Optimum Tuned Mass Dampers for Structures with Different Periods

and Damping

GEBRAIL BEKDAS,SINAN MELIH NIGDELI

Department of Civil Engineering

Istanbul University

34320 Avcılar Istanbul

TURKEY

bekdas@istanbul.edu.trmelihnig@istanbul.edu.tr

Abstract: - Tuned mass dampers are implemented to mechanical systems in order to surpass vibrations. The

usage of tuned mass dampers includes civil structures subjected towind, earthquake and other excitations. For

the best efficiency, properties of tuned mass dampers must be tuned. Since their tuning for civil structures are

depended to properties of superstructure and random characteristics of excitation, numerical optimization

techniques can be used for optimization of tuned mass dampers. In this study, harmony search algorithmwas

employed in order to find optimum mass ratio, period and damping ratio of tuned mass dampers implemented

on structures with different periods and damping ratios. The approach is more effective to reduce structural

displacements comparing to simple expressions used for tuning of tuned mass dampers.

Key-Words: Structural Control, Vibration, Tuned Mass Damper, Seismic Behavior, Harmony Search,

Optimization.

1 Introduction Frahm [1] invented a vibration absorber device in

1909. This device is the oldest known version of

tuned mass damper (TMD), but it consist of

stiffness members and mass without inherent

damping. In order to damp vibrations resulting from

excitations with random frequencies, an inherent

damping must be added to the device [2].

In structural engineering, TMDs are generally used

in high-rise structures, towers and bridges in order

to surpass structural vibrations resulting from

undesired excitations like earthquakes, strong winds

and traffic loads. TMDs can be used as a seismic

retrofit of iconic structures like the LAX Theme

building [3]. Citigroup Center in New York City,

Yokohama Landmark Tower in Yokohama, Burj Al

Arab in Dubai, Trump World Tower in New York

City, Taipei 101 in Taipei and The TV Tower in

Berlin (Fig. 1) are some of the example structures

with TMD.

In order to tune TMD optimally, several

simplified closed form expressions has been

developed. By using these expressions, frequency

and damping ratio of TMDs are calculated for a

preselected mass ratio (µ) [4-7].

In order to find more effective TMD properties,

numerical search methods [6, 8] and metaheuristic

methods [7, 9-18] can be employed by considering

the characteristics of structures and excitations.

Also, the mass of TMD can be optimized.

Fig. 1. The TV Tower in Berlin

In this study, the optimum TMD parameters for

single degree of freedom (SDOF) superstructures

with different period and damping frequency were

found by using HS algorithm. In the optimization

process, six different earthquake records were used.

During optimization, structures were also analyzed

by using optimum TMD parameters calculated

according to the simplified ground motions and the

results were compared and eliminated until a better

combinations of TMD parameters were found.

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ISBN: 978-960-474-337-7 17

2 Optimization of TMDs A SDOF superstructure with a TMD on the top is a

two-degree of freedom (2DOF) system. The

equations of motion of a 2DOF system can be

written as

{} )))) txtxtxtx g (1M(K(C(M ɺɺɺɺɺ −=++ (1)

for ground motion excitation. M, C and K matrices

are mass (Eq. (2)), damping (Eq. (3)) and stiffness

matrices (Eq. (4)), respectively. The x(t), )(tx gɺɺ and

{1} are the vector of structural displacements (Eq.

(5)), ground acceleration and a column of ones with

a dimension of (2,1), respectively. The physical

model is given in Fig. 2.

M=diag[m md] (2)

−

−+=

dd

dd1

cc

cccC

)( (3)

−

−+=

dd

dd1

kk

kkkK

)((4)

x(t)= [x xd]T(5)

In Fig. 2, m, c, k and x is mass, damping

coefficient, stiffness coefficient and horizontal

displacement of structure. The optimized properties

of the TMD are mass (md), damping coefficient (cd)

and stiffness coefficient (kd). The displacement of

the TMD is represented as xd.

The compared expressions for the optimum

TMD parameters are shown in Table 1. The

expressions of Den Hartog [4] are obtained under

harmonic main mass excitation. In Table 1, the

optimum expressions for white noise ground

excitation are given for Warburton [5] and Leung &

Zhang [7]. The optimum frequency ratio (fopt) is the

ratio of the optimum frequency of TMD (wd,opt) and

the frequency of SDOF structure (ws). The optimum

damping ratio of TMD (ξd,opt) is depended to the

mass of TMD (md), optimum frequency of TMD

(wd,opt) and damping coefficient of TMD (cd,opt).

The main aim of the optimization is to minimize

maximum displacement of structures. In order to

reach that goal, harmony search (HS) algorithm was

employed to search best suitable TMD parameters.

Harmony search is an effective metaheuristic

algorithm on finding optimum TMD parameters,

which is inspired from music. The process of

musical performances were imitated by Geem et al.

[19] for solving optimization problems. The main

aim of musical performances and scientific

problems are similar to each other. In scientific

problems, a solution must be found in order to reach

an objective function which is generally related with

maximum gain. The maximum gain of musical

performances is to have ear of audience. In

engineering problems, a mathematical expression

can be used for the objective function but the origin

of this function is the admiration of users.

The methodology of HS used in this study can be

explained in 6 steps.

Step 1: The solution ranges for the HS

parameters are defined. The solution range of TMD

period was taken between 0.8 and 1.2 times of the

critical period of the structure. The mass ratio range

was taken between 0.5% and 5%. The TMD

damping ratio was takenbetween 5%-30%. HS

parameters; Harmony Memory Size (HMS),

Harmony Memory Considering Rate (HMCR) and

Table 1. The frequency and damping ratio expressions

Method s

optd

optw

wf

,= optdd

optd

optdwm2

c

,

,

,=ξ

Den Hartog

1947 [4] µ+1

1

)( µ+

µ

18

3

Warburton

1982 [5] µ+

µ−

1

21 )(

)21)(1(4

)41(

µµµµ−+

−

Sadek et al.

1997 [6]

µ+

µξ−

µ+ 11

1

1

µ+µ

+µ+ξ

11

Leung & Zhang

2009 [7]

200002582874

94193723192094534

1

21

ξµµ+−+

ξµµ−µ+−+

µ+

µ−

)..(

)...(

)(

µξ−

µ−µ+µ−µ

230245

2114

41

.

))((

)(

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ISBN: 978-960-474-337-7 18

Pitch Adjusting Rate (PAR) are defined.

Fig. 2. Physical model

Step 2: In the second step, single degree of

freedom structure without TMD must be analyzed

under several earthquake excitations for the future

comparisons with TMD controlled structure. Time

domain analyses are conducted by using Matlab

with Simulink. Block diagrams for equation of

motion was prepared. For precise analyses,

RungeKutta method with 10-3 s time step is used. A

longer time step may be used for time efficiency but

the used time step is required for accurate solutions.

Step 3: After the initial analyses of uncontrolled

structure, the application of HS algorithm starts in

this step. As many as HMS, harmony vectors are

generated. These vectors contain randomly chosen

TMD mass ratio, period and damping ratio.

Additionally, optimum period and damping ratio of

TMD are found according to simple expressions

given in Table 1 by using randomly chosen mass

ratio value. For the comparison of the proposed

method with simple expressions and structure

without TMD, maximum displacements of TMD

controlled structures occurred under optimization

earthquakes are stored for HS approach and the best

of the simple expressions. The best simple

expression with the maximum reduction of the

displacement can be change due to mass ratio and

period of the superstructure. Detailed information

can be found in reference [18]. The generation of a

vector is repeated until two criteria are satisfied.

One of the criteria is to obtain a reduction on

displacement and the other one is to obtain a

reduction more than compared simple expressions.

If a suitable set of values stored in vector is not

found in 500 iterations, the range of mass ratio and

damping ratio are enlarged. For every 500 iterations,

the maximum bound of mass and damping ratio are

increased with 0.25% and 1%, respectively.

Generated harmony vectors are merged together in

order to obtain initial harmony memory (HM)

matrix.

Step 4: In this step, a new harmony vector is

generated according to the special rules of HS. As it

can be generated from whole solution range, it can

be generating according to previously stored

harmony vectors in HM matrix. HMCR is the

possibility to generate a new vector around the

existing ones. In that case, the range of solution

range is defined according to the PAR.

Step 5: According to amount of reduction of the

displacement, harmony vectors are ranged from

worst to best. If the vector generated in step 4 is

better than the worst one, it is replaced with it.

Step 6: Stopping criteria are checked for all

harmony vectors in HM matrix. Until the stopping

criteria are satisfied, the optimization process

continue form step 4. The optimization ends when

the three criteria which are listed below, are

satisfied.

- The reduction percentage of the

displacements for HS approach must be less

than the other compared methods.

- Maximum acceleration transfer function of

the TMD controlled structure must be less

than uncontrolled one.

- Mainly, maximum reduction of

displacements (for the most critical

earthquake) must be lower than a percentage

of the uncontrolled value. At the start of the

optimization process, this percentage is taken

as zero. After 250 iterations, this value is

updated according to average of the

displacement reductions of the stored

vectorsin HM matrix.

3 Numerical Example The optimum TMD parameters for single degree of

freedom structures with periods (T) between 0.5s-5s

and damping ratios (ξ) between 2.5%-20% were

investigated. In the optimization, HS parameters

HMS, PAR and HMCR were taken as 5, 0.2 and

0.5, respectively. Earthquake records used in

optimization process are given in Table 2.

Table 2. Optimization earthquake records [20]

Earthquake Date Station Component

Kobe 1995 0 KJMA KJM000

Imperial Valley 1940 El Centro I-ELC180

Erzincan 1992 Erzincan ERZ-NS

Landers 1992 Lucerne LCN000

ck

xg..

m

x

xd

md

cd

kd

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ISBN: 978-960-474-337-7 19

Northridge 1994 Sylmar SYL360

Loma Prieta 1989 LGPC LGP000

The optimum TMD parameters, optimum mass

ratio (µopt), damping ratio (ξd,opt) and period (Td,opt)

are given in Table 3. Also, the ratios of the

displacement between TMD controlled (xc) and

Table 3. Optimum TMD parameters

µopt ξ (%)

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

T (

s)

0.5 0.018 0.019 0.024 0.034 0.029 0.040 0.024 0.022

1.0 0.036 0.037 0.045 0.046 0.067 0.070 0.076 0.093

1.5 0.009 0.028 0.031 0.054 0.059 0.074 0.060 0.076

2.0 0.029 0.025 0.044 0.053 0.046 0.076 0.060 0.070

2.5 0.016 0.030 0.016 0.013 0.049 0.014 0.073 0.051

3.0 0.025 0.035 0.030 0.039 0.054 0.066 0.058 0.077

3.5 0.037 0.045 0.051 0.043 0.051 0.058 0.062 0.081

4.0 0.008 0.023 0.040 0.048 0.055 0.063 0.063 0.078

4.5 0.034 0.031 0.041 0.043 0.050 0.049 0.052 0.053

5.0 0.033 0.017 0.032 0.038 0.040 0.038 0.044 0.045

ξd,opt ξ (%)

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

T (

s)

0.5 0.289 0.289 0.284 0.307 0.298 0.302 0.326 0.388

1.0 0.273 0.283 0.332 0.369 0.390 0.436 0.424 0.475

1.5 0.289 0.294 0.317 0.356 0.415 0.445 0.481 0.511

2.0 0.285 0.297 0.302 0.349 0.398 0.422 0.486 0.519

2.5 0.291 0.293 0.307 0.349 0.396 0.445 0.474 0.519

3.0 0.295 0.284 0.319 0.366 0.406 0.449 0.486 0.501

3.5 0.273 0.281 0.319 0.368 0.410 0.439 0.476 0.506

4.0 0.298 0.288 0.309 0.337 0.384 0.387 0.418 0.444

4.5 0.257 0.289 0.287 0.300 0.305 0.318 0.341 0.327

5.0 0.251 0.286 0.280 0.281 0.297 0.287 0.296 0.309

Td,opt ξ (%)

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

T (

s)

0.5 0.491 0.504 0.508 0.509 0.513 0.496 0.513 0.508

1.0 1.044 1.047 1.066 1.056 1.029 1.059 1.093 1.091

1.5 1.527 1.536 1.567 1.642 1.559 1.583 1.587 1.678

2.0 2.043 2.079 2.035 2.061 2.071 2.203 2.111 2.125

2.5 2.674 2.542 2.571 2.550 2.779 2.534 2.860 2.730

3.0 3.066 3.100 3.101 3.084 3.129 3.213 3.145 3.305

3.5 3.654 3.546 3.494 3.573 3.623 3.646 3.609 3.650

4.0 4.022 4.050 4.184 4.054 4.173 4.312 4.143 4.166

4.5 4.654 4.509 4.572 4.451 4.654 4.546 4.644 4.660

5.0 4.795 4.778 4.720 4.861 5.128 4.990 4.912 5.137

xc/x ξ (%)

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

T (

s)

0.5 0.988 0.966 0.964 0.961 0.961 0.980 0.955 0.934

1.0 0.702 0.740 0.748 0.772 0.843 0.831 0.855 0.872

1.5 0.730 0.777 0.786 0.807 0.818 0.850 0.864 0.893

2.0 0.824 0.838 0.888 0.889 0.877 0.889 0.874 0.879

2.5 0.811 0.816 0.815 0.804 0.790 0.780 0.772 0.767

3.0 0.726 0.754 0.757 0.742 0.730 0.720 0.738 0.742

3.5 0.777 0.781 0.775 0.770 0.760 0.759 0.771 0.784

4.0 0.843 0.849 0.857 0.859 0.846 0.863 0.866 0.863

4.5 0.940 0.919 0.919 0.912 0.917 0.916 0.917 0.933

5.0 0.892 0.919 0.925 0.927 0.957 0.960 0.958 0.958

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ISBN: 978-960-474-337-7 20

uncontrolled (x) structures are given in Table 3.

4 Conclusion According to the results, a reduction of structural

displacements is not provided within the initial

range for the structures with inherent damping more

than 5%. For reinforced concrete and steel

structures, the damping ratio is assumed as 5% and

2%, respectively. Structures more than 5% inherent

damping can only represents structures with

additional damping. For that structure high damping

is needed for a TMD. Generally, the optimum

damping ratio of TMD is maximum for structures

with 1.5-4.0s period. By the increase of inherent

damping,ξd,opt is also increasing. Optimum period of

TMD and maximum displacement reduction ratios

are not related to inherent damping of structure. The

best reduction of displacement is observed for

structures with period 1.0-1.5s and 3.0-3.5s. This

situation is resulting from the characteristic of

different earthquakes used in optimization process.

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