Seismic isolation can alleviate the seismic force transmitted to the superstructure at the expense of isolator displacement by increasing the fundamental natural period of the structure. However, there is concern regarding long-period and long-duration ground motions that are amplified by a thick sedimentary subsoil on which many buildings are built and induced by large-scale subduction-zone earthquakes (Architectural Institute of Japan, 2007). Excessive isolator displacement caused by long-period and long-duration ground motion may result in moat wall impact, thereby damaging seismic isolation devices and superstructures.

Increasing the number of conventional dampers to address this challenge presents a dilemma: excessive damping compromises the seismic isolation performance under the effect of ground motions that are dominated by high-frequency components and that occur more frequently than long-period ground motions. Ikago and Inoue (2014) and Luo et al. (2019) reported that rate-independent linear damping (RILD) offers an attractive solution because it realizes direct control of the displacement, thus exploiting its rate-independent resistive force. Other systems that increase resistive force in accordance with response displacement include the skid system (Kelly et al., 1980), modulated homogeneous friction (Inaudi, 1997), and nonlinear displacement-dependent dampers (Ilgeigi et al., 2012).

Rate-independent linear damping (RILD) is a linear damping model used for representing the energy dissipation characteristics of structural materials that are independent of frequency over a wide frequency range. This contrasts with the commonly used linear viscous damping (LVD) model, whose energy dissipation is proportional to the frequency. RILD is also referred to as linear hysteretic or structural damping. Although we confined ourselves to linear damping systems in this study, a comprehensive review of nonlinear damping systems can be found in Lu et al. (2018).

Crandall (1991) reported that structures equipped with RILD respond before an excitation is applied, and RILD is thereby noncausal. Figure 1 shows a displacement time history of a single-degree-of-freedom (SDOF) system containing RILD subjected to impulse excitation at time t = 0. Anticipatory response is observed before an impulse is applied, thus violating the causality.

Owing to this noncausal nature, RILD is physically unsound and impossible to realize as a physical device. The first successful causal model that exhibited an approximated rate-independent energy dissipation behavior was proposed by Biot (1958). Caughey (1962) stated that Biot’s model can be constructed by arranging infinite Maxwell elements in parallel.

The transfer function from displacement to force in the frequency domain has a stiffness dimension and is thus referred to as the dynamic stiffness, which is expressed as a complex number. The real and imaginary parts of the dynamic stiffness are referred to as the storage and loss stiffness, respectively.

The storage stiffness is identical to the conventional stiffness coefficient. Although the loss stiffness has the dimensions of stiffness, it is in phase with the velocity—that is, 90^◦ advanced to the displacement—which results in energy dissipation.

Makris (1997) demonstrated that the modification of RILD to satisfy causality necessitates the addition of a storage stiffness that is proportional to the logarithm of the frequency. This explains why the storage stiffness of Biot’s model increases as the frequency increases. A fractional order derivative model for approximating RILD, proposed by Luo and Ikago (2021) also exhibited increasing storage stiffness with respect to frequency. Thus, the storage and loss stiffnesses are bound to each other by the causality requirement, which results in a trade-off between the accuracy of mimicking RILD and maintaining a low storage stiffness. This is the challenge encountered when exploring the application of a RILD-based damping model for the protection of low-frequency structures.

Keivan et al. (2020) developed a control algorithm for a magneto-rheological damper to causally realize the performance of RILD without the addition of an undesirable stiffness at the fundamental natural frequency of the controlled structure. The transfer function of the control algorithm was subsequently observed to be equivalent to that of a parallel configuration of negative stiffness and Maxwell elements (Luo et al., 2019). Their work further led to an examination of combinations of the Maxwell–Wiechert model, negative stiffness, and inerters to realize causal RILD models (Liu and Ikago, 2021a; b, 2022a; b,c). Wu and Ikago (2024) proposed the addition of two inerters to a Maxwell–Wiechert model in series and parallel to negate the undesirable storage stiffness at two specified frequencies. A combination of a negative-stiffness element and an inerter to negate the added storage stiffness in a causal RILD model was investigated by Liu et al. (2022) and Wu et al. (2023).

In this study, unlike conventional approaches, we propose expanding the order of dynamic stiffness of the components in the proposed damping device to mitigate undesirable storage stiffness. Specifically, an inerter is connected in parallel to the damper to construct a commercially available device called a tuned viscous mass damper (TVMD). Limiting the frequency band in which the loss stiffness of the proposed device is tuned can significantly reduce the required number of TVMD branches, thus improving its feasibility. Another benefit of limiting the bandwidth is that it can reduce undesirable storage stiffness. In this study, the frequency band was determined to include the fundamental natural angular frequency of the controlled structure as well as twice that value.

The novelty of this study lies in the development of a physical model that approximates the energy dissipation behavior of an ideal RILD with lower storage stiffness compared with existing causal models in the literature.

The remainder of this paper is organized as follows: Section 2 expands the dynamic stiffness of Biot’s model to the second order. The second-order model included Biot’s model. Section 3 presents the adopted TVMD, which is a commercially available device with second-order dynamic stiffness, to realize the proposed model. To improve its feasibility, in Section 4, we propose that the frequency band be limited, wherein the target constant loss stiffness is attained, which significantly reduces the number of required TVMD devices. An analytical example using the proposed model with three TVMD branches demonstrated its efficacy in causally approximating the ideal RILD, thus outperforming conventional LVD. Section 5 presents the conclusions of this study.

One of the most successful models to causally approximate an ideal RILD is Biot’s model (Biot, 1958), which comprises infinite branches of Maxwell elements (Figure 2).

The transfer function obtained from the deformation x to the resistive force F _j of the jth branch of Biot’s model is: Z j M i ω = F j x = k j i ω i ω + r j . (1) where k _j , c _j , and r _j = k _j /c _j are the stiffness, damping coefficient, and relaxation parameter of the Maxwell element, respectively. i = − 1 and ω are the imaginary unit and excitation angular frequency, respectively. Z j M ( i ω ) has a stiffness dimension and is thus referred to as the dynamic stiffness. Note that the relaxation parameter r _j is the inverse of relaxation time.

The real and imaginary parts of the dynamic stiffness are referred to as the storage and loss stiffnesses, respectively. The storage stiffness is identical to the conventional stiffness. The product of the loss stiffness and deformation yields the resistive force, the amplitude and phase of which are proportional to the deformation and are in phase with the velocity, respectively. Thus, the loss stiffness is related to energy loss.

[Storage stiffness] R Z j M i ω = ω 2 ω 2 + r j 2 k j (2)

[Loss stiffness] I Z j M i ω = ω r j ω 2 + r j 2 k j (3)

Figure 3A shows how Biot’s model realizes constant loss stiffness characteristics at high frequencies. As indicated by the solid lines, the loss stiffness of the jth branch reaches its maximum value (indicated by red circles) of k _j /2 at angular frequency ω = r _j .

When the stiffness and damping coefficient of each branch are arranged such that Δ r = r j + 1 − r j ≡ constant , r j = ε + j − 1 Δ r , k j = Δ r r j k 1 , (4) the lowest relaxation parameter and jth damping coefficient are r 1 = ε , c j = k j r j . (5)

Accordingly, the dynamic stiffness of Biot’s model Z Δ r B ( i ω ) is Z Δ r B i ω = k 0 + lim n → ∞ k 1 ∑ j = 1 n i ω i ω + r j Δ r r j . (6)

Furthermore, when Δr → 0, Z 0 B i ω = lim Δ r → 0 Z Δ r B i ω = k 0 + k 1 ∫ ε ∞ i ω i ω + r d r r = k 0 + k 1 ln 1 + ω ε 2 + i ⁡ arctan ω ε . (7)

Figure 3A presents the loss stiffness of Biot’s model when the stiffnesses and relaxation parameters are determined as follows: k 1 = 2 π , Δ r = 1 , r 1 = ε = 1 , r j = j Δ r = j , k j = Δ r r j k 1 . (8)

Although the loss stiffness exhibits frequency dependency at low frequencies, it becomes almost independent of the frequency at high frequencies.

Figure 3B presents a comparison of the storage stiffness of Biot’s model and ideal RILD when k ₀ = 0. The storage stiffness of Biot’s model increased with frequency, while that of the ideal RILD was constant, irrespective of the frequency. This is owing to the causality requirement that bounds the real and imaginary parts of the dynamic stiffness, which is known as the Kramers–Kronig relationship (Booij and Thoone, 1982). The causality of Biot’s model results in the addition of stiffness when it is incorporated into a structure.

Herein, we propose a novel causal model to approximate ideal RILD. The model comprises infinite branches of any mechanical device in a second-order physical system connected in parallel, as shown in Figure 4, where s is the Laplace variable.

The dynamic stiffness of a second-order device can be defined as a bi-proper transfer function: Z s = k j a 1 s 2 + a 2 r s b 1 s 2 + b 2 r s + b 3 r 2 , (9) where r and k are the frequency and stiffness dimensions, respectively. a ₁, a ₂, b ₁, b ₂, and b ₃ are the dimensionless parameters to be determined.

Let j denote the jth branch, where the dynamic stiffness of the proposed model written in the frequency domain is the summation of all the branches. Z P s = F s x s = lim n → ∞ ∑ j = 1 n k j a 1 , j s 2 + a 2 , j r j s b 1 , j s 2 + b 2 , j r j s + b 3 , j r j 2 (10)

Here we assume that the relaxation parameters are distributed with a uniform interval of Δr = r _j+1 − r _j (j = 1, 2, ⋯) starting from a small value r ₁ = ɛ. Accordingly, the stiffness of the jth branch is k j = k 1 Δ r r j , r j = ε + j − 1 Δ r . (11)

Based on these definitions, the dynamic stiffness of this model can be expressed as follows: Z P s = lim n → ∞ ∑ j = 1 n a 1 , j s 2 + a 2 , j r j s b 1 , j s 2 + b 2 , j r j s + b 3 , j r j 2 Δ r r j . (12)

The sum is then substituted into the integration as follows: Z P s = k 1 ∫ ε ∞ a 1 s 2 + a 2 r s b 1 s 2 + b 2 r s + b 3 r 2 d r r . (13)

Partial fraction decomposition of the integrand yields Z P s = k 1 ∫ ε ∞ A r + B r − s Δ 1 + C r − s Δ 2 d r , (14) where Δ₁ and Δ₂ are the roots of the denominator Δ 1 = − b 2 + b 2 2 − 4 b 1 b 3 2 b 3 , Δ 2 = − b 2 − b 2 2 − 4 b 1 b 3 2 b 3 , (15) and A, B, and C can be represented using the roots as follows: A = a 1 b 3 b 1 , B = a 1 + a 2 Δ 1 Δ 1 Δ 1 − Δ 2 , C = a 1 + a 2 Δ 2 Δ 2 Δ 2 − Δ 1 . (16)

Subsequent to integration and simplification, Eq. 14 is reduced to Z P s = k 1 − B ln 1 + − Δ 1 s ε + − C ln 1 + − Δ 2 s ε . (17)

Hereafter, we exclusively examine the imaginary part of dynamic stiffness (loss stiffness).

When Δ₁ and Δ 2 ∈ R , the loss stiffness is further reduced to I Z P s = k 1 arctan − Δ 1 ω ε + arctan − Δ 2 ω ε , (18) where we used the following relationship: ln x + i y = ln x 2 + y 2 + i ⁡ arctan y x , x , y ∈ R (19)

Eq. 18 clearly shows that, when ω ≫ɛ or ɛ → 0, the loss stiffness approaches the constant value k ₁ π.

For the case in which Δ₁ and Δ₂ are complex-valued, B and C are also complex-valued. It should be noted that the pairs of Δ₁ and Δ₂ and B and C are conjugate pairs. For further simplification, the following equations are introduced: B = α 1 + i β 1 , C = α 1 − i β 1 , Δ 1 = α 2 + i β 2 , Δ 2 = α 2 − i β 2 , (20) where α ₁, α ₂, β ₁, and β ₂ are real-value constants. Eq. 17 is expanded by separating the real and imaginary parts as follows: I Z P s = I α 1 + i β 1 ln 1 − β 2 ω ε 2 + α 2 ω ε 2 + i ⁡ arctan α 2 ω ε 1 − β 2 ω ε + I α 1 − i β 1 ln 1 + β 2 ω ε 2 + α 2 ω ε 2 + i ⁡ arctan α 2 ω ε 1 + β 2 ω ε , (21) which is further reduced to: I Z P s = β 1 ⁡ ln ε ω 2 − β 2 + α 2 ε ω 2 + β 2 + α 2 + α 1 arctan α 2 ε ω − β 2 + arctan α 2 ε ω + β 2 . (22)

Eq. 22 demonstrates that the loss stiffness approaches a constant value as the excitation angular frequency ω increases, which proves that the proposed model can be an excellent approximation of the ideal RILD in terms of loss stiffness.

As previously mentioned, Biot’s model comprises infinite Maxwell elements arranged in parallel. The dynamic stiffness of Biot’s model in the frequency domain is a summation of those of all branches: Z 0 B s = F s x s = lim n → ∞ ∑ j = 1 n k j s s + r j , (23) where r _j represent the relaxation frequency of the jth branch. It should be noted that the proposed model is reduced to Biot’s model when a ₁, b ₁ = 0 and a ₂, b ₂, b ₃ = 1. Z P s = lim n → ∞ ∑ j = 1 n k j a 1 , j s 2 + a 2 , j r j s b 1 , j s 2 + b 2 , j r j s + b 3 , j r j 2 = lim n → ∞ ∑ j = 1 n k j r j s r j s + r j 2 = lim n → ∞ ∑ j = 1 n k j s s + r j (24)

Because the proposed second-order causal model encompassing Biot’s model has more parameters than Biot’s model, it offers a solution for reducing undesirable storage stiffness, as discussed in the following section.

As shown in Figure 5A, a parallel arrangement of an inerter and a damper yields a compact device that can provide thousands of tons of inertance, which is sufficient for structural control. The connection of a spring element to the device configures a tuned-mass-damper-like (TMD-like) system, designated as a TVMD (Ikago et al., 2012). In contrast to a conventional TMD, a mass element (inerter) is arranged in parallel to the damper in a TVMD, as shown in Figure 5B. To the best of our knowledge, inerter-based seismic control systems have been applied to real-life building structures only in Japan (Sugimura et al., 2012; Architectural Institute of Japan, 2014; Ishii et al., 2014; Ogino and Sumiyama, 2014).

The dynamic stiffness of TVMD can be expressed as follows: Z T s = F s x s = k d r s 2 + ω d 2 s r s 2 + ω d 2 s + r ω d 2 (25) where r = k _d /c _d and ω d = k d / m d represent the relaxation parameter and fundamental angular frequency of the device local vibratory system, respectively.

The dynamic stiffness of a TVMD single branch shown in Eq. 25 is similar to Eq. 9. It is easy to infer that the infinite branches of TVMDs will also be a physical realization of the proposed model. To prove this, we first express the dynamic stiffness of the infinite branches of TVMDs as follows: Z T s = lim n → ∞ ∑ j = 1 n k d , j r j s 2 + ω d , j 2 s r j s 2 + ω d , j 2 s + r j ω d , j 2 (26)

The springs and inerters in the model resulted in the formation of infinite oscillators. Herein, we introduce a damping ratio ζ j = c d , j / 2 k d , j m d , j as the ratio of the damping coefficient to the critical damping coefficient in each branch. If the infinite branches of the TVMDs are arranged in a manner similar to Biot’s model, Eq. 26 can be rewritten as Z T s = lim n → ∞ ∑ j = 1 n k d , j s 2 + 4 ζ 2 r j s s 2 + 4 ζ 2 r j s + 4 ζ 2 r j 2 . (27)

On comparing the above equation with Eq. 10, the parameters can be identified as a ₁, b ₁ = 1 and a ₂, b ₂, b ₃ = 4ζ ² for all branches. The dynamic stiffness of the infinite TVMDs can then be derived as Z T s = k d , 1 − B ln 1 + − Δ 1 s ε + − C ln 1 + − Δ 2 s ε (28)

where B, C, Δ₁, and Δ₂ are defined as B = − 2 ζ 2 1 − ζ ζ 2 − 1 C = − 2 ζ 2 1 + ζ ζ 2 − 1 Δ 1 = − ζ − ζ 2 − 1 2 ζ Δ 2 = − ζ + ζ 2 − 1 2 ζ (29)

B, C, Δ₁, and Δ₂ have complex values when 0 < ζ < 1; however, they have real values when ζ > 1. As previously discussed, in either case, this model can serve as an approximation of an ideal RILD model.

This section presents an example to demonstrate the advantages of the proposed model over Biot’s model. The loss stiffness in Biot’s model monotonically increases over the entire frequency range [0, ∞] and approaches k 1 π 2 . Here, we set k 1 = 2 π such that the loss stiffness function approaches unity at high frequencies. I Z 0 B i ω = k 1 ⁡ arctan ω ε < k 1 π 2 = 1 (30)

The lower bound of the relaxation parameter for Biot’s model ɛ ^B was determined such that the maximum error in the loss stiffness at frequencies greater than 1 rad/s was less than 1%. 1 − I Z 0 B ω ω = 1 < 0.01 (31)

An iteration shows that ɛ ^B = 0.015 satisfies Eq. 31. For the proposed model, the required ζ = 0.5 when we set k _d,1 to 2 π in the same manner as in Biot’s model. Another iteration for realizing the accuracy of Biot’s model in terms of the loss stiffness shows that the value of the lower bound of the relaxation parameter of the proposed model ɛ ^P = 0.352. Figures 6A, B present the storage and loss stiffnesses of the Biot’s and proposed models with these parameters. The loss stiffnesses of both models exhibit frequency-independent characteristics in the frequency region beyond 1 rad/s. The proposed model exhibits an advantage over Biot’s model in maintaining the flexibility of the isolation layer at the expense of a compromised loss stiffness in the region of frequencies less than 1 rad/s.

The loss stiffness of a single TVMD branch is given as Im Z T ω = k d r 3 ζ 4 ω ω 2 ζ 4 r 2 + r 2 ζ 2 − ω 2 4 2 (32)

To solve for ∂ ∂ ω Im [ Z T ( ω ) ] = 0 with respect to ω, we obtain the angular frequency at which the loss stiffness exhibits a peak value. ω = f ζ ⋅ r , f ζ = − 8 ζ 4 + 4 ζ 2 + 8 ζ 2 ζ 4 − ζ 2 + 1 3 (33)

Therefore, the peak value of the loss stiffness is max ω Im Z T ω = k d f ζ f 2 ζ + 1 − f 2 ζ 4 ζ 2 . (34)

Figure 7A presents the shape of the TVMD loss stiffness function and its peak value coordinates.

Eq. 34 indicates that the peak value of the loss stiffness is proportional to the stiffness of the branch k _d . As shown in Figure 7B, the peak value decreases and approaches k _d /2 as the damping ratio of the branch ζ increases.

Figure 7C shows that the shape of the loss stiffness function becomes longer and narrower as ζ decreases.

Herein, we denote a second-order causal RILD device comprising n branches of TVMDs as n-TVMD. For an n-TVMD, the frequency band in which the loss stiffness is specified is limited to [ω _ℓ , ω _u ].

Regarding the distribution of the relaxation parameters, Genta and Amati (2009) adopted geometric progression. Liu and Ikago (2021b) adopted fundamental natural frequencies as the relaxation parameters to avoid the parameters of their proposed device to become negative values. This study adopted a uniform distribution as follows: λ j = ω ℓ + j − 1 ω u − ω ℓ n − 1 , j = 1,2 , … , n (35) The number of branches n should be chosen properly to avoid negative parameters of the branches.

All the branches share the same damping ratio ζ, and the jth branch is designed to exhibit a peak value at frequency λ _j . Thus, from Eq. 33, r j = λ j f ζ . (36)

Let Z _n (ω; ω _ℓ , ω _u ) denote the dynamic stiffness of the n-TVMD with a target frequency range [ω _ℓ , ω _u ], and the loss stiffness is Im Z n ω ; ω ℓ , ω u = ∑ j = 1 n k d , j r j 3 ζ 4 ω ω 2 ζ 4 r j 2 + r j 2 ζ 2 − ω 2 4 2 = ∑ j = 1 n k d , j f ζ λ j 3 ω f 2 ζ ω 2 λ j 2 + λ j 2 − f 2 ζ ω 2 4 ζ 2 2 . (37)

We consider incorporating n-TVMD into a base-isolation layer, the isolator stiffness of which is k _s , and the target loss stiffness to be imparted is ηk _s . Accordingly, the following condition is imposed to realize a constant loss stiffness within the frequency region of interest. Im Z n λ 1 ; ω ℓ , ω u = Im Z n λ 2 ; ω ℓ , ω u = … = Im Z n λ n ; ω ℓ , ω u = η k s (38)

When we define β _j and Q _j (λ _k ) as follows: β j = k d , j k s , Q j λ k = f ζ λ j 3 λ k f 2 ζ λ j 2 λ k 2 + λ j 2 − f 2 ζ λ j 2 4 ζ 2 2 , (39) Eq. 38 can be rewritten as Im Z n λ k k s = ∑ j = 1 n β j Q j λ k = η , j = 1,2 , … , n . (40)

Figure 8 illustrates how constant loss stiffness is realized in a four-TVMD model in the specified frequency range. The dashed lines represent the loss stiffness of each branch. The summation of the loss stiffnesses of all the branches attains a constant value stiffness within the target frequency range, as shown by the solid line.

On solving Eq. 40 with respect to {β _j }, we obtain β 1 β 2 ⋮ β n = Q 1 λ 1 Q 2 λ 1 ⋯ Q n λ 1 Q 1 λ 2 Q 2 λ 2 ⋯ Q n λ 2 ⋮ ⋮ ⋮ Q 1 λ n Q 2 λ n ⋯ Q n λ n − 1 η η ⋮ η . (41)

A greater number of branches of TVMDs causes the loss stiffness to become flatter when ζ is fixed, as shown in Figure 9A.

However, some of the β _j values become negative when λ _j are uniformly distributed and n exceeds a certain value, as shown in Figure 9B. The maximum n required to maintain β _j as positive, thereby ensuring the feasibility of the n-TVMD is denoted as n _u , which can be reduced by increasing ζ.

Fewer branches of TVMD are preferable in practical structural design, which is realized at the expense of accuracy in approximating the constant loss stiffness. Discussions on achieving a compromise between feasibility and accuracy when designing an n-TVMD are presented in the next section.

To elucidate the influence of the target frequency range [ω _ℓ , ω _u ] on the control efficacy of an n-TVMD, we examined an SDOF structure equipped with an n-TVMD, as shown in Figure 10.

The transfer functions from the ground acceleration X ̈ g ( ω ) to the displacement, floor response acceleration, and damper resistive force are X j ω ; ω ℓ , ω u X g ̈ ω = − m s − ω 2 m s + k s + Z j ω ; ω ℓ , ω u , A j ω ; ω ℓ , ω u X g ̈ ω = k s + Z j ω ; ω ℓ , ω u − ω 2 m s + k s + Z j ω ; ω ℓ , ω u , F j ω ; ω ℓ , ω u X g ̈ ω = − Z j ω ; ω ℓ , ω u m s − ω 2 m s + k s + Z j ω ; ω ℓ , ω u , (42) where the superscript j represents the damping element incorporated into the structure; j = 1 and 2 correspond to RILD and proposed models, respectively.

Thus, Z 1 ω ; ω ℓ , ω u = Z RILD ω = i η k s sgn ω , Z 2 ω ; ω ℓ , ω u = Z n ω ; ω ℓ , ω u . (43)

Figures 11A–C, present the peak values with respect to ω of the transfer functions of the SDOF structure containing the n-TVMD system (j = 2) normalized by those of the SDOF structure containing RILD (j = 1). The horizontal axes indicate the lower- and upper-bound frequencies normalized by the fundamental angular frequency. The vertical axes indicate the peak values of (a) displacement, (b) acceleration, and (c) control force transfer functions normalized by those of RILD. The n-TVMD exhibits the best performance with a lower-bound frequency near ω ₀, while the displacement mitigation performance is close to that of RILD as long as the lower-bound frequency is lower than ω ₀. When the upper bound frequency exceeds twice the fundamental frequency, the performance of the n-TVMD approaches that of RILD.

Figures 11D–F, present the peak values of the transfer functions with respect to the lower-bound frequency as the loss factor is varied when the upper-bound frequency is fixed at twice the fundamental natural frequency. The upper bound of the lower-bound frequency ω _ℓ , which exhibits a displacement control performance similar to that of RILD, decreases as the loss factor increases, as shown in Figure 11D. However, the optimal lower-bound frequencies that minimize the acceleration and control force remain almost constant. Utilizing this characteristic can minimize the acceleration and damping force responses, rendering them closer to an ideal RILD. The increased displacement response can be managed with a higher loss factor to satisfy design requirements.

The above observations suggest that setting ω _ℓ = ω ₀ is a viable option for n-TVMD to realize a control performance similar to that of an ideal RILD when the target loss factor η is relatively low. Figure 11G presents the cross-sections of Figures 11A–C along ω _ℓ = ω ₀. As discussed in Section 3.3, the proposed system improves the storage stiffness more than the conventional causal RILD model [see Figure 6A], resulting in a better approximation of the floor response acceleration and damper resistive force, especially when ω _u > 2ω ₀. However, this is attained at the expense of the displacement response owing to the reduced loss stiffness at low frequencies [see Figure 6B].

Considering that a lower value of ω _u necessitates a smaller number of branches, thus resulting in enhanced feasibility and cost efficiency, we suggest setting ω _u = 2ω ₀ for SDOF systems.

Figure 12 presents the parameter design flow for an n-TVMD. The damping ratio ζ and target angular frequency range [ω _ℓ , ω _u ] are given first. Herein, ζ = 0.25, and the target angular frequency range is [ω ₀, 2ω ₀], where ω ₀ is the fundamental angular frequency of the controlled structure. The first iteration is performed to identify the loss factor η with the maximum feasible number of branches n _u that satisfy the design criteria. In this study, we began with an initial estimate of the value of η, determined n _u and β _j , and performed a time–history analysis. The η value was gradually increased until the design criteria were satisfied. In the second iteration, the number of branches n was decremented until the minimum n satisfying the design criteria was reached.

Figure 13 presents the analytical model of an example multi-degree-of-freedom (MDOF) base-isolated structure with its properties listed in Figure 13B. The isolator horizontal stiffness was designed such that the fundamental undamped natural period was 4.0 s, following the base-isolation design practice in Japan.

The mass-proportional portion of the Rayleigh damping imparts undesirable damping to the first mode of the isolated structure beyond that provided by the dampers in the isolation layer, thus resulting in an underestimated seismic response, as reported in Hall (2006). Therefore, stiffness-proportional damping was adopted in this study, with a damping ratio of 2% assigned to the first mode of the superstructure with a fixed base (Ryan and Polanco, 2008; Anajafi et al., 2020). The inherent damping at the isolation layer is ignored. Figure 13C presents the modal damping ratios of the structure with no damping devices incorporated into the isolation layer.

The proposed model comprises multiple branches of TVMDs. A TVMD is a commercially available device that has been applied to real-life building structures in Japan (Sugimura et al., 2012; Ishii et al., 2014; Ogino and Sumiyama, 2014). Supplementary Figure S3A illustrates the installation of a TVMD in a building frame (Sugimura et al., 2012). The inertance, stiffness, and maximum resistive force of the TVMD device are 5,400 tons, 7,300 kNm/s, and 68,600 kN/m, and 1,200 kN, respectively. Thus, the parameters of the TVMD devices listed in Table 2 fall within the feasible range. When incorporating the proposed device into a seismic isolation layer, the rotary inerter dampers are connected to a laminated rubber device to provide a tuned stiffness, as shown in Supplementary Figure S3B.

This study focused exclusively on theoretical and numerical analyses. Therefore, experimental studies using small- or full-scale specimens should be conducted in the future. In addition, because the rotary inerter damper is a relatively new device that has not yet been widely adopted, its cost is considered a challenge.