The analytical model of a SDOF structure equipped with a VeVMD is presented in Figure 2. The mass and stiffness of the primary SDOF structure are m and k , respectively. The SDOF structure is assumed to be undamped for simplicity’s sake. As shown in Figure 2, the VeVMD comprises an inerter-damper and viscoelastic element in series. The inertance and stiffness of the VeVMD are m d and k b , respectively. The damping coefficients of the dampers arranged in parallel with the inerter and spring are c d and c b , respectively. The device topology of a VeVMD encompasses those of TVMD and TID. When the dashpot arranged in parallel with the inerter is removed— c d = 0 —the VeVMD reduces to a TID. Similarly, the VeVMD reduces to a TVMD when the dashpot arranged in parallel with the spring is removed— c b = 0 .

x is the displacement of the primary mass relative to the ground. x b and x d are the deformations of the spring and inerter, respectively. Accordingly, the equations of motion of the SDOF structure equipped with a VeVMD are m x ̈ + k x + m d x ̈ d + c d x ̇ d = − m x ̈ 0 m d x ̈ d + c d x ̇ d = k b x b + c b x ̇ b x = x b + x d . (1)

Table 1 summarizes the notations used in this paper.

Here we consider a harmonic ground excitation x ̈ 0 = − ω 2 X 0 e i ω t = − A 0 e i ω t , where i = − 1 , ω , and t are the imaginary unit, excitation angular frequency, and time, respectively. The responses of the primary structure and VeVMD are expressed in Equation 2. x = X e i ω t , x ̈ = − ω 2 X e i ω t = − A e i ω t , x d = X d e i ω t , x b = X b e i ω t . (2)

Accordingly, Equation 1 can be rewritten as − m ω 2 X + k + K i ω X = m A 0 = m ω 2 X 0 , (3) K i ω = c b i ω + k b c d i ω − m d ω 2 k b − m d ω 2 + c b + c d i ω , (4) where K ( i ω ) represents the dynamic stiffness of the VeVMD.

In this study, we examined two transfer functions, H X ( i ω ) and H A ( i ω ) , which are defined as a transfer function from the ground acceleration ( A 0 ) to the displacement of the primary structure relative to the ground ( X ) and a transfer function from the ground acceleration ( A 0 ) to the absolute response acceleration of the primary structure ( A + A 0 ) , respectively. H X i ω = X i ω A 0 H A i ω = A i ω + A 0 A 0 . (5)

Substituting Equations 3 and 4 into Equation 5 obtains ω 0 2 H X i ω = k k b − m d ω 2 + c b + c d i ω − m ω 2 + k k b − m d ω 2 + c b + c d i ω + c b i ω + k b c d i ω − m d ω 2 , (6) H A i ω = k k b − m d ω 2 + c b + c d i ω + c b i ω + k b c d i ω − m d ω 2 − m ω 2 + k k b − m d ω 2 + c b + c d i ω + c b i ω + k b c d i ω − m d ω 2 . (7)

Introducing the nondimensional parameters are expressed in Equation 8. ω 0 = k m , μ = m d m , η = k b k , h d = c b + c d 2 m k , α = c b c b + c d , γ = ω ω 0 . (8)

Equations 6 and 7 can be rewritten as ω 0 2 H X i ω = η − μ γ 2 + 2 h d γ i 1 − γ 2 η − μ γ 2 − μ η γ 2 − 4 h d 2 α 1 − α γ 2 + 2 1 − γ 2 + η 1 − α − α μ γ 2 h d γ i , (9) H A i ω = η − μ γ 2 − μ η γ 2 − 4 h d 2 α 1 − α γ 2 + 2 1 + η 1 − α − α μ γ 2 h d γ i 1 − γ 2 η − μ γ 2 − μ η γ 2 − 4 h d 2 α 1 − α γ 2 + 2 1 − γ 2 + η 1 − α − α μ γ 2 h d γ i . (10)

Note that ω 0 2 is multiplied to both sides of Equation 9 such that they are dimensionless.

The dimensionless displacement transfer function shown in Equation 9 can be rewritten as Equation 11. | ω 0 2 H X | = B 2 + D 2 ⋅ 4 h d 2 E 2 + G 2 ⋅ 4 h d 2 , (11) where B and E are the real parts of the numerator and denominator of Equation 9, respectively; D and G are the imaginary parts of the numerator and denominator of Equation 9 divided by 2 h d , respectively. There exist fixed points at which the curves of the transfer function pass though regardless of the damping ratio h d 2 when B 2 / E 2 = D 2 / G 2 (Hartog, 1985). The condition can be rewritten as η − μ γ 2 2 1 − γ 2 η − μ γ 2 − μ η γ 2 − 4 h d 2 α 1 − α γ 2 2 = 1 1 − γ 2 + η 1 − α − α μ γ 2 2 . (12)

The denominator of the right-hand side of Equation 12 indicates that E is independent of h d 2 when α = 0 and 1. Thus, α should be 0 (TVMD) or 1 (TID) for fixed-points to exist. Indeed, the fixed-point method can be used to derive H ∞ optimum designs of a SDOF structure equipped with TVMD ( α = 0 ) and TID ( α = 1 ) (Ikago et al., 2012a; Lazar et al., 2014; Hu and Chen, 2015; Lobato and Steffen Jr, 2017). This method utilizes the feature that transfer function curves pass through fixed points regardless of damping, and decreasing the ordinate of one of the fixed points increases that of another in TVMD and TID. Provided that the values of transfer function never fall below those of fixed points, the H ∞ norm can be minimized by equalizing the ordinates of the fixed points and rendering the transfer function to take peak values at the fixed points. It is similar with the acceleration transfer function (Equation 10). Table 2 summarizes the closed-form expressions of the optimum designs obtained by the fixed-point method for TVMD ( α = 0 ) and TID ( α = 1 ) (Ikago et al., 2012a; Saito et al., 2008; Lazar et al., 2014; Hu et al., 2015), where superscript ^o denotes optimum designs. Subscripts X and A denote optimum designs of displacement and absolute acceleration control, respectively.

Since we consider a multi-objective minimization problem in this study, the following definitions apply (Fonseca and Fleming, 1993) (Figure 3).

Here, Equation 14 defines the infinity norms of the transfer functions as follows: ρ X = ω 0 2 max ω H X i ω , ρ A = max ω H A i ω . (14)

The multi-objective optimum design problem is formulated as shown in Equation 15. [MODP] Find ν = μ , η , α to minimize ρ X , ρ A subject to 0 ≤ μ ≤ 1.0 0 ≤ η ≤ 1.0 0 ≤ α ≤ 1.0 . (15)

The solution of this problem is not a single design but a set of non-inferior designs, referred to as the “Pareto-optimal set”.

Here, the total damping coefficient in the VeVMD is preset such that h d = 0.1 ( c b + c d = 0.1 × 2 m k ) , and the ratio α determines the distribution of damping coefficient assigned to c b and c d . The MODP can be converted to a series of single objective optimum design problems with varying weight ξ = [ 0,1 ] as follows (Marler and Arora, 2010; Steuer, 1986; Japan Society of Seismic Isolation, 2013):

[Scalarized optimum design problem] Find ν to minimize σ ξ = 1 − ξ ρ X + ξ ρ A subject to 0 ≤ μ ≤ 1.0 0 ≤ η ≤ 1.0 0 ≤ α ≤ 1.0 , (16) where σ ( ξ ) is a scalarized objective function.

The Pareto-optimal set obtained by solving Equation 16 are functions of ξ and are represented by ν o ( ξ ) = { μ o ( ξ ) , η o ( ξ ) , α o ( ξ ) } . The objective functions given by ν o ( ξ ) are represented by ρ X o ( ξ ) and ρ A o ( ξ ) . Similarly, the value of scalarized objective function yielded by an optimum design for ξ is represented by σ o ( ξ ) = ( 1 − ξ ) ρ X o ( ξ ) + ξ ρ A o ( ξ ) . Figures 4A and B show the optimum designs and the values of the objective functions yielded by them with respect to ξ varying from 0 to 1 with an interval of Δ ξ = 0.002 . ρ A o ( ξ ) naturally decreases at the expense of ρ X o ( ξ ) as the weight ξ increases. While ξ is in between 0 and 0.9, a slight change is observed in the optimum design variables. However, when ξ increases beyond 0.9, α o ( ξ ) suddenly jumps from a value around 0 to that around 1. This indicates that an optimal design of the VeVMD gives a design that is close to the TVMD device topology ( α = 0 ) while ξ is between 0 and 0.9; once ξ exceeds 0.9, it shifts to the TID device topology ( α = 1 ) .

Thus, as shown in Figure 5A, the optimum designs are classified into TVMD, VeVMD, and TID configurations when α o < 0.05 , 0.05 ≤ α o < 0.95 , and 0.95 ≤ α o and identified by red, black, and blue colors, respectively. Figure 5 plots { ρ X o , ρ A o } for a series of ξ varying from 0 to 1 with an interval of Δ ξ = 0.002 , which is referred to as the Pareto front. In Figure 5B, the colors of the solid circle notations represent ξ value as indicated by the color bar. The minima of the peak displacement amplification factor ρ X o ( ξ ) and peak absolute acceleration amplification factor ρ A o ( ξ ) are attained at ξ = 0 and ξ = 1 , respectively. The TVMD and TID configurations thus perform effectively in mitigating displacement and absolute acceleration, respectively. This is because, while the excitation frequency is relatively low, the dynamic stiffness of the inerter is relatively low compared to the spring element in the VeVMD (Figure 2), resulting in larger motion and thereby efficient energy dissipation in the damper arranged parallel to the inerter ( c d ) . As the excitation frequency increases, the dynamic stiffness of the inerter increases to hinder the motion of the damper arranged parallel to it, which lead to the dominated energy dissipation in the damper arranged parallel to the spring ( c b ) .

As shown in Figure 5B, the inclination of the Pareto front curve significantly changes at the inflection point of the curve at ξ = 0.5 . The improvement of ρ X o ( ξ ) demands significant expense of ρ A o ( ξ ) when ξ < 0.5 , whereas the sensitivity of ρ A o ( ξ ) with respect to ξ significantly decreases beyond ξ = 0.5 . Thus, the solution at the inflection point, which is hereafter referred as the inflection point design, is one of the best options for structural engineers to pick from the Pareto-optimal set. Another benefit of choosing the inflection point design is that its device topology is TVMD that has been put to practical use in real-life buildings. The rest of this paper exclusively focuses on the inflection point design for the multi-objective optimum design problem.

Here, we compare the single objective optimum designs and inflection point design derived from the multi-objective optimum design problem. Table 2 summarizes the closed form expressions for single objective optimum designs (Ikago et al., 2012a; Lazar et al., 2014; Saito et al., 2008; Hu et al., 2015). The superscript of the design variables, TVMD and TID, represents the device topologies. The subscript X and A represent displacement and absolute acceleration control designs, respectively. For example, η X TVMD , h d , X TVMD represent the H ∞ control design of TVMD to minimize the peak displacement amplification factor.

To compare the performance of the inflection point design of the multi-objective optimum design problem and the solution obtained from fixed point methods, the values of design variables are determined under the constraint that all systems maintain an identical damping ratio h d = 0.10 .

For the H ∞ control of displacement amplification factor of TVMD system as shown in Equation 17; h d , X TVMD = μ 2 3 μ 1 − μ 2 − μ = 0.10 (17) was solved with respect to μ and accordingly η X TVMD was obtained. μ A TVMD , η A TVMD , μ A TID , and η A TID were calculated in a similar manner. Table 3 summarizes the optimum designs of IVCSs when h d = 0.10 .

Figure 6 plots the H ∞ norms of the IVCSs. Any point on the curve of the Pareto-optimal set in Figure 6 is noninferior to another according to the definition of noninferiority presented in Section 3.2., demonstrating the validity of the scalarization method in deriving multi-objective optimal design. The multi-objective design problem at ξ = 0 yields TVMD configuration minimizing the peak displacement amplification factor, which is equivalent to the single-objective design problem to minimize the peak displacement amplification factor of the TVMD system. However, the H ∞ norm obtained from the fixed-point method represented by red downward triangle notation is dislocated from the upper-left endpoint of the Pareto front. Similarly, the H ∞ norm of the fixed-point solution minimizing the peak absolute acceleration amplification factor represented by blue triangle notation is dislocated from the bottom-right endpoint of the Pareto front. This is because the closed form solutions are approximations.

The inflection point design (solid black circle in Figure 6) improves ρ A o by 27% at the 12% expense of ρ X o compared to the fixed-point TVMD displacement control design (solid red downward-pointed triangle), while it improves ρ X o by 17% at the 6% expense of ρ A o compared to the fixed-point TID acceleration control design (sold blue triangle).

Figure 7A depicts the displacement amplification factors when h d = 0.10 . The H ∞ optimum design to minimize ρ X shown in a red dashed line exhibits two aligned peaks attaining the minimum peak displacement amplification factor among the four optimum designs, and other optimum designs exhibit uneven peaks because they are detuned in terms of displacement control objective. Conversely, the red dashed line exhibits slightly uneven peaks, whereas the other three optimum designs with the acceleration control objective exhibit aligned peaks (Figure 7B).

From Table 2, the single objective H ∞ control design of TVMD to minimize peak acceleration amplification factor ( μ A TVMD = 0.24 , η A TVMD = 0.39 ) , is similar to that of the inflection point design derived from the multi-objective optimum design problem ( μ o = 0.25 , η o = 0.42 ) , which results in similar amplification factor curves shown by black solid and gray dashed lines in Figures 7A and B. This means that the closed form expression of TVMD acceleration control design ( μ A TVMD , η A TVMD , h d , A TVMD ) serves as an excellent initial guess in seeking a numerical solution for the inflection point design. The fixed-point TVMD acceleration control design can be an acceptable alternative to the inflection point design when a high precision in manufacturing the device is not required.

Figure 8 demonstrates that the fixed-point TVMD acceleration control designs give good approximations of the inflection point designs in general cases, including those other than h d = 0.10 .

To examine the robustness of a TVMD with respect to the variation of design variables, Equation 18 defines two indices. ζ X μ , η = ρ X μ , η , α = 0 ρ X o , ζ A μ , η = ρ A μ , η , α = 0 ρ A o . (18)

Since we examine TVMD exclusively, α = 0 .

Figures 9A and B show the contour of ζ X ( μ , η ) and ζ A ( μ , η ) when h d = 0.10 . The solid-colored circles represent the Pareto optimal designs of the VeVMD. The colors represent ξ value as indicated by the color bar. The fixed-point TVMD acceleration control design ( μ A TVMD , η A TVMD ) , represented by solid gray diamond notation and inflection point design ( μ o ( ξ = 0.5 ) , η o ( ξ = 0.5 ) ) represented by a solid black circle are located in an area where contour lines are sparse (less sensitive). On the other hand, the fixed-point TVMD displacement control design ( μ X TVMD , η X TVMD ) , represented by a solid red downward-pointing triangle, is located in an area where contour lines are dense (sensitive), indicating that the fixed-point TVMD acceleration control design ( μ A TVMD , η A TVMD ) , and inflection point design ( μ o ( ξ = 0.5 ) , η o ( ξ = 0.5 ) ) are more robust to variations in design variables than fixed-point TVMD displacement control design ( μ X TVMD , η X TVMD ) .

To examine the performance of inflection point design and fixed point designs in a multi-story structure, we use a ten-story benchmark shear building model presented by the Japan Society of Seismic Isolation (2013). Figure 10 shows the analytical model of the benchmark structure. m j , k j , c j , and x j are the mass, stiffness, damping coefficient, and the displacement relative to the ground, respectively. x ̈ 0 is the ground acceleration. Table 4 lists the mass and stiffness distribution of the structural model. The inherent damping is assumed to be 2% of the critical damping. Thus, c j = 2 × 0.02 ω ( 1 ) k j , where ω ( 1 ) = 3.12 is the fundamental natural angular frequency of the structure. As for conventional viscous damper design, the additional damping is set to 6% of the critical damping and the damping coefficients c a , j = 2 × 0.06 ω ( 1 ) k j . The structure is assumed to remain elastic when subjected to strong ground motions. Table 5 lists the fundamental natural periods and modal effective mass ratio of the first three modes.

The structure is equipped with VeVMDs tuned to the first and second modes. The inertance, stiffness, damping coefficient of the damper parallel to the inerter, and that parallel to the spring in a VeVMD are m d , j ( r ) , k b , j ( r ) , c d , j ( r ) , and c b , j ( r ) where r and j are the target mode and the story where the device is installed, respectively. If the distribution of inertance of the VeVMDs is stiffness proportional and the ratio of the modal effective inertance and modal effective mass of the primary structure for the r ^th mode is μ ( r ) , the inertance of VeVMD tuned to the r ^th mode (Ikago et al., 2012b) can be expressed by Equation 19. m d , j r = μ r ω r 2 k j r = 1,2 . (19)

Similarly, letting the stiffness ratio and damping ratio for the VeVMD tuned to the r ^th mode be η ( r ) and h d ( r ) , respectively, obtains Equations 20, 21. k b , j r = η r k j , (20) c b , j r + c d , j r = 2 h d r μ r ω r k j . (21)

Before designing VeVMDs, the total damping coefficient assigned to the devices is determined as shown in Equation 22. ∑ r = 1 2 ∑ j = 1 10 c b , j r + c d , j r = ∑ r = 1 2 2 h d r μ r ω r ∑ j = 1 10 k j = 2 h e ω 1 ∑ j = 1 10 k j , h e = 0.06 . (22)

Thus, Equation 23 holds. ∑ r = 1 2 2 h d r μ r ω r = 2 h e ω 1 . (23)

The damping distribution factor λ determines the damping allocated to the second mode: 2 h d 1 μ 1 ω 1 = 1 − λ 2 h e ω 1 , (24) 2 h d 2 μ 2 ω 2 = λ 2 h e ω 1 . (25)

For a fixed-point design, substituting h d ( r ) obtained from Equations 24 and 25 into h d (Table 2) and solving them with respect to μ ( r ) obtains the inertance ratio of VeVMD tuned to the r ^th mode. Then, η ( r ) can be obtained. For a flection point design, μ ( r ) , η ( r ) , and α ( r ) are numerically derived for a given h d ( r ) following the procedure discussed in Section 3.3. Table 6 summarizes the parameters obtained through the above procedures.

As for design ground motions, three historic ground motion records, three synthetic ground motions, and three long-period ground motion records (listed respectively in Tables 7–9) are used. The first six ground motions were selected in accordance with the practice in Japan, and three long-period ground motions (Xu et al., 2008) were added to examine the effect of long-period ground motion on the displacement control performance of TVMD. Figure 11 depicts the response velocity spectra of recorded ground motions (No. 1–3, 7–9).

When conducting seismic response analyses using normalized ground motion records, the variability in maximum response tends to be smaller when the records are normalized by PGV rather than by PGA, particularly for structures with natural periods of 2 s or longer (Editorial Committee of Structural Design Practice of High-rise Buildings, 2019). For this reason, it is standard practice in Japan to normalize ground motion records by PGV when designing high-rise buildings. Adopting this approach, the recorded ground motions (No. 1–3 and 7–9) are scaled such that their peak ground velocities (PGVs) are 0.5 m/s.

In Japanese structural design practice, artificial ground motions whose response spectra are compatible with the design response spectrum are used to avoid underestimating responses when the natural period of a building coincides with that of a notch in the spectra of recorded ground motions.

The synthetic ground motions are generated such that their response acceleration spectra are compatible with the design response acceleration spectra of Japan’s building code (Figure 12). They adopt the phase properties of recorded ground motions (Table 8).

The detailed parameters to derive the design spectrum can be found in Supplementary Figures S1 and S2 and Supplementary Table S1. Soil type 2 as the most common soil type in Japan is adopted for the surface subsoil type. The target spectrum shown in black solid line in Figure 12 is obtained by multiplying the response acceleration spectrum at the bed rock shown in Supplementary Figure S1 by the type 2 soil amplification factor shown in Supplementary Figure S2. As the amplitudes of the synthetic ground motions are determined by the target spectrum, their PGVs are not exactly 0.5 m/s (Table 8).

d j q and a j q represent the maximum inter-story drift of the i ^th story and maximum floor response acceleration of the i ^th floor yielded by the No. q ground motion.

We first compare the seismic response analysis results with the damping coefficient allocated to the second mode λ = 0.5 . Figures 13A and B compare the maximum inter-story drift ( { d j q ; i = 1,2 , … , 10 ; q = 1,8 } ) and maximum floor response acceleration ( { a j q ; i = 1,2 , … , 10 ; q = 1,8 } ) yielded by El Centro 1940 N–S (No. 1) and ILA056 1999 E–W records (No. 8). Figures 14A and B compare the maximum response in the entire building through all the ground motions: max j , q { d j q } and max j , q { a j q } . ‘*’ represents the performance of the conventional viscous damper design. These figures suggest that fixed-point TID acceleration control design performs the best in mitigating floor response acceleration response, whereas the fixed-point TVMD displacement control design performs better in mitigating inter-story drift. Inflection point design slightly improves both the inter-story drift and floor response acceleration yielded by the fixed-point TVMD displacement control design except for long-period ground motions.

Figures 15A and B plot ( max j , q = 1,2 , … , 6 { d j q } , max j , q = 1,2 , … , 6 { a j q } ) and ( max j , q = 7,8,9 { d j q } , max j , q = 7,8,9 { a j q } ) with respect to varying λ . The color bars indicate the λ value. As shown in the figure, acceleration responses yielded by inflection point design and fixed-point TVMD displacement control design significantly decrease when λ increases, whereas displacement responses yielded by the fixed-point TID acceleration control design increase significantly. The displacement responses yielded by inflection point design and fixed-point TVMD displacement control design are barely affected by the variation of λ . Similarly, the acceleration responses yielded by the fixed-point TID acceleration control design are barely affected by the variation of λ . Allocating 80 ∼ 90% of damping coefficient to the second mode control yielded excellent performance for TVMD configuration designs, whereas allocating 0 ∼ 10% of damping coefficient to the second mode control yielded excellent performance for TID configuration design. The TVMD topology designs with λ = 0.8 and TID topology designs with λ = 0 are not inferior to each other for the design ground motions (Figure 15A). Nonetheless, the fixed-point TVMD displacement control design with λ = 0.8 represented by a downward-pointing triangle notation is superior to any inflection point designs represented by solid circles for long-period ground motions (Figure 15B). When subjected to design ground motions (No. 1–6), both the TVMD and TID configuration designs outperform the conventional damper design in inter-story drifts with slightly increased absolute accelerations. Nonetheless, for the long-period ground motion cases, the TVMD configuration designs simultaneously suppress inter-story drifts and absolute accelerations whereas the TID configuration designs have no significant advantage.