Seismic control mechanism of tuned viscous mass damper (TVMD) outriggers

Abstract

This study investigates the seismic control mechanism of tuned viscous mass damper (TVMD) outriggers in a core wall-frame system, which can be regarded as providing frequency-dependent equivalent stiffness and damping. Closed-form formulas for equivalent stiffness and damping of TVMD outriggers are derived based on a distributed-parameter model that represents the core-wall frame system. As the frequency increases, the equivalent stiffness transitions from negative to positive values and asymptotically decays to the spring stiffness of TVMD outriggers at infinite frequency. Simultaneously, the equivalent damping increases from the TVMD damping coefficient to a maximum value as the frequency increases from zero to the tuning frequency, and then asymptotically decays to zero along with the increasing frequency. A modal response mitigation ratio is defined and calculated, revealing that equivalent damping plays a dominant role over equivalent stiffness in the control mechanism. An analysis of equivalent damping-frequency curves elucidates that the control mechanism of TVMD outrigger, i.e., tuning effect for the targeted mode and damping supplement effect for the lower-order modes. Finally, seismic control performance of TVMD, viscous damper (VD), and negative stiffness damper (NSD) outriggers is compared based on a 140-m tall core wall-frame structure. Seismic response analysis demonstrates that TVMD, VD and NSD outriggers reduce the maximum inter-story drifts by 26%, 10%, and 17%, and the maximum floor accelerations by 27%, 12%, and 18%, respectively.

Keywords

tuned viscous mass damper (TVMD)damped outrigger vibration control core wall-frame system seismic control mechanism

Introduction

Recent earthquakes highlight the importance of enhancing the seismic resiliency of high-rise buildings (Lu et al., 2013; Malhotra, 1999) by controlling their seismic responses and mitigating the consequences. The core wall-frame system is a prominent structural system commonly adopted in commercial high-rise buildings due to its efficient structural solution and highly economical construction process (Hong et al., 2011). The concept of a damped outrigger, illustrated in Figure 1, was proposed for enhancing the seismic performance of core wall-frame system (Smith and Willford, 2007), where energy dissipation devices (EDDs) are installed to connect the outrigger trusses or beams with the perimeter columns.

Figure 1.

Configuration of a damped outrigger.

In early stage, conventional EDDs such as viscous dampers (VDs) (Smith and Willford, 2007; Zhou and Xing, 2021), metallic yielding devices (Wang, 2017), and friction devices (Lin et al., 2023) were adopted in damped outrigger systems to enhance structural performance. For instance, Smith and Willford (2007) proposed a VD outrigger along with its practical design methodology, which was implemented in the 210-m tall Saint Francis Shangri-La Twin Tower in the Philippines. In a comparative study, Wang et al. (2016) evaluated the seismic performance of three damped outrigger systems, i.e., the VD outrigger, buckling-restrained brace outrigger, and shear metal damper outrigger, for a 41-story high-rise building. The results demonstrated that these systems could reduce the maximum inter-story drift ratio by 20% to 25%. Ding et al. (2021) presented the design of a hybrid outrigger system composed of rigid outriggers and VD outriggers in the China International Silk Center Building (518 m tall). The seismic control effectiveness of these conventional damped outriggers has also been validated in other real projects, including the Chongqing Raffles North Tower, which employed steel damper outriggers (Hu et al., 2021), and Shenzhen China Resources Headquarters, which utilized VD outriggers (Ho, 2021).

While VD outriggers provide additional damping to structures, their achievable additional damping ratio could be limited by the axial stiffness of perimeter columns (Zhou and Xing, 2021). The vertical displacement of the VD outriggers comprises both the VD deformation and the axial deformation of the perimeter columns. If the axial stiffness of the columns is inadequate, the resulting non-negligible axial deformation of columns reduces the VD deformation, thereby limiting damping supplement. To overcome this difficulty, scholars have proposed to incorporate a negative stiffness element to connect in parallel with the VD (Liu et al., 2018; Wang et al., 2020), forming a negative stiffness damper (NSD). The introduction of negative stiffness reduces the dynamic stiffness of the damper, leading to increased VD deformation and enhanced damping supplement. The negative stiffness can be realized using precompression springs (Sarlis et al., 2013; Sun et al., 2023a, 2023b), negative stiffness metamaterials (Giri and Mailen, 2021), magnets (Yuan et al., 2021), and buckled beams (Zhang et al., 2020). Nagarajaiah et al. (2022) revealed that NSD outriggers effectively provide damping to multiple modes of high-rise buildings and achieve superior seismic control performance compared with VD outriggers.

Except for the negative stiffness element, the inerter is another element which can enhance the efficiency of damped outriggers. The inerter (Smith, 2002; Smith and Wang, 2004) is a mechanical element that generates an inertial force proportional to the relative acceleration between its two terminals. Primary approaches to obtaining the inerter include the ball-screw system (Arakaki et al., 1999a, 1999b), the rack-and-pinion system (Saitoh, 2012; Mirza and Mercan, 2016), the hydraulic motor system (Wang et al., 2011), and the helical pipe system (Swift et al., 2013). Among various types of inerter-based dampers, the viscous mass damper (VMD, also known as a rotational inertia damper, RID) (Hwang et al., 2007) and the tuned viscous mass damper (TVMD) (Ikago et al., 2012b) are very promising. The VMD is composed of an inerter element and a dashpot element connected in parallel. Liu et al. (2018) proposed a VMD outrigger, and indicated that this outrigger significantly reduced the peak responses of inter-story drift and floor acceleration of high-rise buildings. Nagarajaiah et al. (2022) reported that the VMD outrigger provides supplementary damping for a specific structural mode because of its frequency-dependent characteristic. Connecting the VMD with a spring in series, a TVMD (Ikago et al., 2012b; Cheng and Ji, 2022) can be formed. Various methods based on $H_{\infty}$ optimization (Ikago et al., 2012b) and $H_{2}$ optimization (Cheng et al., 2022; Jia and Ji, 2026) have been proposed for tuning design of the TVMD. Asai et al. (2015) first proposed the TVMD outriggers, demonstrating their seismic control performance when tuned to the first structural mode. Subsequently, Asai and Watanabe (2017) developed an innovative TVMD outrigger system, where damping was achieved through an electromagnetic motor capable of converting mechanical vibration energy into electrical energy. The current authors (Jia et al., 2023) developed a simplified model of a high-rise building with TVMD outriggers and evaluated the seismic control performance of TVMD outriggers tuned to different modes of high-rise buildings. The results indicated that when tuned to higher-order modes of structure, TVMD outriggers achieved better control performance than outriggers tuned to the first mode. However, although the seismic control performance of TVMD outriggers has been investigated in prior research, their underlying control mechanism remains insufficiently understood.

This study investigates the seismic control mechanism of TVMD outriggers. A Kelvin-Voigt model is proposed to represent the TVMD outrigger system, through which frequency-dependent equivalent stiffness and damping are quantitatively formulated. A modal response mitigation ratio is defined and calculated, revealing that equivalent damping plays a dominant role over equivalent stiffness in the control mechanism. An analysis of the equivalent damping-frequency curves elucidates the control mechanism of TVMD outrigger, i.e., tuning effect for the targeted mode and damping supplement effect for lower-order modes. Finally, a case study of a 140-m tall building is performed to compare the seismic control performance of three damped outrigger systems, i.e., the TVMD, VD and NSD outriggers.

Distributed-parameter model of a core wall with damped outriggers

A distributed-parameter model is developed to represent a core wall equipped with damped outriggers. Figure 2 illustrates the model, which comprises a core wall with k pairs of damped outriggers along the structural height. Note that while in practice only a limited number of outriggers are typically installed, the use of k pairs in this formulation preserves generality for theoretical derivation. Due to the flexural-dominated deformation mode of the core wall, it is simplified as an Euler-Bernoulli cantilever beam with a height of H and a sectional flexural stiffness of EI. The jth pair of outriggers is located at a height of $α_{j} H (j = 1, 2, \dots, k)$ , with a length $r$ of each side of the outriggers. Dampers are vertically installed between outrigger trusses/beams and the perimeter columns. The shear stiffness of the jth pair of outrigger trusses/beams and the axial stiffness of the perimeter columns are denoted as $k_{O}^{j}$ and $k_{C}^{j}$ , respectively. Under lateral loading, deformation of the cantilever beam induces outrigger rotation, generating vertical deformation at the end of the outriggers. This vertical deformation comprises: (1) shear deformation of jth outrigger trusses/beams $u_{O}^{j}$ , (2) axial deformation of the jth damper $u_{D}^{j}$ , and (3) axial deformation of perimeter columns at the jth floor height $u_{C}^{j}$ . Consequently, a counter bending moment, which is equal to the damper force multiplying the length of outrigger arms, is generated to mitigate the seismic responses of high-rise buildings. Note that the modeling approach adopted here follows well-established distributed-parameter formulations for damped outrigger systems, as detailed in prior studies (Deng et al., 2013; Jia et al., 2023).

Figure 2.

Simplified model of a core wall with damped outriggers.

Uncontrolled structure

For an uncontrolled core wall without outriggers, the lateral displacement y is expressed as:

y = y (x, t) = \sum_{i = 1}^{\infty} φ_{i} (x) u_{i} (t)

(1)

where

x

denotes the X-directional coordinate,

t

represents the time, and

φ_{i} (x)

and

u_{i} (t)

denote the ith mode shape and the corresponding modal coordinate, respectively. The equation of motion of the Euler-Bernoulli cantilever beam is given by:

m (x) \frac{\partial^{2} y (x, t)}{\partial t^{2}} + \frac{\partial^{2}}{\partial x^{2}} [E I (x) \frac{\partial^{2} y (x, t)}{\partial x^{2}}] = f (x, t)

(2)

where

m (x)

and

E I (x)

denote the mass per unit length and sectional flexural stiffness along the cantilever beam, respectively, and

f (x, t)

denotes the external lateral load. In the simplified model,

m (x)

and

E I (x)

are assumed to be uniformly distributed along the height. The ith modal mass

M_{i}

, modal stiffness

K_{i}

and modal viscous damping coefficient

C_{i}

are given by Equation (3).

{\begin{cases} M_{i} = \int_{0}^{H} m {φ_{i}}^{2} (x) d x = m H \\ K_{i} = ω_{i}^{2} M_{i} \\ C_{i} = 2 ω_{i} ξ_{i} M_{i} \end{cases}

(3)

where

ω_{i}

and

ξ_{i}

denote the ith natural vibration frequency and modal damping ratio, respectively. For the Euler-Bernoulli cantilever beam, the expressions for

ω_{i}

and

φ_{i}

can be found in Deng et al. (2013). It is important to note that in this study, the modal mass

M_{i}

is normalized to

m H

by scaling the mode shape vector

φ_{i} (x)

When subjected to seismic motions, the equivalent external loads and the ith modal force are expressed by Equations (4) and (5).

f (t) = - m {\ddot{x}}_{g} (t)

(4)

F_{i} (t) = \int_{0}^{H} φ_{i} (x) f (t) d x = - m {\ddot{x}}_{g} (t) \int_{0}^{H} φ_{i} (x) d x

(5)

where

{\ddot{x}}_{g} (t)

represents the acceleration time history of seismic motion. Therefore, the equation of motion of the ith mode can be formulated and further simplified into Equation (6), and the ith load coefficient

q_{i}

is expressed as shown in Equation (7).

{\ddot{u}}_{i} (t) + 2 ω_{i} ξ_{i} {\dot{u}}_{i} (t) + ω_{i}^{2} u_{i} (t) = - q_{i} {\ddot{x}}_{g} (t)

(6)

q_{i} = \frac{\int_{0}^{H} φ_{i} (x) d x}{H}

(7)

By combining the first n modal equations of motion, the simplified model of a core wall system is represented by Equations (8) through (11). Further details regarding this distributed-parameter model can be found in previous work by the authors (Jia et al., 2023).

M_{0} \ddot{u} (t) + C_{0} \dot{u} (t) + K_{0} u (t) = - q_{0} {\ddot{x}}_{g} (t)

(8)

u (t) = {[u_{1} (t) u_{2} (t) \dots u_{n} (t)]}^{T}

(9)

q_{0} = {[q_{1} q_{2} \dots q_{n}]}^{T}

(10)

{\begin{cases} M_{0} = {[\begin{array}{l} 1 \\ ⋱ \\ 1 \end{array}]}_{n \times n} \\ C_{0} = {[\begin{array}{l} 2 ω_{1} ξ_{1} \\ ⋱ \\ 2 ω_{n} ξ_{n} \end{array}]}_{n \times n} \\ K_{0} = {[\begin{array}{l} {ω_{1}}^{2} \\ ⋱ \\ {ω_{n}}^{2} \end{array}]}_{n \times n} \end{cases}

(11)

Structures with TVMD, VD, and NSD outriggers

We assume that k pairs of damped outriggers are installed at the height of $α_{1} H, α_{2} H \dots α_{k} H$ of the core wall structure. Based on Equation (1), the rotation angle of the jth pair of outriggers installed at the height of $α_{j} H (j = 1, 2 \dots k)$ is presented in Equation (12).

θ^{j} (t) = {(\frac{\partial y (x, t)}{\partial x})}_{| x = α_{j} H} = \sum_{i = 1}^{\infty} {(\frac{\partial φ_{i} (x)}{\partial x})}_{| x = α_{j} H} u_{i} (t)

(12)

where

θ^{j}

denotes the rotation angle of the jth outrigger. Assuming that the jth pair of damped outriggers provides a damper force of

F_{EDD}^{j}

, the generated distributed bending moment is represented by Equation (13).

m_{b}^{j} (x, t) = δ (x - α_{j} H) 2 r F_{EDD}^{j} (t)

(13)

where

F_{EDD}^{j}

denotes the damper force of the jth EDD and

δ (x - α_{j} H)

denotes the Dirac-delta function (Jia et al., 2023). The ith modal force provided by the jth pair of damped outriggers,

F_{i}^{j}

, is calculated by the product of the equivalent distributed lateral force

- \frac{\partial m_{b}^{j} (x, t)}{\partial x}

and the ith modal shape vector. Using integration by parts, this can be further expressed as Equation (14).

F_{i}^{j} (t) = - \int_{0}^{H} φ_{i} (x) \frac{\partial m_{b}^{j} (x, t)}{\partial x} d x = 2 r {(\frac{\partial φ_{i} (x)}{\partial x})}_{| x = α_{j} H} F_{EDD}^{j} (t)

(14)

Therefore, the ith modal equation of motion for a core wall equipped with k pairs of damped outriggers can be formulated as Equation (15).

{\ddot{u}}_{i} (t) + 2 ω_{i} ξ_{i} {\dot{u}}_{i} (t) + ω_{i}^{2} u_{i} (t) = - q_{i} \ddot{x_{g}} (t) - \frac{1}{m H} \sum_{j = 1}^{k} F_{i}^{j} (t)

(15)

The damper force $F_{EDD}^{j}$ is related to the type of damped outriggers. In this study, three types of damped outriggers are considered: VD, NSD, and TVMD outriggers. Figure 3 illustrates the mechanical models for the core wall equipped with these damped outriggers, while Table 1a presents the $F_{EDD}^{j}$ for these outriggers. Furthermore, as shown in Figure 2, the spring representing the shear stiffness of the jth outrigger beams/trusses $k_{O}^{j}$ is in-series connected to the spring that represents the axial stiffness of perimeter columns $k_{C}^{j} = E A_{c} / α_{j} H$ , where ${E A}_{c}$ denotes the axial sectional stiffness of the perimeter columns. These can be integrated into one equivalent spring with a stiffness of $k_{eq, 1}^{j}$ . In addition, the TVMD itself consists of a spring element, and then the jth TVMD spring stiffness, $k_{b}^{j}$ , can be further integrated into an equivalent spring with a stiffness of $k_{eq, 2}^{j}$ . The two stiffnesses of the equivalent springs can be expressed by Equation (16).

{\begin{cases} k_{eq, 1}^{j} = \frac{k_{C}^{j} k_{O}^{j}}{k_{C}^{j} + k_{O}^{j}} \\ k_{eq, 2}^{j} = \frac{k_{C}^{j} k_{O}^{j} k_{b}^{j}}{k_{C}^{j} k_{O}^{j} + k_{C}^{j} k_{b}^{j} + k_{O}^{j} k_{b}^{j}} \end{cases}

(16)

Figure 3.

Mechanical models for the core wall equipped with damped outriggers. (a) VD outriggers. (b) NSD outriggers. (c) TVMD outriggers.

Table 1.

Derivation for damped outrigger force.

Outrigger type	$F_{EDD}^{j} (t)$
(a) $F_{EDD}^{j}$ for damped outriggers
jth pair of VD outriggers	${}_{VD}c_{d}^{j} {\dot{u}}_{d}^{j} (t)$
jth pair of NSD outriggers	$- \| {}_{NSD}k_{b}^{j} \| u_{d}^{j} (t) + {}_{NSD}c_{d}^{j} {\dot{u}}_{d}^{j} (t)$
jth pair of TVMD outriggers	${}_{TVMD}m_{r}^{j} {\ddot{u}}_{d}^{j} (t) + {}_{TVMD}c_{d}^{j} {\dot{u}}_{d}^{j} (t)$

Outrigger type	Equation (17)
(b) Force equilibrium for damped outriggers
jth pair of VD outriggers	${}_{VD}c_{d}^{j} {\dot{u}}_{d}^{j} (t) = k_{eq, 1}^{j} (\sum_{i = 1}^{\infty} ({(\frac{\partial φ_{i} (x)}{\partial x})}_{\| x = α_{j} H} u_{i} (t)) r - u_{d}^{j} (t))$
jth pair of NSD outriggers	$- \| {}_{NSD}k_{b}^{j} \| u_{d}^{j} (t) + {}_{NSD}c_{d}^{j} {\dot{u}}_{d}^{j} (t) = k_{eq, 1}^{j} (\sum_{i = 1}^{\infty} ({(\frac{\partial φ_{i} (x)}{\partial x})}_{\| x = α_{j} H} u_{i} (t)) r - u_{d}^{j} (t))$
jth pair of TVMD outriggers	${}_{TVMD}m_{r}^{j} {\ddot{u}}_{d}^{j} (t) + {}_{TVMD}c_{d}^{j} {\dot{u}}_{d}^{j} (t) = k_{eq, 2}^{j} (\sum_{i = 1}^{\infty} ({(\frac{\partial φ_{i} (x)}{\partial x})}_{\| x = α_{j} H} u_{i} (t)) r - u_{d}^{j} (t))$

The force equilibrium between the EDD and the equivalent springs can be expressed by $F_{ED D}^{j} (t) = k_{e q}^{j} u_{k}^{j} (t)$ . Moreover, the sum of deformation of the equivalent spring $u_{k}^{j} (t)$ and the EDD $u_{d}^{j} (t)$ is equal to the vertical displacement $θ^{j} (t) r$ at the end of the damped outrigger. Consequently, the $F_{ED D}^{j}$ can be calculated as Equation (17). The force equilibrium for the aforementioned three damped outriggers is listed in Table 1b.

F_{EDD}^{j} (t) = k_{eq}^{j} (θ^{j} (t) r - u_{d}^{j} (t))

(17)

By combining the first n modal equations of motion (Equation (15)) and k supplementary equations of force equilibrium (Equation (17)), the simplified model of a core wall system with k pairs of damped outriggers can be formulated as Equations (18) through (23). The additional matrices used in the simplified model are listed in Table 2.

(M^{'} + \sum_{j = 1}^{k} M_{add}^{j}) \ddot{u} (t) + (C^{'} + \sum_{j = 1}^{k} C_{add}^{j}) \dot{u} (t) + (K^{'} + \sum_{j = 1}^{k} K_{add}^{j}) u (t) = - q^{'} {\ddot{x}}_{g} (t)

(18)

u (t) = {[u_{1} (t) u_{2} (t) \dots u_{n} (t) u_{d}^{1} (t) \dots u_{d}^{k} (t)]}_{1 \times (k + n)}^{T}

(19)

q^{'} = {[q_{1} q_{2} \dots q_{n} 0 \dots 0]}_{1 \times (k + n)}^{T}

(20)

{\begin{array}{c} M^{'} = [[\begin{array}{c} {M_{0}}_{n \times n} & 0_{n \times k} \\ 0_{k \times n} & 0_{k \times k} \end{array}] \\ C^{'} = [\begin{array}{c} {C_{0}}_{n \times n} & 0_{n \times k} \\ 0_{k \times n} & 0_{k \times k} \end{array}] \\ K^{'} = [\begin{array}{c} {K_{0}}_{n \times n} & 0_{n \times k} \\ 0_{k \times n} & 0_{k \times k} \end{array}] \end{array}

(21)

θ_{j}^{'} = \sqrt{\frac{2 r^{2}}{m H}} \times {[\begin{array}{c} \frac{\partial φ_{1} (x)}{\partial x} & \begin{array}{c} \dots & \frac{\partial φ_{n} (x)}{\partial x} & \begin{array}{c} 0 & \dots & \begin{array}{c} 0 & \begin{array}{c} - \frac{1}{r} & \begin{array}{c} 0 & \dots \end{array} \end{array} & 0 \end{array} \end{array} \end{array} \end{array}]}_{| x = α_{j} H, 1 \times (k + n)}

(22)

L_{j}^{'} = {[\begin{array}{c} 0 & \dots & \begin{array}{c} \sqrt{\frac{2}{m H}} & \dots & 0 \end{array} \end{array}]}_{1 \times (k + n)}

(23)

where

- \frac{1}{r}

and

\sqrt{\frac{2}{m H}}

are located at the (n + j) column of

θ_{j}^{'}

and

L_{j}^{'}

, respectively.

Table 2.

Additional matrices of damped outriggers.

Outrigger type	VD outrigger	NSD outrigger	TVMD outrigger
$M_{add}^{j}$	${[0]}_{(n + k) \times (n + k)}$	${[0]}_{(n + k) \times (n + k)}$	$_{TVMD} m_{r}^{j} {L_{j}^{'}}^{T} L_{j}^{'}$
$K_{add}^{j}$	$k_{eq, 1}^{j} {θ_{j}^{'}}^{T} θ_{j}^{'}$	$k_{eq, 1}^{j} {θ_{j}^{'}}^{T} θ_{j}^{'} - \|_{NSD} k_{b}^{j} \| {L_{j}^{'}}^{T} L_{j}^{'}$	$k_{eq, 2}^{j} {θ_{j}^{'}}^{T} θ_{j}^{'}$
$C_{add}^{j}$	$_{VD} c_{d}^{j} {L_{j}^{'}}^{T} L_{j}^{'}$	$_{NSD} c_{d}^{j} {L_{j}^{'}}^{T} L_{j}^{'}$	$_{TVMD} c_{d}^{j} {L_{j}^{'}}^{T} L_{j}^{'}$

Equivalent stiffness and damping coefficient of TVMD outrigger system

This section proposes an equivalent Kelvin–Voigt model for representing the TVMD outrigger system, which consists of frequency-dependent equivalent stiffness and damping. The term “equivalent” here indicates that the force-displacement response of the Kelvin-Voigt model is equal to that of the TVMD outrigger under dynamic excitation. Besides, the authors’ prior research (Jia et al., 2023) has demonstrated that the installation of TVMD outriggers nearly does not change the mode shape of uncontrolled structure. Therefore, the mode shape of uncontrolled structure is used in the subsequent derivations. Assuming that the ith modal coordinate $u_{i} = U_{i} e^{i ω t}$ and the jth VMD displacement $u_{d}^{j} (t) = U_{d}^{j} e^{i ω t}$ , the following equation can be derived from force equilibrium of the jth pair of TVMD outriggers (see Table 1b):

u_{d}^{j} = \frac{k_{eq, 2}^{j}}{k_{eq, 2}^{j} - ω^{2} {}_{TVMD}m_{r}^{j} + i ω {}_{TVMD}c_{d}^{j}} \sum_{i = 1}^{\infty} ({(\frac{\partial φ_{i} (x)}{\partial x})}_{| x = α_{j} H} u_{i}) r

(24)

We then substitute the following dimensionless parameters: $γ^{j} = \frac{ω}{ω_{p_{j}}}$ , $ρ_{p_{j}}^{j} = r {(\frac{\partial φ_{p_{j}} (x)}{\partial x})}_{| x = α_{j} H}$ , $μ_{p_{j}}^{j} = \frac{2 {}_{TVMD}m_{r}^{j}}{m H} {ρ_{p_{j}}^{j}}^{2}$ , where $p_{j}$ denotes the target tuning mode of jth pair of TVMD outriggers. Using the $H_{\infty}$ -norm-based closed-form formulas for TVMD design based on fixed-point theory (Ikago et al., 2012a), the optimal frequency ratio $β_{opt}^{i}$ and damping ratio $ξ_{opt}^{i}$ of the ith mode are expressed as:

{\begin{array}{c} β_{opt}^{i} = \frac{1}{\sqrt{1 - μ_{total}^{i}}} \\ ξ_{opt}^{i} = \frac{1}{2} \sqrt{\frac{3 μ_{total}^{i}}{2 - μ_{total}^{i}}} \end{array}

(25)

where

μ_{total}^{i}

denotes total inertance ratio of TVMD outriggers tuned to the ith mode. Substituting Equation (25) into Equation (24) gives:

u_{d}^{j} = \frac{{β_{opt}^{p_{j}}}^{2} ({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2}) - 2 i {β_{opt}^{p_{j}}}^{3} ξ_{opt}^{p_{j}} γ^{j}}{{({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2})}^{2} + 4 {β_{opt}^{p_{j}}}^{2} {ξ_{opt}^{p_{j}}}^{2} {γ^{j}}^{2}} \sum_{i = 1}^{\infty} ρ_{i}^{j} u_{i}

(26)

Substituting Equation (26) into the force of the jth pair of TVMD outriggers $F^{j} = {}_{TVMD}m_{r}^{j} {\ddot{u}}_{d}^{j} + {}_{TVMD}c_{d}^{j} {\dot{u}}_{d}^{j}$ , the force $F^{j}$ is then derived as:

{\begin{array}{c} F^{j} = k_{eq}^{j} (μ_{p_{j}}^{j}, γ^{j}) u_{t}^{j} + c_{eq}^{j} (μ_{p_{j}}^{j}, γ^{j}) {\dot{u}}_{t}^{j} \\ k_{eq}^{j} (μ_{p_{j}}^{j}, γ^{j}) = \frac{m H {ω_{p_{j}}}^{2}}{{ρ_{p_{j}}^{j}}^{2}} \frac{- μ_{p_{j}}^{j} {γ^{j}}^{2} {β_{opt}^{p_{j}}}^{2} ({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2}) + 4 μ_{p_{j}}^{j} {β_{opt}^{p_{j}}}^{4} {ξ_{opt}^{p_{j}}}^{2} {γ^{j}}^{3}}{{2 ({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2})}^{2} + 8 {β_{opt}^{p_{j}}}^{2} {ξ_{opt}^{p_{j}}}^{2} {γ^{j}}^{2}} \\ c_{eq}^{j} (μ_{p_{j}}^{j}, γ^{j}) = \frac{m H ω_{p_{j}}}{{ρ_{p_{j}}^{j}}^{2}} \frac{μ_{p_{j}}^{j} β_{opt}^{p_{j}} ξ_{opt}^{p_{j}} {γ^{j}}^{2} + μ_{p_{j}}^{j} {β_{opt}^{p_{j}}}^{3} ({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2}) ξ_{opt}^{p_{j}}}{{({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2})}^{2} + 4 {β_{opt}^{p_{j}}}^{2} {ξ_{opt}^{p_{j}}}^{2} {γ^{j}}^{2}} \end{array}

(27)

where

u_{t}^{j}

denotes total displacement of the jth pair of TVMD outriggers, and

k_{eq}^{j}

and

c_{eq}^{j}

denote the equivalent stiffness and damping coefficient of jth pair of TVMD outriggers, respectively. Equation (27) reveals that seismic control of TVMD outriggers can be viewed as providing frequency-dependent equivalent stiffness and damping coefficient in parallel. The total equivalent parameter value of multiple TVMD outriggers at a specific frequency can be considered as the sum of the equivalent parameters of each TVMD outrigger with different

μ_{p_{j}}^{j}

and

γ^{j}

, which are expressed as:

{\begin{cases} k_{eq}^{total} (μ_{p_{j}}^{j}, γ^{j}) = \sum_{j = 1}^{j = k} \frac{m H {ω_{p_{j}}}^{2}}{{ρ_{p_{j}}^{j}}^{2}} \frac{- μ_{p_{j}}^{j} {γ^{j}}^{2} {β_{opt}^{p_{j}}}^{2} ({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2}) + 4 μ_{p_{j}}^{j} {β_{opt}^{p_{j}}}^{4} {ξ_{opt}^{p_{j}}}^{2} {γ^{j}}^{3}}{{2 ({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2})}^{2} + 8 {β_{opt}^{p_{j}}}^{2} {ξ_{opt}^{p_{j}}}^{2} {γ^{j}}^{2}} \\ c_{eq}^{total} ({μ_{p_{j}}^{j}, γ}^{j}) = \sum_{j = 1}^{j = k} \frac{m H ω_{p_{j}}}{{ρ_{p_{j}}^{j}}^{2}} \frac{μ_{p_{j}}^{j} β_{opt}^{p_{j}} ξ_{opt}^{p_{j}} {γ^{j}}^{2} + μ_{p_{j}}^{j} {β_{opt}^{p_{j}}}^{3} ({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2}) ξ_{opt}^{p_{j}}}{{({β_{opt}^{p_{j}}}^{2} - {γ^{j}}^{2})}^{2} + 4 {β_{opt}^{p_{j}}}^{2} {ξ_{opt}^{p_{j}}}^{2} {γ^{j}}^{2}} \end{cases}

(28)

Seismic control mechanism of TVMD outriggers

Equivalent parameter curves

For a core wall system equipped with one pair of TVMD outriggers tuned to the pth mode, the supplemental equivalent stiffness $k_{eq}$ and damping coefficient $c_{eq}$ are expressed as:

{\begin{array}{c} k_{eq} (μ_{p}, γ) = \frac{m H {ω_{p}}^{2}}{{ρ_{p}}^{2}} \frac{- μ_{p} γ^{2} {β_{opt}^{p}}^{2} ({β_{opt}^{p}}^{2} - γ^{2}) + 4 μ_{p} {β_{opt}^{p}}^{4} {ξ_{opt}^{p}}^{2} γ^{3}}{{2 ({β_{opt}^{p}}^{2} - γ^{2})}^{2} + 8 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2} γ^{2}} \\ c_{eq} (μ_{p}, γ) = \frac{m H ω_{p}}{{ρ_{p}}^{2}} \frac{μ_{p} β_{opt}^{p} ξ_{opt}^{p} γ^{2} + μ_{p} {β_{opt}^{p}}^{3} ({β_{opt}^{p}}^{2} - γ^{2}) ξ_{opt}^{p}}{{({β_{opt}^{p}}^{2} - γ^{2})}^{2} + 4 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2} γ^{2}} \end{array}

(29)

where

γ = \frac{ω}{ω_{p}}

denotes the ratio of excitation frequency to the natural vibration frequency of this pair of TVMD outriggers.

Figure 4(a) and (b) present the relationships between equivalent parameters ( $k_{eq}$ and $c_{eq}$ ) and both the inertance ratio $μ_{p}$ and excitation frequency ratio $γ$ . The relationships between equivalent parameters and excitation frequency ratio $γ$ exhibit consistent trends across varying inertance ratios $μ_{p}$ (see Figure 4), as demonstrated in Figure 4(c).

Figure 4.

Relationship between equivalent parameters with $μ_{p}$ and $γ$ . (a) Equivalent stiffness $k_{eq}$ . (b) Equivalent damping coefficient $c_{eq}$ . (c) Key points of equivalent parameter curves of TVMD outriggers ( $μ_{p} = 0.2)$ .

Figure 4(c) reveals several key points of equivalent parameter curves: (1) At

γ = 0

(Key point A); the TVMD outriggers provide an equivalent damping coefficient of

c_{eq} = {}_{TVMD}c_{d}

while providing zero equivalent stiffness (

k_{eq} = 0

). (2) As excitation frequency increases, the TVMD outriggers provide an equivalent negative stiffness, reaching a minimum value at

γ = \sqrt{1 - \sqrt{\frac{3 μ_{p}}{4 - μ_{p}}}}

(Key point B), while the equivalent damping coefficient increases to a peak value at

γ = \sqrt{\frac{8 - 5 μ_{p}}{8 - 2 μ_{p}}}

(Key point C). (3) At

γ = 1

(Key point D), the TVMD outrigger system reaches resonance, providing an equivalent damping coefficient slightly below the peak value and with a positive equivalent stiffness. (4) Beyond resonance, the equivalent stiffness reaches its peak value at

γ = \sqrt{1 + \sqrt{\frac{3 μ_{p}}{4 - μ_{p}}}}

(Key point D), while the equivalent damping coefficient continues to decrease. (5) At

γ = \infty

(Key point F), the equivalent stiffness approaches

{}_{TVMD}k_{b}

, with the equivalent damping coefficient decreasing to 0. The key point values of equivalent parameter curves are presented in Table 3.

Table 3.

Key point values of the equivalent parameter curves of TVMD outriggers.

Point	$γ$	Value of TVMD outrigger equivalent parameters
A	0	$k_{eq}$	0
A	0	$c_{eq}$	${}_{TVMD}c_{d}$
B	$\sqrt{1 - \sqrt{\frac{3 μ_{p}}{4 - μ_{p}}}}$	$k_{eq}$	$\frac{m H {ω_{p}}^{2}}{{ρ_{p}}^{2}} \frac{(- 2 a - 2) {μ_{p}}^{3} + (13 a + 3 c + 3) {μ_{p}}^{2} - (22 a - 12 c + 2) μ_{p} + 8 a}{(4 a + 4) {μ_{p}}^{3} + (- 22 a - 18) {μ_{p}}^{2} + (26 a + 44) μ_{p} - 8 a - 36}$
B	$\sqrt{1 - \sqrt{\frac{3 μ_{p}}{4 - μ_{p}}}}$	$c_{eq}$	$\frac{m H ω_{p}}{{ρ_{p}}^{2}} \frac{\sqrt{3} b ((1 - a) {μ_{p}}^{2} + (c - 2 (1 - a) - a c) μ_{p} + c a + 1 - a)}{4 e f {μ_{p}}^{3} + (- 22 a + 18) f {μ_{p}}^{2} + (26 a + 44) f μ_{p} - (8 a + 36) f}$
C	$\sqrt{\frac{8 - 5 μ_{p}}{8 - 2 μ_{p}}}$	$k_{eq}$	$\frac{m H {ω_{p}}^{2}}{{ρ_{p}}^{2}} \frac{- 441 {μ_{p}}^{5} + (3969 - 3 g) {μ_{p}}^{4} + (24 g - 12662) {μ_{p}}^{3} - (16560 - 48 g) {μ_{p}}^{2} - 7168 μ_{p}}{882 {μ_{p}}^{5} - 8484 {μ_{p}}^{4} + 30398 {μ_{p}}^{3} - 50708 {μ_{p}}^{2} + 40384 μ_{p} - 12544}$
C	$\sqrt{\frac{8 - 5 μ_{p}}{8 - 2 μ_{p}}}$	$c_{eq}$	$\frac{m H ω_{p}}{{ρ_{p}}^{2}} \frac{- \sqrt{3} b μ_{p} (42 {μ_{p}}^{3} + (21 g - 212) {μ_{p}}^{2} + (298 - 83 g) μ_{p} + 56 g - 128)}{882 e {μ_{p}}^{5} - 8484 e {μ_{p}}^{4} + 30398 e {μ_{p}}^{3} - 50708 e {μ_{p}}^{2} + 40384 e μ_{p} - 12544 f}$
D	1	$k_{eq}$	$\frac{m H {ω_{p}}^{2}}{{ρ_{p}}^{2}} \frac{{μ_{p}}^{2} + μ_{p}}{6 - 2 {μ_{p}}^{2} - 2 μ_{p}}$
D	1	$c_{eq}$	$\frac{m H ω_{p}}{{ρ_{p}}^{2}} \frac{d {μ_{p}}^{3} - 3 d {μ_{p}}^{2} + 3 d μ_{p} - 2 d}{2 {μ_{p}}^{2} \sqrt{1 - μ_{p}} + 2 μ_{p} \sqrt{1 - μ_{p}} - 6 \sqrt{1 - μ_{p}}}$
E	$\sqrt{1 + \sqrt{\frac{3 μ_{p}}{4 - μ_{p}}}}$	$k_{eq}$	$\frac{m H {ω_{p}}^{2}}{{ρ_{p}}^{2}} \frac{(- 2 a - 2) {μ_{p}}^{3} + (13 a - 3 e - 3) {μ_{p}}^{2} - (22 a - 12 e + 2) μ_{p} + 8 a}{(4 a + 4) {μ_{p}}^{3} + (- 22 a + 18) {μ_{p}}^{2} + (26 a - 44) μ_{p} - 8 a + 36}$
E	$\sqrt{1 + \sqrt{\frac{3 μ_{p}}{4 - μ_{p}}}}$	$c_{eq}$	$\frac{m H ω_{p}}{{ρ_{p}}^{2}} \frac{- \sqrt{3} b ((1 + a) {μ_{p}}^{2} + (e - 2 (1 + a) + a e) μ_{p} - e a + 1 + a)}{- 4 c f {μ_{p}}^{3} + (- 22 a + 18) f {μ_{p}}^{2} + (26 a - 44) f μ_{p} - (8 a + 36) f}$
F	∞	$k_{eq}$	${}_{TVMD}k_{b}$
F	∞	$c_{eq}$	0

Note: $a = \sqrt{\frac{3 μ_{p}}{4 - μ_{p}}}, b = (1 - μ_{p}) (2 - μ_{p}) (μ_{p} - 4) \sqrt{\frac{μ_{p}}{2 - μ_{p}}}, c = {(1 - a)}^{1.5}, d = \sqrt{\frac{3 μ_{p}}{2 - μ_{p}}}, e = {(1 + a)}^{1.5}, f = {(1 - μ_{p})}^{1.5}, g = \sqrt{2} {(\frac{21 μ_{p} - 64}{μ_{p} - 4})}^{1.5}$ .

Figure 5 presents the transfer functions from TVMD outrigger total displacement $U_{t}$ to spring displacement $U_{s}$ and VMD displacement $U_{d}$ . Figure 6 presents the steady-state hysteretic response characteristics of TVMD outriggers components and equivalent parameters under different $γ$ . The control mechanism of TVMD outriggers is revealed through four cases:

Figure 5.

Transfer functions of TVMD outriggers.

Figure 6.

Hysteretic curves of TVMD outrigger components and equivalent parameters. (a) Case 1 $(γ ≪ \frac{\sqrt{16 {β_{opt}^{p}}^{4} {ξ_{opt}^{p}}^{4} + 4 {β_{opt}^{p}}^{2}} - 4 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2}}{2})$ . (b) Case 2 $(γ = \frac{\sqrt{16 {β_{opt}^{p}}^{4} {ξ_{opt}^{p}}^{4} + 4 {β_{opt}^{p}}^{2}} - 4 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2}}{2})$ . (c) Case 3 $(γ = 1)$ . (d) Case 4 $(γ ≫ \frac{\sqrt{16 {β_{opt}^{p}}^{4} {ξ_{opt}^{p}}^{4} + 4 {β_{opt}^{p}}^{2}} - 4 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2}}{2})$ .

(1) Case 1: When $γ ≪ \frac{\sqrt{16 {β_{opt}^{p}}^{4} {ξ_{opt}^{p}}^{4} + 4 {β_{opt}^{p}}^{2}} - 4 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2}}{2}$ , the TVMD outriggers provide negative stiffness and damping coefficients, as shown in Figure 6(a). Under low-frequency excitation, the dynamic stiffness of the VMD $(-_{TVMD} m_{r}^{} ω^{2} +_{TVMD} c_{d}^{} i ω)$ is lower than the spring stiffness $_{TVMD} k_{b}^{}$ , resulting in large VMD displacement (see Case 1 in Figure 5). As illustrated by the red curve in Figure 6(a), the VMD’s cyclic responses with large magnitude can dissipate energy and thus supply additional damping to low-frequency vibration (e.g., the lower-order modes).

(2) Case 2: When $γ = \frac{\sqrt{16 {β_{opt}^{p}}^{4} {ξ_{opt}^{p}}^{4} + 4 {β_{opt}^{p}}^{2}} - 4 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2}}{2}$ , the equivalent stiffness approaches zero (see Figure 6(b)). Both spring and VMD displacements are amplified when compared with TVMD outrigger total displacement (see Case 2 in Figure 5). Cyclic responses of TVMD outriggers at large magnitudes enable a significant reduction in the responses of target tuning-frequency vibration.

(3) Case 3: When $γ = 1$ , resonance occurs and the dynamic behavior of TVMD outriggers is similar to that in Case 2 (see Figures 5 and 6). For Cases 1, 2, and 3, the equivalent force provided by equivalent damping coefficient dominates TVMD total force compared with that provided by equivalent stiffness (see purple and orange lines in Figure 6).

(4) Case 4: When $γ ≫ \frac{\sqrt{16 {β_{opt}^{p}}^{4} {ξ_{opt}^{p}}^{4} + 4 {β_{opt}^{p}}^{2}} - 4 {β_{opt}^{p}}^{2} {ξ_{opt}^{p}}^{2}}{2}$ , the TVMD outriggers demonstrate positive stiffness spring and dashpot elements in parallel (see Figure 6(d)). As excitation frequency increases, the dynamic stiffness of the VMD $(-_{TVMD} m_{r}^{} ω^{2} +_{TVMD} c_{d}^{} i ω)$ is larger than the spring stiffness $_{TVMD} k_{b}^{}$ , resulting in a decrease of VMD displacement (see Case 4 in Figure 5). The VMD with cyclic responses at low magnitude cannot provide substantial damping to high-frequency vibrations (e.g., the higher-order modes).

In summary, near the tuning frequency, the anti-phase motion between the spring and the VMD becomes most pronounced (Figure 6(c)). Correspondingly, the equivalent damping of TVMD outriggers attains its maximum (see Figure 4(c)), which enables significant suppression of the tuning frequency vibration. Under low-frequency vibration, the VMD displacement dominates the TVMD total displacement (Figure 6(a)). The VMD’s cyclic responses with large magnitude supply additional damping to low-frequency vibration. Under high-frequency vibration, the equivalent damping is close to 0 (see Figure 4(c)), rendering the TVMD outrigger ineffective in controlling high-frequency vibration.

Control contribution of equivalent parameters

To further investigate the control contribution of equivalent parameters, a 140-m tall core wall-frame structure is considered as a case study. One pair of TVMD outriggers with an inertance ratio of 0.05 is tuned to the structural second mode. More detailed information about this 140-m structure can be found in ‘Prototype structure and structural design' section. Figure 7(a) and (b) present transfer functions from base acceleration to structural displacement and acceleration at the top floor, which are calculated using the proposed simplified model. The transfer functions clearly reveal the tuning effect for target mode (i.e., second mode) and damping supplement effect for lower-order mode (i.e., first mode). It should be emphasized that this observed control mechanism stems from the inherent dynamic characteristics of the TVMD itself (Ji et al., 2020; Jia et al., 2023). Besides, the TVMD outriggers exhibit negligible control effectiveness for higher-order modes. For scenarios requiring simultaneous control across a broader frequency range (including higher modes), alternative strategies such as the frequency-independent outrigger system proposed by Wang et al. (2023) may be considered. It should be noted that the control effectiveness of TVMD outrigger is related to its location and structural mode shapes. For instance, when a TVMD outrigger is situated at a floor corresponding to the stationary point of one mode shape, it cannot provide control forces to this specific mode.

Figure 7.

Transfer functions of a high-rise building structure. (a) From base acceleration to structural top displacement. (b) From base acceleration to structural top acceleration.

Using Equation (29), the equivalent stiffness and damping of the TVMD outriggers are calculated for this case study. The transfer functions from base acceleration to top floor acceleration are calculated using these equivalent parameters for three configurations: considering only the equivalent stiffness (blue line), only the equivalent damping (green line), and both parameters simultaneously (red line), as presented in Figure 8(a). The modal peak values are identified by solid dots. The equivalent damping, $c_{eq}$ , serves as the predominant control mechanism, yielding a peak value reduction almost equivalent to that achieved when both $c_{eq}$ and $k_{eq}$ are considered (as indicated by the close agreement between the red and green lines). Figure 8(b) presents the equivalent parameter-frequency curves in this case study. Near the tuning frequency ( $γ \approx 1$ ), the equivalent damping reaches its maximum value, leading to a substantial reduction in the transfer function amplitude of the second structural mode (see Figure 8(a)). At the first modal frequency ( $γ \approx 0.25$ ), the equivalent damping approaches the TVMD damping coefficient, which contributes to providing supplemental damping to the first mode.

Figure 8.

Control contribution of equivalent parameters. (a) Transfer functions of structures with separately considered equivalent parameters. (Note: solid dots denote modal peak values of transfer function, and dashed vertical lines indicate modal frequencies). (b) Equivalent stiffness and damping-frequency curves for $μ = 0.05$ .

The TVMD outriggers exhibit negative equivalent stiffness in the low-frequency range ( $γ ≪ 1$ , as shown in Figure 8(b)), which reduces the dynamic stiffness of TVMD outriggers and amplifies the total displacement. Although this amplified displacement enhances the control effect of the TVMD outriggers, the contribution of equivalent stiffness remains insignificant (see Figure 8(a)), both for the target mode (i.e., the second mode) and for the lower-order mode (i.e., the first mode).

To quantitatively investigate the control contribution of equivalent stiffness and damping of TVMD outriggers, this study defines a modal response mitigation ratio, as expressed in Equation (30). This ratio represents the proportional reduction in the ith modal peak value of transfer functions for the structure with equivalent parameters compared to the uncontrolled structure. Figure 9 presents the mitigation ratios for the first two modal acceleration responses within an inertance ratio range of [0, 0.1]. The results align consistently with the conclusions drawn from the transfer function analysis in Figure 8. Over the examined inertance ratio range, the contribution of equivalent damping to the control effect of the TVMD outrigger consistently exceeds 90%.

R_{i} = \frac{P_{i}^{U} - P_{i}^{E}}{P_{i}^{U}} \times 100 %

(30)

where

P_{i}^{U}

and

P_{i}^{E}

represent transfer function values at the ith modal frequency for the uncontrolled structure and the structure with additional equivalent parameters, respectively.

Figure 9.

Modal response mitigation ratios of TVMD outriggers. (a) First modal response mitigation ratio. (b) Second modal response mitigation ratio.

Comparison of the seismic control performance of damped outriggers

Prototype structure and structural design

A prototype structure located in Beijing was selected as a case study. The building site had a peak ground acceleration of 0.2 g for a design basis earthquake (DBE, with a 10% probability of exceedance in 50 years) and a characteristic site period of 0.55 s. The building adopted a reinforced concrete (RC) core wall-frame system. As displayed in Figure 10(a), the first three floors had story heights of 5 m, 4.5 m, and 4.5 m, respectively, and other floors had a story height of 4.2 m, resulting in a total structural height of 140 m. The building structure had a plan measuring 40.8 m × 30.6 m, with a 16 m × 12 m core wall at its center (see Figure 10(b)).

Figure 10.

Prototype building. (a) Elevation view. (b) Plan view.

The building utilized HRB400 grade steel reinforcement for all structural components, while concrete strength grades varied by floor level, as specified in Table 4a. The sectional dimensions of structural components are presented in Table 4b and 4c. The structural design incorporated a dead load of 5.5 kN/m² for floors and 8.5 kN/m² for the roof (both including slab self-weight), along with a uniformly distributed live load of 3.5 kN/m² applied to each floor. Additionally, a line dead load of 4.5 kN/m was applied to the peripheral frame beams to account for the weight of exterior cladding systems. In accordance with ASCE/SEI 7-16 (American Society of Civil Engineers, 2016), the damping ratio of the structure was assumed to be 0.025.

Table 4.

Material and member size of prototype building.

Member type	1-10 floor			11-18 floor			19-26 floor			27-33 floor
Member type	Grade	$f_{cu, n}$	$f_{ck}$	Grade	$f_{cu, n}$	$f_{ck}$	Grade	$f_{cu, n}$	$f_{ck}$	Grade	$f_{cu, n}$	$f_{ck}$
(a) Concrete strength for all structural components (units in MPa)
Beam	C35	35	23.4	C35	35	23.4	C35	35	23.4	C35	35	23.4
Shear wall	C50	50	32.4	C50	50	32.4	C45	45	29.6	C40	40	26.8
Column	C60	60	38.5	C50	50	32.4	C45	45	29.6	C40	40	26.8
Coupling beam	C50	50	32.4	C50	50	32.4	C45	45	29.6	C40	40	26.8
Slab	C35	35	23.4	C35	35	23.4	C35	35	23.4	C35	35	23.4

Floor number	Column	Beam
Floor number	C1	B1/B2	B3	B4/B5	B6/B7
(b) Member size of beams/columns (units in mm)
1–10 floor	1200 × 1200	500 × 900	500 × 900	400 × 900	350 × 750
11–18 floor	1000 × 1000	500 × 900	500 × 900	400 × 900	350 × 750
19–26 floor	900 × 900	400 × 900	500 × 900	400 × 900	350 × 750
27–33 floor	800 × 800	400 × 900	500 × 900	400 × 900	350 × 750

Floor number	Wall (thickness)		Coupling beam		Slab (thickness)
Floor number	WX	WY	CBX	CBY	Slab (thickness)
(c) Member size of walls/coupling beams/slabs (units in mm)
1–10 floor	600	720	600 × 1200	720 × 1200	120
11–18 floor	500	600	500 × 1200	600 × 1200	120
19–26 floor	400	500	400 × 1200	500 × 1200	120
27–33 floor	400	500	400 × 1200	500 × 1200	120

^a $f_{c u, n}$ denotes nominal cubic compressive strength; $f_{ck}$ denotes nominal axial compressive strength.

The abbreviations of structural members (e.g., C1, B1, WX, CBX) correspond to the labels in Figure 10(b).

The structure was designed according to the Chinese code for the seismic design of buildings (GB 50011-2010) (CMC, 2010b) and Chinese technical specifications for the concrete structures of tall buildings (JGJ 3-2010) (CMC, 2010a). Linear response spectrum analysis of a three-dimensional structural model was performed to determine the design forces of the structural components and the deformation of the structure under a service level earthquake (SLE, with a probability of exceedance of 63% in 50 years) with a peak ground acceleration (PGA) of 0.07 g. The first three natural periods of the building were 3.33 s, 2.68 s, and 2.11 s, corresponding to the vibration modes of translation in the Y-direction (i.e., transverse direction), translation in the X-direction (i.e., longitudinal direction), and the torsional mode, respectively. Under SLE, the maximum inter-story drift ratios of the structure were calculated to be 1/1055 in the X-direction and 1/884 in the Y-direction, both below the upper limit of 1/800 required in GB 50011-2010 (CMC, 2010b).

TVMD outrigger design

The design of Y-directional TVMD outriggers is presented. As illustrated in Figure 11, the TVMD outriggers were positioned at the locations highlighted by red lines in the plan view. The outrigger trusses adopted Q420 (nominal yield strength = 420 MPa) steel with box sections. The chord members had cross-sectional dimensions of 500 × 500 × 60 × 60 (units in mm), while the diagonal members had cross-sectional dimensions of 500 × 500 × 50 × 50 (units in mm). The shear stiffness of outrigger trusses was calculated as 326 kN/mm.

Figure 11.

Placement of TVMD outriggers.

The process for designing TVMD outriggers was as follows:

(1) Based on the prototype building, calculate the structural dynamic properties, including the modal frequency $ω_{i}$ , and the modal shape vector ${φ_{i}}$ .

(2) Determine the number of damped outriggers k. Select the installation position of each damped outrigger $α_{j} H (j = 1, 2 \dots k)$ . Determine the total inertance ratio of the k pairs of TVMD outriggers tuned to a selected ith mode. The total inertance ratio $μ_{i}$ denotes the ratio of ith modal inertial mass of TVMD outriggers to the ith modal mass of the structure, as calculated by Equation (31).

μ_{i} = \sum_{j = 1}^{k} \frac{2 {}_{TVMD}φ_{j}^{i} {}_{TVMD}m_{r}^{j} {}_{TVMD}φ_{j}^{i}}{{φ_{i}} M_{0} {φ_{i}}^{T}}

(31)

where

{}_{TVMD}m_{r}^{j}

denotes the inertial mass of a single TVMD in the jth pair of TVMD outriggers and

M_{0}

denotes the structural mass matrix. The

{}_{TVMD}φ_{j}^{i}

represents the ith modal shape coordinate of the jth pair of TVMD outriggers, which is expressed by Equation (32).

{}_{TVMD}φ_{j}^{i} = {(\frac{\partial φ_{i} (x)}{\partial x})}_{| x = α_{j} H} r

(32)

where r denotes the length of outrigger arms. Based on Equations (31) and (32), the inertial mass of each pair of TVMD outriggers is calculated for a given inertance ratio.

(3) Based on fixed-point theory (Ikago et al., 2012b), calculate the ith modal optimal frequency ratio $β_{i}^{opt}$ and the optimal damping ratio $ξ_{i}^{opt}$ according to Equation (25). It is important to note that these two parameters are calculated based on the total inertance ratio of a selected tuning mode.

For the jth pair of TVMD outriggers, the stiffness and damping coefficient are calculated using Equation (33).

{\begin{array}{c} {}_{TVMD}k_{b}^{j} = {(β_{i}^{opt} ω_{i})}^{2} {}_{TVMD}m_{r}^{j} \\ {}_{TVMD}c_{d}^{j} = 2 ω_{i} β_{i}^{opt} ξ_{i}^{opt} {}_{TVMD}m_{r}^{j} \end{array}

(33)

In this study, two pairs of TVMD outriggers were considered, tuned to the first (T-1) and second (T-2) translational modes in the Y direction, respectively. These outriggers were assumed to be located at heights of 0.6H and H, respectively, each with an inertance ratio of 0.05. Table 5 presents the design results for the TVMD outriggers. Note that the TVMD outriggers positioned at the same height shared the identical set of design parameters. To ensure fair comparison, the damping coefficients of VD and NSD outriggers were assumed to be identical to those of TVMD outriggers, i.e., 9550 kNs/m at the floor of 0.6H and 2470 kNs/m at the top floor. The equivalent negative stiffness values for NSD and TVMD outriggers were designed to be identical, expressed as

{}_{NSD}k_{b} = - {}_{TVMD}m_{r} ω^{2}

, where

ω

represents the tuning frequency of the TVMD outriggers. Consequently, the negative stiffness values of the two NSD outriggers are determined as −43811 and −41666 kN/m, respectively.

Table 5.

Design parameters of TVMD outriggers.

	Floor height	Tuning mode	${}_{TVMD}c_{d}^{j}$ (kNs/m)	${}_{TVMD}k_{b}^{j}$ (kN/m)	${}_{TVMD}m_{r}^{j}$ (ton)
T-1	0.6H	1st mode	9550	47805	12085
T-2	H	2nd mode	2470	46268	836

OpenSees numerical model

A numerical model was developed in OpenSees to represent the prototype structure. For unconfined concrete, the Kent-Park model (Kent and Park, 1971) was adopted to define the uniaxial compressive stress-strain relationship. For confined concrete, the Saatcioglu-Razvi model (Saatcioglu and Razvi, 1992) was adopted. The Giuffré-Menegotto-Pinto model (OpenSees Wiki, nd) was utilized to represent the uniaxial stress-strain relationship of the steel reinforcement. The beams and columns were modeled using displacement-based beam-column elements. Multi-layer shell elements (Lu et al., 2015) were employed to model the shear walls. For each L-shaped wall panel at every story level, the element mesh was discretized into four divisions along the story height, eight divisions along the X-direction and six divisions along the Y-direction. The coupling beams were also modeled using the multi-layer shell elements, with a discretization scheme consisting of two elements along the height and three elements along the length. Additionally, the Rayleigh damping model was adopted, where the parameters were determined based on the assumption that the damping ratios of the first and third translational modes in the Y direction were equal to 0.025.

Seismic control performance comparison

A comparative analysis was performed to evaluate the seismic control performance of the three types of damped outriggers. A third-party tool named GimmeMCK, based on OpenSeesPy (Zhu et al., 2018), was adopted to extract the mass, stiffness, and damping matrices of the OpenSees FE model of uncontrolled structure. By combining these matrices of the uncontrolled structure with the additional matrices listed in Table 2, the mass, stiffness, and damping matrices of the structures equipped with damped outriggers were obtained. The seismic responses of high-rise buildings with damped outriggers were then calculated based on linear response spectrum analysis using MATLAB program. The design spectrum specified by the Chinese code (CMC, 2010b) was adopted and was scaled to the design basis earthquake (DBE). Modal responses were combined using the complex complete quadratic combination (CCQC) method (Ikago et al., 2010; Zhou and Yu, 2006). Note that a non-linear time history analysis in OpenSees demonstrated the structure considered in this study behaved in nearly elastic stage when subjected to seven DBE-level ground motions, with maximum inter-story drifts less than 0.32%. The non-linearity of the structure that may occur when subjected to strong seismic motions is not considered in this paper.

As illustrated in Figure 12, compared with the uncontrolled building, use of VD, NSD and TVMD outriggers led to reductions of 9.5%, 16.7%, and 26.2% in the maximum inter-story drift ratio, and reductions of 12.4%, 18.0%, and 27.0% in the maximum floor acceleration, respectively. The control effectiveness of TVMD outriggers stems predominantly from its equivalent damping coefficient, as evidenced by the blue and cyan lines in Figure 12.

Figure 12.

Seismic responses of VD, NSD, and TVMD outriggers. (a) Inter-story drift ratio (%). (b) Floor acceleration (m/s²).

Table 6 presents the maximum inter-story drift ratio and floor acceleration of structures equipped with different damped outriggers. These include cases where the equivalent stiffness and equivalent damping of the TVMD outriggers were considered separately, as well as cases in which the first-mode tuning and second-mode tuning of the TVMD outriggers were individually considered. The following observations are obtained from Table 6.

(1) For TVMD outriggers, first-mode tuning produced a substantial reduction in inter-story drift (a 22% decrease from 0.42% to 0.33%) but limited acceleration control due to second-mode dominance of acceleration responses. Second-modal tuning reduced accelerations by 21% (5.08 m/s² to 4.02 m/s²), while maintaining a 10% inter-story drift reduction (0.42% to 0.38%) by providing additional damping for lower-order modes (i.e., first mode).

(2) For NSD outriggers, implementation of negative stiffness ( ${}_{NSD}k_{b}^{1}$ and ${}_{NSD}k_{b}^{2}$ ) enhanced supplemented damping, resulting in an 8% inter-story drift response reduction (from 0.38% to 0.35%) and a 7% acceleration response reduction (from 4.45 m/s² to 4.17 m/s²) compared with ${}_{NSD}c_{d}^{1}$ and ${}_{NSD}c_{d}^{2}$ only (i.e., ${}_{VD}c_{d}^{1}$ and ${}_{VD}c_{d}^{2}$ ). This demonstrates the damping amplification effect of negative stiffness.

(3) For VD outriggers, the vibration control relied entirely on supplementary damping for modal response mitigation across all vibration modes. Control performance comparison demonstrated inferior effectiveness relative to the TVMD outrigger system, exhibiting 19% and 17% smaller reductions in displacement and acceleration responses, respectively.

Table 6.

Comparison of the control performance of TVMD, VD and NSD outriggers.

Conclusion

This study investigates the seismic control mechanism of tuned viscous mass damper (TVMD) outriggers. A Kelvin-Voigt model is proposed to represent the TVMD outrigger system, through which frequency-dependent equivalent stiffness and damping are quantitatively formulated. The dominant parameter governing the vibration control performance of TVMD outriggers is identified. An analysis of equivalent damping-frequency curves elucidates the control mechanism of TVMD outriggers. Finally, the seismic control performance of TVMD, viscous damper (VD), and negative stiffness damper (NSD) outriggers is compared based on a 140-m tall core wall-frame structure. The main findings are listed below.

(1) Closed-form formulas for equivalent stiffness and damping coefficient of TVMD outriggers are derived based on a distributed-parameter model and are functions of inertance ratio and excitation frequency ratio. The relationships between equivalent parameters and excitation frequency ratio exhibited similar trends consistent across varying inertance ratios.

(2) As the frequency increases, the equivalent stiffness transitions from negative to positive values and asymptotically decays to the spring stiffness of TVMD outriggers at infinite frequency. Simultaneously, the equivalent damping increases from the TVMD damping coefficient to a maximum value as the frequency increases from zero to the tuning frequency, and then asymptotically decays to zero along with the increasing frequency. The modal response mitigation ratio reveals that the equivalent damping plays a dominant role over the equivalent stiffness in the control mechanism.

(3) An analysis of the equivalent damping-frequency curves elucidates the control mechanism of TVMD outriggers, i.e., tuning effect for the targeted mode and damping supplement effect for lower-order modes. Near the tuning frequency, the equivalent damping reaches its maximum value, leading to a substantial reduction of the targeted modal response. Under low-frequency excitation, the equivalent damping approaches the TVMD damping coefficient, which contributes to providing supplemental damping.

(4) A case study of a 140-m tall core wall-frame structure demonstrates that the TVMD, VD and NSD outriggers reduce maximum inter-story drifts by 26%, 10%, and 17%, and decrease maximum floor accelerations by 27%, 12% and 18%, respectively. Due to the aforementioned seismic control mechanism of the TVMD outriggers and the damping amplification effect of the NSD outriggers, they achieve superior control performance compared to the VD outriggers.

Footnotes

ORCID iDs

Ruofan Jia

Xiaodong Ji

Tianwei Sun

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work presented in this paper was supported by funding from the National Natural Science Foundation of China (Grant No. 52425808), Science Research Project of Hebei Education Department (Grant No. BJ2025196) and Hebei Natural Science Foundation (Grant No. E2025105058). The authors are sincerely grateful for this financial support.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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