Post on 15-May-2020
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.
Recent Advances in Civil and Mining Engineering
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
.
))((
)(
Recent Advances in Civil and Mining Engineering
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
Recent Advances in Civil and Mining Engineering
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
Recent Advances in Civil and Mining Engineering
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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Recent Advances in Civil and Mining Engineering
ISBN: 978-960-474-337-7 21
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Recent Advances in Civil and Mining Engineering
ISBN: 978-960-474-337-7 22