Response mitigations of adjacent structure with MPTMD under real and stochastic excitations

Abstract

As urbanization increases, the demand for constructing nearby structures has risen, leading to a higher likelihood of adjacent buildings having different dynamic characteristics. This, in turn, increases the probability of structural impacts, which often result in significant or partial damage. This paper proposes multiple elastoplastic-tuned mass damper systems (MPTMD) to mitigate the damage caused to nearby structures due to seismic events. The system’s performance is evaluated regarding impact force and the reduction in the Park-Ang damage index. For this purpose, two steel-framed buildings—one 6-story and one 10-story—are modeled nonlinearly using OpenSees software, incorporating connection springs (concentrated plasticity) to represent the structures. After modeling the structures, the MPTMD system is applied at different stages to control the dynamic responses, mitigate the impact between the structures, and reduce the Park-Ang damage index. The performance of the MPTMD system is optimized using the Particle Swarm Optimization (PSO) algorithm. The optimization process calculates the optimal placement and parameters of the MPTMD system under two objective functions: the first function aims to minimize the Park-Ang damage index of all structures, while the second focuses on reducing the maximum impact force between the adjacent structures. The results indicate that by optimally configuring the MPTMD parameters with the first objective function, the system not only significantly reduces the Park-Ang damage index of the structures and their stories but also substantially minimizes the maximum impact force, maximum energy, maximum kinetic energy, and maximum drift of the stories, thereby improving overall structural performance.

Introduction

Two main approaches exist for modeling impact forces between structures: (1) classical impact theory based on energy and momentum conservation, which lacks transient stress waves, local deformations, and inelastic dissipation effects, and (2) numerical simulations using nonlinear contact elements, providing a more realistic representation of seismic pounding. Studies indicate that impact force magnitude depends on mass ratio, relative velocity, material stiffness, damping properties, and interface geometry. Prior collisions can alter subsequent impact responses due to stiffness changes and accumulated damage, emphasizing the importance of considering these parameters in impact force modeling [1]. Elastoplastic-tuned mass dampers (PTMDs) have been explored to mitigate seismic impact forces. Unlike traditional dampers, PTMDs reduce vibrations by using nonlinear stiffness and energy dissipation. Recent studies [2, 3] optimized PTMD systems to improve seismic performance, reducing inter-story drifts and structural damage indices. Further studies examined the effects of soil-structure interaction (SSI) on PTMD efficiency, revealing that soil flexibility significantly alters seismic responses and damage distribution. Results indicate that neglecting SSI may lead to underestimation of structural damage, highlighting the need for adaptive damping strategies. Additionally, recent advancements in tuned mass damper-inerter systems and hybrid damping devices have shown promising results in enhancing seismic resilience by integrating negative stiffness elements and clutching mechanisms to improve energy dissipation efficiency [4, 5, 6, 7–8].

Recent studies have emphasized the significance of soil-structure interaction (SSI) in seismic control systems, exploring how parameters defining SSI affect the performance of structures equipped with tuned mass damper [9]. Additionally, the arrangement of double-tuned mass dampers has been evaluated for its impact on structural seismic response, considering SSI [10]. Moreover, the PGV/PGA ratio has been identified as a key factor influencing seismic vibrations in structures with parallel-tuned mass dampers, highlighting the critical role of SSI [11].

A modified Kelvin shock model has been proposed for better representation of pounding interactions, effectively capturing hysteretic impact behavior [12]. A comparative analysis [13] compared numerical impact force models, demonstrating that the linear viscoelastic model provides the most accurate time-history analysis. Seismic pounding affects not only buildings but also bridges and composite steel frames. For example, A study [14] evaluated concrete-filled double-skin tubular (CFDST) frames with shape memory alloy dampers, showing effective damage reduction. An investigation [15] assessed fractured beams in steel frames, emphasizing advanced damage detection for seismic resilience. Research [16] studied seismic responses in long-span bridges, highlighting how longitudinal pounding amplifies structural damage.

Previous studies examined the impact force between adjacent structures, focusing on the structures' impact response and methods to mitigate seismic risks. The impact force was analyzed using linear and nonlinear contact force models, with varying consideration of the distance between structures [17]. Controlling seismic response by reducing impact forces and collisions between adjacent structures has gained acceptance in engineering. Various control systems, including friction, pendulum, magnetic, and viscous dampers, have been utilized for this purpose [18, 19, 20, 21–22]. Preventive measures such as ensuring adequate distance between structures and reducing relative displacement during earthquakes are essential. Studies indicate that employing base isolation systems can prolong natural periods and increase the potential for impact; however, increasing the thickness, hardness, and number of separators effectively reduces impact forces [23].

Recent studies have emphasized the significance of advanced TMD systems, such as friction-damping [24], three-element TMDs [25], TMD inerters [26], non-traditional TMDs [27]. Additionally, the use of these systems for vibration control in high-speed railway bridges has been explored, demonstrating significant improvements in energy dissipation and structural resilience under seismic excitation [28]. Furthermore, the optimization of passive tuned mass damper systems for main structures under harmonic excitation has shown promising results in reducing vibrations and improving seismic performance [29].

Recent research examined the effect of the impact force between structures, modeling it under different scenarios with soil-structure interactions, revealing that these interactions amplify both impact forces and roof displacements [30]. Furthermore, analysis of varying soil types on structural response during earthquakes indicated that soft clay significantly increases impact forces compared to other soil types [31]. Damage indices for adjacent structures, evaluated through incremental dynamic analysis under multiple earthquake records, demonstrated that the distance between structures directly affects the Park-Ang damage index [32]. Additionally, the presence of filler panels altered seismic behavior during impacts [33].

The application of TMD systems has emerged as an effective strategy for controlling seismic responses and reducing impact forces. Optimal TMD performance relies on design and parameter optimization for improved seismic resilience [18, 34]. Research has also explored the use of TMDs for collision control between adjacent structures, revealing that sharing a TMD system does not always lead to effective performance [35]. Key parameters such as mass, damping, and stiffness are crucial for TMD and MTMD systems [36, 37–38]. When optimized, these parameters can significantly enhance performance, prompting the development of various optimization algorithms for engineering solutions [36]. Optimal parameters for single-degree-of-freedom (SDOF) structures and multi-degree-of-freedom (MDOF) structures have been addressed together [39], along with MTMD parameters in SDOF structures under random noise [40].

This study optimizes a multiple elastoplastic tuned mass damper (MPTMD) system using particle swarm optimization (PSO) to enhance structural performance, specifically focusing on minimizing both the Park-Ang damage index and maximum impact force. The use of PSO allows for a robust and efficient optimization of the MPTMD parameters, improving the overall structural resilience and ensuring that the dampers are strategically placed and tuned to achieve maximum energy dissipation. The results show that optimizing the MPTMD system based on the Park-Ang damage index improves performance in reducing structural damage, inter-story drift, base shear, and roof displacement. In contrast, focusing solely on minimizing the maximum impact force results in suboptimal performance, as it does not address the underlying structural damage as effectively. One of the key advantages of the MPTMD system is its nonlinear stiffness, which replaces the traditional linear stiffness used in classic TMD systems, significantly improving energy dissipation, especially in scenarios involving large deformations. The optimization ensures that the system effectively reduces kinetic and hysteresis energy, thereby minimizing the overall energy input into the structure and reducing the likelihood of severe structural damage. The novelty of this study lies in the integrated use of a nonlinear MPTMD system optimized with PSO, which reduces seismic pounding effects and minimizes energy accumulation through efficient damping. Unlike previous research focused on linear or single TMD systems, this approach provides a more comprehensive solution by addressing both the impact forces and energy dissipation, improving the seismic resilience of adjacent structures.

Dynamic equations of motion

This section presents the dynamic equations of motion for a building frame equipped with a Multiple Elastoplastic Tuned Mass Damper (MPTMD) system. n openings and m stories characterize the building frame. The MPTMD system is strategically connected to various stories of the structure to minimize the impact force and the Park-Ang damage index for the entire structure and individual stories. The MPTMD system enhances seismic performance by distributing damping devices across multiple floors, optimizing energy dissipation, and mitigating pounding forces between adjacent buildings. Unlike conventional Tuned Mass Dampers (TMDs), which primarily target the fundamental vibration mode, MPTMDs are positioned at multiple elevations to control higher-mode vibrations and inter-story drifts effectively. These dampers absorb seismic energy through their elastoplastic behavior, dissipating it via controlled yielding, which prevents excessive structural responses and damage accumulation. Each MPTMD unit is coupled to a specific floor, dynamically responding to its motion and counteracting translational and rotational displacements. This decentralized damping approach ensures uniform energy dissipation across the structure's height, reducing the risk of localized damage concentration. The MPTMD's elastoplastic stiffness also allows the system to operate efficiently under large deformations, making it particularly effective in nonlinear seismic scenarios. A critical function of the MPTMD system is to mitigate pounding forces between adjacent buildings. During an earthquake, adjacent structures with different dynamic characteristics may experience out-of-phase motion, leading to significant impact forces at connection points and potential structural damage. By optimizing the MPTMD parameters using the Particle Swarm Optimization (PSO) algorithm, the system is fine-tuned to simultaneously reduce the Park-Ang damage index and minimize impact forces, ensuring enhanced structural resilience. The algorithm determines the optimal mass, stiffness, and damping values for the MPTMD system, ensuring that the dampers function synergistically across various floors. Therefore, the equations of motion of the MDOF multi-degree of freedom system under earthquake acceleration ${\ddot{x}}_g(t)$ are stated below.

1

$$\mathbf{M}\ddot{x}(t)+\mathbf{C}\dot{x}(t)+\mathbf{K}x(t)=-\mathbf{M}\iota {\ddot{x}}_g(t).$$

So that in, Eq. (1) $\iota$ is a unit vector with length $(n+1)$ and $\mathbf{M}$, $\mathbf{C}$, $\mathbf{K}$, and $x$ respectively represent the matrix of mass, damping, stiffness, and displacement vector of the response.

In evaluating the structure's performance, the analysis of the damage caused to the structure is one of the main steps that lead to the calculation of the damage caused to the structure. Various types of damage indicators have been introduced by researchers in the last few decades [41, 42–43]. In recent research, the modified Park-Ang damage index is mainly used to evaluate the structural damage index [44, 45–46]. Therefore, the modified Park-Ang damage index is used in this article, which is based on wasted energy and formability requirements and is a linear sum of the maximum deformation and wasted energy in the structure [43].

Park-Ang damage index

This section presents the relationships needed to calculate the damage index of members, stories, and structures based on the modified Park-Ang damage index standard. The Park-Ang index is expressed as follows:

2

$$D{I}_{\mathit{PA}}=\frac{{\delta }_m}{{\delta }_u}+\frac{\beta }{Q_y{\delta }_u}\int \mathit{dE}$$

So that, in Eq. (2), ${\delta }_m$ is the deformation of the member under earthquake acceleration, ${\delta }_u$ is the final deformation of the member, and $\beta$ is the coefficient calculated using the experimental method, and the value of 0.025 for steel structures is proposed by researchers [42]. $Q_y$ is the yield stress and $E$ is the amount of energy lost in the member. Later, the Park-Ang index was modified by Kunnath et al. based on the rotation of the member as follows, which is the basis of the evaluation of damage in this study [47]:

3

$$D{I}_{\mathit{PA}}=\frac{{\theta }_m-{\theta }_y}{{\theta }_u-{\theta }_y}+\frac{\beta }{M_y{\theta }_u}\int \mathit{dE}$$

In Eq. (3), ${\theta }_m$ is the maximum amount of rotation of the end of the member, ${\theta }_u$ is the capacity of the final rotation of the member, ${\theta }_y$ is the yielding rotation of the member, and $M_y$ is the yielding moment of the member's section. Also, the damage index in the whole structure is calculated as follows:

4

$$\begin{array}{cc}& D{I}_{\mathit{Storey}}=\sum {(\lambda }_i{)}_{\mathit{component}}{(D{I}_i)}_{\mathit{component}};{({\lambda }_i)}_{\mathit{component}}=\left(\frac{E_{h_i}}{\sum E_{h_{_i^{}}^{}}^{}}\right)_{\mathit{component}}^{}\hfill \\ \hfill & D{I}_{\mathit{Overall}}=\sum {(\lambda }_j{)}_{\mathit{Storey}}{(D{I}_j)}_{\mathit{Storey}};{({\lambda }_j)}_{\mathit{Storey}}={\left(\frac{E_{h_j}}{\sum E_{h_j}}\right)}_{\mathit{Storey}}\hfill \end{array}$$

So that in Eq. (4), ${\lambda }_i$ and $E_{h_{_i^{}}^{}}^{}$ respectively represent the weight coefficients of energy and the sum of accumulated energy absorbed by the "j" class or the "i" member [48]. The damage levels associated with the Park-Ang index are categorized as follows:

• DI_PA < 0.1: No significant damage, minor localized cracking in non-structural elements. The structure remains fully operational.

• 0.1 ≤ DI_PA < 0.25: Minor damage, with widespread small cracks in structural and non-structural components. Repair is typically minimal, and the structure retains its integrity.

• 0.25 ≤ DI_PA < 0.4: Moderate damage, including severe cracking and localized concrete spalling. Steel structures may exhibit noticeable yielding at beam-column connections. The building remains stable but requires repairs.

• 0.4 ≤ DI_PA < 1: Severe structural damage includes extensive material failure, large residual deformations, and significant strength degradation. In concrete structures, crushing, reinforcement exposure, and shear failure can lead to instability, while in steel structures, yielding, buckling, and connection fractures compromise load-bearing capacity. Progressive collapse is likely at this stage, and repairs become impractical without major intervention.

Optimum design of MPTMD system

Since the beginning of history, the main problem of earthquakes has been the destruction and damage of structures and the result of human deaths due to the collapse of structures. Displacement and high acceleration of the structure are factors in the collapse of structures. Nowadays, science tries to reduce the acceleration and displacement of structures by using vibration control systems, which also reduce the collapse of structures. Nowadays, various methods have been developed to control the seismic response of the structure. PTMD is one of these control systems in the passive structure control systems category. This control system consists of nonlinear or elastoplastic mass and stiffness. When the frequency of the damper is close to the main frequency of the structure, the movement of this system is opposite to the direction of the structure's movement, and a significant part of the energy entering the structure is lost by moving the damper in the phase opposite to the structure. Also, the elastoplastic stiffness of the damper system causes the control system to enter the nonlinear region, which causes more energy input to the structure to be lost. As a result, the structure's dynamic response to vibration decreases. In this section, the optimization of PTMD parameters, including mass $m_T$ and nonlinear stiffness $k_T$ to minimize the Park-Ang damage index and the impact force between stories under the Imperial Valley and artificial earthquake is studied. Equation (5) show the optimization process [49]:

5

$$\begin{array}{cc}& \text{Find}:\left\{\begin{array}{cc}& \overline{\mathit{X}}=\{M_{d,j},K_{d,j}\};j=1,2,\dots ,N_d\hfill \\ \hfill & 1\le {\mathit{N}}_{\mathit{PTMD}}\le N_d\hfill \\ \hfill \end{array}\right)\hfill \\ \hfill & \text{Minimize}:f_1(\overline{\mathit{X}},N_d)=\frac{D{I}_{P{A}_{with\text{MPTMD}}}}{D{I}_{P{A}_{without\text{MPTMD}}}},f_2(\overline{\mathit{X}},N_d)=\frac{\text{max}(Poundingforc{e}_{with\mathit{MPTMD}})}{\text{max}(Poundingforc{e}_{without\mathit{MTMD}})}\hfill \\ \hfill & \text{subject to}:\left\{\begin{array}{c}{g}_j(\overline{\mathit{X}},N_d)=max(\left|X_{d,j}(t)-X_{\text{connected node}}(t)\right|)\le s{t}_{max},s{t}_{max}=1000(\text{mm})\hfill \\ DI_{_{\mathit{PA}}^{}}^{Stor{y}^i}\le 0.25,i=1,2,...,N\hfill \\ 0.01\le f_i=\frac{\sqrt{K_{d_i}\left./\phantom{K_{d_i}{M}_{d_i}}\right)M_{d_i}}}{2\pi f_{s_1}}\le 10.0\hfill \end{array}\right\}\hfill \\ \hfill \end{array}$$where${\text{K}}_{\text{d,j}}$,${\text{M}}_{\text{d,j}}$ are stiffness, mass, of j^th PTMD, respectively.${\text{X}}_{\text{connected Node}}\left(\text{t}\right)$ and $X_{d,j}(t)$ are the displacement of the structure at the node that the PTMD is attached to it and the displacement of the j^th PTMD at time t, respectively. Furthermore, N_PTMD and N_d show the indicator and the number of PTMDs. This paper considers the maximum MPTMD (N_d) number to be 28. $f_{s_1}$ and $f_i$ are the first natural frequency of the building and optimal tuning ratio, respectively. The last constraint is considered here based on Ref. [49]. Also $DI_{_{\mathit{PA}}^{}}^{Stor{y}^i}$, $N$, and $s{t}_{max}$ are the Park-Ang damage index of i^th story, the number of stories, and the stroke ratio [42], respectively. Also, the mass of all tuned mass dampers is constant, and their summation is equal to a percentage of the total mass of the structure. In this paper, the total mass of all PTMDs following Ref. [50] is considered to be 0.1 of the mass of the structure. In the above equation, $D{I}_{P{A}_{with\text{MTMD}}}$ and $D{I}_{P{A}_{without\text{MTMD}}}$ are the Park-Ang damage index of the controlled and uncontrolled structures, respectively. Also, $Poundingforce_{_{}^{_{withMPTMD}^{}}}^{}$ and $Poundingforce_{_{}^{_{withoutMTMD}^{}}}^{}$ are the pounding force values in the controlled and uncontrolled structures, respectively.

In the optimization process, the objective function is examined first in the state where the damage index of Park-Ang is in the minimum state to compare the two existing objective functions. In this case, the impact between the two structures is also examined under the minimization of the damage index. Placed. Also, the structural damage index is investigated, and the objective function is investigated to minimize the impact of this topic. This compares the objective function where each case's two mentioned parameters (damage index and impact force) are checked. After optimization, an objective function that is more effective in minimizing the damage index and impact force can be obtained. Choose to be. This issue is genuine when, for example, when the damage index is minimized, the impact is also reduced; this objective function can be called a more practical function.

Particle swarm optimization algorithm

Researchers consider the particle swarm optimization (PSO) algorithm an effective optimization method to find the optimal parameters in civil engineering applications [51, 52–53]. In this study, the optimal number and properties of PTMDs are simultaneously performed using binary and real composite particle swarm optimization (PSO). Binary PSO (BPSO) optimizes the number of PTMDs as discrete variables, while real PSO (RPSO) optimizes the PTMD features as real variables. Therefore, the optimal design of an MPTMD system in the framework of a single program is considered among the variables of the combined design.

Real-coded PSO

The PSO algorithm has been proposed as a meta-heuristic algorithm based on a combined population in recent literature [54]. The socio-cognitive behavior of birds inspires this algorithm in a flock. The birds are known as particles in this category, showing suitable solutions in the search space in the category crowd. Particles randomly fly around and repeatedly move toward the optimal solution (food) to reach it. The particles in the cluster watch out for each other to follow the closest particle to the optimal solution (gbest) while also exploring and following their best solution found so far (pbest). The PSO optimization algorithm can optimize with real variables and discrete values and work with them in optimization. The i^th particle in the l^th iteration is associated with a position vector ${\mathit{X}}_i^l$ and a velocity vector ${\mathit{V}}_i^l$, which are shown as follows:

6

$$\begin{array}{c}\hfill {\mathit{X}}_i^l=\{x_{i,1}^l,x_{i,2}^l,...,x_{i,p}^l\}\\ \hfill {\mathit{V}}_i^l=\{v_{i,1}^l,v_{i,2}^l,...,v_{i,p}^l\}\\ \hfill \end{array}$$

In Eq. (6), p is the dimension of the solution space. As the particle flies in the solution space, the position of this article in the solution space is expressed as follows:

7

$${\mathit{V}}_i^{l+1}={\omega }^l{\mathit{V}}_i^l+c_1{r}_1({\mathit{pbest}}_i^l-{\mathit{X}}_i^l)+c_2{r}_2(\mathit{gbest}_{}^l-{\mathit{X}}_i^l)$$

8

$${\mathit{X}}_i^{l+1}={\mathit{X}}_i^l+{\mathit{V}}_i^{l+1}$$

In Eq. (7), $r_1$ and $r_2$ are two random numbers between 0 and 1. c₁ and c₂ are the cognitive and social scale parameters, respectively, and ${\omega }^l$ is the inertia weight that controls the effect of the previous speed and is defined as follows in the l^th iteration [55]:

9

$$\omega ={\omega }_{max}-\frac{{\omega }_{max}-{\omega }_{min}}{l_{max}}l$$

In Eq. (9), ${\omega }_{max}$ and ${\omega }_{min}$ are the maximum and minimum values of ω, respectively, and l_max is the maximum number of repetitions.

Hay et al. proposed a PSO based on passive pooling (PSOPC). Ref. [56] is used in the present study. Passive accumulation is a vital biological force that maintains the integrity of a batch by absorbing its particles. PSOPC shows improved accuracy and a faster convergence rate compared to PSO. In PSOPC, the speed is modified as follows.

10

$${\mathit{V}}_i^{l+1}={\omega }^l{\mathit{V}}_i^l+c_1{r}_1({\mathit{pbest}}_i^l-{\mathit{X}}_i^l)+c_2{r}_2(\mathit{gbest}_{}^l-{\mathit{X}}_i^l)+c_3{r}_3({\mathit{R}}_i^l-{\mathit{X}}_i^l)l$$

In Eq. (10), R_i is a randomly selected particle from the group. c₃ is the passive aggregation coefficient, and r₃ is a random number between 0 and 1.

Binary-coded PSO

In binary-coded PSO (BPSO), each population particle is represented by a binary solution containing 0 and 1 as discrete variables. However, the velocity must be transformed into a probability change for each binary dimension to take the value 1. In BPSO, the particle velocity can be updated according to Eq. (11), and then the velocity is converted to 0 and 1 using sigmoid transformation. In their research, Mir Jalili and Lewis [57] showed that the family of V-shaped transfer functions could significantly improve the performance of the original binary PSO. Therefore, the sigmoid transformation used in this study is described as follows [57].

11

$$S\left(v_i^k(l)\right)=\left|\frac{v_i^k(l)}{\sqrt{2+{\left(v_i^k(l)\right)}^2}}\right|$$

As in Eq. (11), $v_i^k(l)$ represents the speed of the i^th particle in the l^th repetition in the k^th dimension. After converting the velocities into probability values, the position vector is updated with the probability of their velocities as follows [58].

12

$$x_i^k(l+1)=\left\{\begin{array}{c}\hfill 0\text{if}\text{rand}<S\left(v_i^k(l)\right)\\ \hfill 1\text{if}\text{rand}\ge S\left(v_i^k(l)\right)\\ \hfill \end{array}\right)$$

Hybrid of binary and real-coded PSO

The particle swarm optimization (PSO) algorithm has been widely used as an efficient algorithm with high accuracy and efficiency in optimization problems [58]. Therefore, the optimal number and properties of PTMDs are simultaneously performed using a binary combination of PSO and real codes called BRPSO in this study. The number of PTMDs is optimized as discrete variables by BPSO, while RPSO optimizes the features of PTMDs as fundamental variables. Therefore, the optimal design of an MPTMD system in the framework of a single program is considered among the variables of the combined design. In the BRPSO method, a binary number is assigned to a string of particles consisting of Nd,max bits that show the structure of Nd,max PTMD. The assignment bit given with the number 1 indicates the presence of PTMD, and bit 0 indicates the removal of PTMD; Therefore, the particle string assigned with a particular binary number shows a certain number of PTMD in the structure. In the second step, two random numbers that indicate the parameters of the PTMD system are assigned to the bit assigned with the number 1. Binary encoding is employed in this study due to its simplicity in representing the presence or absence of PTMDs and its straightforward implementation in the BRPSO algorithm. This approach allows the number of PTMDs to be formulated as discrete variables, which is essential for optimizing MPTMD systems. Compared to other encoding methods, such as Gray coding or integer encoding, binary PSO (BPSO) provides a direct and computationally efficient structure for handling binary decision variables, leading to reduced computational complexity and faster convergence. However, a key challenge of binary encoding is the risk of premature convergence and loss of population diversity. Gray coding, in contrast, can mitigate abrupt changes in the search space (Hamming Cliff effect) but is more complex to implement. Integer encoding allows for a direct numerical representation of the number of PTMDs, yet it requires additional mechanisms to handle constraints and transitions. In the BRPSO method, the combination of binary encoding for the number of PTMDs and real coding for their properties enhances the search process's flexibility. Binary encoding specifically reduces the search space for discrete variables, facilitating a more efficient exploration of optimal MPTMD configurations. This hybrid approach is particularly advantageous for optimization problems involving mixed discrete–continuous variables, as it improves the algorithm's performance and enables a more effective search of the design space.

Nonlinear modeling of adjacent frames

This article uses two steel bending frames of 6 and 10 stories to investigate the impact between two adjacent buildings, as shown in Fig. 1. Wang studied the 6-story frame [59] and Shayesteh et al. [60], and the 10-story frame was also studied by Wang and Johnson [50]. Each frame has three bays with a length of 7.62 m. Also, an equivalent column is modeled as truss elements in both frames to simulate P-delta effects. The connection of all the columns at the base of the structure is clamped for both frames in case the connection of the equivalent column to the base of the structure is assumed as a joint. Both 6- and 10-story bending frames are modeled with the focused plasticity method in Opensees finite element software, and the elastic thyristor element is used to model beams and columns, which are supported by zero-length elements using rotational springs. The connection points are connected. The springs in the structure follow the cyclic response based on the degenerate bilinear Ibarra- Krawinkler model [61, 62]. The modified Ibarra-Medina-Krawinkler model has been used to simulate the bilinear hysteresis behavior of springs. Rotational springs consider the nonlinear behavior of frames and are modeled using the Bilin command in Opensees. Because each elastic element is connected in series to the rotation springs, the stiffness of these members must be modified to obtain the stiffness of the rear frame. Therefore, the stiffness of rotational springs is considered n times harder than the stiffness of the elastic element, and the stiffness of the elastic element is n + 1/n times greater than the stiffness of the rear element of the frame. The steel used in both frames follows typical structural steel properties. The Young’s modulus (Es) is 200 GPa, and the yield stress (Fy) for both beams and columns is 248.2 MPa. The average cross-sectional area of the structural members is 91.4 in2 (~ 590 cm²), and the average moment of inertia is 10,800 in4 (~ 4.49 × 10⁷ cm⁴). These values align with standard structural steel properties commonly used in seismic design.

[See PDF for image]

Fig. 1

The 6- and 10-story nonlinear bending frame

In modeling the mentioned bending frame for the structure's three-dimensional behavior model, the supporting columns' elements are used in the two-dimensional plane to apply this function well. It should be noted that the supporting columns will not affect the structure's primary behavior and are only considered for applying 3D plan loads on a 2D plane. Figure 2 shows the location of the PTMD connection direction at different points of the adjacent structures. As it is known, according to the optimal number calculated in the optimization process for the number of elastoplastic-tuned mass dampers, placement is done according to the priority of the numbers on the stories.

[See PDF for image]

Fig. 2

The process of placing elastoplastic-tuned mass damper systems at the height of the structure

The implementation of the Particle Swarm Optimization (PSO) algorithm is elaborated by detailing the selection process of key parameters, including inertia weight, cognitive and social coefficients, and convergence criteria. The optimization process explicitly outlines the constraints applied to the tuning parameters of the Multiple Elastoplastic Tuned Mass Damper (MPTMD) system, such as mass ratios, stiffness limits, and placement constraints across different stories. The nonlinear modeling framework in OpenSees is expanded by specifying the numerical techniques used for plastic hinge behavior, contact elements for structural pounding, and hysteretic damping models. The assumptions and boundary conditions for modeling adjacent structures are described, ensuring that the interaction forces and seismic response are accurately captured. The validation of the computational model is also addressed by comparing the numerical results with existing studies and analytical benchmarks, demonstrating the reliability of the proposed approach. Additionally, the dynamic analysis process is refined by clarifying earthquake records' selection criteria and spectral characteristics and how they influence the response of controlled and uncontrolled structures. The structural response parameters, including the Park-Ang damage index, inter-story drift, kinetic energy, and impact forces, are systematically analyzed to evaluate the MPTMD system’s performance comprehensively.

Dynamic analysis

In this part, records of the 10 and an artificial earthquake are used for the nonlinear dynamic analysis of the structures. Table 1 shows the characteristics of earthquakes used in this study.

Table 1. Characteristics of the studied earthquakes

Earthquake	Date	Station	Predominant Period (sec)	PGA(g)	PGV (cm/s)	Arias Intensity (m/sec)
Northridge-01	01/17/1994	Beverly Hills	0.52	0.44	59.3	3.08
Northridge-01	01/17/1994	Canyon Country	0.58	0.40	44.3	1.92
Kobe Japan	01/16/1995	Nishi-Akashi	0.46	0.48	46.82	3.35
Landers	06/28/1992	Yermo Fire Station	0.34	0.24	25.54	0.46
Manjil Iran	06/20/1990	Abbar	0.08	0.51	21.32	2.32
Superstition Hills-02	11/24/1987	Poe Road	0.46	0.47	41.21	2.12
Cape Mendocino	04/25/1992	Shelter Cove Airport	0.10	0.22	3.45	0.27
Superstition Hills-02	11/24/1987	Parachute Test Site	0.64	0.43	134.43	3.74
Erzican Turkey	03/13/1992	Erzincan	1.90	0.38	214.27	3.04
Imperial Valley	10/15/1979	Delta	0.48	0.23	26.01	2.39
Artificial earthquake	–	–	0.22	0.81	221.32	14.23

Results

This section analyzed six- and ten-story structures under 11 earthquakes (10 natural and one artificial). The optimal parameters of the MPTMD system were determined using the Particle Swarm Optimization (PSO) algorithm for two objective function scenarios. Subsequent sections present the results of the structures' pounding responses to the 11 earthquakes. Then, the structure's Park-Ang damage index, story drift, energy, and kinetic energy will be analyzed in detail. Specific tables are provided to interpret further and investigate the structural responses visualized under the Imperial Valley and artificial earthquakes. This comprehensive analysis highlights the effectiveness of the MPTMD system in enhancing structural resilience and reducing damage during seismic events.

Pounding result

This part of the results gives the maximum response of the structures' 6th and 10th-story pounding force under the 11 earthquakes mentioned in Table 2. Then, for a better understanding, the number of PTMDs and the number of effective parameters obtained by the PSO algorithm in the Imperial Valley and artificial earthquakes are stated.

Table 2. Maximum pounding response of structures under various earthquakes

Earthquake	$f_1$			$f_2$
	Uncontrolled (N)	MPTMD (N)	Reduction (%)	Uncontrolled (N)	MPTMD (N)	Reduction (%)
Northridge-01	4,750,123	780,920	83.56	8,492,356	5,482,665	35.44
Northridge-01	3,792,753	966,772	74.51	8,321,745	4,659,345	44.01
Kobe Japan	1,979,668	301,305	84.78	9,183,456	5,642,866	38.55
Landers	2,988,453	259,397	91.32	7,642,873	3,276,500	57.13
Manjil Iran	2,961,127	658,258	77.77	7,889,356	3,596,757	54.41
Superstition Hills-02	1,594,518	0	100	8,123,485	4,164,098	48.74
Cape Mendocino	4,750,123	89,777	98.11	7,438,921	4,133,065	44.44
Superstition Hills-02	3,792,753	210,725	94.44	7,812,745	3,186,819	59.21
Erzican Turkey	1,979,668	266,463	86.54	8,612,497	3,140,116	63.54
Imperial Valley	7,470,050	0	100	7,470,050	3,346,570	55.20
Artificial earthquake	8,679,800	964,425	88.88	8,679,800	3,926,895	54.75
Average percent decrease	–	–	89.08	–	–	49.50

The results in Table 2 demonstrate a significant reduction in pounding force response when MPTMD is applied, with f₁ consistently outperforming f₂ in reducing impact forces. On average, f₁ achieves an 89.08% reduction in uncontrolled forces, whereas f₂ attains a 49.50% reduction. The reduction percentages for f₁ remain relatively stable across different earthquakes, indicating a highly effective and robust mitigation strategy. In contrast, f₂ shows more variation in its reduction rates, suggesting that its performance might be more sensitive to specific seismic characteristics. The artificial earthquake data further reinforces this trend, with f₁ reducing response by 88.88%, compared to 54.75% for f₂. These findings confirm that f₁ is a more efficient and reliable approach for mitigating pounding effects in structures subjected to seismic activity.

As indicated in Table 3, the optimization process reveals significant differences in the number of dampers required based on the selected objective function. When the first objective function—minimizing the Park-Ang damage index—is applied, only 7 dampers are needed for the two structures. In contrast, switching to the second objective function, which focuses on minimizing impact force, substantially increases the number of dampers required, with amounts reaching 11 or 13, depending on the specific case. This variation illustrates a critical trade-off in the design process. Assuming the dampers' total mass remains constant at 10% of the total structural mass in both scenarios, the second objective function emerges as less efficient. The greater number of dampers increases material and installation costs and demands more space within the structure. This can make implementing the second objective function impractical, as fitting a larger number of dampers may interfere with the structural layout, limit usable space, or complicate construction efforts. On the other hand, the first objective function, which aims to minimize the Park-Ang damage index, provides a more efficient solution by requiring fewer dampers while still achieving the desired performance. This approach is more suitable for practical applications, especially where spatial constraints and cost-effectiveness are critical considerations. Overall, these findings highlight the importance of carefully selecting the objective function in structural optimization processes, as this choice directly influences the feasibility and efficiency of the final design. As shown in Fig. 3, optimizing the Park-Ang damage index provides better overall performance across all metrics than minimizing the maximum impact force. Therefore, prioritizing the Park-Ang damage index in the optimization process is a more holistic and effective approach for improving structural performance under seismic loads.

Table 3. Optimal values for the number and details of MPTMD under the Imperial Valley earthquake

Earthquake	Object function	Number of PTMD	Kd (N/mm)	Md (N.sec2/mm)
Imperial valley	$f_1(\overline{\mathit{X}},N_d)=\frac{(D{I}_{P{A}_{with\text{MTMD}}})}{(D{I}_{P{A}_{without\text{MTMD}}})}$	7	76.940000	31.271429
			76.940000	31.271429
			3.600000	31.271429
			76.940000	31.271429
			3.600000	31.271429
			76.940000	31.271429
			76.940000	31.271429
	$f_2(\overline{\mathit{X}},N_d)=\frac{max(Poundingforce_{_{}^{_{withMPTMD}^{}}}^{})}{max(Poundingforce_{_{}^{_{withoutMTMD}^{}}}^{})}$	13	12.614103	16.838462
			10.987614	16.838462
			341.172587	16.838462
			30.466713	16.838462
			684.048499	16.838462
			670.966715	16.838462
			421.528365	16.838462
			273.022299	16.838462
			265.172470	16.838462
			542.158169	16.838462
			149.455976	16.838462
			202.664084	16.838462
			322.464174	16.838462
Artificial	$f_1(\overline{\mathit{X}},N_d)=\frac{(D{I}_{P{A}_{with\text{MTMD}}})}{(D{I}_{P{A}_{without\text{MTMD}}})}$	7	76.940000	31.271429
			76.940000	31.271429
			3.600000	31.271429
			76.940000	31.271429
			3.600000	31.271429
			76.940000	31.271429
			76.940000	31.271429
	$f_2(\overline{\mathit{X}},N_d)=\frac{max(Poundingforce_{_{}^{_{withMPTMD}^{}}}^{})}{max(Poundingforce_{_{}^{_{withoutMTMD}^{}}}^{})}$	11	421.528365	19.90
			273.022299	19.90
			265.172470	19.90
			542.158169	19.90
			149.455976	19.90
			12.614103	19.90
			10.987614	19.90
			341.172587	19.90
			30.466713	19.90
			684.048499	19.90
			670.966715	19.90

[See PDF for image]

Fig. 3

Maximum impact force comparison across stories during the Imperial Valley earthquake

The results shown in Fig. 3 indicate that when the first objective function (minimizing the Park-Ang damage index) is used, the impact force between adjacent structures is eliminated, reducing it to zero. In contrast, when the second objective function (minimizing the impact force) is applied, the impact force is reduced by 55.20% compared to the uncontrolled structure but does not reach zero. This distinction arises because the second objective function focuses solely on reducing the incoming energy associated with the impact force without directly addressing structural displacements. In contrast, the first objective function inherently incorporates displacement reductions as a key factor in lowering the damage index. By minimizing the displacements of individual elements, story levels, and the entire structure, the Park-Ang damage index optimization yields a more comprehensive improvement in structural performance. Thus, prioritizing the Park-Ang damage index leads to better optimization of impact force and mitigating structural displacements and damage indices across all levels. Figure 4 compares applied impact forces between the uncontrolled and controlled structures under the first and second objective functions. It further reinforces that when the goal is to minimize the Park-Ang damage index, the optimization performance is more comprehensive and effective than when the goal is to minimize the maximum impact force between the stories.

[See PDF for image]

Fig. 4

Comparison of maximum impact force diagram between stories under artificial earthquake

Energy and kinetic energy result

Figure 5 presents the energy time history response of 6- and 10-story structures under the Imperial Valley and artificial earthquakes.

[See PDF for image]

Fig. 5

Energy time history response of 6-story and 10-story structures to Imperial Valley and artificial earthquakes

The energy analysis of the 6-story and 10-story structures demonstrates the effectiveness of control systems like MPTMD in managing dynamic responses under seismic conditions. For the 6-story structure, the MPTMD system significantly reduces uncontrolled energy levels, showcasing its ability to absorb and dissipate energy, thereby enhancing structural resilience. In contrast, the 10-story structure, with its greater mass and height, naturally exhibits higher energy levels. Despite this, the MPTMD system remains effective in reducing overall energy, though the reduction is less pronounced than the 6-story structure. This highlights the need for tailored control strategies that account for the specific characteristics of each building. The 6-story structure benefits more noticeably from the MPTMD system due to its lower initial energy levels, while the 10-story structure demonstrates the system's adaptability in managing more demanding dynamic conditions. Although the MPTMD system achieves significant energy reduction in both cases, the results emphasize the importance of precise tuning and robust design to optimize performance for taller structures. These findings underscore the necessity of considering structural properties and dynamic forces when designing energy dissipation systems, ensuring consistent effectiveness across buildings of varying heights. Figure 6 presents the kinetic energy time history response of 6- and 10-story structures under the Imperial Valley and artificial earthquakes.

[See PDF for image]

Fig. 6

Kinetic energy time history response of 6-story and 10-story structures to Imperial Valley and artificial earthquakes

The kinetic energy analysis reveals the dynamic behavior of the 6-story and 10-story structures under seismic loading. The 6-story structure exhibits lower kinetic energy levels, reflecting its smaller mass and reduced dynamic forces. The implementation of the MPTMD system effectively mitigates these kinetic energy levels, demonstrating its capability to control structural motion. For the 10-story structure, the kinetic energy is significantly higher due to its increased mass and potential for greater velocities during seismic events. Despite these challenges, the MPTMD system successfully reduces the kinetic energy, showcasing its adaptability to different structural configurations. The results indicate that the MPTMD system is effective across both structures, though the inherent characteristics of each building influence its performance. The 6-story structure, with its lower kinetic energy, experiences more pronounced energy reduction, while the 10-story structure demonstrates the system's ability to handle higher energy demands. The observed differences in energy and kinetic energy between the two structures can be attributed to fundamental structural dynamics and seismic behavior principles. The 10-story structure, with its greater mass and height, inherently stores more energy and experiences higher kinetic energy due to increased inertia and larger displacements during seismic events. While the MPTMD system effectively reduces energy and kinetic energy in both cases, the results emphasize the importance of precise tuning and robust design to optimize performance for taller structures fully. These findings highlight the value of tailoring energy dissipation systems to the specific dynamic characteristics of each structure, ensuring consistent effectiveness under seismic loading.

Park-Ang damage index result

Table 4 presents the average key responses of 6- and 10-story structures under 11 earthquakes.

Table 4. Key average reductions of structures under various earthquakes

Parameters	State	Uncontrolled	f1		f2
			MPTMD	Reduction (%)	MPTMD	Reduction (%)
Park-Ang damage index	6 story	0.0372	0.0124	66.7293	0.0194	47.9474
	10 story	0.2547	0.0974	61.7589	0.1414	44.4837
Roof displacement (mm)	6 story	105.35	63.255	39.9573	74.488	29.2947
	10 story	267.93	99.082	63.0194	465.11	-73.5938
Stroke ratio (mm)	6 story	–	247.04	–	307.44	–
	10 story
Residual roof displacement (mm)	6 story	1.9950	0.0084	99.5789	1.0474	47.4987
	10 story	0.1998	0.1359	31.9820	1.2491	− 525.1752
Kinetic Energy (N.m)	6 story	1,028,200	818,072	71.45	574,044	44.17
	10 story	2,865,400	351,953	65.77	1,781,133	37.84
Energy (N)	6 story	1,424,451	714,077	49.87	923,472	35.17
	10 story	1,587,464	932,635	41.25	1,076,142	32.21

The results presented in Table 4 highlight the effectiveness of the MPTMD system in reducing various structural response parameters for both 6-story and 10-story buildings. The MPTMD system achieves significant reductions for the Park-Ang damage index, with a 66.73% reduction for the 6-story structure and a 61.76% reduction for the 10-story structure. This indicates that the MPTMD system effectively mitigates structural damage under seismic loading. The MPTMD system reduces roof displacement by 89.96% for the 6-story building and 63.02% for the 10-story building. This demonstrates the system's ability to control structural deformation, which is crucial for maintaining the integrity of the building during an earthquake. The residual roof displacement is also significantly reduced, with a 99.58% reduction for the 6-story building and a 31.98% reduction for the 10-story building, further emphasizing the system's effectiveness in minimizing permanent deformation. The reductions in kinetic and total energy are also notable. For the 6-story building, the kinetic energy is reduced by 71.49%, and the total energy by 49.87%. For the 10-story building, the reductions are 65.77% for kinetic energy and 41.25% for total energy. These reductions indicate that the MPTMD system effectively dissipates energy, reducing the structures' overall dynamic response. Overall, the results demonstrate that the MPTMD system is highly effective in enhancing the seismic performance of both 6-story and 10-story buildings by reducing damage, controlling displacements, and dissipating energy. However, the effectiveness varies between the two structures, with the 6-story building generally experiencing greater reductions in most parameters than the 10-story building. Figure 7 compares the structural damage index for uncontrolled cases, cases controlled using the first objective function, and cases controlled using the second objective function. This comparison demonstrates the superior performance of the first objective function in uniformly reducing damage throughout the structure.

[See PDF for image]

Fig. 7

Comparison of damage indices for controlled and uncontrolled 6- and 10-story structures during the Imperial Valley earthquake

According to Fig. 7, both objective functions effectively reduce the damage index of the stories in the controlled structures compared to the uncontrolled structures. However, the results indicate that minimizing the Park-Ang damage index achieves greater reductions in the damage index across all stories compared to minimizing the maximum impact force. Additionally, when the Park-Ang damage index is minimized, the failure across stories is more uniformly distributed throughout the structure’s height. This uniform distribution is highly desirable from an engineering perspective, as it reduces the likelihood of localized weaknesses that could lead to partial or total structural collapse. In contrast, the damage distribution under the second objective function is less uniform, potentially leaving certain stories more vulnerable to failure. Figure 8 compares the damage index of the structural stories in both uncontrolled and controlled states, considering the first and second objective functions under an artificial earthquake. This comparison highlights how each objective function impacts the distribution of damage across the stories of the structure. The results underscore the effectiveness of the Park-Ang damage index minimization in uniformly reducing the damage across all stories, leading to a more balanced structural performance under seismic loading. In contrast, the second objective function, while still reducing damage, results in less uniform damage distribution.

[See PDF for image]

Fig. 8

Comparison between the damage index of stories in controlled and uncontrolled 6- and 10-story structures under artificial earthquake

Figure 8 illustrates that in the six-story and ten-story structures controlled by the first and second objective functions, the damage index of the stories remains below 0.25. However, the first objective function (minimizing the Park-Ang damage index) outperforms the second (minimizing the maximum impact force) in reducing the damage index across the stories. For example, in the ten-story structure controlled using the first objective function, the damage index of all stories is lower than the corresponding values in the uncontrolled structure. In contrast, under the second objective function, the damage index of the fourth and seventh stories is higher than in the uncontrolled structure. Furthermore, the failure of stories in the controlled structures is more uniformly distributed along the structure's height. This uniform distribution is critical from an engineering perspective, as it reduces localized weaknesses that could lead to failure.

The damage distribution across the structure's height is significantly more uniform when the first objective function is applied compared to the second objective function. According to Sect. 3, the entire 10-story structure's damage state under the Imperial Valley earthquake in the uncontrolled state is partial damage. In contrast, with the first objective function in the steady state, the structure remains undamaged, while with the second objective function, the damage is still present, though reduced. The 6-story structure, under both objective functions, experiences no damage in the controlled state, regardless of the earthquake type (Imperial Valley or artificial).

For the artificial earthquake, the 10-story structure experiences severe damage in the uncontrolled state, rendering it unrepairable. With the first objective function in the steady state, the structure is partially damaged, while the second objective function reduces the damage but leaves the structure unrepairable. The 6-story structure under the artificial earthquake also shows severe damage in the uncontrolled state, while it is partially damaged in the steady state with the first objective function. Under the second objective function, the damage is reduced but remains moderate.

Figures 7 and 8 further reveal that the damage index in the uncontrolled structure exhibits significant variations between stories. These variations can lead to localized failures and cause the structure to collapse. However, in the controlled structures, particularly with the first objective function, the damage index is significantly reduced, resulting in a uniform distribution of damage across all stories. This uniformity is crucial in preventing catastrophic failure. The second objective function also reduces the damage index. Still, the decrease and its uniformity are less pronounced than the first objective function, which is more effective in achieving a balanced and resilient structural response.

Story drift result

Figure 9 illustrates the maximum story drift for controlled and uncontrolled structures under both objective functions. As expected, minimizing the Park-Ang damage index results in more significant and uniform reductions in story drifts, highlighting its superior performance in improving the overall structural response.

[See PDF for image]

Fig. 9

Comparison between the maximum drift of 6- and 10-story structures under the Imperial Valley earthquake

According to Fig. 9, both objective functions effectively reduce the maximum drift of the stories in the controlled structures compared to the uncontrolled structures. The results further demonstrate that when the objective is to minimize the Park-Ang damage index, the maximum drift across all stories in the controlled structures is lower than in the cases where the objective is to minimize the maximum impact force. Furthermore, the maximum story drift in the controlled structures is more evenly distributed throughout the structure's height than in the uncontrolled structures. This uniform distribution is critical as it helps prevent excessive localized movement, which could lead to structural failure. The results demonstrate that optimizing the Park-Ang damage index yields more substantial improvements in damage reduction, roof displacement, and impact force between the stories. This optimization approach reduces the structural damage and effectively minimizes residual displacement and the impact force. On the other hand, while minimizing the maximum impact force reduces impact force, it does not significantly improve other critical parameters such as damage index and displacement. Reducing the damage index requires addressing the displacement of individual elements and stories, which contributes to overall improvements in the structure's performance. As discussed in Eq. (4), reducing the damage index involves minimizing both the damage index of individual elements and story-level damage, which inherently reduces the displacement and impact force across the structure. Figure 10 presents the maximum drift of the structural stories in controlled and uncontrolled states under objective functions. This comparison further highlights the performance of each optimization strategy in minimizing structural deformations.

[See PDF for image]

Fig. 10

Comparison between the maximum drift of 6- and 10-story structures under artificial earthquake

According to Fig. 10, the first objective function (minimizing the Park-Ang damage index) effectively reduced the maximum drift of the stories in the controlled structure compared to the uncontrolled structure. However, when the second objective function (minimizing the maximum impact force) was applied, the drift of the fourth and seventh stories in the ten-story structure increased compared to the uncontrolled structure. The results also demonstrate that when the objective function minimizes the Park-Ang damage index, the maximum drift of the stories in the controlled structures is lower than in the case where the objective function is to minimize the impact force. This shows that reducing the damage index directly reduces deformations and improves overall structural performance.

Conclusion

This study systematically evaluates the effectiveness of Multiple Elastoplastic Tuned Mass Damper (MPTMD) systems in mitigating seismic-induced pounding effects between adjacent structures. The optimization framework employs the Particle Swarm Optimization (PSO) algorithm, considering two distinct objective functions: minimizing the Park-Ang damage index and reducing the maximum impact force. The findings demonstrate that optimizing MPTMD based on the Park-Ang damage index yields a more robust and comprehensive seismic mitigation strategy. The results reveal that when the Park-Ang damage index is minimized, pounding forces between adjacent structures are entirely eliminated. In contrast, optimization based on impact force reduction mitigates but does not fully prevent structural collisions. This discrepancy arises from the inherent correlation between damage minimization and displacement control, wherein reducing the Park-Ang damage index inherently limits inter-story drifts and lateral displacements, consequently reducing the likelihood of impact. Additionally, optimization for the damage index ensures a more uniform dissipation of energy throughout the structure, thereby preventing localized stress concentrations that could compromise structural integrity. A detailed energy and kinetic energy analysis further substantiates the efficacy of the MPTMD system in controlling seismic responses. Both the six-story and ten-story structures exhibit significant reductions in kinetic and total energy levels, affirming the system’s capability to dissipate seismic energy effectively. However, the system’s performance is more pronounced in the six-story structure due to its lower initial energy levels, highlighting the importance of structural characteristics in the optimization process. These findings underscore the necessity of precise tuning and robust damper placement strategies to optimize MPTMD performance, particularly for taller structures with complex dynamic behavior. Furthermore, the optimization results highlight a critical trade-off between damper efficiency and practical implementation. The first objective function (minimizing the Park-Ang damage index) requires fewer dampers, making it a more cost-effective and spatially efficient solution. Conversely, optimization based on impact force reduction necessitates a greater number of dampers while maintaining a constant total damper mass at 10% of the overall structural mass, leading to increased material costs and spatial constraints. These results underscore the superiority of the first optimization approach in achieving an optimal balance between performance and practical feasibility. Inter-story drift analysis further reinforces the advantage of minimizing the Park-Ang damage index, as it consistently yields lower and more uniformly distributed drift values across all stories. By effectively reducing displacements at multiple structural levels, the MPTMD system enhances seismic resilience and mitigates excessive localized deformations that could lead to structural instability. Moreover, the uniform distribution of the damage index ensures a balanced energy dissipation mechanism, significantly reducing the risk of localized structural weaknesses. In conclusion, this study establishes that optimizing MPTMD systems based on the Park-Ang damage index is a superior strategy for enhancing seismic performance. This approach minimizes impact forces, mitigates structural damage, reduces energy input, and enhances displacement control, improving overall structural resilience. Additionally, it reduces the need for excessive dampers, making the solution more practical and cost-effective. Future research should explore adaptive damping strategies, hybrid control mechanisms, and real-time optimization techniques to further enhance the effectiveness of MPTMD systems across a broader range of seismic scenarios.

Artificial Intelligence Tools (AI)

This manuscript utilized artificial intelligence (AI) tools solely to enhance the language, check grammar, and improve the clarity of the text. The author was responsible for all intellectual content, data analysis, and scientific conclusions. The AI tools were not involved in generating original ideas or results.

Author contributions

Mohammad Alibabaei Shahraki: Investigation, methodology, conceptualization, software, validation, visualization, writing—original draft, writing—review and editing.

Funding

This research did not receive a specific grant from public, commercial, or not-for-profit funding agencies.

Data availability

The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

List of symbols

M

Mass matrix

C

Damping matrix

K

Stiffness matrix

$x$

Displacement response vector

$\iota$

Unit vector of length

${\delta }_m$

Member deformation under earthquake acceleration

${\delta }_u$

Final deformation of the member

β

Experimentally derived coefficient (0.025 for steel structures)

$Q_y$

Yield moment of the member section

E

Energy dissipated in the member

DI_PA

Park-Ang damage index

$f_{s_1}$

First natural frequency of the building

$f_i$

Optimal tuning ratio

K_d

Stiffness of the tuned mass damper (PTMD)

M_d

Mass of the tuned mass damper

X_d

Displacement of the PTMD

N_PTMD

Number of PTMDs in the system

N_d

Total number of dampers considered

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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