Adaptive multiple tuned mass dampers based on modal parameters for earthquake response reduction in multi-story buildings

Abstract

The primary objective of this research is to find the effectiveness of an adaptive multiple tuned mass damper distributed along with the story height to control the seismic response of the structure. The seismic performance of a 10-story building was investigated, which proved the efficiency of the adaptive multiple tuned mass damper. Structures with single tuned mass damper and multiple tuned mass dampers were also modeled considering the location of the dampers at the top of the structure, whereas adaptive multiple tuned mass damper of the structure was modeled based on the story height. Selection of the location of the adaptive multiple tuned mass damper along with the story height was dominated by the modal parameters. Participation of modal mass directly controlled the number of the modes to be considered. To set the stage, a comparative study on the displacements and modal energies of the structures under the El-Centro, California, and North-Ridge earthquakes was conducted with and without various types of tuned mass dampers. The result shows a significant capability of the proposed adaptive multiple tuned mass damper as an alternative tool to reduce the earthquake responses of multi-story buildings.

Keywords

adaptive multiple tuned mass dampers modal energy mode shapes multi-story buildings passive control seismic response reduction structural control tuned mass dampers

Introduction

To reduce the undesirable building vibration, tuned mass damper (TMD) works as a passive energy-absorbing device. A TMD can be defined as the combination of a mass, a spring, and a viscous damper attached to the vibration system. Its reliability, effective performance, inexpensiveness, and easy installation are the main reasons for the popularity of TMD among the vibration control methods. Moreover, TMD can be installed into the existing buildings with less difficulty compared to the other methods. Simply transferring some of the vibration energy of the structure to the TMD is the working principle of this passive vibration control method (Rana and Soong, 1998).

A lot of researchers found TMD as an efficient vibration control method. Wang and Lin (2007) found that TMD systems are effective in the structural response reduction. Tezcan and Uluca (2003) studied three different kinds of structures and concluded that viscoelastic dampers reduce the seismic response of the structures significantly. Multiple tuned mass dampers (MTMDs) were also studied on a large scale to achieve effective response control. Use of two TMDs was found more effective in the study of Iwanami and Seto (1984). In the study of Xu and Igusa (1992), it is found that for an equal total mass, an optimally designed MTMD performs better than a single tuned mass damper (STMD). Chey et al. (2013) showed the innovative seismic retrofitting strategy of added stories isolation (ASI) system. These systems can effectively manage seismic response for multi-degree-of-freedom (MDOF) systems across a broader range of ground motions without requiring burdensome extra mass. Moon (2010) showed in a study that the installation of a massive STMD requires a large space, whereas MTMD can easily be installed with a comfortably smaller space. Chey et al. (2010a,b) showed the design and application of semi-active TMD for building system. In the proposed system, the stiffness of the resettable (semi-active) device was combined with the stiffness of rubber bearing. Luigi and Massimiliano (2008) showed an efficient formula of a new robust design of STMD for controlling the torsional response of an asymmetric-plan system. Chunxiang and Yanxia (2008) proved the effectiveness of MTMDs having identical stiffness and damping coefficient along with unequal mass. In their study, optimum frequency spacing was reflecting the robustness; the average damping ratio and the tuning frequency ratio were the factors to optimize the MTMD. Chey et al. (2015a,b) showed building isolation systems under multi-level earthquake excitations. In part 1, a conceptual design and nonlinear analysis procedure were described. An approach about energy dissipation and damage reduction system was shown in part 2. Mohebbi et al. (2015) tried to assess different design effects on the performance of MTMDs. Mohebbi et al. showed the effect of MTMDs’ mass ratio, TMDs’ number, and stroke length. Efficiency in floor vibration control using distributed multiple viscoelastic TMDs was studied in the research of Nguyen et al. (2012). Horst (2013) showed the effectiveness of the detuned TMD for a footbridge in his research. Yuxin and Zhitao (2014) demonstrated the advantage of suspended floor section as a massive TMD system in improving the seismic performance and structural stability. In a study of Chey and Kim (2012), a parametric control of structural responses using an optimal passive TMD under stationary Gaussian white noise excitations was represented. The study presented a process for designing optimal TMD systems highlighting the optimized parametric configuration to enable optimal displacement and acceleration control of structures. In the research of Domizio et al. (2015), performance of TMDs on the nonlinear structures subjected to near-fault earthquakes was observed. Bakre and Jangid (2007) showed in their research the relationship between the damping ratio of the TMD and damping of the main system. Benefits of MTMD over STMD in an approach of the experiment and further comparison with the numerical results were shown in the research of Chen and Wu (2003). Lin et al. (2010) also studied on MTMDs with limited stroke. In their optimum design and experimental study, the superiority of the MTMDs over STMDs was observed. Xiang and Nishitani (2014) showed seismic vibration control of the building structures using integrated floor system as an MTMD. Floors can serve as a TMD system with a large mass ratio without additional masses, and the proposed system was found very effective to control the building response in their study. A comparative study of different control strategies was shown by Chunwei Zhang (2004). In his study, a 76-story building was analyzed with different control systems titled as structural adjacent reaction wall control (STA) and mass semi-active damper (AMD-2) individually. The study was concluded with the understanding of STA control system being efficient at low cost with other attractive advantages. However, the STMD is generally designed and installed to control fundamental vibration mode. It is very effective in controlling narrow band frequency earthquake force which is relatively closer to the fundamental frequency of a structure. But for controlling the wide-band earthquake frequency, multi-mode controlled approach should be considered. MTMD is one kind of multi-mode controlling device which is capable of suppressing wide-spectrum frequency component of an earthquake force.

This study presents a new type of multiple mode vibration controlling system which is named as adaptive multiple tuned mass damper (AMTMD). This AMTMD is designed considering different vibration modes and installed in the selected locations according to the mode shapes to control wide-banded earthquake frequency. To evaluate the efficiency of AMTMD, a 10-story building is modeled and excited as per El-Centro, California, and North-Ridge ground motions. The maximum displacement of the top floor is considered for performance evaluation. Moreover, modal energy is also studied to find out the energy absorption of the structural system. Comparisons with the STMD and MTMD are conducted to establish the effectiveness of the proposed system. It is observed that STMD and MTMD perform better during the free vibration (after the earthquake) which is controlled by the first mode, whereas during earthquake excitation if the response is governed by a broad range of exciting frequencies, AMTMD is more effective as it controls a broad range of frequencies including various higher modes. Therefore, it can be said that AMTMD is more efficient in response reduction comparing to the STMD and MTMD installed on the top of the building for a wide earthquake frequency range.

Structural model with TMD system

A 10-story building is analyzed for the observation of earthquake response without any TMD and also with TMDs installed at various locations. A stick model is considered for the ease of the analysis and representation. Table 1 represents the details of story mass and stiffness for the analyzed structure.

Table 1.

Structural properties.

Story ID	Mass (×10³ kg)	Stiffness (×10³ kN/m)
First	179	62.47
Second	170	52.26
Third	161	56.14
Fourth	152	53.02
Fifth	143	49.91
Sixth	134	46.79
Seventh	125	43.67
Eighth	116	40.55
Ninth	107	37.43
Tenth	98	34.31

Rayleigh damping approach has been adopted to calculate the structural damping. Where 2% damping ratio has been considered for all modes. Damping matrix is calculated using equation (1)

C = a_{o} M + a_{1} K

(1)

where C is the damping matrix, M is the mass matrix, and K is the corresponding stiffness matrix. $a_{o}$ and $a_{1}$ are the constants which depend on the natural frequency of the structure.

In Figure 1, the simplified building model is represented. Figure 1(a) represents a 10-story building with STMD installed on the top floor. Figure 1(b) illustrates a 10-story building having MTMD installed at the top of the building and Figure 1(c) shows adaptive TMDs installed along with the story height. The buildings are modeled in the MATLAB. All the structural cases are subjected to time history analysis.

Figure 1.

Structural models with TMDs installed: (a) STMD, (b) MTMD, and (c) AMTMD.

Equation of motion

The representative model of the building structure is shown in Figure 1 which shows the installation location of different TMDs in the system. It is also considered that N-story building structure has N-DOF. TMD masses are connected to the main structural system through linear stiffness (spring) and linear viscous damping (dashpot). $M_{n} (n = 1, 2, \dots, N)$ in the figure represents the floor lump mass of the main structural system and $m_{dj} (j = 1, 2, \dots, p)$ represents the TMDs mass. If p is the number of installed TMDs on a N-DOF lumped mass structure, which is excited by a ground acceleration of ${\overset{\cdot\cdot}{x}}_{g} (t)$ , the equation of motion for passively controlled system can be described as per Xiang and Nishitani (2014)

M \overset{\cdot\cdot}{x} + C \overset{\cdot}{x} + Kx = - MH {\overset{\cdot\cdot}{x}}_{g}

(2)

where x, $\overset{\cdot}{x}$ , and $\overset{\cdot\cdot}{x}$ , respectively, represent the displacement, velocity, and acceleration vectors of the system relative to the base point. The dimension of the matrix can be presented as $(N + p) \times 1$ . $M$ , $C$ , and $K$ are the mass, damping, and stiffness matrices, respectively. Where the matrix dimension is $(N + p) \times (N + p)$ .

The global forms of $M$ , $C$ , and $K$ are as follows

M = [\begin{matrix} M_{sN \times N} & 0_{N \times p} \\ 0_{p \times N} & 0_{p \times p} \end{matrix}] + \sum_{j = 1}^{p} m_{dj} a_{j} a_{j}^{T}

(3)

C = [\begin{matrix} C_{sN \times N} & 0_{N \times p} \\ 0_{p \times N} & 0_{p \times p} \end{matrix}] + \sum_{j = 1}^{p} c_{dj} b_{j} b_{j}^{T}

(4)

K = [\begin{matrix} K_{sN \times N} & 0_{N \times p} \\ 0_{p \times N} & 0_{p \times p} \end{matrix}] + \sum_{j = 1}^{p} k_{dj} b_{j} b_{j}^{T}

(5)

where $M_{s}$ , $C_{s}$ , and $K_{s}$ are the mass, damping, and stiffness matrices of the structure, respectively, having a matrix dimension of $N \times N$ . In addition, $T$ denotes the transpose of the matrix or vector, and $m_{dj}$ , $c_{dj}$ , and $k_{dj}$ are the mass, damping, and stiffness of the jth TMD, respectively. The full matrices with AMTMD are given in Appendix 1. The location vectors for the TMD are as follows

a_{j} = [0_{1 \times (N + j - 1)} 10_{1 \times (p - j)}]^{T}

(6)

and

b_{j} = [0_{1 \times (lo c_{j} - 1)} - 10_{1 \times (N + j - lo c_{j} - 1)} 1 0_{1 \times (p - j)}]^{T}

(7)

The dimension of the location vector is $(N + p) \times 1$ , $lo c_{j}$ represents the location of jth TMD, and H is a unit vector of $(N + p) \times 1$ matrix dimension.

To simulate the equation of motion by MATLAB, a representative form of equations of motion is used as state-space equation or time-domain approach mentioned by Kim (2013). The changing of the equation of motion into state-space form only requires one set of variable substitution. Using the variable substitution, equation (8) can be written as

\begin{matrix} \overset{\cdot}{X} = A_{ss} X + B_{ss} U \\ Y = C_{ss} X + D_{ss} U \end{matrix}

(8)

where $A_{ss}$ , $B_{ss}$ , $C_{ss}$ , and $D_{ss}$ are the matrices defined as

A_{ss} = [\begin{matrix} \begin{matrix} 0 \\ - M^{- 1} K \end{matrix} & \begin{matrix} I \\ - M^{- 1} C \end{matrix} \end{matrix}]

(9)

B_{ss} = [\begin{matrix} 0 \\ - {[M]}^{- 1} H \end{matrix}]

(10)

C_{ss} = [\begin{matrix} - M^{- 1} K & - M^{- 1} C \end{matrix}]

(11)

D_{ss} = [- {[M]}^{- 1} H]

(12)

where X is a state vector, Y is an output vector, U is the input vector, H is the force location matrix, and ${\overset{\cdot\cdot}{x}}_{g}$ is the ground acceleration. Using state-space equation, all structural cases for lateral excitation are solved.

Design of TMDs

TMD works with a simple process of associating a mass, a spring, and a damper to reduce the peak amplitude as minimum as possible. The TMDs’ design parameters and installation locations are selected based on the modal parameters of the building structure. The modal analysis is carried out in several cases of modeling. Modal analysis of the structure is performed without any TMD to get the natural frequencies and mode shapes of the structure. Moreover, effective modal mass ( $M_{r}$ ) is also obtained from the modal analysis. More than 90% of the total mass of the building was considered to determine the modes that needed to be controlled using TMDs. From the obtained modal analysis, it is clear that the first three modes contribute to more than 95% of the building mass, 80.57% from the first mode, 11.28% from the second mode, and 3.85% from the third mode (Figure 2). The mass ratio of TMD has a strong and direct effect in controlling structural vibration. In general, the mass ratio should not exceed 10% of the total structural mass. In this research, the corresponding mass ratio (µ) is assumed 5% of the total structural mass. In all the cases, the total mass of TMD $(m_{d})$ is kept the same. The optimum frequency ratio $(α_{opt})$ is the ratio of TMDs’ frequency $(ω_{j})$ and natural frequency of the building structure $(Ω_{j})$ . The TMD mass $(m_{d})$ is obtained from equation (13). The design of the STMD is executed considering the first mode of the uncontrolled structure. Optimum frequency ratio $α_{opt}$ and optimal damping $ξ_{opt}$ of the STMD are determined using equation (14) by Zahrai and Ghannadi-Asl (2008). Using the obtained values of spring stiffness $K_{d}$ , damping $C_{d}$ , and natural frequency of STMD, $ω_{d}$ are calculated by equation (15)

m_{d} = μ M_{s}

(13)

α_{opt} = \frac{1}{1 + μ} \sqrt{\frac{2 - μ}{2}}, ξ_{opt} = \sqrt{\frac{3 μ}{8 (1 + μ)}} \sqrt{\frac{2}{2 - μ}}

(14)

\begin{array}{l} K_{d} = 4 π^{2} μ α_{o p t}^{2} \frac{M_{s}}{T s^{2}}, C_{d} = 4 π μ ξ_{o p t} α_{o p t} \frac{M_{s}}{T s}, \\ ω_{d} = \sqrt{\frac{K_{d}}{m_{d}}} \end{array}

(15)

where $M_{s}$ is the total structural mass and $T_{s}$ is the natural period of the uncontrolled structure.

Figure 2.

First three mode shapes of the structure.

For controlling several structural vibrations modes, it is needed to design several TMDs according to the corresponding modes. The mass of each TMD is obtained from the modal mass participation factor. After the determination of the masses of TMDs, the first TMD has been designed and installed according to the first mode of the uncontrolled structure. Then the modal analysis is conducted again with the installed first TMD to find out the next frequency mode which is needed to be controlled. This procedure continues up to the design and installation of the last TMD of the system. The design procedure of AMTMD is illustrated in Figure 3 in the form of a flowchart.

Figure 3.

Flowchart for selecting location and design of the AMTMD parameters.

The optimal frequency ratio $(α_{opt})$ and optimal damping ratio $(ξ_{opt})$ for the MTMD or AMTMD are kept the same as the STMD system for an effective comparison between the AMTMD system and other control systems as approached in this study. The first three $(j = 1, \dots, 3)$ modes are needed to be controlled according to the modal participation factors, and the frequency of each TMD is calculated as per the frequency ratio of the corresponding frequency mode. The TMDs properties are provided in Table 2 where $TM D_{1}$ , $TM D_{2}$ , and $TM D_{3}$ are the TMDs properties for controlling the first, second, and third vibrational modes of the structure.

Table 2.

TMDs’ properties.

Cases	Items	Mass $m_{dj}$ (tons)	Stiffness $K_{dj}$ (kN m)	Damping $C_{dj}$ (kN s/m)
STMD	TMD	69.25	562.25	53.40
MTMD	TMD ₁	58.30	475.98	45.09
	TMD ₂	8.16	499.67	17.29
	TMD ₃	2.79	455.70	9.64
AMTMD	TMD ₁	58.30	473.02	45.09
	TMD ₂	8.16	499.67	17.29
	TMD ₃	2.79	453.68	9.62

TMD: tuned mass damper; STMD: single tuned mass damper; MTMD: multiple tuned mass damper; AMTMD: adaptive multiple tuned mass damper.

The MTMD or AMTMD mass $(m_{dj})$ for different mode control TMD is determined as

m_{dj} = m_{d} \times \frac{M_{rj}}{\sum_{j = 1}^{p} M_{rj}}

(16)

The frequency $(ω_{j})$ of each TMD is calculated by the following equation

ω_{j} = α_{opt} Ω_{j}

(17)

The stiffness $(K_{dj})$ of each TMD for MTMD or AMTMD is calculated as

K_{dj} = m_{dj} ω_{j}^{2}, j = 1, \dots, p

(18)

Damping $C_{dj}$ of the TMD is calculated as below

C_{dj} = 2 ξ_{opt} m_{dj} ω_{j}, j = 1, \dots, p

(19)

In this study, a suitable location for the AMTMD has been selected depending on the mode shapes of the structure. The mode shapes are demonstrated in Figure 2. TMDs have been installed at the largest amplitude of the corresponding mode shapes. In this study, TMDs are installed on the top floor, ninth floor, and sixth floor of the structure based on the modal shapes. The natural time period of the building from all the analysis cases is shown in Table 3.

Table 3.

Natural period (s) of the structure.

Mode no.	Without TMD	With STMD	With MTMD	With AMTMD
First mode	2.022	1.799	1.831	1.613
Second mode	0.779	0.753	0.730	0.716
Third mode	0.476	0.465	0.455	0.457

TMD: tuned mass damper; STMD: single tuned mass damper; MTMD: multiple tuned mass damper; AMTMD: adaptive multiple tuned mass damper.

Performance of AMTMD

Modal energy and ground motions

The total energy of the system represented by the model parameter is utilized by Bigdeli and Kim (2014) and can be stated as equation (20)

\begin{array}{l} E_{t o t a l} (t) = \sum_{j = 1}^{n} P E_{j} + \sum_{j = 1}^{n} K E_{j} \\ = \sum_{j = 1}^{n} \frac{1}{2} k_{j} q_{j}^{2} (t) + \sum_{j = 1}^{n} \frac{1}{2} m_{j} {\dot{q}}_{j}^{2} (t) \end{array}

(20)

where $P E_{j}$ and $K E_{j}$ are the j-th modal potential and kinetic energies at time $t$ , respectively. $k_{j}$ , $m_{j}$ , and $q_{j}$ are the modal stiffness, modal mass, and modal coordination corresponding displacement.

In this study to evaluate the performance of AMTMD, three earthquake loads are applied, and for each earthquake, the response is evaluated. El-Centro earthquake is applied with 2500 load steps, having a time interval of 0.02 s and peak ground acceleration (PGA) was 0.348g. California and North-Ridge ground motion both have 2000 load steps and the time interval of 0.01 s. PGA is 0.156g and 0.343g, respectively. Figures 4 and 5 represent the time histories and Fourier amplitude of ground motions applied in the analysis.

Figure 4.

Time history of input ground motions: (a) time history of El-Centro earthquake, (b) time history of California earthquake, and (c) time history of North-Ridge earthquake.

Figure 5.

Fourier amplitude of input ground motions: (a) FFT of El-Centro earthquake, (b) FFT of California earthquake, and (c) FFT of North-Ridge earthquake.

Result and discussion

To check the performance efficiency of the proposed system, the displacement of the top floor and modal energy are evaluated in this study. Figures 6 to 8 show the top floor displacement comparison for the ground excitation of El-Centro, California, and North-Ridge earthquake, respectively. In addition, Figures 9 to 11 show the modal energy for first three modes for the previously mentioned earthquakes.

Figure 6.

Results obtained from El-Centro ground motion: (a) top floor displacement comparison between AMTMD and without TMD, (b) top floor displacement comparison between AMTMD and STMD, and (c) top floor displacement comparison between AMTMD and MTMD.

Figure 7.

Results obtained from the California ground motion: (a) top floor displacement comparison between AMTMD and without TMD, (b) top floor displacement comparison between AMTMD and STMD, and (c) top floor displacement comparison between AMTMD and MTMD.

Figure 8.

Results obtained from North-Ridge ground motion: (a) top floor displacement comparison between AMTMD and without TMD, (b) top floor displacement comparison between AMTMD and STMD, and (c) top floor displacement comparison between AMTMD and MTMD.

Figure 9.

Modal energy for El-Centro ground motion: (a) modal energy for first mode, (b) modal energy for second mode, (c) modal energy for third mode, and (d) summation of modal energy for first three modes.

Figure 10.

Modal energy for California ground motion: (a) modal energy for first mode, (b) modal energy for second mode, (c) modal energy for third mode, and (d) summation of modal energy for first three modes.

Figure 11.

Modal energy for North-Ridge ground motion: (a) modal energy for first mode, (b) modal energy for second mode, (c) modal energy for third mode, and (d) summation of modal energy for first three modes.

Figure 6 shows the displacement at the top floor of the structure subjected to El-Centro ground motion with an STMD installed at the top floor, MTMD installed at the top floor, and MTMD installed along with the story height. From the obtained results, it is observed that the uncontrolled maximum displacement of the structure is 36.18 cm, 22.84 cm for STMD, 23.47 cm for MTMD, and 16.94 cm for AMTMD. This demonstrates that STMD is capable of reducing 36.87% of the maximum floor displacement, whereas MTMD reduces 35.12% of the maximum displacement. Meanwhile, AMTMD reduces 53.32% of the uncontrolled displacement which shows significant efficiency compared to STMD and MTMD.

Figure 7 shows the displacement at the top floor of the structure subjected to California ground motion with an STMD installed at the top floor, MTMD installed at the top floor, and MTMD installed along with the story height. From the obtained results, it is observed that the uncontrolled maximum displacement of the building is 4.01 cm, 3.36 cm for STMD, 3.42 cm for MTMD, and 2.74 cm for AMTMD. This means that STMD is capable of reducing 16.15% of the maximum floor displacement while MTMD reduces 14.73% of the maximum displacement. Meanwhile, AMTMD reduces 41.66% of the uncontrolled displacement which is considerably more efficient than STMD or MTMD.

Figure 8 shows the result of the displacement of the top floor of the structure subjected to North-Ridge ground motion with the STMD installed on the top floor, MTMD installed on the top floor, and MTMD installed along with the story height. From the obtained results, it is observed that the uncontrolled maximum displacement of the building is 12.78 cm, 5.26 cm for STMD, 5.04 cm for MTMD, and 3.51 cm for AMTMD. This means that STMD is capable of reducing 58.76% of the maximum floor displacement while MTMD reduces 60.54% of the maximum displacement. Meanwhile, AMTMD reduces 72.49% of the uncontrolled displacement which proves the efficiency of AMTMD over STMD or MTMD.

Figure 9 illustrates the modal energy of the building corresponding to the El-Centro ground motion excitation. The total energy of the uncontrolled building for the first three modes is 8985.88 kJ. The first three modes’ energy of the controlled building with STMD, MTMD, and AMTMD are 459.83, 449.43, and 402.93 kJ, respectively. This indicates that STMD, MTMD, and AMTMD reduce 94.88%, 94.99%, and 95.51% of uncontrolled energy, respectively. This clearly indicates that AMTMD is 12.0% more effective than STMD and 10% more efficient compared to MTMD.

Figure 10 illustrates the modal energy of the building corresponding to the California ground motion excitation. The total energy of the uncontrolled building for the first three modes is 364.27 kJ. The first three modes’ energy of the controlled building with STMD, MTMD, and AMTMD are 69.91, 68.78, and 59.31 kJ, respectively. This indicates that STMD, MTMD, and AMTMD reduce 80.80%, 81.11%, and 83.71% of uncontrolled energy, respectively. This clearly indicates that AMTMD is 15.16% more effective than STMD and 13.76% more efficient compared to MTMD.

Figure 11 represents the modal energy of the building corresponding to the North-Ridge ground motion excitation. The total energy of the uncontrolled building for the first three modes is 2121.271 kJ. The first three modes’ energy of the controlled building with STMD, MTMD, and AMTMD are 165.82, 147.57, and 171.38 kJ, respectively. This indicates that STMD, MTMD, and AMTMD reduce 92.18%, 93.05%, and 91.92% of uncontrolled energy. These results represent that during earthquake excitation if the response is governed by a broad range of exciting frequencies, AMTMD will be more effective as it will control the broad range of frequency including higher modes.

Conclusion

The earthquake responses of a 10-story building are analyzed using the AMTMD as a vibration control method. The purpose is to investigate the efficiency of the AMTMD in controlling the maximum displacement of the building as well as provide sufficient energy dissipation of the structural system during earthquakes. Comparing to the other tuned mass control methods, the performance of the AMTMD is found efficient. The following conclusions are drawn from the trend of the results of this study:

The AMTMD is more efficient than the STMD and MTMD in the response reduction in the building. The results also indicate a noticeable reduction in the maximum top displacement for AMTMD equipped building.

Selection of locations to install the TMDs is largely determined by the mode shapes of the building. Installing TMDs considering the modal characteristics of the structure is found more effective compared to TMDs installed on the top floor of the structure.

The AMTMD is found more effective than the STMD and MTMD to dissipate energy especially in the broadband of frequencies. Therefore, it could be used to mitigate the vibration responses of high-rise building structures where multiple modes play important roles in the structural responses.

Footnotes

Appendix 1 Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The National Research Foundation of Korea Grant funded by the Korean Government (NRF-2014R1A2A1A10049538) supported this work.

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