Metaheuristic-Based Optimum Design Methods for Multiple Tuned Mass Dampers under Seismic Excitations

1  Introduction

One of the greatest challenges facing buildings is their exposure to forced vibrations caused by undesirable excitations such as earthquakes and winds. This situation becomes even more dangerous in high-rise buildings due to their thinness and flexibility [1]. In many cases, forced vibrations cause significant failure in the main structural elements, ultimately leading to the complete collapse of the structure [2]. Therefore, structural controls play a crucial role in reducing vibration generated by both earthquakes and winds [3]. To mitigate the impact of earthquakes and winds on structures, researchers have sought to find optimal solutions for reducing the structure’s response. They have focused on adding elements with specific mass, stiffness, and damping properties to the floors of the building. The primary function of these elements is to control vibration by dissipating the energy. This type is called passive vibration control and is the most popular among other types. The first person to use this concept was the German scientist Frahm [4], who invented and patented a device that dissipates vibration energy. One of the most important conclusions of his research was that this device is only effective when the natural frequency of the structure is close to the excitation frequency. This device was later modified by using damping elements, creating the Tuned Mass Damper (TMD) [5].

Although the tuned mass damper offers significant advantages in damping the energy vibration in structures, it also has significant drawbacks that cannot be ignored. Among these is that this system only affects control of a specific mode and therefore may not be effective in controlling the other modes [6]. Furthermore, the mass required for production is very large, requiring adequate space, and the large mass makes maintenance difficult. All these factors have led to significant interest in developing a system similar to the tuned mass damper that is both effective in continuous damping across different modes and easy to install and maintain. This system is a multiple-tuned mass damper. It was first used in the John Hancock Building in the United States in the 1970s. Studies have demonstrated the significant contribution of the Multiple Tuned Mass Damper (MTMD) in damping the vibration of the buildings [7].

Many studies have described the effectiveness of multiple-tuned mass dampers in damping building vibration compared to tuned mass dampers. Igusa and Xu investigated the effectiveness of multiple-tuned mass dampers in damping the vibration of a building due to wind loads [8]. They designed the mass parameters using the optimum solution and then compared the results with the single-tuned mass damper case for the same building. From the results, it was found that multiple-tuned mass dampers were more effective and had higher efficiency than the single-tuned mass damper. Li and Liu compared tuned mass damper and multiple tuned mass damper systems by considering soil-structure interaction in irregular structures and found that the MTMD was more effective than the TMD [9]. Wang and Lin investigated the effect of multiple tuned mass dampers on an irregular torsional structure subjected to seismic-induced vibration [10]. The results showed that multiple tuned mass dampers were more efficient in damping energy than single tuned mass dampers when the effect of soil interaction was considered. Moon investigated the distribution of vertically tuned mass dampers along a 60-story high-rise building [11]. He discussed the effect of this distribution on vibration energy absorption. The results showed that the energy absorption efficiency was significantly higher than that of the conventional installation of a single tuned mass damper at the highest point of the high-rise building. Chen and Wu, in their study of tuned mass dampers and multiple tuned mass dampers, confirmed that the effectiveness of multiple tuned mass dampers in absorbing energy was higher than that of the tuned mass damper, even when the mass values were equal in both cases [12]. Brandao and Miguel investigated a steel building subjected to multiple seismic excitations [13]. They then examined the building in three different cases. In the first case, a tuned mass damper was installed on the top floor. In the second case, multiple tuned mass dampers were installed on all floors. In the third case, multiple tuned mass dampers were installed on the top floor. They calculated the mass damper parameters using a metaheuristic optimization algorithm. The results showed that all cases performed well in reducing floor displacements. Djerouni et al. [14] proposed MTMDs for adjacent structures and a hybrid metaheuristic method was used to find optimum design values. As a result of the study, MTMDs that are distributed along the height of the adjacent buildings perform better. Wang et al. proposed an Adaptive-Passive Variable-Mass Multiple Tuned Mass Damper (APVM-MTMD) system to mitigate detuning effects in MTMD applications for large-span floor structures. In their approach, each damper can retune its frequency by adjusting its mass based on dominant structural frequencies identified via the Short Time Fourier Transformation (STFT), resulting in improved vibration control compared to mistuned passive MTMD systems. Although effective, the system requires sensing, control hardware, and mechanical mass-adjustment mechanisms, which increase complexity relative to fully passive optimized MTMD solutions [15].

Tuned mass dampers and multiple-tuned mass dampers do not perform well without tuning. Determining the optimum values of these parameters is crucial for achieving good vibration energy absorption results. Numerous researchers have conducted extensive research on how to determine these values, using various equations, called algorithms, to calculate optimal solutions. Mathematical programming and algorithms have proven highly effective in complex optimization processes. These algorithms have been developed by researchers to find general solutions to solve complex problems. The algorithm was tested for 10-story shear frames as well as for controlling the response of a 76-story concrete building under wind excitation. The results showed great effectiveness in vibration damping due to the optimum calculation of the tuned mass damper. Singh et al. presented a study on the optimal design of tuned mass dampers using genetic algorithms to mitigate the seismic response of torsional building systems to seismic excitations [16]. The study provided important insights into the use of genetic algorithms in calculating the optimum parameters of tuned mass dampers. Frans and Arfiadi conducted a study to determine the parameters of multiple-tuned mass dampers using Hybrid Coded Genetic Algorithms (HCGAs) [17]. The study also focused on determining the optimal location of the mass dampers. For this purpose, they examined three different cases: the first is a three-story building, the second is a 10-story building, and the third is a 40-story building. The results showed great effectiveness in absorbing vibration energy after determining the optimal values and location of the mass dampers. Bekdaş et al. used the new Bat algorithm to calculate the mass, period, and damping ratio of a 10-story building [18]. They then calculated the previous parameters using analytical methods and other methods and compared the results between the previous methods. The results showed that the new Bat algorithm yielded good results, leading to good building behavior. Cetin and Ayden calculated the parameters of a tuned mass damper using the transfer function [19]. They calculated the mass, damping, and stiffness of the tuned mass damper using this function. Thus, they reduced the response of the structure in terms of displacement and acceleration caused by seismic excitations. The results showed that the transfer function was highly effective in reducing the response of the structure. Ozturk et al. discussed how to calculate the optimum parameters of tuned mass dampers for a 10-story building and how to determine the optimum location of the tuned mass dampers [20]. Kaveh et al. determined the optimum parameters of a tuned mass damper using a novel Chaotic Optimization Algorithm (COA) [21]. Wang et al. investigated a seismic multi-objective stochastic optimization strategy for MTMD systems applied to a large podium–twin towers structure. Using a Kanai–Tajimi-based stochastic earthquake model and the artificial fish swarm algorithm, they demonstrated that optimally distributed MTMDs can effectively mitigate coupled translational and torsional seismic responses under bi-directional excitations [22]. Ghojehbiglou et al. investigated the efficiency of a hybrid nature-inspired optimization algorithm for tuning multiple tuned mass dampers in buildings considering soil–structure interaction. By combining the Marine Predators Algorithm with Particle Swarm Optimization and applying it to a 10-story building under various earthquake spectra, they showed that optimally distributed TMDs can significantly reduce seismic responses, with performance strongly influenced by soil conditions [23].

Several recent studies have investigated the use of inerter-based multiple tuned mass damper systems for improving the seismic performance of building structures. For instance, Shahrouzi et al. proposed an optimization approach for multiple tuned mass damper inerter (MTMDI) systems using the escaping bird search algorithm to enhance seismic control of buildings [24]. Similarly, Djerouni et al. examined the effectiveness of MTMDI systems in mitigating earthquake-induced vibrations in structures [25]. A multi-objective optimization framework for inerter-based mass dampers was presented by Garrido et al. [26]. In another study, Wang et al. developed a root-locus-based optimal formulation for structures equipped with tuned mass damper inerters enhanced with negative stiffness while accounting for non-resonant modal contributions [27]. Additionally, Kiran et al. investigated the optimization of tuned mass damper inerter systems for both base-isolated and fixed-base structures under seismic excitations [28].

The aim of this study is to compare the effects of tuned mass dampers (TMD) and multiple tuned mass dampers (MTMD) in reducing floor displacements and vibration energy in a 10-story building subjected to seismic activity. A 10-story building was examined for this purpose. Three main cases were then examined. The first case involved a tuned mass damper installed on the tenth floor. The second case involved multiple-tuned mass dampers installed on all floors. The third case involved multiple tuned mass dampers installed on all floors. However, the difference between this case and the previous one is that the damper mass was assumed to be 50,000 kg. The stiffness and damping ratio values of the masses connected to each floor were also calculated using the Jaya algorithm, and the optimum solution was achieved. The vibration energy and floor displacements resulting from the seismic effect were calculated for each case. Finally, a comparison was made between the previous three cases.

Despite extensive studies on TMD and MTMD systems, the practical implications of using optimized vs fixed damper mass distributions within MTMD configurations have not been sufficiently quantified under a unified optimization framework. The main contribution of this paper is a systematic comparison of TMD, optimized MTMD, and fixed-mass MTMD systems using the same Jaya-based optimization approach, providing clear insights into the trade-off between seismic performance, energy dissipation efficiency, and practical applicability.

2  Conceptual Framework

2.1 Tuned Mass Damper System (TMD)

A tuned mass damper is a device that reduces the amplitude of vibrations resulting from forced movements of structures. A tuned mass damper operates by adjusting its frequency to match the frequency of the structure. In this case, the tuned mass damper absorbs the vibration energy, creating a force equal and opposite to the vibration force, which damps the vibration. The dynamic behavior of a tuned mass damper is expressed by the equation of motion for a system with two degrees of freedom. Fig. 1 shows the tuned mass damper system.

Figure 1: Tuned mass damper system (TMD).

The equation includes the displacement, velocity, and acceleration of the tuned mass damper. The equation of motion of the Tuned Mass Damper (TMD) system associated with a Single Degree of Freedom (SDOF) structure under seismic excitation is as follows:

[M]X¨+[C]X˙+[K]X=−[M]1X¨g(1)

here:

X¨: Acceleration vector, X˙: Velocity vector, X: Displacement vector, X¨g: Ground acceleration.

The mass matrix is given by the following equation:

[M]=[m100md](2)

here:

m1: Mass of the structure, md: Mass of the tuned mass damper.

The damping matrix is given by the following equation:

[C]=[c100cd](3)

here:

c1: Damping of the structure, cd: Damping of the tuned mass damper.

The stiffness matrix is given by the following equation:

[K]=[k100kd](4)

here:

k1: Stiffness of the structure, kd: Stiffness of the tuned mass damper.

Eq. (7) can be written as follows:

[m100md]{x1¨xd¨}+[c100cd]{x1˙xd˙}+[k100kd]{x1xd}=−[m1md]Xg¨(5)

2.2 Motion Equation of a Multiple-Storey Building with Tuned Mass Damper

In this system, the tuned mass damper is placed at the highest point of the building. Furthermore, each floor is assumed to have one degree of freedom. The tuned mass damper also has one degree of freedom. Therefore, this system consists of (n + 1) degrees of freedom. Fig. 2 shows the tuned mass damper system:

Figure 2: System of multiple-storey building with TMD.

The equation of motion of the previous system is similar to Eq. (1). The mass matrix of the system is diagonal matrix. This matrix is written as follows:

M=[m100…0000m20…00000m3…000⋮⋮⋮⋱⋮⋮⋮000…mn−100000…0mn0000…00md](6)

here:

m1,m2….mn−1,mn: Masses of the floors of the structure.

md: Mass of the tuned mass damper. The system’s stiffness matrix is written as follows:

K=[k1+k2−k20…000−k2k2+k3−k3…0000−k3k3+k4…000⋮⋮⋮⋱⋮⋮⋮000…kn−1+kn−kn0000…−knkn+kd−kd000…0−kdkd](7)

here:

k1,k2….kn−1,kn: Stiffnesses of the floors of the structure.

kd: Stiffness of the tuned mass damper. The damping matrix of the system is written as follows:

C=[c1+c2−c20…000−c2c2+c3−c3…0000−c3c3+c4…000⋮⋮⋮⋱⋮⋮⋮000…cn−1+cn−cn0000…−cncn+cd−cd000…0−cdcd](8)

here:

c1,c2…,cn−1,cn: Damping of the floors of the structure.

cd: Damping of the tuned mass damper.

2.3 Motion Equation of a Multiple-Storey Building with a Multiple-Tuned Mass Damper

A multiple-tuned mass damper (MTMD) is more effective than a single-tuned mass damper because each layer is connected to a small mass damper. This system contributes to the effective damping of vibrations induced by seismic excitations. Fig. 3 shows a multiple-tuned mass damper system:

Figure 3: System of multiple-storey building with MTMD.

The equation of motion of the previous system is similar to Eq. (1). The mass matrix of the system is diagonal. This matrix is written as follows:

M=[m100…00000…000m20…00000…0000m3…00000…00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋮⋮⋮000…mn−10000…00000…0mn000…00000…00md100…00000…000md20…00000…0000md3…00⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋮⋮000000000…mdn−10000000000…0mdn](9)

here:

m1,m2,…,mn−1,mn: Masses of the floors of the structure.

md1,md2,…,mdn−1,mdn: Mass of the tuned mass damper.

The system’s stiffness matrix is written as follows:

K=[k1+k2+kd1−k2…0−kd10…00−k2k2+k3+kd2…00−kd2…000−k3…000…00⋮⋮⋱⋮⋮⋮⋮⋮⋮00…000…−kdn−1000…kn−1+kn+kd−kn0…00−kd10…−knkd10…0−kdn0−kd2…00kd2…0000…000…00⋮⋮⋮⋮⋮⋮⋱⋮⋮000−kdn−100…kdn−100000−kdn0…0kdn](10)

here:

k1,k2,…,kn−1,kn: Stiffnesses of the floors of the structure.

kd1,kd2,…,kdn−1,kdn: Stiffness of the tuned mass damper.

The damping matrix of the system is written as follows:

C=[c1+c2+cd1−c…0−cd10…00−cc2+c3+cd2…00−cd2…000−c3…000…00⋮⋮⋱⋮⋮⋮⋮⋮⋮00…000…−cdn−1000…cn−1+cn+cd−cn0…00−cd10…−cncd10…0−cdn0−cd2…00cd2…0000…000…00⋮⋮⋮⋮⋮⋮⋱⋮⋮000−cdn−100…cdn−100000−cdn0…0cdn](11)

here:

c1,c2,…,cn−1,cn: Damping of the floors of the structure.

cd1,cd2,…,cdn−1,cdn: Damping of the tuned mass damper.

3  Method

Metaheuristics are regarded as a higher-level procedure to find the optimum of a problem or tune a system or algorithm. These algorithms use iterative and random methods that can be used with problems that cannot be fully mathematically formalized. Generally, an inspiration is used in the metaheuristics to reach optimum solutions.

The Jaya algorithm was proposed by Rao [29]. It is considered simple yet powerful because it converges to the best solutions and avoids the worst ones. The Jaya equation is an optimization equation designed for both constrained and unconstrained solutions that depend on the population. In the Jaya equation phase, the design constants, design variables, and population size must be defined. Then, the design variables of the initial solution matrix must be randomly determined according to their lower and upper limits. The following equations are used for this purpose:

md=mdmin+rand(mdmax−mdmin)(12)

Td=Tdmin+rand(Tdmax−Tdmin)(13)

Cr=Crmin+rand(Crmax−Crmin)(14)

here: md: Mass of the tuned mass damper. Td: Period of the tuned mass damper. Cr: Damping ratio. The basic equation of the Jaya algorithm is as follows:

Xi,new=Xi,j+r1(Xi,gthebest−|Xi,j|)−r2(Xi,gtheworst−|Xi,j|)(15)

here: Xi,new: The updated value of the design variable. Xi,j: The current value of the design variable. Xi,gthebest: The value of the variable for the best candidate solution. Xi,gtheworst: The value of the variable for the worst candidate solution. r1,r2: Random numbers between 0 and 1.

To ensure reproducibility, the implementation details of the Jaya algorithm are explicitly defined in this study. The optimization procedure was coded in MATLAB, and the design variables (damper mass, period, and damping ratio) were bounded according to Eqs. (16)–(19). A population size of 20 was adopted and the algorithm was terminated after a predefined number of iterations once convergence of the objective function was observed. At each iteration, candidate solutions were updated using Eq. (15), where the current solution is moved toward the best solution and away from the worst solution in the population. Random numbers uniformly distributed between 0 and 1 were generated at each step to maintain exploration capability.

The Jaya algorithm, proposed by Rao, was selected because it is a parameter-less metaheuristic algorithm that does not require algorithm-specific control parameters such as crossover rate, mutation rate, inertia weight, or acceleration coefficients, which are required in algorithms like Genetic Algorithms or Particle Swarm Optimization. This simplicity improves transparency and reproducibility while reducing the risk of biased tuning of algorithmic parameters. Furthermore, Jaya simultaneously promotes convergence toward the best solution and divergence from the worst solution, enhancing both exploitation and exploration capabilities. Since the MTMD optimization problem involves multiple continuous design variables and nonlinear dynamic structural responses, a robust and computationally efficient population-based optimizer without additional calibration requirements is particularly suitable. Therefore, the Jaya algorithm provides a reliable and unbiased framework for the optimal tuning of MTMD systems.

3.1 Definition of Structural Building

This study focuses on improving the behavior of a 10-story building subjected to seismic excitation using a tuned mass damper for the first case and a multiple-tuned damper for the second. The acceleration, velocity, and displacement of the floors were obtained using MATLAB and Simulink. The Jaya algorithm was also used to tune the mass damper parameters. Each floor in the building has the same properties for mass (50,000 kg), stiffness (900 MN/m), and damping coefficient (7 MNs/m). The building is subjected to seismic excitation and it is the El-Centro earthquake.

Figs. 4 and 5 show the TMD & MTMD system for the two basic cases of 10-story structure.

Figure 4: TMD system on 10-story structure.

Figure 5: MTMD system on 10-story structure.

where:

m1,m2,m3,…,m9,m10: Floors masses.

c1,c2,c3,…,c9,c10: Floors damping.

k1,k2,k3,…,k9,k10: Floors stiffness.

md: The mass of tuned mass damper (TMD case).

cd: The damping of tuned mass damper (TMD case).

kd: The stiffness of tuned mass damper (TMD case).

md1,md2,md3,…,md9,md10: The mass of tuned mass dampers (MTMD case).

cd1,cd2,cd3,…,cd9,cd10: The damping of tuned mass dampers (MTMD case).

kd1,kd2,kd3,…,kd9,kd10: The stiffness of tuned mass dampers (MTMD case).

x1,x2,x3,…,x9,x10: Floors displacements.

xd1,xd2,xd3,…,xd9,xd10: Diplacements of tuned mass dampers (MTMD case).

3.2 1st Case: Tuned Mass Damper (TMD)

In this case, a tuned mass damper is connected to the 10th floor. Because of the TMD connection, the number of degrees of freedom in this system is 11. The mass, stiffness, and damping matrix must have rows and columns of 11 × 11.

3.2.1 Definition of Design Variables and Population Numbers

The design variables are related to the parameters of the tuned mass damper, including the mass, period, and damping ratio. The values of these parameters are adopted according to the following:

mdmax=0.1×(∑n=10n=1mn)(16)

mdmin=0.005×(∑n=10n=1mn)(17)

Tdmax=3πωmin(18)

Tdmin=0.25πωmin(19)

μmax=0.5, μmin=0.01.

where:

md: Mass of the tuned mass damper (kg).

mn: Mass of the floors (kg), n = 1, 2, 3, …, 10.

Td: Period of the tuned mass damper (s).

μ: Damping ratio.

ω: Building frequency (rad/s).

Pn = 20: Population number.

3.2.2 Definition of the Objective Function

The objective function is to reduce the displacement of the floors. It can be expressed as follows:

f(X)=max(|x1x2x3…x9x10|)(20)

here:

x1,x2,x3,…,x9,x10: Floors displacements.

3.2.3 Definition of Design Constraints

The adopted design constraints can be defined as follows:

g(x)=max(|xd−x10|)withTMD|x10|withoutTMD≤stmax(21)

where:

xd: Displacement of the tuned mass damper,

x10: Displacement of the tenth floor,

stmax: Maximum limit of the design constraint (stmax = 2).

3.3 2nd Case: Multiple-Tuned Mass Dampers (MTMD)

This case involves attaching a small tuned mass damper to each floor. The tuned mass dampers have a specific stiffness, damping, and mass, resulting in 20 degrees of freedom. The mass, stiffness, and damping matrix should have a 20 × 20 row and column size.

3.3.1 Definition of Design Variables and Population Numbers

Tdmax=3πωmin(22)

Tdmin=0.25πωmin(23)

μmax=0.5, μmin=0.01

Pn=20: Population number.

The lower and upper limit values of the tuned mass dampers were investigated for two cases:

•   Case A: The variables of the masses are as follows:

mdmax=0.1×(∑n=10n=1mn),mdmin=0.005×(∑n=10n=1mn)

•   Case B: Using a constant value of mass:

md=50,000 kg

3.3.2 Definition of the Objective Function

The equation of the objective function for this case is the same equation mentioned in Section 3.2.2.

3.3.3 Definition of Design Constraints

g(x)=max(|xdn−xn|)withTMD|xn|withoutTMD≤stmax(24)

4  Results

In this study, the effectiveness of a tuned mass damper (TMD) and a multiple-tuned mass damper (MTMD) system in dissipating vibration energy and reducing floor displacements was investigated for a 10-story building under the seismic excitation, which is the El-Centro earthquake. Each floor in the building has a specific stiffness, mass, and damping coefficient. The building has been analyzed using MATLAB and Simulink. Three different cases have been examined as follows:

1.   The first case is the use of the TMD system connected to the 10th floor. The mass, stiffness, and damping parameters of the TMD are calculated using the Jaya algorithm.

2.   The second case is the use of the MTMD system connected to all floors. The mass, stiffness, and damping parameters of the MTMD are calculated using the Jaya algorithm. This case is called MTMD-A.

3.   The third case is the use of the MTMD system. However, in this case, the difference from the previous case is that the MTMD masses are set at 50,000 kg. The remaining MTMD parameters are calculated using the Jaya algorithm. This case is called MTMD-B.

4.1 The First Case: The Building with TMD

4.1.1 Stiffness, Mass and Damping Matrices of the System

The ideal properties of the tuned mass damper can be obtained from the above matrices as shown in Table 1.

4.1.2 The Floors Displacements

The results show a significant improvement in floor displacements compared to the floor displacements in buildings without any damping system. The following tables (Tables 2 and 3) show the floor displacement values for all floors every 5 s:

Table 4 shows the improvement in displacement resulting from adding the TMD system.

Fig. 6 shows that the TMD system provides an average displacement reduction of about 53%–54% compared to the uncontrolled building.

Figure 6: Average improvement in displacement for TMD case (%).

4.1.3 Dissipation of Vibrational Energy

Kinetic and potential energy were calculated for all floors of the building, with and without TMD. The resulting energy values are shown in the tables below (Tables 5–11).

The energy dissipation rate of the building with the TMD system reached 75% compared to the same building without the damping system. Fig. 7 presents the average improvement in energy dissipation for the TMD case. The results indicate a significant increase in the total dissipated energy compared to the uncontrolled structure. This confirms that TMD not only reduces displacements but also enhances the overall damping capacity of the system by absorbing and dissipating a portion of the input seismic energy.

Figure 7: Average improvement in energy dissipation for TMD case (%).

4.2 Second Case: The Building with MTMD System (MTMD-A)

4.2.1 Stiffness, Mass and Damping Matrices of the System

The ideal properties of multiple-tuned mass dampers can be obtained from the above matrices as shown in Table 12.

4.2.2 The Floors Displacements

The floor displacements for the building with MTMD-A is shown in Table 13 for different times of simulation.

By comparing the previous table with Table 3, the improvement rate of floor displacements can be determined according to Table 14.

According to the previous table, the improvement rate in floor displacements has reached 76%. Fig. 8 presents the average displacement improvement for the MTMD-A case and allows direct comparison with the single TMD results (Fig. 6). Compared to the TMD, MTMD-A provides higher displacement reduction.

Figure 8: Average improvement in displacement for MTMD-A case (%).

4.2.3 Dissipation of Vibrational Energy

Kinetic and potential energy (Tables 15–17) with MTMD have been calculated for all floors of the building.

By comparing the previous table with Table 8, the improvement rate of dispersion vibration energy can be determined according to Table 18.

The energy dissipation rate of the building with the MTMD-A system reached 93% compared to the same building without the MTMD system. The effect of reduction is clearly seen for all stories as seen in Fig. 9.

Figure 9: Average improvement in energy dissipation for MTMD-A case (%).

4.3 Third Case: The Building with MTMD System (MTMD-B)

4.3.1 Stiffness, Mass and Damping Matrices of the System

The ideal properties of multiple-tuned mass dampers can be obtained from the above matrices as shown in Table 19.

4.3.2 The Floors Displacements

The floor displacements for the building with MTMD-B are shown in Table 20 for different times of simulation.

By comparing the previous table with Table 3, the improvement rate of floor displacements can be determined according to Table 21.

The improvement rate in floor displacements has reached 35% and Fig. 10 shows the average improvement in displacement for the MTMD-B case (%).

Figure 10: Average improvement in displacement for MTMD-B case (%).

4.3.3 Dissipation of Vibrational Energy

Kinetic and potential energy (Tables 22–24) with MTMD have been calculated for all floors of the building.

By comparing the previous table with Table 7, the improvement rate of dispersion vibration energy can be determined according to Table 25. The average improvement in energy dissipation for the MTMD-B case is plotted in Fig. 11.

Figure 11: Average improvement in energy dissipation for MTMD-B case (%).

The minus sign in the previous table represents the increase in vibration energy in the MTMD-B case compared to the building without the MTMD system. According to the previous table, the energy dissipation ratio of the building with the MTMD-B system reached 33% compared to the same building without the MTMD system.

4.4 Comparison between TMD System and MTMD Case-A System

4.4.1 The Floors Displacements

The difference in floor displacements between the two cases can be calculated from Tables 3 and 13 considering that the minus sign indicates that the improvement rate of the floor displacements in the TMD case is greater than that in the MTMD case.

According to Table 26, the MTMD-A system provides 60% more improvement in floor displacements than the TMD system.

4.4.2 Comparison of Maximum Displacement Values

This comparison provides important information about the percentage improvement in maximum floor displacements between the two cases. Table 27 considers the maximum values at each floor for both cases, regardless of the time at which the displacements occurred.

According to Table 27, the MTMD-A system provides 11% more improvement in maximum floor displacements than the TMD system.

4.4.3 Dissipation of Vibrational Energy

The difference in floor vibration energy distribution between the two cases can be calculated from Tables 10 and 17. The minus sign indicates that the improvement rate of the vibration energy distribution at a floor in the TMD case is greater than that in the MTMD-A case. According to Table 28, the MTMD-A system provides 95% more improvement in floor vibration energy distribution than the TMD system.

4.5 Comparison between TMD System and MTMD Case-B System

4.5.1 The Floors Displacements

Similarly, the difference in floor displacements between the two cases can be calculated from Tables 3 and 20. The minus sign indicates that the improvement rate of the floor displacements in the TMD case is greater than that in the MTMD case. According to Table 29, the TMD system provides 17% more improvement in floor displacements than the MTMD-B system.

4.5.2 Comparison of Maximum Displacement Values

In Table 30, the maximum values at each floor are considered for both cases, regardless of the time at which the displacements occur.

According to Table 30, the TMD system provides 10% more improvement in maximum floor displacements than the MTMD-B system.

4.5.3 Dissipation of Vibrational Energy

From Tables 10 and 24, the difference in floor vibration energy distribution between the two cases can be calculated. The minus sign indicates that the improvement rate of the vibration energy distribution at a floor in the TMD case is greater than that in the MTMD-B case.

According to Table 31, the TMD system provides 21% more improvement in floor vibration energy distribution than the MTMD-B system.

4.6 The Maximum Displacement & Story Drift Values

In this section, a comparison is presented between the three cases studied in terms of the maximum values of displacement and story drift. Fig. 12 shows the maximum values of story displacements for all cases.

Figure 12: Maximum displacment values (m).

Fig. 13 shows the story drift for all cases (assuming a story height of 3.5 m):

Figure 13: Story drift (Unitless).

5  Discussion

The results of all cases have been compared with the building without a damping system. The results have then been compared with the TMD and MTMD systems separately for each case. The results demonstrate the following key points:

1.   The building with the TMD system demonstrated significant effectiveness compared to the building without the damping system. It reduced floor displacements by 54% and reduced the building’s vibration energy by 75%.

2.   The building with the MTMD-A system demonstrated a very high effectiveness in reducing floor displacements, reaching a percentage equal to 76%, compared to the same building without the damping system, and the percentage of vibration energy dissipation reached 93%.

3.   The building with the MTMD-B system was found to be less effective in reducing floor displacements and dissipating building vibration energy compared to the previous two cases. Compared to a building without a damping system, the displacement reduction reached 35%, and the vibration energy dissipation rate reached 33%.

4.   Comparing the TMD with MTMD-A cases, the MTMD-A system was found to be more effective than the TMD system in reducing displacements and energy dissipation by 60% and 95%, respectively. Conversely, the TMD system was found to be more effective than the MTMD-B system in reducing displacements and energy dissipation by 17% and 21%, respectively.

5.   Although the MTMD-A system yielded better results than other systems, it has the disadvantage that the damper masses obtained using the Jaya algorithm are much larger than those used in other systems. The total mass used across all floors in the MTMD-A system is 3,229,058 kg. The total mass across all floors in the MTMD-B system is 500,000 kg. This value is the same as the value used in the TMD system; that is, there is a single mass weighing 500,000 kg on the tenth floor. Therefore, the mass weight of the dampers in the MTMD-A system is approximately 85% greater than in the TMD and MTMD-B systems. This result is considered a shortcoming of the MTMD-A system. This makes the MTMD-B system more practical and applicable than the MTMD-A system in terms of cost, installation, maintenance, and the cross-sectional area of the structural elements supporting the mass dampers.

6.   This study emphasizes the importance of the MTMD system’s overall effect on the behavior of buildings under seismic excitation compared to the TMD system. However, it should be noted that the results obtained in this study are limited to the cases examined.

6  Conclusion

This study focuses on the building’s behavior when a tuned mass damper is added to the tenth floor. The second case focuses on the behavior when a tuned mass damper is added to each floor. Acceleration and displacement graphs for the floors are plotted using MATLAB. The accelerations and displacements of the tuned mass damper are also shown in this section. The cases are then compared. This information will help understand the effect of the tuned mass damper in damping vibration energy.

The significance of this study is focused on understanding the effectiveness of using tuned mass dampers and multiple-tuned mass dampers to improve the structural behavior of buildings. This provides designers and engineers with the flexibility to choose the best option based on the project and its specifications. Furthermore, the optimization results presented in this study provide the characteristics of the tuned mass damper that should be adopted for similar situations. Furthermore, the study focuses on optimizing the parameters of the tuned mass damper system using the Jaya algorithm within the scope of MATLAB simulation.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding.

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Sinan Melih Nigdeli, Gebrail Bekdaş; data collection: Omer Elselumi; analysis and interpretation of results: Omer Elselumi, Sinan Melih Nigdeli, Gebrail Bekdaş; draft manuscript preparation: Omer Elselumi. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

References

1. Choi SW, Seo JH, Lee HM, Kim Y, Park HS. Wind-induced response control model for high-rise buildings based on resizing method. J Civ Eng Manag. 2015;21(2):239–47. doi:10.3846/13923730.2013.802742. [Google Scholar] [CrossRef]

2. Farshidianfar A, Soheili S. Ant colony optimization of tuned mass dampers for earthquake oscillations of high-rise structures including soil–structure interaction. Soil Dyn Earthq Eng. 2013;51(3):14–22. doi:10.1016/j.soildyn.2013.04.002. [Google Scholar] [CrossRef]

3. Spencer BF Jr, Nagarajaiah S. State of the art of structural control. J Struct Eng. 2003;129(7):845–56. doi:10.1061/(asce)0733-9445(2003)129:7(845). [Google Scholar] [CrossRef]

4. Frahm H, inventor. Device for damping of bodies. United States patent US 989,958. 1911 Apr 18. [Google Scholar]

5. Ormondroyd J, Den Hartog JP. The theory of the dynamic vibration absorber. Trans Am Soc Mech Eng. 1928;49-50(2):021007. doi:10.1115/1.4058553. [Google Scholar] [CrossRef]

6. Sadek F, Mohraz B, Taylor AW, Chung RM. A method of estimating the parameters of tuned mass dampers for seismic applications. Earthq Eng Struct Dyn. 1997;26(6):617–35. doi:10.1002/(SICI)1096-9845(199706)26:. [Google Scholar] [CrossRef]

7. Yamaguchi H, Harnpornchai N. Fundamental characteristics of Multiple Tuned Mass Dampers for suppressing harmonically forced oscillations. Earthq Eng Struct Dyn. 1993;22(1):51–62. doi:10.1002/eqe.4290220105. [Google Scholar] [CrossRef]

8. Igusa T, Xu K. Vibration control using multiple tuned mass dampers. J Sound Vib. 1994;175(4):491–503. doi:10.1006/jsvi.1994.1341. [Google Scholar] [CrossRef]

9. Li C, Liu Y. Optimum multiple tuned mass dampers for structures under the ground acceleration based on the uniform distribution of system parameters. Earthq Eng Struct Dyn. 2003;32(5):671–90. doi:10.1002/eqe.239. [Google Scholar] [CrossRef]

10. Wang JF, Lin CC. Seismic performance of multiple tuned mass dampers for soil–irregular building interaction systems. Int J Solids Struct. 2005;42(20):5536–54. doi:10.1016/j.ijsolstr.2005.02.042. [Google Scholar] [CrossRef]

11. Moon KS. Vertically distributed multiple tuned mass dampers in tall buildings: performance analysis and preliminary design. Struct Des Tall Spec Build. 2010;19(3):347–66. doi:10.1002/tal.499. [Google Scholar] [CrossRef]

12. Chen G, Wu J. Optimal placement of multiple tune mass dampers for seismic structures. J Struct Eng. 2001;127(9):1054–62. doi:10.1061/(asce)0733-9445(2001)127:9(1054). [Google Scholar] [CrossRef]

13. da Silva Brandão F, Miguel LFF. Vibration control in buildings under seismic excitation using optimized tuned mass dampers. Frat Ed Integrità Strutturale. 2020;14(54):66–87. doi:10.3221/igf-esis.54.05. [Google Scholar] [CrossRef]

14. Djerouni S, Bekdaş G, Nigdeli SM. Optimization and performance assessment of Multi-Tuned Mass Dampers (MTMD) to mitigate seismic pounding of adjacent buildings via a novel hybrid algorithm. J Build Eng. 2025;103(3):112168. doi:10.1016/j.jobe.2025.112168. [Google Scholar] [CrossRef]

15. Wang L, Shi W, Zhang Q, Zhou Y. Study on adaptive-passive multiple tuned mass damper with variable mass for a large-span floor structure. Eng Struct. 2020;209(1):110010. doi:10.1016/j.engstruct.2019.110010. [Google Scholar] [CrossRef]

16. Singh MP, Singh S, Moreschi LM. Tuned mass dampers for response control of torsional buildings. Earthq Eng Struct Dyn. 2002;31(4):749–69. doi:10.1002/eqe.119. [Google Scholar] [CrossRef]

17. Frans R, Arfiadi Y. Designing optimum locations and properties of MTMD systems. Procedia Eng. 2015;125(1):892–8. doi:10.1016/j.proeng.2015.11.079. [Google Scholar] [CrossRef]

18. Bekdaş G, Nigdeli SM, Yang XS. A novel bat algorithm based optimum tuning of mass dampers for improving the seismic safety of structures. Eng Struct. 2018;159(3):89–98. doi:10.1016/j.engstruct.2017.12.037. [Google Scholar] [CrossRef]

19. Cetin H, Aydin E. A new tuned mass damper design method based on transfer functions. KSCE J Civ Eng. 2019;23(10):4463–80. doi:10.1007/s12205-019-0305-x. [Google Scholar] [CrossRef]

20. Ozturk B, Cetin H, Aydin E. Optimum vertical location and design of multiple tuned mass dampers under seismic excitations. Structures. 2022;41(1):1141–63. doi:10.1016/j.istruc.2022.05.014. [Google Scholar] [CrossRef]

21. Kaveh A, Javadi SM, Mahdipour Moghanni R. Optimal structural control of tall buildings using tuned mass dampers via chaotic optimization algorithm. Structures. 2020;28(3):2704–13. doi:10.1016/j.istruc.2020.11.002. [Google Scholar] [CrossRef]

22. Wang L, Zhou Y, Shi W. Seismic multi-objective stochastic parameters optimization of multiple tuned mass damper system for a large podium twin towers structure. Soil Dyn Earthq Eng. 2024;177(5):108428. doi:10.1016/j.soildyn.2023.108428. [Google Scholar] [CrossRef]

23. Ghojehbiglou AM, Mousavi Ghasemi SA, Ferdousi A. Efficiency of a hybrid nature-inspired algorithm for tuning multiple tuned mass dampers in buildings with soil–structure interaction. J Struct Des Constr. 2026;31(2):04026016. doi:10.1061/jsdccc.sceng-1668. [Google Scholar] [CrossRef]

24. Shahrouzi M, Fahimi-Farzam M, Gholizadeh J. Optimization of multiple tuned mass damper inerter by escaping bird search for seismic control of buildings. Int J Dyn Control. 2025;13(10):362. doi:10.1007/s40435-025-01875-4. [Google Scholar] [CrossRef]

25. Djerouni S, Elias S, Abdeddaim M, Rupakhety R. Multi-tuned mass damper inerter (MTMDI) system for earthquake-induced vibration control of buildings. Eng Struct. 2025;322(10):119139. doi:10.1016/j.engstruct.2024.119139. [Google Scholar] [CrossRef]

26. Garrido H, Domizio M, Curadelli O, Ambrosini D. Multi-objective optimization of inerter-based building mass dampers. J Vib Control. 2025;31(19–20):4273–87. doi:10.1177/10775463241280997. [Google Scholar] [CrossRef]

27. Wang M, Chen JL, Nagarajaiah S, Du XL. Root-locus optimal formula for seismic control of multi-story structures equipped with tuned mass damper inerter enhanced by negative stiffness considering non-resonant mode contributions. J Build Eng. 2025;103(7):112203. doi:10.1016/j.jobe.2025.112203. [Google Scholar] [CrossRef]

28. Kiran KK, Al-Osta MA, Ahmad S, Bahraq AA. Optimization of tuned mass damper inerter systems for seismic control of base-isolated and fixed-base structures. Arab J Sci Eng. 2025;50(20):17153–87. doi:10.1007/s13369-025-10352-1. [Google Scholar] [CrossRef]

29. Rao RV. Jaya: a simple and new optimization algorithm for solving constrained and unconstrained optimization problems. Int J Ind Eng Comput. 2016;7:19–34. doi:10.5267/j.ijiec.2015.8.004. [Google Scholar] [CrossRef]