and total control force, demonstrating its cost efficiency.

A C K N O W L E D G M E N T S
This work was supported by JST SPRING, Grant Number JPMJSP2114.We would like to thank Editage (www.editage.jp)for English language editing.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.

F I G U R E 1 F I G U R E 2
Photograph of the iRDT.iRDT, inertial rotary damping tube.Schematic representation of the rotary inerter damper.

Figure 4
Figure4shows a TVMD element in which an inerter   , represented by a double circle, and a damping element   in a parallel arrangement are connected in series with a spring   . a ,  b , and  c represent the displacements of nodes A, B, and C, respectively. a ,  b , and  c represent the forces acting on nodes A, B, and C, respectively.

F I G U R E 3
Elevation of a frame containing TVMD.45 TVMD, tuned viscous mass damper.F I G U R E 4 TVMD element.TVMD, tuned viscous mass damper.

F I G U R E 6
DAF of an SDOF structure containing a TVMD.DAF, dynamic amplification factor; SDOF, single degree of freedom; TVMD, tuned viscous mass damper.F I G U R E 7 MDOF structure containing a concentratedly arranged TVMD.MDOF, multidegree-of-freedom; TVMD, tuned viscous mass damper.

F I G U R E 1 4
Relationship between the mass ratios and the peak amplitude of the DAF around  = 1.DAF, dynamic amplification factor.TA B L E 3Optimum damper parameters and corresponding damping ratios ( = 0.1).

F I G U R E 1 8
Deformations of the viscous element and the TVMD (closed-form method).TVMD, tuned viscous mass damper.TA B L E 5The resistive forces of concentratedly arranged and distributed TVMD systems ( = 0.1).
Fundamental periods and angular frequencies of the bare frame.

Total damping coefficient (kN/m)
F I G U R E 1 6 Input ground motions.