Ranking of design scenarios of TMD for seismically excited structures using TOPSIS

Abstract

In this paper, design scenarios of a tuned mass damper (TMD) for seismically excited structures are ranked. Accordingly, 10 design scenarios in two cases, namely unconstrained and constrained for the maximum TMD, are considered in this study. A free search of the TMD parameters is performed using a particle swarm optimization (PSO) algorithm for optimum tuning of TMD parameters. Furthermore, nine criteria are adopted with respect to functional, operational, and economic views. A technique for order performance by similarity to ideal solution (TOPSIS) is utilized for ranking the adopted design scenarios of TMD. Numerical studies are conducted on a 10-story building equipped with TMD. Simulation results indicate that the minimization of the maximum story displacement is the optimum design scenario of TMD for the seismic-excited structure in the unconstrained case for the maximum TMD stroke. Furthermore, H₂ of the displacement vector of the structure exhibited optimum ranking among the adopted design scenarios in the constrained case for the maximum TMD stroke. The findings of this study can be useful and important in the optimum design of TMD parameters with respect to functional, operational, and economic perspectives.

Keywords

seismic-excited building / TMD / optimum design / PSO / design scenario / TOPSIS

Introduction

Optimum design of tuned mass damper (TMD) parameters for a single-degree-of-freedom (SDOF) main system under external harmonic force, harmonic, and white-noise base acceleration has been considered in different studies [¹–⁷]. A review on formulation of the TMD tuning has been reported in Ref. [⁸]. However, the optimum design of TMD parameters is still an open research topic for seismic-excited structures [⁹]. Recently, some certain meta-heuristic optimization algorithms, such as ant colony optimization [¹⁰], charged system search [¹¹], improved harmony search algorithm [¹²], a hybrid method using harmony search and flower pollination algorithm [¹³], and a novel bat algorithm [¹⁴], have also been utilized for the optimum design of TMD parameters. Furthermore, machine learning method [¹⁵] and least-squares support-vector machine [⁷] are successfully applied for estimation of optimum TMD parameters. Moreover, Nigdeli and Bekdas [¹⁶] studied the optimum tuned mass damper approaches for adjacent structures. By considering the axial force capacity constraint in a structure, Nigdeli and Bekdas [¹⁷] investigated optimum design of multiple positioned TMDs. Lu et al. [¹⁸] developed an equivalent reduced-order model for the optimal tuned impact damper (TID) parameters for a nonlinear building subjected to earthquake excitations. The effects of soil-structure interaction on optimal tuning of TMD parameters for seismic-excited structures have been recently considered in Refs. [¹⁰,¹⁹–²⁵]. Additionally, some studies considered the optimum design of TMD parameters as a multi-objective optimization problem [²¹,²⁵–²⁷].

In extant studies, different cost functions are considered for the optimum design process of TMD parameters in multiple-degree-of-freedom (MDOF) structures subjected to earthquake excitations. The difference between the damping of the first two modes of the structure equipped with TMD is minimized and considered as a target function to determine the optimum TMD parameters [²⁸]. The H₂ norm of the transfer function of the structure is considered as an objective function in [²⁹]. Lee et al. [³⁰] defined a performance index of structural response in the frequency domain as an objective function. Farshidianfar and Soheili [¹⁰] defined an overall objective function, including the maximum displacement and acceleration as a cost function. Kaveh et al. [¹¹] investigated a weighted sum of the normalized responses comprising the maximum displacement of the first story and acceleration transfer function as a cost function. Bekdaş and Nigdeli [¹⁹] considered the maximum displacement of the structure with TMD to a user-defined value as an objective function. The maximum amplitude of the acceleration transfer function of the top story was investigated as an objective function in Refs. [³¹,³²]. The maximum floor displacement was adopted as a cost function in the studies conducted by Heidari et al. [³³] and Shahi et al. [²³].

Recently, multi-criteria decision-making methods were considered for ranking construction technologies or design scenarios in problems pertaining to building structures. technique for order performance by similarity to ideal solution (TOPSIS) is a well-known and useful approach for multiple-criteria decision-making methods. The selection of a pile-column technology was considered in Ref. [³⁴] using the TOPSIS. Some researchers applied the TOPSIS for selecting a strategy for seismic retrofitting of structures [³⁵–³⁸]. Furthermore, Terracciano et al. [³⁹] utilized the TOPSIS for comparing thin-walled steel structures by considering structural, economic, and environmental parameters.

Studies on optimum design of TMD parameters studies were conducted for a preselected TMD mass ratio. Furthermore, the limitation of the maximum stroke of TMD was not considered as a constraint in the design process. Although selecting a smaller TMD mass, stiffness, and damping coefficients can be important from practical and economic perspectives, these points are not considered in the design scenarios of TMD. Different scenarios are proposed for the optimal design of TMD parameters. However, to the best of the author’s knowledge, no studies have been conducted on the selection of an appropriate scenario for the optimum design of TMD parameters. The aim of the present study involves finding the optimum design scenario of the TMD parameters for seismically excited structures with respect to functional, operational, and economic perspectives. Accordingly, 10 design scenarios of TMD for two cases, namely unconstrained and constrained, are considered for the maximum stroke of TMD. Additionally, free search of TMD parameters is adopted for tuning mass, damping, and stiffness coefficients of TMD. A design process based on a PSO algorithm is proposed for the optimum design of TMD parameters. Subsequently, a TOPSIS is utilized to compute the rank of the design scenarios of TMD parameters for structures subjected to earthquake excitations.

The remainder of this paper is organized as follows: A mathematical model of a seismic-excited structure equipped with TMD is introduced in Section 2. An overview of the PSO algorithm and TOPSIS are provided in Sections 3 and 4. Numerical studies are conducted on a benchmark building in Section 5. The simulation results are discussed in Section 6, and the concluding remarks are provided in Section 7.

Mathematical model of a structure equipped with TMD

The equation of motion of an N-degree-of-freedom shear-type building equipped with a TMD on the top floor, subjected to seismic excitation, can be given as follows:

(1)\mathit{M}\ddot{\mathit{U}}(t)+\mathit{C}\dot{\mathit{U}}(t)+\mathit{K}\mathit{U}(t)=\mathit{P}{\ddot{x}}_g(t),

where

\mathit{M}=\left[\begin{array}{ll}{\mathit{M}}_s& \mathrm{0}\\ \mathrm{0}& m_{\mathrm{TMD}}\end{array}\right],\mathit{C}=\left[\begin{array}{ll}{\mathit{C}}_s+\mathit{T}{c}_{\mathrm{TMD}}{\mathit{T}}^{\mathrm{T}}& -\mathit{T}{c}_{\mathrm{TMD}}\\ -c_{\mathrm{TMD}}{\mathit{T}}^{\mathrm{T}}& c_{\mathrm{TMD}}\end{array}\right],\mathit{K}=\left[\begin{array}{ll}{\mathit{K}}_s+\mathit{T}{k}_{\mathrm{TMD}}{\mathit{T}}^{\mathrm{T}}& -\mathit{T}{k}_{\mathrm{TMD}}\\ -k_{TMD}{\mathit{T}}^{\mathrm{T}}& k_{\mathrm{TMD}}\end{array}\right],

and

(2)\mathit{P}=\left\{\begin{array}{l}-{\mathit{M}}_{s}e\\ -m_{\mathrm{TMD}}\end{array}\right\},\mathit{U}(t)=\left\{\begin{array}{l}{\mathit{X}}_s(t)\\ x_{\mathrm{TMD}}(t)\end{array}\right\},

where M_s, C_s, and K_s denote N× N mass, damping, and stiffness matrices of the main structure, respectively. Furthermore, m_TMD, c_TMD, and k_TMD denote the TMD parameters that are mass, damping, and stiffness coefficients, respectively. In particular, X_s(t) and x_TMD(t) are the displacement vectors of the main structure and TMD stroke, respectively. The N×1 vectors T and e are considered as T = [0 0 ⋯ 0 1]^T and e = [1 1 ⋯ 1 1]^T, respectively. Equation (1) can be written in the state-space form as follows:

(3)\begin{array}{l}\dot{\mathit{Z}}(t)=\mathit{A}\mathit{Z}(t)+\mathit{B}{\ddot{x}}_{\mathrm{g}}(t)\hfill \\ Y(t)=\mathit{L}\mathit{Z}(t)+\mathit{D}{\ddot{x}}_{\mathrm{g}}(t)\hfill \end{array},

where

(4)\mathit{A}=\left[\begin{array}{ll}\mathbf{0}& \mathbf{0}\\ -{\mathit{M}}^{-\mathrm{1}}\mathit{K}& -{\mathit{M}}^{-\mathrm{1}}\mathit{C}\end{array}\right],\mathit{B}=\left[\begin{array}{l}\mathbf{0}\\ -{\mathit{M}}^{-\mathrm{1}}\mathit{P}\end{array}\right],\mathit{Z}(t)=\left\{\begin{array}{l}\mathit{U}(t)\hfill \\ \dot{\mathit{U}}(t)\hfill \end{array}\right\}\mathrm{.}

Furthermore, L is a vector of a custom response, such as displacement, velocity, and drift vectors, of the structure. Additionally, D is a matrix with zero elements. The transfer function of the system is as follows:

(5)\mathit{G}(s)=\mathit{L}{(sI-\mathit{A})}^{-\mathrm{1}}\mathit{B}\mathrm{.}

Particle swarm optimization algorithm

PSO is a population-based stochastic optimization method, which is introduced based on the behavior of certain swarms, such as birds and fish, in nature [⁴⁰]. In PSO, the position and velocity of a particle can represent the status of a particle in the search space. The position and velocity of the ith particle can be denoted by the vectors x_i = (x_i1, x_i2,…,x_id) and v_i = (v_i1,v_i2,…,v_id) in a d-dimensional search space. The best position, p_i = (p_i1, p_i2,…,p_id), of each particle corresponds to the individual best objective value at time t. Furthermore, g denotes the global best particle, which is represented by the best position at time t in the entire swarm. Subsequently, the new velocity and position of the ith particle are updated using the following equations [⁴¹]:

(6)v_{ij}(t+\mathrm{1})=w{v}_{ij}(t)+c_{\mathrm{1}}{r}_{\mathrm{1}}[p_{ij}-x_{ij}(t)]+c_{\mathrm{2}}{r}_{\mathrm{2}}[g_j-x_{ij}(t)],\hspace{1em}j=\mathrm{1},\mathrm{2},\dots ,d,

(7)x_{ij}(t+\mathrm{1})=x_{ij}(t)+v_{ij}(t+\mathrm{1}),\hspace{1em}j=\mathrm{1},\mathrm{2},\dots ,d,

where c₁ and c₂ denote the acceleration coefficients and influence on the convergence speed of each particle, respectively. Based on the previous studies, they are often adopted as constant values, i.e., c₁ = c₂ = 1.8. Furthermore, r₁ and r₂ are independent random numbers that are uniformly distributed in the range [0,1]. The inertia weight factor, w, is typically considered in the range [0.1,0.9]. To enhance the convergence of the PSO, the inertia weight factor can be updated by the following equation:

(8)w=w_{\max }-n_i\left({\displaystyle \frac{w_{\max }-w_{\min }}{n_{\max }}}\right),

where w_max, w_min, n_i, and n_max denote the maximum and minimum weights and the current and maximum generation number, respectively [⁴²].

TOPSIS

Hwang and Yoon [⁴³] introduced TOPSIS as a practical and useful approach to multi-criteria decision-making problems. It can rank numerous possible alternatives by measuring Euclidean distances. The aim of this approach is to find a scenario, which is closest to the ideal solution and farthest from the negative ideal solution. A TOPSIS based on information entropy is proposed as a decision support tool. This methodology can be implemented by the following steps [⁴⁴].

Step 1: Create a decision matrix for m possible alternatives and c criteria as follows:

\begin{array}{cccccc}{C}_1& C_2& \mathrm{...}& C_g& \mathrm{...}& C_c\end{array}

(9)X_{m\times c}=\begin{array}{c}{A}_1\\ A_2\\ \mathrm{...}\\ A_i\\ \mathrm{...}\\ A_m\end{array}\left[\begin{array}{cccccc}{x}_{11}& x_{12}& \mathrm{...}& x_{1g}& \mathrm{...}& x_{1c}\\ x_{21}& x_{22}& \mathrm{...}& x_{2g}& \mathrm{...}& x_{2c}\\ \mathrm{...}& \square & \square & \square & \square & \mathrm{...}\\ x_{i1}& \square & \square & x_{ig}& \square & x_{ic}\\ \mathrm{...}& \square & \square & \square & \square & \mathrm{...}\\ x_{m1}& x_{m2}& \mathrm{...}& x_{mg}& \mathrm{...}& x_{mc}\end{array}\right]

where x_ij indicates the value of alternative A_i for criterion C_g.

Step 2: Normalize the decision matrix by determining the weight of the criteria based on the entropy technique:

(10)q_{ig}={\displaystyle \frac{x_{ig}}{{\displaystyle \sum _{i=\mathrm{1}}^m{x}_{ig}}}};\forall g\in \{\mathrm{1},\dots ,C\}\mathrm{.}

Calculate the information entropy of criterion {\mathrm{\Delta }}_gby defining information entropy as per the following equation.

(11){\mathrm{\Delta }}_g=-{\displaystyle \frac{\mathrm{1}}{\ln (m){\displaystyle \sum _{i=\mathrm{1}}^m{q}_{ig}}}}\ln q_{ig};\hspace{1em}\forall g\in \{\mathrm{1},\dots ,c\}\mathrm{.}

The entropy technique is an objective weight method based on data statistical properties for calculating the weights of criteria. This method was first introduced by Shannon and Behzadian et al. [⁴⁵,⁴⁶]. The degree of divergence (d_g) of the average information contained by each criterion can be given as follows:

w_g={\displaystyle \frac{d_g}{d_i+\cdots +d_c}},

(12)d_g=\mathrm{1}-{\mathrm{\Delta }}_g,\hspace{1em}g=\mathrm{1},\dots ,c\mathrm{.}

Finally, the weight of criteria by the entropy technique can be given as follows:

(13)w_g={\displaystyle \frac{d_g}{d_i+\cdots +d_c}}\mathrm{.}

Additionally, the following equations are applied to aggregate the weight vector {\lambda }_g of the energy manager and obtain the aggregated weight{\bar{w}}_g:

(14){\overline{w}}_g={\displaystyle \frac{{\lambda }_g{w}_g}{{\displaystyle \sum _{i=\mathrm{1}}^c{\lambda }_i}{w}_i}},

(15){\overline{w}}_g=\left\{{\overline{w}}_{\mathrm{1}},{\overline{w}}_{\mathrm{2}},\dots ,{\overline{w}}_c\right\}\mathrm{.}

Step 3: Calculate the normalized value r_ig and formation of the normalized decision matrix N_{m × c} using the following equations:

(16)r_{ig}={\displaystyle \frac{x_{ig}}{\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^m{x}_{ig}^{\mathrm{2}}}}}},

(17)N_{m\times c}={[r_{ig}]}_{m\times c},(i=\mathrm{1},\dots ,m;g=\mathrm{1},\dots ,c)\mathrm{.}

Step 4: Calculate the weighted normalized decision matrix as follows:

(18)V=N_{m\times c}\cdot {\overline{w}}_{c\times c}={[v_{ig}]}_{m\times c},(i=\mathrm{1},\dots ,m;g=\mathrm{1},\dots ,c)\mathrm{.}

Step 5: Determine the ideal and nadir ideal solutions using the following equations:

(19)A^{+}=\left\{\left(\max v_{ig}|g\in G\right)\mathrm{;}\hspace{0.17em}\left(\min v_{ig}|g\in G^{\prime }\right)\right\}=\left(v_{\mathrm{1}}^{+},v_{\mathrm{2}}^{+},...,v_c^{+}\right),

(20)A^{\mathrm{1}}=\left\{\left(\min v_{ig}|g\in G\right)\mathrm{;}\hspace{0.17em}\left(\max v_{ig}|g\in G^{\prime }\right)\right\}=\left(v_{\mathrm{1}}^{-},v_{\mathrm{2}}^{-},...,v_c^{-}\right)\mathrm{.}

where G and G′ represent the subsets of positive and negative criteria, respectively.

Step 6: Compute the distance of each ideal and nadir ideal solutions:

(21)d_i^{+}=\sqrt{{\displaystyle \sum _{g=\mathrm{1}}^c{(v_{ig}-v_g^{+})}^{\mathrm{2}}}}

(22)d_i^{-}=\sqrt{{\displaystyle \sum _{g=\mathrm{1}}^c{(v_{ig}-v_g^{-})}^{\mathrm{2}}}}

Step 7: Calculate the closeness coefficient of each alternative:

(23)C{C}_i^{+}={\displaystyle \frac{d_i^{-}}{d_i^{-}+d_i^{+}}};i=\mathrm{1},\mathrm{2},\dots ,m\mathrm{.}

Step 8: Calculate the rank of the alternatives using Eq. (24) such that the highest and lowest values represent the best and worst ranks, respectively.

(24)v=\{v_i|{\max }_{\mathrm{0}\le i\le m}(C{C}_i^{+})\}\mathrm{.}

Numerical studies

To determine the best design scenario of the TMD parameters based on functional, operational, and economic perspectives, a decision matrix for m possible alternative design scenario and c criteria must be created. In this study, m = 10 alternative design scenarios ({A₁,A₂,…,A₁₀}) are considered for the optimum design of TMD device for a benchmark structure subjected to earthquake excitations. These scenarios are defined for two cases, namely unconstrained and constrained, for the maximum stroke of the TMD. Table 1 lists the design scenarios of the TMD for the unconstrained case. For all design scenarios defined in Table 1, the optimization problem involves determining design variables of TMD, including m_TMD, c_TMD, and k_TMD, and the optimization problem constraints are as follows:

(25)

Scenario 1 considers the 2-norm (H₂) of the displacement vector of the structure equipped with TMD normalized to its corresponding value in the structure without TMD. Similarly, scenario 2 represents the 2-norm (H₂) of the drift vector of the structure equipped with TMD normalized to its corresponding value in the structure without TMD. Scenarios 3 and 4 refer to the infinity-norm (H_∞) of the displacement and drift vectors of the structure equipped with TMD normalized to the corresponding values in the structure without TMD, respectively. Furthermore, the maximum story displacement, acceleration, and drift of the structure equipped with TMD, which is normalized to the corresponding values in the structure without TMD under an artificial earthquake excitation, are considered in the design scenarios 5, 8, and 9, respectively. Moreover, scenarios 6, 7, and 10 denote the maximum base shear, damage energy, and input energy of the structure equipped with TMD, which is normalized to the corresponding values in the structure without TMD, respectively.

For the design scenarios of TMD in constrained cases, a limitation for the maximum TMD stroke is added to the corresponding scenarios of the unconstrained cases as follows:

(26){\max }_t\Vert x_{\mathrm{T}\mathrm{M}\mathrm{D}}\left(t\right)-x_{\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{f}}\left(t\right)\Vert \ll x_{\mathrm{L}}\mathrm{.}

The complicated constrained optimization problem can be converted to an unconstrained optimization problem using a penalty method as follows:

(27)J_{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}}=J_{\mathrm{U}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}}\times (1+50\max \left([0,\alpha ]\right)),\alpha =\frac{{\max }_t\Vert x_{\mathrm{T}\mathrm{M}\mathrm{D}}\left(t\right)-x_{\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{f}}\left(t\right)\Vert }{x_{\mathrm{L}}}-1,

where \alpha represents the aforementioned limitation for the maximum stroke of the TMD. In addition, J_{Unconstrained} denotes the cost function defined for the design scenarios of TMD for the unconstrained case presented in Table 1, and J_{Unconstrained} is the corresponding cost function defined for design scenarios of TMD for the constrained case. Based on Eq. (27), the limitation of the maximum TMD stroke is satisfied for the design scenarios of TMD in constrained cases.

Nine criteria ({C₁, C₂,…,C₉}) are considered for utilization in the TOPSIS, as listed in Table 2. The negative criteria, C₁ to C₃, which should be decreased economically, refer to the percentage of optimum TMD mass, damping, and stiffness values when compared to the maximum allowable values. The positive criteria, C₄ to C₉, which should be increased to mitigate risk, address the reduction in percentage of peak responses of the structure equipped with TMD when compared to those of the structure without TMD in terms of the maximum story displacement, drift, acceleration, base shear, damage, and input energies. To investigate the criteria C₄–C₉ with respect to different earthquake excitations, 44 far-field ground acceleration records (two components of 22 stations), which are defined by the federal emergency management agency (FEMA) [⁴⁷], are used. Detailed information on these earthquake excitations is provided in Table 3. By considering these seismic records, the average reductions for 44 earthquake excitations are assigned to the criteria C₄ to C₉.

To conduct numerical studies, a ten-story building equipped with a TMD, located on the top floor, is considered. The mass, damping, and stiffness coefficients of each story are 360 t, 6.2 MNs/m, and 650 MN/m, respectively. Practically, the upper bounds of the TMD mass, damping, and stiffness coefficients of the TMD are defined as (m_TMD)_max = 108 t, (c_TMD)_max = 1000 kNs/m, and (k_TMD)_max = 5000 kN/m. Furthermore, in the constrained case for the TMD movement, the maximum stroke length of the TMD relative to the top floor acceleration is assumed as x_L = 150 cm.

A design process based on the PSO algorithm is utilized for a free search of the TMD parameters and optimum design of mass, damping, and stiffness coefficients of the TMD. In this study, the population size of the PSO is adopted as n = 50. The initial and final inertia weights are considered as w_min = 0.4 and w_max = 0.9, respectively. The acceleration constants are adopted as c₁ = c₂ = 1.8. The maximum number of function evaluations is considered as 1000. Optimum tuning of TMD parameters for different alternative design scenarios ({A₁, A₂,…,A_m}) is conducted using the PSO algorithm. Thereafter, the values of the criteria ({C₁, C₂,…,C_c}) of the structure subjected to 44 earthquake excitation (listed in Table 3) are obtained for different alternative design scenarios ({A₁, A₂,…,A_m}). Subsequently, the decision matrix of the TOPSIS can be created using Eq. (1). The implementation of steps 2–8 of the TOPSIS leads to the ranking of the design scenarios of TMD parameters for the structure subjected to earthquake excitations.

Results and discussions

The PSO algorithm is applied to tune the optimum TMD parameters considering the alternative design scenarios presented in Table 1. The convergence histories of the PSO to minimize the cost function adopted for different alternative design scenarios in the unconstructed cases are illustrated in Fig. 1. Similarly, Fig. 2 shows the convergence histories of the PSO to minimize the cost function adopted for different alternative design scenarios in the constructed cases. The cost function for different design scenarios of TMD parameters in unconstrained and constrained cases are shown in Fig. 3. Additionally, the corresponding optimum damping and stiffness coefficients of the TMD obtained from the PSO algorithms for different alternative design scenarios in two unconstrained and constrained cases are indicated in Figs. 4 and 5. Notably, the optimum TMD mass for all cases is given as (m_TMD)_Opt = 108 t.

In this study, three additional design scenarios, which are introduced in the extant literature for the unconstrained case study, are considered for comparison purposes. Design scenario 11 was proposed by Hadi and Arfiadi [²⁹], in which a genetic algorithm was utilized for minimizing H₂ norm of the transfer function from the external disturbance to a certain regulated output as an optimization criterion, thereby tuning the optimum TMD parameters as (m_TMD)_Opt = 108 t, (c_TMD)_Opt = 115.15 kN·s/m, and (k_TMD)_Opt = 3750 kN/m. Lee et al. [³⁰] defined a performance index of a structural response in the frequency domain as a design scenario of TMD parameters, which is considered as design scenario 12. They reported the optimum TMD parameters for the case study as (m_TMD)_Opt = 108 t, (c_TMD)_Opt = 271.79 kN·s/m, and (k_TMD)_Opt = 4126.93 kN/m. Keshtegar and Etedali [⁴⁸] considered the mean square displacement response of the equivalent SDOF system as an objective function and proposed nonlinear mathematical models for an optimum design of TMD. For this case study, they optimized the TMD parameters as (m_TMD)_Opt = 108 t, (c_TMD)_Opt = 144.6 kN·s/m, and (k_TMD)_Opt = 3586.6 kN/m, which is considered as design scenario 13.

To find the best design scenario of TMD for seismic-excited structure with respect to functional, operational, and economic perspectives, the TOPSIS, as a powerful decision support tool, is utilized in this study. The TOPSIS can design a scenario, which is closest to the ideal solution and farthest from the negative ideal solution. Accordingly, the corresponding optimum TMD parameters for each alternative design scenario are assigned to the TMD, and the values of the criteria ({C₁, C₂,…,C₉}) are obtained for the structure subjected to 44 earthquake excitations. Thereafter, the decision matrix of the TOPSIS is created using Eq. (1).

Table 4 presents the decision matrix for the unconstrained case. Moreover, the decision matrix for the constrained case is shown in Table 5. The ranking of the design scenarios of TMD parameters for the structures subjected to earthquake excitations is obtained via the implementation of steps 2–8 of the TOPSIS. The criteria of weight obtained via the entropy method are listed in Table 6. Furthermore, the distance from ideal and nadir solutions and the closeness coefficient of each alternative is presented in Table 7. Consequently, Figs. 6 and 7 display the ranking of the design scenarios of TMD parameters for the unconstrained and constrained cases, respectively. The best design scenarios for the optimum TMD design based on the performance, practical, and economic perspectives are A₈ for the unconstrained case and A₁ for the constrained case. This implies that the maximum story displacement is the best cost function for the optimum design of TMD parameters in the unconstrained case i.e., the case without a limitation for the maximum stroke of TMD.

By considering a limitation for the maximum TMD stroke, the best result is obtained for the design scenario that considers the H₂ of the displacement vector of the structure as a cost function. Meanwhile, the worst design scenarios are A₁₀ and A₁₂ for the constrained and unconstrained cases, respectively. These results can be essential for the optimum design of TMD parameters for seismic-excited structures.

In Figs. 8–11, the time responses of the structure without TMD during the Northridge earthquake (NORTHR/MUL009) are compared with those for the structure equipped with TMD optimized based on the best design scenario of TMD for the unconstrained and constrained cases. The results indicate that the optimized TMDs in both cases effectively reduce the maximum displacement of the top floor, drift of the first floor, acceleration of the top floor, and base shear of the structure. Furthermore, a significant reduction in the aforementioned time responses were observed during the earthquake.

Conclusions

In this study, different design scenarios in two cases, namely unconstrained and constrained for the maximum TMD stroke, were evaluated to tune TMD parameters in seismic-excited structures. By considering a 10-story building, a design process based on PSO algorithm was utilized for the optimum design of TMD parameters. Subsequently, nine criteria were defined based on functional, operational, and economic perspectives, and a TOPSIS was implemented to compute the ranking of the design scenarios of TMD parameters for the structures subjected to earthquake excitations. Consequently, the minimization of the displacement of the highest story corresponded to the best design scenario of TMD for the seismic-excited structure in the unconstrained case. However, minimization of the H₂ of the displacement vector of the structure corresponded to the best ranking design scenario among the adopted scenarios in the constrained case. These conclusions can guide designers in the optimum design of TMD parameters with respect to functional, operational, and economic perspectives for the seismic-excited structures.

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