Passive energy devices are well known due to their performance for vibration control in buildings subjected to dynamic excitations. Tuned mass damper (TMD) is one of the oldest passive devices, and it has been very much used for vibration control in buildings around the world. However, the best parameters in terms of stiffness and damping and the best position of the TMD to be installed in the structure are an area that has been studied in recent years, seeking optimal designs of such device for attenuation of structural dynamic response. Thus, in this work, a new methodology for simultaneous optimization of parameters and positions of multiple tuned mass dampers (MTMDs) in buildings subjected to earthquakes is proposed. It is important to highlight that the proposed optimization methodology considers uncertainties present in the structural parameters, in the dynamic load, and also in the MTMD design with the aim of obtaining a robust design; that is, a MTMD design that is not sensitive to the variations of the parameters involved in the dynamic behavior of the structure. For illustration purposes, the proposed methodology is applied in a 10story building, confirming its effectiveness. Thus, it is believed that the proposed methodology can be used as a promising tool for MTMD design.
The development of damping devices dates back to the beginning of the twentieth century when Hermann Frahm invented a device for damping vibrations in bodies, which was patented, as presented by Frahm [
Recently, a rapid increase in the development and application of passive energy dissipation devices, such as base isolation systems [
The TMD considered in this paper is a conventional one, which is a passive control device consisting of a mass, a spring, and a viscous damper attached to a vibrating system to reduce undesirable vibrations. Due to its performance to reduce the response of structures to harmonic or random excitations, a large number of TMDs has been installed in highrise buildings to reduce windinduced vibrations, such as the 244 m high John Hancock Tower in Boston with a TMD consisting of two 270,000 kg lead and steel blocks, the 280 m high Citicorp Center Office Building in New York, with a TMD using a 360,000 kg concrete block, and the Terrace on the Park Building in New York City, in which a TMD was installed to reduce the vibration induced by dancing [
A single TMD performs well in reducing the dynamic response of a structure under external excitation when the device is tuned to the first vibration mode of the structure [
The performance of MTMD depends on its parameters such as mass, stiffness, and damping. However, determining the number of devices to be installed and the best position in the structure, as well as optimum parameters in terms of spring stiffness and damping constant for each TMD, is a problem of great interest to the engineer designer.
In order to solve the problem mentioned above, optimization algorithms are used to minimize an objective function and to find an optimal solution of the problem. On the other hand, it is well known that in a dynamic engineering problem, there are a high number of uncertainties involved. This leads to represent these uncertainties through probability distribution functions and involve them in the optimization process of passive dampers. Thus, the optimization process becomes more complex, and it is necessary to implement an optimization methodology capable of dealing with dynamic problems that involve uncertainties in the structural properties, in the MTMD properties, and in the seismic load.
Thus, this work presents a methodology of optimization under uncertainty to determine the optimal parameters of MTMD and its best positions in a single stage, i.e., simultaneously, in buildings subjected to earthquakes, with the aim of improving dynamic structural response in terms of minimizing maximum interstory drift. It is interesting to highlight that the optimization problem proposed in the present work is complex because (i) it is a problem of optimization of a dynamic system that involves uncertainties, (ii) it is a mixedvariable optimization problem, i.e., that involves discrete (position of each TMD) and continuous (parameters of each TMD) variables at same time, and (iii) its objective function is not convex.
Consequently, the problem of optimization under uncertainty of MTMD proposed in this work must be solved with the help of optimization methods able to deal with the complexity of this problem. In this case, the most appropriate is the implementation of a metaheuristic optimization technique, and some of its most important advantages are follows: (i) they do not require gradient information, (ii) they are not trapped in local minimums if they are adjusted correctly, (iii) they can be applied to nonconvex or discontinuous objective functions, (iv) they provide a set of optimal solutions, and (v) they can be implemented to solve optimization problems of mixed variables [
Among the heuristic algorithms, the Search Group Algorithm (SGA), recently proposed by the last author of this paper [
This section presents the methodology proposed for the simultaneous optimization of MTMD taking into account the uncertainties. It presented the equations and procedures adopted to the problem formulation.
The motion equation of a multidegreeoffreedom building with MTMD possibly located in all floors of the structure (Figure
The TMDs contribution to
To solve equation (
In this work, it is adopted the parametric probabilistic approach to model uncertainties. This methodology is similar to the used by the authors in [
In addition, to consider uncertainties in the installed MTMD, their parameters of spring stiffness and damping constant are also assumed to be independent Lognormal random variables with known coefficients of variation and mean values given by the design variables.
It is necessary to define the seismic loading to solve equation (
Nevertheless, the optimal solution possibly will be different if the ground parameters of the Kanai–Tajimi spectrum are altered. Therefore, uncertainties in the ground excitation should be considered. Thus, to take into account the random nature of the dynamic excitation, the ground frequency
In this paper, the objective function used to evaluate the effectiveness of MTMD installed in buildings under seismic excitation is the expected value of the maximum interstory drift
The design variables are the MTMD parameters, i.e., spring and damping constants, considered as continuous design variables, and the positions in the structure of the MTMD, considered as discrete design variables.
Therefore, given the possible positions (
This optimization problem may be solved through the Search Group Algorithm summarized in the next section.
As explained previously, the optimization problem presented in this work is complex, involving uncertainties and mixed variables and not convex objective function. Therefore, this sort of optimization problem must be solved by methods capable of handling these characteristics. Within optimization methods, heuristics techniques are best suited to solve such optimization problems.
The Search Group Algorithm (SGA), developed by the last author of this paper in 2015 [
The SGA has a good balance between the exploration (the search of promising regions on the domain at the first iterations of the optimization process) and exploitation (the algorithm refines the best design in each of these promising regions at each iteration).
The first step in the optimization process is the random generation of the initial population PP on the search domain; the second step is the objective function evaluation for each individual of the PP population, and after that, the search group R is constructed by selecting
For more details about the SGA, refer [
To illustrate the effectiveness of the proposed method in optimum design of MTMD, as well as to evaluate the capacity of MTMD in improving the performance of structures under seismic excitation, a 10story building, modeled as shear building (Figure
As explained previously in Sections
Mean value and coefficient of variation of each input random variable.
Random variable  Mean value  Coefficient of variation (%) 

Mass per story  360 t  5 
Stiffness per story  650 MN/m  5 
Damping per story  6.2 MNs/m  5 
Spring constant for each TMD  Design variable  5 
Damping constant for each TMD  Design variable  5 
PGA  0.475 g  10 

18 rad/s  10 

0.6  10 
This work proposes a methodology for robust optimization of MTMD installed in structures subjected to artificial seismic excitation taking into account the uncertainties present in both structure and excitation, in order to minimize the expected value of the maximum interstory drift. Thus, as explained previously, many parameters are modeled as random variables.
In this context, in order to reduce computational cost, the Latin hypercube sampling (LHS) is used, which provides an efficient way of generating variables from their multivariate distributions, taking samples from equally probable intervals [
The robust design optimization of MTMD in order to minimize the expected value of the maximum interstory drift
Considering the 10story building, modeled as shear building, i.e., 10 degree of freedom, there are ten possible locations to install a maximum of ten TMD (one for each story). Thus, constraints are the number of possible locations for the dampers (
Regarding the parameters of the SGA, it is considered that the population
Robust design of MTMD.
Run  Positions 





0.03941  
1  [0000000111]  1313.857; 915.187; 1468.914  43.358; 200.407; 11.058  0.01588 
2  [0000000111]  1439.044; 914.560; 1426.387  43.400; 205.440; 11.314  0.01595 
Table
Statistical moments of maximum interstory drift.
Uncontrolled  Robust design  Reduction (%) 

 
0.03941  0.01588  59.71 
 
2.3971 
4.7513 – 6  80.17 
The percentage reduction of the expected value and the variance of
In addition, Figure
Probability density function of maximum interstory drift
Looking at Figure
To demonstrate the effectiveness of the proposed method in different ways, the optimum solution obtained in simulation 1 of Table
Comparison between robust design and alternative methods.
Method  Positions 




Robust design  [0000000111]  1313.857; 915.187; 1468.914  43.358; 200.407; 11.058  0.01588 
Alternative 1  [0010010010]  1313.857; 915.187; 1468.914  43.358; 200.407; 11.058  0.01828 
Alternative 2  [1111111111]  369.796 for each one of the 10 TMDs  25.482 for each one of the 10 TMDs  0.02712 
Alternative 3  [0000000001]  3697.958  254.823  0.02121 
Alternative 4  [0000000001]  4296.981  115.874  0.01617 
Alternative 5  [0000000111]  1419.331; 1448.020; 1447.742  39.478; 312.519; 9.683  0.01603 
As can be seen in Table
For purposes of illustration, considering only the expected value of the structural properties, that is, coefficient of variation equal to zero for all parameters, and a seismic excitation generated using the expected value of the parameters and assuming that the coefficient of variation is zero, Figure
Maximum interstory drift per story for uncontrolled structure (red curve) and controlled structure (blue curve), for coefficient of variation equal to zero for all parameters.
Next, in Table
Comparison between maximum interstory drift.
Story  Uncontrolled structure (m)  With control (m)  Reduction (%) 

1  0.0383  0.0148  61.35 
2  0.0381  0.0147  61.39 
3  0.0374  0.0141  62.30 
4  0.0357  0.0132  63.05 
5  0.0330  0.0120  63.50 
6  0.0294  0.0115  60.80 
7  0.0248  0.0107  56.74 
8  0.0194  0.0091  53.29 
9  0.0133  0.0068  49.32 
10  0.0068  0.0037  45.65 
Thus, as can be seen in Table
Interstory drift at first story for uncontrolled structure (red curve) and controlled structure (blue curve).
Displacement at top of the structure for uncontrolled structure (red curve) and controlled structure (blue curve).
It is well known that passive dampers increase the energy dissipation capacity in buildings. Thus, in recent years, engineers have been concerned with the optimal implementation of passive energy dissipation devices and among the most used passive devices is the TMD.
However, until nowadays, several research works have not considered the uncertainties present in the structure and in the parameters of the device. For this reason, the main contribution of this research is a methodology that provides an optimal and robust design of multiple tuned mass dampers (MTMDs). The methodology developed considers the uncertainties in the mechanical properties of the structure, in the mechanical properties of the MTMD, and also in the properties used for the generation of artificial earthquakes.
The proposed methodology is constituted by the SGA optimization algorithm that is able to provide in a single stage, i.e., simultaneously, the optimum values of the mechanical parameters of MTMD and their positions in the structure. On the other hand, the performance of the proposed methodology is evaluated with a computational routine developed by the authors based on the Newmark method that allows computing the structural response of buildings subjected to seismic excitation and equipped with MTMD. To consider uncertainties in the parameters involved, the Monte Carlo simulation was used to determine the expected value of the maximum interstory drift in the structure, that is, the objective function to be minimized.
It is interesting to note that the response reduction performance was expressed in terms of reduction of the expected value of the maximum interstory drift of the building; however, the proposed methodology is flexible, allowing the user to change the objective function.
Additionally, the methodology proved to be robust, since, after two independent runs, it delivered two very similar solutions, that is, the same number of TMDs with similar mechanical parameters and located in the same positions (floors 8, 9, and 10). Both solutions allowed reducing the objective function around 60%.
Moreover, the comparison of the proposed methodology with five alternative methods showed that the proposed method resulted in the lowest maximum interstory drift in all cases. The second and third best results were obtained with alternative methods 5 and 4, respectively; it is important to note that these two alternative methods (4 and 5) perform a robust optimization following the proposed methodology, only changing the SGA by GA (in the case of alternative method 5) and fixing only 1 TMD at the top and optimizing its parameters with the proposed methodology for the case of the alternative method 4. Therefore, these two alternative methods (4 and 5) also serve to prove the effectiveness of the methodology proposed in this work.
It is also interesting to highlight that, for a usual PC (an Intel Core i74700MQ 2.4 GHz CPU and 12 GB RAM), the computational cost required to carry out the proposed robust optimization was satisfactory for this sort of dynamic problem, highlighting another advantage of the developed methodology.
Finally, due to its performance, the proposed methodology can be recommended as an effective tool to carry out the optimum design of MTMD. Thus, this work showed that the design of passive devices for the vibration control as MTMD can be accomplished in an economic and safe way, reducing costs and optimizing the resources.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge the financial support of CAPES and CNPq, Brazil.