Multiple Tuned Mass Dampers and Double Tuned Mass Dampers for
Soft Story Structures: A Comparative Study
FARAH ALNAYHOUM
Civil Engineering Department
Istanbul University-Cerrahpaşa
34320 Avcılar, Istanbul
TURKEY
IBRAHIMA KALIL CAMARA
Civil Engineering Department
Istanbul University-Cerrahpaşa
34320 Avcılar, Istanbul
TURKEY
SİNAN MELİH NİGDELİ
Civil Engineering Department
Istanbul University-Cerrahpaşa
34320 Avcılar, Istanbul
TURKEY
GEBRAİL BEKDAŞ
Civil Engineering Department
Istanbul University-Cerrahpaşa
34320 Avcılar, Istanbul
TURKEY
Abstract: Recent seismic events have highlighted the vulnerability of reinforced concrete buildings, especially
soft-story structures, to damage and collapse during strong earthquakes due to the ground's vibrational response.
This study aims to mitigate these adverse vibrations using passive control mechanisms, particularly tuned mass
dampers (TMDs). Conventional TMDs require a substantial mass to effectively influence the structure's lateral
reactions. The research explores the use of multi-tuned mass dampers (MTMDs) and double-tuned mass dampers
(DTMDs) in a soft-story building. The study uses MATLAB to estimate TMD parameters and subject the models
to seismic loading. The comparative assessment of these models reveals the potential benefits of using MTMD
and DTMD systems to enhance the seismic resilience of soft-story structures.
Key-Words: Den Hartog, multiple TMD, Double TMD, soft story structures, passive control, optimum design.
Received: June 24, 2022. Revised: August 24, 2023. Accepted: October 11, 2023. Published: November 14, 2023.
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DOI:10.37394/232025.2023.5.19
Farah Alnayhoum, Ibrahima Kalil Camara,
Si
nan Meli
h Ni
gdeli
, Gebrai
l Bekdaş
E-ISSN: 2692-5079
167
Volume 5, 2023
1 Introduction
Rapid population growth in major cities worldwide
has led to the construction of tall buildings and
closely spaced structures to accommodate the
increasing population. However, natural disturbances
like earthquakes and severe winds can lead to
excessive structural vibrations, affecting people's
comfort. [1]
To maintain safety and technological
competitiveness, structural designers have
implemented vibration control technologies to
control excessive vibrations and reduce their impact
on structural response. These measures are
particularly aimed at keeping structures within
acceptable limits, especially during unpredictable
events like earthquakes or winds. [2] Earthquakes
have shown that some reinforced concrete buildings
are vulnerable to damage or collapse, particularly
soft-story buildings or flexible-story buildings,
which are approximately 70% less rigid than other
floors. [3,4]
Various vibration control technologies have been
implemented to reduce damage and alter structural
performance, including dampers, vibration isolators,
control of excitation forces, and vibration
absorbers.[5] Vibration absorbers like tuned mass
dampers (TMD), Active Mass Dampers (AMD),
Semi-Active Mass Dampers (SAMD), and Hybrid
Mass Dampers (HBD)[5,6] have been studied and
installed in skyscrapers to control the behavior of
structures under vibration forces. [5]
In the realm of structural engineering, it is imperative
to acknowledge that the initial methodologies
employed to address environmental challenges,
including base isolations and tuned mass dampers
(TMDs), predominantly employed passive control
mechanisms.[7,8] To further enhance the
management of structural vibrations, the
optimization of various parameters, such as mass,
stiffness, and damping, has emerged as a viable
strategy. With the progression of scientific
knowledge, the successful implementation of specific
control methods and practices has significantly
contributed to the heightened effectiveness of
vibration control in this context.[9] The placement of
these dampers in civil engineering structures is
carefully done to avoid causing harm to the
structure.[6]
This study delves into the application of passive
vibration control systems, particularly emphasizing
the utilization of multiple-tuned mass dampers
(MTMDs) and double-tuned mass dampers
(DTMDs) as viable solutions for minimizing
vibrations in structures subjected to dynamic loads.
The MTMD system is characterized by the
incorporation of multiple smaller dampers
strategically distributed within the structure, with
distribution patterns, whether uniform, linearly
varying, or designer-assessed. Each damper within
the MTMD system is meticulously tuned to a specific
frequency, tailored to mitigate vibrations occurring at
that particular frequency. In contrast, tuned mass
dampers (TMDs) are meticulously tuned to the
natural frequency of the host structure.[7,10,11,12]
The central objective of the DTMD system is to
introduce effective damping mechanisms into the
primary structure, thereby significantly reducing
structural oscillations.[13] The configuration
comprises two TMDs: a larger undamped unit
(TMD1) and a smaller, conventionally tuned unit
(TMD2). Notably, TMD1 is strategically deployed to
suppress vibrations within the primary structure,
while TMD2 is dedicated to addressing vibrations
within TMD1. [13,14]
For this study, the optimization and design of TMD
systems are achieved through the application of the
Den-Hartog equation. Den-Hartog technique is an
enduring yet highly effective method in TMD system
design. Over the years, the Den-Hartog technique has
served as an efficient analytical approach, frequently
employed to create and fine-tune TMD systems, all
while maintaining structural integrity by not
introducing additional damping into the primary
structure.[15,16]
The principal objective of this study is to assess the
effectiveness and response of both multiple-tuned
mass dampers (MTMDs) and double-tuned mass
dampers (DTMDs) when applied in soft-story
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Farah Alnayhoum, Ibrahima Kalil Camara,
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, Gebrai
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buildings under seismic loading conditions. This
research involves the application of MTMD and
DTMD systems to a soft-story building, with a focus
on determining the optimal parameters for the tuned
mass dampers, including mode shapes, period, and
frequency. The analysis will be executed through the
utilization of a two-story model: in the MTMD case,
each story is equipped with a single-tuned mass
damper, while in the DTMD case; a double-tuned
mass damper will be placed on the top story.
Subsequently, these models will be subjected to the
El Centro earthquake and analyzed using the Den-
Hartog equation in conjunction with MATLAB. The
study will conclude with a comparative assessment of
the models featuring soft-story configurations and
those without, shedding light on the potential
advantages of employing MTMD and DTMD
systems to enhance seismic resilience.
2 Problem Formulation
The present investigation focuses on the dynamic
response analysis of a two-story shear building
subjected to El-Centro seismic excitation. This study
encompasses two main cases; the first case examines
a system that does not consist of a soft story, while
the second case consists of a soft story as a second
story. Both cases will be examined twice. The first
will be a system featuring multiple-tuned mass
dampers (TMDs) installed on each story, while the
second case explores a system with double TMDs
located on the top story, as shown in Figure 1.
In this analysis, the structural parameters are
represented by (m, k, c) denoting the mass, stiffness,
and damping coefficients, respectively.
Simultaneously, the TMD parameters are expressed
as (mdi, kdi, cdi) representing the mass, stiffness, and
damping coefficients of the TMDs. The structural
responses are characterized by (xi) indicating the
displacement of each story concerning the ground,
and (ẍg) representing the ground acceleration.
Similarly, the TMD responses are denoted as (xdi)
signifying the displacement of the TMDs relative to
the ground.
Fig. 1 Shear structure model.
The governing equations of motion for both scenarios
are derived based on the principle of equilibrium of
forces at each degree of freedom, as illustrated in
Equation (1):
( ) ( ) ( ) ( )
1
t t t g t
Mx Cx Kx M x
(1)
In Equation (1), the symbols M, C, and K correspond
to the mass, damping, and stiffness matrices for both
the structure and the tuned mass damper (TMD).
Specifically, the term "−𝑀{1}𝑥𝑔(𝑡)" represents the
inertial forces arising from ground accelerations.
In the current study, the Den-Hartog equation was
employed as a foundational framework. The design
characteristics of the TMDs encompass their optimal
damping ratio, frequency, mass, and stiffness. These
design parameters are determined using the following
equations:
1
1
opt
f
(2)
,
3
8(1 )
d opt
(3)
Within this context, the symbols "𝒇𝒐𝒑𝒕, 𝝁, 𝝃𝒅,𝒐𝒑𝒕"
stand for the optimal frequency, mass ratio, and
optimum damping ratio, respectively.
For the scenario where a tuned mass damper (TMD)
is not employed, the system's Ms, Ks, and Cs matrices,
which pertain to mass, stiffness, and damping, are
expressed as follows:
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1
2
s
N
m
m
M
m
(4)
1 2 2
2 2 3 3
3
s
N
NN
k k k
k k k k
Kkk
kk
(5)
1 2 2
2 2 3 3
3
s
N
NN
c c c
c c c c
Ccc
cc
(6)
For systems with multi TMDs, Mm, Km, and Cm
matrices of the system are shown as:
0
0
s
d
m
M
MM
(7)
1
2
0
0
d
d
d
m
Mm
(8)
s d d
dd
m
K K K
KKK
(9)
1
2
0
0
d
d
d
k
Kk
(10)
s d d
dd
m
C C C
CCC
(11)
1
2
0
0
d
d
d
c
Cc
(12)
For systems with double TMDs, Mdb, Kdb, and Cdb
matrices of the system are shown as:
0
0
s
d
db
M
MM
(13)
1 2 2
2 2 3 3
3
11
1 1 2 2
22
N
db
N N d d
d d d d
dd
k k k
k k k k
kk
Kk k k k
k k k k
kk
(14)
1 2 2
2 2 3 3
3
11
1 1 2 2
22
N
db
N N d d
d d d d
dd
c c c
c c c c
cc
Cc c c c
c c c c
cc
(15)
2.1 Case study
2.1.1 Case 1
A structure subjected to El Centro earthquake load
was evaluated in this example. The structure is made
up of two stories, each having equal masses
connected to it, equivalent damping coefficients, and
equal stiffness. In the present case:
(a) Multi-tuned mass dampers (MTMDs) were placed
at each story of the structure.
(b) Double-tuned mass dampers (DTMDs) were
placed at the top of the structure.
Both cases (a) and (b) result in the structure acting as
a 4DOF system. The Den-Hartog approach was used
to derive the TMD parameters based on the
structure's initial mode. TMD was considered to have
a mass ratio of 5% of the structure's two masses.
2.1.2 Case 2
The same structure with the same characteristics
but with different stiffness values was evaluated. In
this case:
(a) Multi-tuned mass dampers (MTMDs) were placed
at each story of the structure.
(b) Double-tuned mass dampers (DTMDs) were
placed at the top of the structure.
By assuming the same mass ratio of 5% and by using
the Den-Hartog technique TMD parameters have
been evaluated. The structure characteristics are
shown in Table 1.
Table 1. Characteristics of structures
Case
Story
m
(kg)
k
(N/m)
C
(Ns/m)
1
1
2924
1390000
1581
2
2924
1390000
1581
2
1
2924
1390000
1581
2
2924
2780000
1581
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3 Problem Solution
The mass, stiffness, and damping matrices were
successfully derived. The optimum values of the
TMDs were listed in Table 2, Table 3, Table 4,
and Table 5. Derived values have been used to
calculate the frequencies and periods, which are
presented in Table 6 and Table 7. Additionally,
the mode shapes for both cases were extracted
and are visualized in Figure 2 and Figure 3. With
normalized values detailed in Table 8, Table 9,
Table 10, and Table 11. Furthermore, the
simulation results for roof displacement and total
acceleration are depicted in Figure 4 and Figure
5, with maximum values of displacement and
acceleration detailed in Table 12 and Table 13.
These findings constitute an integral part of the
study's results, providing insights into the
structural behavior and response under the
applied seismic loading conditions.
Table 2. TMD values for case 1. a
Case 1. a
With Multiple TMD
𝝁 = 𝟓%
𝒇𝒐𝒑𝒕
𝝃𝒅,𝒐𝒑𝒕
𝝎𝒅,𝒐𝒑𝒕
𝑪𝒅,𝒐𝒑𝒕
𝒎𝒅,𝒐𝒑𝒕
𝒌𝒅,𝒐𝒑𝒕
0.952
0.133
12.833
501.44
146.2
24078.58
0.952
0.133
33.598
1312.8
146.2
165037.06
Table 3. TMD values for case 1. b
Case 1. b
With Double TMD
𝝁 = 𝟓%
𝒇𝒐𝒑𝒕
𝝃𝒅,𝒐𝒑𝒕
𝝎𝒅,𝒐𝒑𝒕
𝑪𝒅,𝒐𝒑𝒕
𝒎𝒅,𝒐𝒑𝒕
𝒌𝒅,𝒐𝒑𝒕
0.952
0.133
12.833
501.44
146.2
24078.58
0.952
0.133
33.598
1312.8
146.2
165037.06
Table 4. TMD values for case 2. a
Case 2. a
With Multiple TMD
𝝁 = 𝟓%
𝒇𝒐𝒑𝒕
𝝃𝒅,𝒐𝒑𝒕
𝝎𝒅,𝒐𝒑𝒕
𝑪𝒅,𝒐𝒑𝒕
𝒎𝒅,𝒐𝒑𝒕
𝒌𝒅,𝒐𝒑𝒕
0.952
0.133
13.749
537.24
146.2
27639.07
0.952
0.133
44.349
1732.88
146.2
287553.66
Table 5. TMD values for case 2. b
Case 2. b
With Double TMD
𝝁 = 𝟓%
𝒇𝒐𝒑𝒕
𝝃𝒅,𝒐𝒑𝒕
𝝎𝒅,𝒐𝒑𝒕
𝑪𝒅,𝒐𝒑𝒕
𝒎𝒅,𝒐𝒑𝒕
𝒌𝒅,𝒐𝒑𝒕
0.952
0.133
13.749
537.24
146.2
27639.07
0.952
0.133
44.349
1732.88
146.2
287553.66
Table 6. Frequency and period values for case 1
Without TMD
Case 1. a
Multiple TMD
Case 1. b
Double TMD
Frequency
(Hz)
Period
(s)
Frequency
(Hz)
Period
(s)
Frequency
(Hz)
Period
(s)
2.1446
0.4662
1.9513
0.5124
1.3797
0.7247
5.6146
0.1781
2.1917
0.4562
2.2014
0.4542
5.2324
0.1911
5.6197
0.1779
5.8769
0.1701
7.7045
0.1297
Table 7. Frequency and period values for case 2
Without TMD
Case 2. a
Multiple TMD
Case 2. b
Double TMD
Frequency
(Hz)
Period
(s)
Frequency
(Hz)
Period
(s)
Frequency
(Hz)
Period
(s)
2.2977
0.4352
2.0751
0.4818
1.4923
0.6700
7.4113
0.1349
2.3782
0.4204
2.3515
0.4252
6.8194
0.1466
7.4173
0.1348
7.8154
0.1279
10.1045
0.0989
(a) Without TMD
(b) MTMD
(c) DTMD
Fig. 2 Mode shapes for case 1. (a) Without TMD (b)
MTMD (c) DTMD
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Table 8. Mode shape values case 1. a
Mode 1
Mode 2
Mode 3
Mode 4
0.0872
-0.1514
-0.1678
0.2449
0.1310
-0.2624
0.0425
-0.2078
1
1
0.0301
-0.0336
0.1512
-0.3153
1
1
Table 9. Mode shape values case 1. b
Mode 1
Mode 2
Mode 3
Mode 4
0.0277
0.6259
1
0.0016
0.0511
1
-0.6227
-0.0048
0.9334
-0.6408
-0.0094
1
1
-0.7716
0.0904
-0.9293
(a) Without TMD
(b) MTMD
(c) DTMD
Fig. 3 Mode shapes for case 2. (a) Without TMD (b)
MTMD (c) DTMD
Table 10. mode shape values case 2. a
Mode 1
Mode 2
Mode 3
Mode 4
0.1007
-0.1811
-0.1585
0.2203
0.1241
-0.2408
0.0665
-0.2260
1
1
0.0181
-0.0187
0.1358
-0.2717
1
1
Table 11. mode shape values case 2. b
Mode 1
Mode 2
Mode 3
Mode 4
0.0325
0.7871
1
0.0012
0.0458
1
-0.7845
-0.0034
0.9552
-0.6878
-0.0078
1
1
-0.7737
0.0754
-0.9529
Table 12. Comparison between displacements and
accelerations of the top story for case 1
Without TMD
Case 1. a
With Multiple TMD
Case 1. b
With Double TMD
Max
displacement
(m)
0.0096
Max
displacement
(m)
0.0045
Max
displacement
(m)
0.0020
Max
acceleration
(m/s2)
11.9639
Max
acceleration
(m/s2)
6.1634
Max
acceleration
(m/s2)
4.8716
Table 13. Comparison between displacements and
accelerations of the top story for case 2
Without TMD
Case 2. a
With Multiple TMD
Case 2. b
With Double TMD
Max
displacement
(m)
0.0038
Max
displacement
(m)
0.0025
Max
displacement
(m)
0.0012
Max
acceleration
(m/s2)
8.4017
Max
acceleration
(m/s2)
6.2051
Max
acceleration
(m/s2)
5.0391
(a) Displacement
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(b) Acceleration
Fig. 4 Responses of the structure for case 1. (a)
Displacement. (b) Acceleration.
(a) Displacement
(b) Acceleration
Fig. 5 Responses of the structure for case 2. (a)
Displacement. (b) Acceleration.
4 Conclusion
In conclusion, the effectiveness of a multiple TMD
system and a Double TMD system as passive
vibration control systems when applied to a soft story
building subjected to El-Centro seismic excitation,
considered as a 4DOF system, was investigated and
compared. The parameters of the MTMDs and
DTMDs, as well as mode shapes and responses, were
thoroughly examined, leading to the following
conclusions:
1. The utilization of a double-tuned mass
damper (DTMD) at the top of a structure
demonstrates superior effectiveness in
reducing peak responses, including
acceleration and displacement, as compared
to the deployment of multiple-tuned mass
dampers (TMDs) distributed across the
structure's stories. The DTMD achieves a
displacement reduction of 125% and an
acceleration reduction of 26.51% for
structures without soft stories. In the case of
structures with soft stories, the results
indicate a displacement reduction of
108.33% and an acceleration reduction of
23.14%. This underscores the enhanced
performance and applicability of the DTMD
system, thereby offering valuable insights
into optimizing vibration control strategies in
structural engineering.
2. Regarding the frequency aspect, it is evident
that employing a double-tuned mass damper
(DTMD) leads to higher frequency values
when compared to the use of multiple TMDs
for both cases.
3. The periods associated with the double TMD
model are consistently shorter than those of
the multiple TMD model for both cases.
4. It has been ascertained that the parameters
governing the tuned mass dampers (TMDs)
exhibit uniform characteristics across both
the multiple TMD and double TMD
configurations. Notably, these parameters
reveal a marked increase in values in the
context of soft story structures, irrespective
of whether we examine the multiple-tuned
mass dampers (MTMD) or double-tuned
mass dampers (DTMD).
In summary, the primary objective in soft story
buildings, where lower stories are more flexible and
vulnerable to lateral motion during seismic events, is
to enhance lateral stability and mitigate the risk of
structural damage or collapse. In this context, a
double-tuned mass damper (DTMD) system can
prove more effective, as it offers supplementary
damping to the primary structure, reducing lateral
vibration amplitudes. While multiple-tuned mass
damper (MTMD) systems can effectively control
vibrations, they may not provide the same level of
damping to the primary structure, particularly in soft
story buildings characterized by significant
flexibility discrepancies between stories. In addition,
because the Den Hartog method is used for
calculation, the effectiveness of the DTMD is
somewhat reduced. To find more suitable DTMD
parameters, optimization methods such as Harmony
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Search (HS), Teaching-Learning-Based
Optimization (TLBO), and Enhanced Teaching-
Learning-Based Optimization (ETLBO) can be
employed in future work.
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Engineering World
DOI:10.37394/232025.2023.5.19
Farah Alnayhoum, Ibrahima Kalil Camara,
Si
nan Meli
h Ni
gdeli
, Gebrai
l Bekdaş
E-ISSN: 2692-5079
174
Volume 5, 2023
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that are relevant to the content of this article.
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