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Designing optimal tuned mass dampers for nonlinear frames bydistributed genetic algorithms

Mohtasham Mohebbi1,*, and Abdolreza Joghataie2

1Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran2Civil Engineering Department, Sharif University of Technology, Tehran, Iran

SUMMARY

In this paper, the capabilities of tuned mass dampers (TMDs) for the mitigation of response of nonlinearframe structures subjected to earthquakes have been studied. To determine the optimal parameters of aTMD, including its mass, stiffness and damping, we developed an optimization algorithm based on theminimization of a performance index, defined as a function of the response of the nonlinear structure to becontrolled. Distributed genetic algorithm has been used to solve the optimization problem. For illustration,

the method has been applied to the design of a linear TMD for an eightstory nonlinear shear building withbilinear hysteretic material behavior subjected to earthquake. The results have shown that the method hasbeen successful in determining the TMD parameters to reduce the structure response. The simplicity anddesirable convergence behavior of the method have also been two important results of the method. Twoperformance indices have been defined: (a) the minimization of the maximum drift and (b) the accumulatedhysteretic energy. It has also been shown that the efficiency of the TMD has been influenced by the massratio of the TMD, the maximum TMD stroke length and the TMD design earthquake. Copyright 2011John Wiley & Sons, Ltd.

Received 17 August 2010; Revised 4 January 2011; Accepted 3 March 2011

KEY WORDS: tuned mass damper (TMD); distributed genetic algorithm (DGA); accumulated hysteretic energy(AHE); nonlinear; optimization; passive control

1. INTRODUCTION

Tuned mass dampers (TMDs) are passive control devices that are added to the structures to absorb

energy from the primary structure, consequently reducing the response (displacement, acceleration

or internal force) of the main structure (McNamara, 1977; Soong and Dargush, 1997). In some

cases, TMDs have been installed in actual buildings (Spencer and Nagarajaiah, 2003). Since the

effectiveness of a TMD relies on its parameters including its mass, damping and stiffness, the

optimization of TMDs has been studied in structural control for many years, and different methods

have been proposed for the determination of the optimal values of TMD parameters. Den Hartog

(1956) studied the response of a TMDcontrolled structure subjected to sinusoidal excitation and

developed the basic principles and procedures to select the optimal parameters of a TMD when themain structure is undamped. Warburton and Ayorinde (1980), Tsai and Lin (1993) and Tsai and Lin

(1994) investigated the design of optimum TMDs when the main structure is damped and subjected

to different harmonic excitations. Warburton (1982) discussed the determination of the optimal

parameters of a TMD to minimize the root mean square (RMS) of the structure displacement or

acceleration under white noise excitation. Warburton and Ayorinde (1980) and Ayorinde and

Warburton (1980) idealized the main multidegree of freedom system as an equivalent single degree

* Correspondence to: Mohtasham Mohebbi, Assistant Professor, Engineering Faculty, University of MohagheghArdabili, Daneshghah Avenue, Ardabil 5619 911 367, Iran Email: [email protected]

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGSStruct. Design Tall Spec. Build. (2011)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/tal.702

Copyright 2011 John Wiley & Sons, Ltd.

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of freedom (SDOF) system in designing the optimal TMDs for elastic structures. Other methods

have been proposed for designing TMDs with different objective functions such as the

maximization of the effective damping of a structure equipped with a TMD (Luft, 1979),

minimization of the difference between the damping of the first two modes of the structureTMD

system (Sadek et al., 1997) by extending the work of Villaverde (1985), minimization of the H2norm of the transfer function of the external excitation to a regulated output (Hadi and Arfiadi,

1998), minimization of various mean square of responses to develop explicit formulas for TMDparameters attached to viscously damped main systems under various external loads (Bakre and

Jangid, 2007) and optimization of TMD for structures subjected to nonstationary base excitation by

using particle swarm optimization method (Leung et al., 2008). Also, Wong and Chee (2004)

studied the effect of applying TMDs in reducing various forms of energy of structures subjected to

earthquakes and the process of structural energy transfer. The results of numerical simulations have

shown that using TMDs is more effective in reducing the energy response of the structures with

moderate to long period. Most of the proposed methods have been developed for application to

linear structures with constant properties so that the tuning of the TMD could be performed based

on the constant properties of the structure. However, in reality, many buildings undergo large

deformations or even yielding when subjected to ground shaking, where they exhibit nonlinear

elastic or inelastic behavior. Consequently, the control system should be capable of dealing with

nonlinear structures. For structures under moderate and severe earthquakes, which show nonlinear

behavior, the stiffness degradation or uncertainty in the stiffness estimation may cause the TMD to

be detuned, which leads to the decreasing of the efficiency of the TMD. Application of TMDs to

nonlinear structures has been studied in some researches. Jagadish et al. (1979) studied the

reduction in the response of twostory inelastic buildings subjected to Taft (1952) earthquake where

the top story acts like a vibration absorber. They concluded that a significant reduction in ductility

demand on the lower story could be obtained. Kaynia et al. (1981) made a preliminary assessment

of the seismic effectiveness of TMDs for elasticperfectly plastic SDOF structures under earthquake

motion. The results have shown a small reduction in the cumulative yielding ductility, whereas the

reduction in the ductility ratio has been insignificant. Sladek and Klinger (1983) studied the effect

of TMD in reducing the seismic response of a 25story building with stiffness degradation under

the 1940 El Centro excitation. It was concluded that the maximum response of the structure was

almost unaffected and the TMD could not help eliminate the yielding at the base of the structure.

SotoBrito and Ruiz (1999) investigated the effect of ground motion characteristics on theeffectiveness of TMDs in reducing the maximum displacement of a 22story nonlinear structure

under moderate and highintensity earthquakes. The results have shown that although the

installation of a TMD on a nonlinear structure could not provide desirable results in reducing the

maximum displacement under severe earthquakes, it has been successful for moderate and low

density earthquakes. Lukkunaprasit and Wanitkorkul (2001) and Pinkaew et al. (2003) have

proposed that to indicate the accumulated damage induced in nonlinear structures, a measure based

on the ratio of the accumulated hysteretic energy (AHE) absorbed in the structure with TMD to that

of the same structure without TMD should be used in conjunction with the maximum displacement

of the structure to assess the effectiveness of the TMDs. Wong (2008) studied the effectiveness of

TMDs on inelastic structures subjected to earthquake ground motion from an energy perspective.

Based on numerical simulations, it has been concluded that using TMDs could improve the ability

of the structures to withstand strong earthquakes by storing a larger amount of energy at the criticalmoments and transferring this energy to the structure in the form of damping energy at a later time

when the response of structure is not at critical state. In previous researches on the application of

TMD to nonlinear structures, the studies focused on the effectiveness of TMDs by considering

specific values for the TMD parameters or performing sensitivity analysis to assess the effect of the

parameters on the performance of the TMD. There has not been any systematic method to design

the optimal TMDs for nonlinear frames. Therefore, in this paper, a method has been developed to

design the optimal TMDs for nonlinear frames based on defining an optimization problem and

minimizing a specified objective function to determine the parameters of TMDs.

Genetic algorithms (GAs) have been applied successfully to the optimization of various problems in

structural engineering such as design of trusses (Rajeev and Krishnamoorthy, 1997) and optimal

M. MOHEBBI AND A. JOGHATAIE

Copyright 2011 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. (2011)

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controllers (Ahlawat and Ramaswamy, 2002). Hadi and Arfiadi (1998) used GA to find the optimal

values of TMD parameters by minimizing the H2 norm of response of linear structures under white

noise excitation.

Joghataie and Mohebbi (2007) have used distributed GA (DGA), which is an improved version of

the conventional GA, in a preliminary study on the optimization of TMDs on nonlinear frames and

also the optimal design of active controllers for nonlinear frames (Joghataie and Mohebbi, 2011). In

this paper, too, DGA has been used to solve the optimization problem that has been defined todetermine the parameters of TMD.

In the following sections, first, the equations and algorithm for the DGAbased optimal design of

TMDs for nonlinear frames are derived. Next, a brief explanation of the DGA is presented followed by

an application example to explain the procedure of the proposed algorithm and conclusions.

2. SOLVING COUPLED STRUCTURETUNED MASS DAMPER EQUATIONS OF MOTION

In this section, Newmark integration method has been used to solve the coupled equations of

dynamics of a structure and its TMD. The equation of motion of a controlled nonlinear n degree of

freedom structure equipped with a TMD placed on its top can be considered as follows:

MX::

t FD X t

FS X t MeX::

g (1)

where t= time; X::

g = ground acceleration; X, X:

and X::

= displacement, velocity and acceleration vectors

relative to the ground, respectively; M= (n + 1 ) (n + 1), mass matrix; FD = (n + 1)dimensional vector of

damping forces, which is a function of velocity; FS = (n + 1)dimensional vector of restoring forces, which

is a function of displacement; e = [1,1,,1]T = (n + 1)dimensional ground accelerationmass

transformation vector.

The equation of motion during the time interval ((k1)t, (k)t) can be written as follows:

MX::

t C* X t K*X t F t (2a)

where

X::

t X::

kX::

k1 (2b)

X t Xk Xk1 (2c)

Xt XkXk1 (2d)

F t Me X::

gkX

::g

k1

(2e)

Also, k= integration time step and C*k1 and K*k1 = tangential damping and stiffness matrices at

t= (k 1)t, respectively. The elements of these matrices at time step k 1 are as follows:

C*ijk1 FDik1

Xjk1

i;j 1; 2;; n (3a)

OPTIMAL TMDS FOR NONLINEAR FRAMES


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k*ijk1 FSik1Xjk1

i;j 1; 2;; n (3b)

In discrete numerical analysis, the values of the elements of C* and K* matrices at each time step

can be obtained from the hysteresis curve based on the values of the relative displacement and velocity

vectors at that time step.

2.1. Newmark method

The response of a nonlinear structure can be obtained by solving the set of Equations (2ae) according

to Newmark method as follows:

Xk

Xk1

Xk

(4a)

X:

k 1a5 X

:k1a6X

::k1

a4Xk (4b)

X::

k 1a3 X::

k1a2X

:k1

a1Xk (4c)

Xk K*n

1

kFk (5)

K*nk a1M a4C*k1 K

*k1 (6)

Fk FkFk1 M a2X:

k1 a3X::

k1

C*k1 a5Xk1 a6X

::k1

(7)

a1 1

t 2; a2

1

t; a3

1

2; (8a ; b; c)

a4

t; a5

; a 6 t

21

; (8d; e; f)

where K*n k varies at each time step and , = Newmark parameters. In this study, = 0.5 and = 0.25have been used for nonlinear analysis of the system to assure the stability of numerical integration

(Bathe, 1996), where the integration time interval has been t= 0.002 seconds to achieve the required

accuracy.

3. OPTIMAL DESIGN OF TUNED MASS DAMPERS

In this paper, the optimum values of TMD parameters have been determined by minimizing an

objective function. Noticing that the purpose of using TMDs has been to minimize the maximum

relative displacement (drift), acceleration, internal force or AHE for a given structure, we have



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developed a constrained optimization problem that includes the constraints on the TMD displacement

and its parameters (mass, stiffness and damping), too, as follows:

Find

md; cd; kd (9a)

Minimize

FV fv1; v2;; vp (9b)

Subject to

giv1; v2;; vp0:0 i 1; 2;; q (9c)

hjv1; v2;; vp 0:0 j 1; 2;; r (9d)

where md, cd and kd are the mass, damping and stiffness of the TMD, respectively; V is the

pdimensional vector used to define the objective function and constraints of the optimization problem;

and gi and hi are the inequality and equality constraints, respectively. q and r show the number of

inequality and equality constraints. Because the optimization problem in the case of nonlinear structures

is sophisticated and nonlinear, using traditional optimization techniques such as gradientbased methods

is considerably complicated, especially for problems with a large number of variables. Hence, a

powerful algorithm is needed to solve the problem. Both GA and its improved version, the DGA, could

be used. In this paper, DGA (Starkweatheret al., 1990; Mhlenbein et al., 1991), which has shown better

convergence characteristics than GA, has been used for better convergence.

In some of the previous researches on the application of TMDs to nonlinear structures, the

parameters of optimal TMDs were determined through minimization of the maximum relative

displacement and AHE (Lukkunaprasit and Wanitkorkul, 2001; Pinkaew et al., 2003). The same idea

has been followed in this paper; however, first, the problem of designing TMDs has been categorized

based on the design criteria used as either belonging to case a, minimizing the maximum relative

displacement (drift), or case b, minimizing the maximum AHE, as explained below.

3.1. Case aminimizing the maximum relative displacement (drift)

The most commonly used criterion to assess the effectiveness of a TMD installed on a structure is its

capability to reduce the maximum relative displacement (drift) of the structure. In this case, defining t0and tf as the initial and final times of control, the optimization problem consists of an objective

function, which includes minimizing the maximum relative displacement, umax, and correspondingconstraints on the TMD stroke length (drift) and its parameters as follows:

Find

md; cd; kd (10a)

Minimize

umax maxjukij; k 1; 2; ::::; kmax i 1; 2;:; n (10b)



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Subject to

umaxtmduL (10c)

0< md < mdmax; 0< kd < kdmax; 0< cd < cdmax (10d; e; f)

where t= sampling time interval, kmax = (tf t0)/t= total number of time steps and uk(i) = relative

displacement (drift) of ith story at time step k defined as follows:

uki XkiXki1 i 2; 3;; n (11a)

uk1 Xk1 (11b)

uL, mdmax, kdmax and cdmax are the maximum stroke length and upper limit of mass, stiffness and

damping of the TMD, respectively. These values can be defined by the designer considering practical

limitations. The problem was first reformulated as an unconstrained optimization problem following

the penalty method (Rao, 1984) where the complicated constrained optimization problem is reduced to

an unconstrained optimization problem by introducing a new objective function, which is the sum of

two terms: the original objective function plus a penalty term for the violation of the constraints of the

original problem. Through the minimization of the new objective function, both the minimum of the

original problem is obtained, whereas no constraint has been violated. The optimization problem has

been as follows:

Find

md; cd; kd (12a)

Minimize

FT umax max 0; g1 (12b)

g1 umax tmd

uL1 (12c)

and are penalty parameters that can be determined either through parametric study or by trial

and error. Joghataie and Mohebbi (2008) have studied the effect of choosing different values for

and and have concluded that although they can change the convergence speed, they have nosignificant effect on the final value of the optimal answer. Different unconstrained optimization

algorithms can be used to solve the problem where DGA has been used in this paper.

By assuming different values for the TMD mass ratio, =md/mtot, where mtot denotes the total

mass of the structure, the optimum parameters of the TMD have been determined so that the

maximum drift of the uncontrolled structure can be minimized.

3.2. Case bminimizing the maximum accumulated hysteretic energy

Although in many researches, the most commonly used index to assess the effectiveness of application

of TMD to nonlinear structures is the reduction of the maximum relative displacement (drift), this



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index cannot account for the effect of the accumulation of damage that occurs in a nonlinear structure

under earthquake excitation (Lukkunaprasit and Wanitkorkul, 2001; Pinkaew et al., 2003). Therefore,

another index to reflect the amount of cumulative damage, called the AHE absorption, has been

defined for a nonlinear hysteretic structure as follows:

AHE i 1;j AHE i;j

fs i fs i1 2 ui1 j ui j (13)

where AHE(i,j), fs(i) and ui(j) are the AHE, shear force and relative displacement (drift) of the story j

at time it. The total hysteretic energy can be determined by the summation of the hysteretic energy

over the excitation time t. To design the TMD to minimize the maximum AHE, we can write the

optimization problem by applying the constraint as penalty as follows:

Find

md; cd; kd (14a)

Minimize

FT AHEmax max 0; g1 (14b)

where the maximum hysteretic energy, AHEmax, is defined as follows:

AHEmax max j kmax

i1

AHE i;j j j 1; 2;:; n (14c)with , and g1 similarly defined as in case a.

4. DISTRIBUTED GENETIC ALGORITHM

The algorithm of traditional GA has been well explained in many references (Goldberg, 1989;

Michalewicz, 1996) and used in many engineering applications successfully such as structural control

systems (Hadi and Arfiadi, 1998; Joghataie and Mohebbi, 2007, 2008). When traditional GA is

applied to problems with a large number of variables, the convergence is generally slow, and the final

answer is achieved mostly after a large number of generations. In such problems, it is suggested to

divide the population of individuals into Nsub subpopulations of smaller sizes, where the traditional

GA is executed on each subpopulation separately. In this process, which is called the DGA, a smaller

number of individuals are expected to lead to quicker convergence and higher searching capability as

compared with the conventional GAs (Starkweather et al., 1990; Mhlenbein et al., 1991). In DGA,some individuals migrate from one subpopulation to the others periodically according to a migration

strategy, such as the ring topology, neighborhood and unrestricted migration. In this paper,

unrestricted migration strategy has been used in which at each migration interval, the immigrant

individuals from a subpopulation can migrate to any other subpopulation simultaneously. The number

of generations between each two successive migrations and the percentage of individuals selected for

migration from each subpopulation at the time of migration are determined based on m_in = migration

interval and m_rate = migration rate parameters of DGA. Because of the smaller number of variables

in this study, both GA and DGA could be used to solve the optimization problem, but for better

convergence, DGA has been selected. Since for realvalued numerical optimization problems, such as

the design of control system in this paper, it has been preferred to use the realvalued coding



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representation, which has certain advantages such as simpler programming, less memory required and

more freedom to use different genetic operators (Arfiadi and Hadi, 2001; Jenkins, 2002), it has been

decided to use realvalued coding for representing the variables.

For the fitness function that is used to transform the value of the objective function into a measure

of relative fitness, the method proposed by Baker (1985) has been adopted where a fitness value has

been assigned to each individual based on its rank in the population.

In this paper, the stochastic universal sampling method (Baker, 1987) has been used for selectingthe chromosomes for mating, based on theirfitness values in the current population. The probability

that a chromosome can be selected has been as follows:

Pind xi Find xi

Nind

i1

Find xi

; i 1; 2;; Nind (15)

where Find(xi) is the fitness of chromosome xi, Pind(xi) is the probability of its selection and Nind is the

number of individuals.

The intermediate recombination method (Mhlenbein and SchlierkampVoosen, 1993) has been

used for crossover, where the values of genes of the newborns are obtained through linearinterpolation and extrapolation on the values of genes of the parents as follows:

G P1 P2P1 (16)

where G is the value to be determined of a gene of a newborn, P1 and P2 are the values of the

corresponding genes of the parent individuals and is a scale factor chosen randomly over [0.25,

1.25] typically. This method uses a new for each pair of genes.

For the mutation that is an operator for providing a guarantee that the probability of searching any

given string will never be zero, also helps GA to escape local minima (Goldberg, 1989) by randomly

selecting individuals, the algorithm presented by Mhlenbein and SchlierkampVoosen (1993) has

been used.

4.1. Elite strategy

To maintain the size of the original population, we have to reinsert the new individuals into the old

population. The insertion rate, , is used to determine the number of newly produced individuals

(Nins), which are to be inserted into the old population according to the following:

Nins Nnew (17)

where Nnew = number of newborns and is considered within 0.800.90. Nelites of the best individuals

in the current population are selected as elites of the current generation to go to the next generationwithout modification. The rest of the individuals in the population are replaced by inserted newborns:

Nelites NindNins (18)

5. NUMERICAL EXAMPLE

An eightstory shear building frame (Yang et al., 1988), as shown in Figure 1, has been considered to

examine the success of the proposed TMD optimal design algorithm presented in this paper in

improving the performance of structures under earthquake excitation. It has been assumed that the



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structure has been made of a bilinear hysteretic material, the behavior of which is as shown in Figure 2

where the elastic stiffness K1 =3.404105 kN/m and the postelastic stiffness K2 =3.40410

4 kN/m.

The floor mass is 345.6 tons (1 ton = 9810 N), and the natural frequency of the structure based on its

initial stiffness is finitial = 0.92 Hz. The linear viscous damping coefficient c has been assumed to be

734.3 kN s/m, which corresponds to a 0.5% damping ratio of the first vibration mode of the structure.

Yielding occurs at a lateral relative displacement of uyielding = 2.4 cm for each story.

Figure 1. Structuretuned mass damper model.

Figure 2. Nonlinear bilinear elasticplastic stiffness model.



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To focus on the main problem and to avoid complexity in the system at this stage of the research,

the proposed method has been developed for 2D frames and under design criteria a and b. The

extension of the method to include 3D structures and using other criteria are important issues that are

under study by the authors.

The uncontrolled structure was subjected to a white noise ground acceleration with peak ground

acceleration (PGA) = 0.4 g, denoted by W(t) and shown in Figure 3. This excitation has been selected

as a sample excitation to explain the proposed method and also for the assessment of the efficiency ofthe optimally designed TMD on the nonlinear structure. To design a controller for a speci fic site, we

can use in the analysis and optimization the design record for the site with a proper acceleration scale

factor or the multiple ground acceleration records.

The maximum drift, acceleration and AHE for each story of the uncontrolled structure have been

shown in Table 1, noticing that the AHE has been defined only for the stories that experience

nonlinearity. UnderW(t) excitation, the structure undergoes nonlinear behavior in stories 1, 2 and 3 as

reflected in Table 1 where the maximum relative displacement (drift) has exceeded the yielding

relative displacement, uyielding = 2.4 cm. Also, the first story has experienced the maximum relative

displacement (drift) in the structure where umax(uncon) = 4.75 cm 200% uyielding.

In the following sections, the optimal parameters of the TMD have been determined using DGA and

based on both criteria explained before, including minimizing (a) the maximum drift and (b) the

maximum AHE of the structure as defined under cases a and b.

5.1. Designing optimal tuned mass damper for = 0.7% based on case a criterion

It was desired to obtain the set of TMD parameters T* = (c*d,k*d) so that the maximum uncontrolled

drift was minimized and also the maximum stroke length of the TMD and its parameters T= (cd,kd)

Figure 3. Time history of white noise ground acceleration, W(t), with peak ground acceleration = 0.4 g.

Table 1. Response of uncontrolled and controlled structures when subject to a white noise and controlledby tuned mass damper with = 0.7%.

Storyno.

Uncontrolled controlled Controlled

Max. drift (cm) Max. acc. (cm/s2) AHE (N m) Max. drift (cm) Max. acc. (cm/s

2) AHE (N m)

1 4.75 632.7 396 348 2.90 615.1 54 8892 3.52 676.4 91 491 2.35 641.5 3 2.47 678.4 1314 2.15 673.8 4 2.21 683.2 2.02 707.1 5 1.78 762.3 1.71 741.8 6 1.46 838.5 1.28 775.8 7 1.12 847.6 0.94 690.1 8 0.65 807.5 0.55 749.0

AHE, accumulated hysteretic energy; max., maximum; acc., acceleration.



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remained within specified limits. By assuming uL =0.5m, kdmax =3.404105 kN/m and cdmax = 734.3

kN s/m for the parameters of Equations (10cf), the DGA is used to solve the optimization problem

when the parameters of the DGA has been selected as follows:

Nsub = number of subpopulations = 2, Nind = number of individuals in each subpopulation = 25,

Nelites = 3, Nnew = number of the newborns = 25, = insertion rate = 0.9, mr= mutation rate = 0.05,

m_in = migration interval = 10 and m_rate = migration rate = 0.20.

Also, the values of and in Equation (12b) have been selected as follows:

1

uyielding

1

2:4 cm; 200 (19a ; b)

Two subpopulations each with 25 randomly generated vectors of TMD parameters T= (cd,kd)

were generated as the initial population. The maximum drift of the coupled structureTMD

system were recorded. The objective function F(T) in Equation (12b) was calculated for each T.

Iteratively, the subpopulations were modified according to the DGA so that new populations were

generated until convergence was achieved. By monitoring the controlled response of the nonlinear

system from the individual Ts in every generation, the fittest individual of that generation was

identifi

ed. To guarantee the validity of the optimum vector T

*

= (c

*

d,k

*

d), different runs have beenperformed in DGA for = 0.7%. Figure 4(a) shows the convergence behavior of DGA toward the

optimum answer for four runs. All the runs have the same optimum answer but were obtained

with different convergence speeds. Also, the fitness value of the individuals at the final generation

has been shown in Figure 4(b), which shows that most individuals have the same value. The

Figure 4. (a) Convergence toward optimum answer for four runs; (b) fitness value of individuals at

final generation.



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optimum value has been T* = (c*d,k*d) = (17.06kN s/m, 5.6072 10

2 kN/m) with F(T*) = 1.208,

umax = 2.90 cm and umax(tmd) = 50 cm. Hence, it can be concluded that the proposed method has

been successful in determining the optimum values of TMD parameters. Table 1 shows the

maximum drift, acceleration and AHE of the controlled structure for = 0.7%. Results show that

about 39%, 8.5% and 86% reduction in the maximum drift, acceleration and AHE, respectively,

of uncontrolled structure has been achieved. The controlled and uncontrolled drift obtained for

stories 1, 2 and 3 has been shown in Figure 5, whereas the corresponding hysteresis loops havebeen compared in Figure 6. According to the results, it is clear that by the installation of a TMD

with = 0.7%, the nonlinearity observed at story 1 of the uncontrolled structure has been reduced.

Figure 5. Twenty seconds of uncontrolled and controlled drift of structure shown for the following:

(a) firstfloor; (b) second floor; (c) third floor.



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Also, the nonlinearity and consequently the AHE of the second and third stories of the

uncontrolled structure have been completely diminished.

5.2. Assessment of the effect of tuned mass damper maximum stroke length on tuned mass

damper performance

Following the same procedure explained for = 0.7% and uL = 0.5 m, the optimal sets of TMD

parameters have been determined to minimize the maximum drift under W(t) excitation, for different

values of maximum stroke length of TMD, uL when = 0.5%. The peaks of controlled response

corresponding to each uL value have been normalized by division to the peaks of the uncontrolled

response and plotted in Figure 7, where the maximum RMS of drift denoted by RMS max has been

defined as follows:

Figure 6. Hysteresis loops for the first, second and third floors of the uncontrolled and controlled

frames subject to a white noise and controlled by tuned mass damper when = 0.7%: (a) firstfloor;

(b) second floor; and (c) third floor.



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RMS u i kmax

k1

uk i 2

kmax

0BBB@

1CCCA

1

=2

i 1; 2;; n (20a)

RMSmax maxjRMSuij i 1; 2;; n (20b)

where k= 1, 2, , kmax represents the time step number.

It is clear that the reduction in the structure response has been affected significantly by the

maximum stroke length of TMD especially for smaller values of TMD stroke lengths. Results showthat increasing uL leads to more reduction in structural response so that even the response of the

structure can be kept in the linear domain. Figure 8(a, c) show the drift and hysteresis loops for the

first story of the uncontrolled and controlled frames for uL = 75cm and = 0.5%, which shows about

51% reduction in the maximum drift so that nonlinearity has been eliminated from all the stories.

Hence, noticing the significant effect of the maximum TMD stroke length on the TMD performance, it

is suggested that for real applications, the maximum acceptable value of uL should be selected as

precisely as possible.

The values of TMD parameters (kd and cd) obtained from optimization procedure have been shown

in Figure 9(a) for different TMD stroke length (uL) but the same TMD mass ratio where = 0.5%.

Based on the results, it can be said that the stiffness of the TMD has remained almost constant, which

shows that the optimum TMDs have approximately a constant frequency while the TMD damping has

been changing with uL.

5.3. Designing optimum tuned mass dampers for different values of mass ratio

Other optimum TMDs have been designed under W(t) excitation for different values of the mass ratio,

, where a maximum stroke length of uL = 1.5 m has been defined and the optimization problem of

Equations (12ac) has been solved. The reduction in maximum drift and acceleration for stories 1, 2, 3

and 4 has been shown in Figure 10 for different values of . It can be seen that the firststory

maximum drift has been reduced by about 57%, 63% and 58% for = 0.3%, 0.5% and 1%,

respectively. The results have shown that under W(t) excitation, for smaller values of TMD mass

ratio, by increasing the mass ratio of the TMD, its effectiveness to reduce the drift and AHE has increased,

whereas for 2%, the reduction in the maximum drift and AHE has been approximately constant.

Figure 7. Normalized maximum response of controlled structure for different tuned mass damper

stroke lengths when = 0.5%. AHE, accumulated hysteretic energy; RMS, root mean square.



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Assuming uL = 1.5 m, for mass ratios greater than = 0.3%, the reduction in the maximum

firststory drift has been greater than 50%, where the nonlinearity and consequently the AHE of

the structure have been eliminated. Figure 8(b) shows the maximum drift of the controlled and

uncontrolled structures for = 0.3%, which shows about 57% reduction in the maximum drift,

whereas the hysteresis loops of the controlled and uncontrolled structures have been compared

in Figure 8(c). Based on these results, it can be concluded that for the structure studied in this

paper, the performance of the TMD under W(t) excitation has been more sensitive to the mass

ratio for smaller values of TMD mass ratio, whereas larger values of TMD mass ratio have not

been of significant effect on the TMD performance. Also, although the TMDs have been

Figure 8. Twenty seconds of drift of uncontrolled and controlled structures shown for the firstfloor:

(a) = 0.5% and uL = 75cm; (b) = 0.35% and uL = 150 cm; (c) hysteresis loops of uncontrolled and

controlled structures.



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designed based on the minimization of the maximum drift, the maximum acceleration has been

reduced relatively, too.

Figure 9(b) shows the variation of TMD stiffness and damping for different values of . It can be

said that the stiffness of optimum TMDs has approximately a linear relation with TMD mass ratio for

most values of TMD mass ratio, whereas TMD damping does not vary linearly with .

5.4. Designing optimal tuned mass damper based on case b criterion

It was desired to assess the effectiveness of TMD in reducing the cumulative damage in the eightstory

frame of Figure 1. Hence, optimal TMDs have been designed based on minimizing the maximum

AHE. By assuming = 0.3%, 0.5%, 0.7% and uL = 0.5 m, the optimum TMDs underW(t) excitation

have been designed where, in Table 2, the corresponding values of TMD parameters (kd and cd) have

been reported. Also, Table 2 contains some details of the results where the maximum drift and AHE

both for the uncontrolled and controlled structures have been reported. Designing the optimal TMD

for a specified value of , based on minimizing the maximum AHE or maximum drift, has

approximately led to the same reduction in the maximum response.

5.5. Designing optimal tuned mass damper for real earthquakes based on case a criterion

To assess the effect of input excitation in designing the optimal TMDs for nonlinear frames, we have

designed optimal TMDs based on case a design criterion for different values of TMD mass ratio while

assuming uL = 1.5 m. The structure was subjected to El Centro (1940, PGA = 0.348 g), Hachinohe (1968,

PGA = 0.23g) and 200% San Fernando (1971, PGA = 0.315 g) excitations, which are farfield

earthquakes. The maximum response values of the controlled frame divided by their corresponding

Figure 9. Stiffness and damping values of optimal TMDs for (a) different values of TMD stroke

length when = 0.5%;(b) different values of TMD mass ratio when uL = 150 cm. TMD, tuned

mass dampers.



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maximum response values of the uncontrolled frame have been calculated and plotted in Figure 11. The

results show that using TMD has significantly mitigated the maximum response of the structure under

different earthquakes, although the amount of reduction in the maximum structural response depends on

the characteristics of the earthquake. Also, similar to the results obtained for white noise excitation,

increasing the value of the TMD mass ratio has led to its better performance under all applied excitations.

Figure 10. The reduction in maximum (a) drift and (b) acceleration offirst, second, third and fourth

stories (St. 1, St. 2, St. 3 and St. 4) plotted versus different values of TMD mass ratios, under W(t).

TMD, tuned mass damper.

Table 2. Response of uncontrolled and controlled structures and tuned mass damper parameters when usingcases a and b criteria for the optimal design of tuned mass damper.

Response and TMDparameters

Controlled structure

Uncontrolled

=0.3 =0.5 =0.7

Case a Case b Case a Case b Case a Case b

Max. AHE (N m) 243 771 238 113 153 958 146 846 54 889 57 529 396 348Max. drift (cm) 4.21 4.13 3.53 3.55 2.9 2.86 4.75Max. acc. (cm/s2) 832.0 837.8 814.2 793.4 775.8 769 847.6

Max. drift RMS (cm) 1.25 1.24 1.09 1.08 0.93 0.95 1.55kd (kN/m) 213 210 367 390 561 576 cd (kNs/m) 5.81 3.63 10.13 13.07 17.06 18.65

TMD, tuned mass damper; RMS, root mean square; AHE, accumulated hysteretic energy; max., maximum;acc., acceleration.



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6. CONCLUSIONS

In this study, a method to design the optimal TMDs for the mitigation of response of nonlinear

structures has been developed. The method is based on defining an optimization problem that

considers TMD parameters as variables and reduction of the structural maximum drift and/or the

maximum AHE as the objective function with constraints defined on the maximum TMD stroke

length and TMD parameters for the purpose of application limitations. DGA has been applied

Figure 11. Normalized maximum response of controlled frame plotted versus different values of

tuned mass damper mass ratios () under (a) El Centro (1940), (b) Hachinohe (1968) and (c)

200% San Fernando (1971) earthquakes. AHE, accumulated hysteretic energy; RMS, root mean square.



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successfully to solve the optimization problems to find the optimum set of TMD parameters, including

its stiffness and damping. The method has been applied to an eightstory hysteretic bilinear elastic

plastic structure under white noise excitation and real earthquakes. The simplicity and desirable

convergence behavior associated with applying DGA in designing the optimal TMDs for nonlinear

structures have been noteworthy. Also, it has been shown that under white noise earthquake, which

can induce large nonlinearity in the structure up to 200% of its yielding drift, by application of the

proposed method, it has been possible to design optimal TMDs to keep the structures from yielding aswell as to decrease their AHE and the RMS of their drift and acceleration. Also, it has been found that

the performance of the TMDs has been improved with increasing the maximum stroke length of TMD,

especially for smaller values of the stroke length. The results of numerical simulation for different

values of the TMD mass ratios and under different excitations have shown that, generally and for most

of its values, increasing the TMD mass ratio leads to its better performance. Also, according to the

numerical simulations under different input excitations, it has been found that by using TMD, it has

been possible to effectively mitigate the maximum response of the structure under different

earthquakes, although the amount of reduction in the maximum structural response has depended on

the characteristics of the TMD design earthquake. Comparison of the results obtained from the two

performance criteria of (a) minimization of the maximum drift and (b) minimization of the maximum

AHE, as the objective functions to design optimal TMDs, shows that both cases can approximately

lead to the same results in reducing the response of the nonlinear structures; however, more studies

should be conducted on 2D frames and 3D structures of different geometric properties as well as

considering different combinations of design criteria before this conclusion can be generalized.

ACKNOWLEDGEMENT

The authors would like to thank the committees of higher education and deputies of research of University

of Mohaghegh Ardabili and Sharif University of Technology for partially supporting this research.

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AUTHORS BIOGRAPHIES

Dr. Mohtasham Mohebbi received his B.Sc., M.Sc. and Ph.D. in 1997, 1999 and 2008 respectively form SharifUniversity of Technology (SUT), Tehran, Iran. He currently works as an assistant professor in engineeringdepartment of University of Mohaghegh Ardabili, Ardabil, Iran. He has participated in writing a number of paperspublished in refereed journals and conference proceedings in the structural control field.His current research interest is in the area of structural control systems (passive, active and semi-active systems)and optimization in civil engineering.

Also Dr. Mohebbi is a professional engineer and works in designing of earthquake resistant structures field.

Abdolreza Joghataie received his Ph.D. from the University of Illinois at Urbana-Champaign, Illinois, US, in1994 and is now a faculty member of the structural and earthquake engineering groups at the civil engineeringdepartment of Sharif University of Technology, Tehran, Iran. His research is within the fields of structuraloptimization, intelligent computation, structural control and concrete structures.


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