Engineering Structures 29 (2007) 1548–1560
www.elsevier.com/locate/engstruct
A bidirectional and homogeneous tuned mass damper: A new device for
passive control of vibrations
José L. Almazán, Juan C. De la Llera, José A. Inaudi, Diego López-Garcı́a ∗ , Luis E. Izquierdo
Departamento de Ingenierı́a Estructural y Geotécnica, Pontificia Universidad Católica de Chile, Macul, Santiago 690441, Chile
Received 12 March 2006; received in revised form 4 September 2006; accepted 7 September 2006
Available online 23 October 2006
Abstract
Passive tuned-mass dampers (TMDs) are a very efficient solution for the control of vibrations in structures subjected to long-duration, narrowband excitations. In this study, a Bidirectional and Homogeneous Tuned Mass Damper (BH-TMD) is proposed. The pendular mass is supported
by cables and linked to a unidirectional friction damper with its axis perpendicular to the direction of motion. Some advantages of the proposed
BH-TMD are: (1) its bidirectional nature that allows control of vibrations in both principal directions; (2) the capacity to tune the device in each
principal direction independently; (3) its energy dissipation capacity that is proportional to the square of the displacement amplitude, (4) its low
maintenance cost. Numerical results show that, under either unidirectional or bidirectional seismic excitations, the level of response reduction
achieved by the proposed BH-TMD is similar to that obtained from an “ideal” linear viscous device. Moreover, experimental shaking table tests
performed using a scaled BH-TMD model confirm that the proposed device is homogeneous, and, hence, its equivalent oscillation period and
damping ratio are independent of the motion amplitude.
c 2006 Elsevier Ltd. All rights reserved.
Keywords: Tuned mass damper; Passive control; Structural dynamics; Bi-directional control; Homogeneous device; Frictional damping; Low-cost TMD
implementation
1. Introduction
Passive Tuned Mass Dampers (TMDs) are used in vibration
reduction of flexible structures subjected to long-duration
narrow-band excitations [1–3]. While a TMD does not
necessarily reduce the peak deformation demand in an
inelastic structure subjected to ground motion, it reduces the
corresponding level of damage [5,6].
In the TMD literature, there are publications that deal
with the bidirectional behavior of a structure. Most of this
research aims to control the lateral–torsional response of the
bare structure by means of multiple unidirectional TMDs [7,
8]. In order to use the total weight of the supplemental mass,
a typical design would consider one or multiple bidirectional
TMDs, with frequencies tuned independently in each principal
∗ Corresponding address: Departamento de Ingenierı́a Estructural y
Geotécnica, Pontificia Universidad Católica de Chile, Av. Vicuna Mackenna
4860, 782-0436 Santiago, RM, Chile. Tel.: +56 2 354 7684; fax: +56 2 354
4243.
E-mail address: dlg@ing.puc.cl (D. López-Garcı́a).
c 2006 Elsevier Ltd. All rights reserved.
0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.09.005
direction of the structure. As far as the authors know,
the behavior of nominally symmetric structures with TMDs
subjected to bidirectional excitations has not been considered
in the literature.
Since the implementation of TMDs is often restricted by
budget and technical constraints, it is important to devise a
low cost TMD solution that is simple, robust, and of simple
installation and maintenance. Motivated by that, a novel device
whose design is intended to overcome the aforementioned
constraints is presented in this paper. One of its innovative
features is the structural layout in which the mass is attached
to the main structure, a simple implementation that makes the
tuning process of the device easy and inexpensive, and allows
the device to be tuned independently in each principal direction.
Another innovative feature of the proposed device is the use
of a friction damper instead of a viscous damper, attached to
the TMD mass in a direction perpendicular to the plane of
motion of the mass. This approach follows the idea presented
earlier by Inaudi and Kelly [9] that results in energy dissipation
quadratic in amplitude, and hence, an equivalent damping ratio
independent of the motion amplitude. This is in contrast to
J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
1549
Fig. 1. Schematic representation of a BH-TMD: (a) 3D-view of the device in the undeformed position; (b) x z-plane motion; and (c) yz-plane motion.
the equivalent damping ratio of a friction damper acting in the
direction of motion of the mass which is inversely proportional
to the deformation amplitude, thus leading to an efficiency of
the damper that depends on the excitation level.
2. Description and analysis of the proposed device
The proposed Bidirectional and Homogeneous Tuned Mass
Damper (BH-TMD) has a pendular mass attached to a friction
damper with its original axis perpendicular to the plane of
motion (Fig. 1). As stated above, this geometric configuration
leads to energy dissipation quadratic in the displacement
amplitude [9]. Further, if a first-order approximation of the
motion is considered, the equivalent damping ratio of the device
becomes independent of the displacement amplitude.
The device may be designed either as an isotropic (i.e.,
identical oscillation period in all directions) or as an orthotropic
(i.e., different oscillation period in the two principal directions)
pendulum (Fig. 1). The orthotropic characteristics are obtained
by hanging the pendular mass from a Y-shape cable system.
Thus, as the pendular mass moves in the x-direction, the system
behaves as a pendulum of length L x (Fig. 1(b)). If, on the
other hand, the pendular mass oscillates in the y-direction,
the system behaves as a pendulum of length L y (Fig. 1(c)),
and cable C D rotates around point C as long as cables AC
and BC are in tension. Please notice that the cables might be
substituted by metallic rods, thus, preventing buckling. Cables
have one important advantage, which is to tune the TMD
by adjusting the cable lengths L x and L y . Next, a detailed
description of the kinematics of the proposed device, along with
the corresponding equations of motion, are presented.
2.1. Kinematics
A schematic 3D representation of the displaced Y-shape
cable system is shown in Fig. 2(a). In order to simplify the
analytical representation of the kinematic relationships, it is
assumed that the cables are axially rigid, and that the motion of
the pendular mass m d is purely translational. With respect to the
x − y−z coordinate system shown in Fig. 2(a), the displacement
of the mass m d is given by r = [u, v, w]T , where u, v, and w
are the x-, y- and z-components of the position of the mass r,
respectively. From Fig. 2(a), it follows that:
  

u
1L + L y cos β sin θ

L y sin β
r(θ, β) =  v  = 
(1)
w
L x − 1L + L y cos β cos θ
where 1L = L x − L y is the difference in TMD lengths; θ
is the angle (measured in the x z plane) between QC and the
vertical direction; and β is the angle (measured in the ABC
plane) between the height of triangle ABC, QC, and cable
C D. For convenience, displacement components u and v are
set as the independent coordinates, and grouped in a degree-of
T
freedom (DOF) vector q = u v . The relationship between
the dependent coordinate w and q can be found from Eq. (1).
An example of contour lines of w(u, v) can be seen in Fig. 2(b),
along with the direction and magnitude of the gradient of
w(u, v), which is related to the restoring force acting on m d
due to the gravitational field.
The engineering axial deformation s along the direction of
the friction damper is given by:
q
s(u, v) = ld (u, v) − lo = u 2 + v 2 + (w + lo )2 − lo
(2)
where ld (u, v) and lo are the deformed and undeformed lengths
of the device, respectively. The corresponding axial force in the
friction damper is approximated by a rigid-plastic model:
!
∂s T
q̇
(3)
f µ = po sign (ṡ) = po sign
∂q
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
(small deformation geometry), can be obtained by following the
procedure presented in Appendix B. It can be shown that if the
restraint restoring force fr is omitted, the equation of motion of
the BH-TMD can be approximated by:
m d q̈ + Cq q̇ + f̄d (q, q̇) = −m d ah
(6)
where
f̄d (q, q̇) =
Fig. 2. (a) Deformed state of the Y-shape cable system; (b) contour lines of
vertical displacement w = w(u, v) indicating direction and magnitude of the
gradient of w for an orthotropic BH-TMD (L x = 100 cm, L y = 60 cm).
T
∂s
where po is the slip force; ṡ = ∂q
q̇ is the rate of the damper
axial deformation; and sign represents the signum function.
Finally, a restrainer was incorporated into the friction damper
in order to limit the lateral displacement of the pendular mass.
The magnitude of the corresponding force is:
0
if s < slim
fr =
(4)
kr (s − slim ) + cr ṡ if s > slim
where slim is the threshold deformation level beyond which the
restraint engages, and kr and cr are its stiffness and damping,
respectively.
2.2. TMD dynamic equilibrium
Assuming that the BH-TMD is a 2-DOF system and that the
external excitation is applied simultaneously at all the supports
(points A, B and E in Fig. 1 which are assumed to be rigidly
linked to each other), the corresponding equation of motion can
be derived using the Euler–Lagrange equations. The detailed
derivation is shown in Appendix A, and leads to the following
nonlinear matrix differential equation:
M
(q)
q̈ + Cq q̇ + f d (q, q̇) = −J m d a − Q̂d (q, q̇)
T
(5)
where M(q) is the generalized (coordinate dependent) mass
matrix; Cq is the assumed intrinsic viscous damping matrix
that accounts for the energy dissipated at the TMD connections;
∂s
fd (q, q̇) = ∂w
∂q m d g + ∂q ( f µ + f r ) is the generalized restoring
force vector including the pendular as well as the frictional
(q)
˙ ∂∂qT is a
and restraint force components; Q̂d (q, q̇) = Ṁ q−
∂r
second-order term that couples q and q̇; J = ∂q
is the Jacobian
matrix of the kinematic transformation (i.e., ṙ = Jq̇); and
T
a = aTh az is the vector of support accelerations, where ah =
T
ax a y and az are the horizontal and vertical components,
respectively.
Eq. (5) takes into account the actual kinematics of the
BH-TMD and is highly nonlinear. A first-order approximation
az
1+
g
K̄ p + K̄ f (ṡ) q
m g

d
0
k px
0


K̄ p =
=  Lx
md g 
0 k py
0
Ly
"
#
2
ω px
0
= md
0
ω2py
k̄
0
K̄ f (ṡ) = sign(ṡ) f x
0 k̄ f y

 1
1
+
0


lo
= sign(ṡ) po  L x
1
1
0
+
Ly
lo
(7a)
(7b)
(7c)
where f̄d (q, q̇) is the first-order approximation of the
generalized nonlinear force vector; K̄ p is the (constant)
pendular stiffness matrix, with k px = m d g/L x = m d ω2px
and k py = m d g/L y = m d ω2py the apparent pendular
stiffness√in the local directions
x and y, respectively, and
p
ω px = g/L x and ω py = g/L y the corresponding nominal
pendular frequencies; and K̄ f (ṡ) is the (variable) frictional
matrix representing the projection of the friction force in the
local directions, where k̄ f x = po (1/L x + 1/lo ) and k̄ f y =
po 1/L y + 1/lo .
Some observations on the linearized expressions (6) and (7)
are interesting. First, the inertial and pendular stiffnesses turn
out to be uncoupled. Second, matrix K̄ f is diagonal, but the
dissipative effect nevertheless remains coupled due to the term
sign(ṡ). Finally, if γ is an arbitrary factor of q and q̇, Eq. (7a)
satisfies the following relationship:
f̄d (γ q, γ q̇) = γ f̄d (q, q̇)
(8)
which indicates that for small deformations the proposed BHTMD behaves as a first-order nonlinear but homogeneous
system. Because of their simplicity, Eqs. (6) and (7) will be
used later for the design of the proposed BH-TMD.
2.3. Experimental validation
In order to experimentally validate the first-order approximation of the constitutive relationship of the proposed BHTMD (Eqs. (7a)–(7c)), a scaled model of an isotropic BHTMD was constructed and tested on a shaking table at the
Structural Engineering Laboratory of the Pontificia Universidad Católica de Chile. As shown in Fig. 3, the model has a
cylindrical pendular mass of weight Wd = 100 N supported
by
√ three vertical cables of length L = 10 cm (ωd x = ωdy =
g/L = 2.6π rad/s). The initial, undeformed length of the
J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
Fig. 3. Experimental setup of a shaking table test of a 1:4 scaled isotropic BHTMD model.
friction damper is lo = 25 cm and the magnitude of the slip
force is po = 0.20Wd = 20 N. The resulting equivalent damping ratio is approximately 0.08. Accelerometers as well as linear potentiometers were used to measure accelerations of the
table and accelerations and displacements of the mass (Fig. 3).
The model was subjected to a series of unidirectional
harmonic excitations ax (t) = ao sin 2π fˆt cm/s2 having
different frequencies fˆ = [1/4, 1/2, 1, 2, 4] Hz. The base
excitation history ax (t) (input) and the displacement response
history of the pendular mass u(t) (output) are shown in Fig. 4.
The normalized hysteresis loops determined experimentally are
shown in Fig. 5; it is apparent that the constitutive relationship
of the BH-TMD is essentially triangular. The smooth shape of
1551
Fig. 5. Experimentally inferred hysteretic cycles of a scaled BH-TMD
model. Normalized measured force (λx (t)/Wd ) versus normalized measured
displacement (u(t)/L x ) of the pendular mass, under unidirectional harmonic
excitation.
the loops and the non-zero forces at zero displacements indicate
the presence of some degree of viscous damping, which is
due primarily to the energy dissipated at the hinges and other
connections of the device.
3. Coupled motion equations of structure and TMD
Let us assume that the primary structure is a linear n-DOF
system subjected to ground excitations. When equipped with a
BH-TMD, the corresponding differential equation of motion is
given by:
Ms ÿ + Cs ẏ + K s y + L T λ = −Ms Rs üg
(9)
Fig. 4. Experimental results for a scaled BH-TMD model: a displacement of the pendular mass with respect to the base (top); and measured base acceleration
(bottom).
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
where y{n × 1} is the vector of DOFs of the primary structure;
Ms , Ks , and Cs are the mass, stiffness and damping matrices
(of order n × n); λ is the interaction force between the pendular
mass and the primary structure, L{3 × n} being a kinematic
T
transformation matrix; üg = ẍ g (t) ÿg (t) z̈ g (t) is a vector
of ground accelerations; and Rs {n × 3} is the input influence
vector that relates the components of üg with the structure’s
DOFs y.
The interaction force λ can be expressed as:
T
λ = λx , λ y , λz = m d r̈t = m d (r̈ + a)
(10)
where r̈t = r̈ + a is the total (or absolute) acceleration of the
pendular mass, and
d
(Jq̇) = Jq̈ + J̇q̇
dt
a = Lÿt = L ÿ + Rs üg
r̈ =
(11a)
(11b)
where ÿt = ÿ + Rs üg is the vector of total accelerations in the
primary structure.
Finally, combining Eqs. (9), (10), (11a) and (11b) and Eq.
(5), the final equations of motion of the structure and TMD are
given by:
Ms + LT m d L LT m d J ÿ
Cs 0
ẏ
+
q̈
0 Cq q̇
JT m d L
M(q)
Ks 0 y
+ ···
+
0 0 q


T
Ms + LT m d L
L m d J̇q̇
0


=−
Rs üg (t) −
fd (q, q̇)
Q̂d (q, q̇)
JT m L
d
(12)
4. Design of the proposed BH-TMD
It is well-known that the efficiency of a TMD is sensitive
primarily to the tuning of the fundamental frequency ωd , and to
a lesser extent of the damping ratio ξd . Optimal values of these
parameters for an undamped linear SDOF system subjected to
a white-noise excitation are given by [10]:
√
ωd
1 − µ/2
(14)
Ωop =
=
ωs
1+µ
s
µ (1 − µ/4)
ξop =
(15)
4 (1 + µ) (1 − µ/2)
where µ is the ratio between the mass of the TMD and that
of the primary structure; and ωd and ωs are the fundamental
nominal frequencies of the TMD and that of the structure in the
direction considered, respectively. Based on these equations (or
on any of the equivalent equations proposed in the literature
[11–13]), valid for linear behavior, simple design equations for
the BH-TMD can be easily derived. Because of its orthotropic
properties, the BH-TMD can be tuned in each principal
direction independently, and the pendular lengths are given by
(Eq. (7b)):
g
g
= 2 2
ω2px
Ωop ωsx
g
g
Ly = 2 = 2 2 .
ω py
Ωop ωsy
(16)
Lx =
(17)
An expression for the optimal value of the slip force po
can be obtained by considering the equivalent viscous damping
ratio ξeq . Assuming harmonic motion of amplitude u o in the
x-direction, the total equivalent viscous damping is given by:
ξeq = ξo + ξ f = ξo +
1 Ed
4π E s
where all second-order terms are on the right-side of the
equation. Eq. (12) can be greatly simplified by using the firstorder approximation indicated earlier in Eq. (6). Omitting again
fr , the equations of motion are given by:
"
# Ms + LTh m d Lh m d LTh ÿ
ẏ
Cs 0
+
+ ···
(q)
q̈
0 Cq q̇
m d Lh
M̄


Ks 0
 y

az
q
0
1+
K̄ p + K̄ f (ṡ)
g
"
#
Ms + LTh m d Lh Rs
=−
üg (t)
(13)
m d L h Rs
where E d = 2k f x u 2o is the energy dissipated by friction in one
cycle; E s = 21 k px u 2o is the maximum potential energy stored
1 Ed
in the system; ξ f = 4π
E s is the so called frictional damping
ratio; and ξo is the intrinsic viscous damping ratio that takes
into account the energy dissipated in the connections.
Setting ξeq = ξop , and substituting k px and k f x by the
corresponding expressions indicated in Eqs. (7b) and (7c), it
can be shown that:
ξop − ξo π
po
p̂o =
=
(19)
md g
(1 + L x /lo )
where Lh = L(1 : 2, :) are the first two columns of
L. Please note that Eq. (13) does not include second-order
terms. Furthermore, the only nonlinear term is the function
sign(ṡ) contained in K̄ f (ṡ), which considerably reduces
the computational effort necessary to perform numerical
integrations. It will be shown later, however, that strong ground
motions induce large deformations in the BH-TMD and, hence,
the exact equations of motion should be used in such cases to
obtain an accurate solution.
where p̂o is the optimal slip force, normalized by the weight
of the pendular mass. Notice that the optimal value for the
y-direction is obtained by substituting L x by L y in Eq. (19),
which gives a greater value of p̂o (remember that L x > L y ). A
possible solution for this inconsistency is to adopt a value of p̂o
for the direction in which a higher degree of control is required.
In a true building, however, this inconsistency is essentially
irrelevant, since the performance of TMDs is rather insensitive
to the damping ratio in the neighborhood of the optimal value.
(18)
J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
1553
(a) Plan view of a typical building story.
Fig. 6. Thin-walled cylindrical steel chimney considered in this study (model
M1).
Based on this observation, the smallest of the values of p̂o given
by Eq. (19) is adopted in this study.
5. Structures, response quantities, and ground motions
Two nominally symmetric structural models are considered
in this study. The first one, denoted as model M1 (Fig. 6) is
a steel chimney typically found in copper processing plants.
The height of the structure is 80 m, the diameter is 3 m,
and the average mantle thickness is 0.02 m. The fundamental
frequencies are ωx = ω y = 1.37π rad/s, where perpendicular
directions X and Y may have any orientation. It is assumed that
an isotropic TMD is incorporated at the top of the structure,
as shown in Fig. 6. Torsional effects due to the eccentric
location of the TMD will not be taken into account. The second
model, denoted M2 (Fig. 7) is a 25-story reinforced concrete
building designed to the current Chilean seismic code. The
corresponding fundamental frequencies are ωx = 1.05π and
ω y = 1.4ωx in the X and Y directions, respectively. It is
assumed that an orthotropic TMD has been attached to the roof
level at the center of mass of the structure.
For comparison, two types of TMDs are included in each
of the structural models: (i) the proposed BH-TMD; and
(ii) an “ideal” bidirectional linear TMD with viscous energy
dissipation. The latter, denoted BLV-TMD, is shown in Fig. 8.
The practical implementation of the BLV-TMD requires that
both the springs and the viscous dampers behave linearly
even when subjected to large deformations. The dynamic
properties of the structural models and corresponding TMDs
are summarized in Table 1. The properties of the TMDs were
selected using design equation (14)–(19).
The efficiency of the TMDs in reducing an arbitrary response
quantity r (t) is evaluated through the following reduction
factors:
Ψ1 = 1 −
(r̄ + σr )controlled
(r̄ + σr )uncontrolled
(20)
(b) Resisting planes 1, 2, and 3.
Fig. 7. 25-story R/C building considered in this study (model M2).
Fig. 8. Schematic plan view of the Bidirectional Linear Viscous Tuned Mass
Damper (BLV-TMD) used as benchmark device.
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
Table 1
Dynamical parameters of the models considered in this study
Parameters
M1 model
M2 model
X -Dir. Frequencies (rad/s)
ω1
ω2
ω3
1.37π
8.45π
24π
1.05π
3.66π
7.21π
Y -Dir. Frequencies (rad/s)
ω1
ω2
ω3
1.37π
8.45π
24π
1.47π
5.12π
10.1π
Damping ratio
ξs
0.02
0.05
Bidirectional Linear Viscous (BLV-TMD)
ωd x (rad/s)
ωdy (rad/s)
ξd x
ξdy
µ
1.32π
1.32π
0.086
0.086
0.03
1.01π
1.42π
0.086
0.086
0.03
Bidirectional Homogeneous (BH-TMD)
L x (cm)
L y (cm)
po /Wd
lo /L x
ξo
µ
57
57
0.12
1.0
0.02
0.03
97
49
0.10
1.0
0.02
0.03
Primary structure
TMD
Fig. 9. 5%-damped pseudo-acceleration response spectra of the earthquake records considered in this study.
and
max (kr (t)k)controlled
(21)
max (kr (t)k)uncontrolled
PN
r (t), N being the number of time
where r̄ (t) = N1 t=1
PN
discretization points of r (t); and σr = ( (N 1−1) t=1
(r (t) −
r̄ (t))2 )1/2 .
Since the efficiency of TMD devices is sensitive to the intensity, duration and frequency content of the excitation [1–5],
ground acceleration histories from different events and soil
were selected: (1) El Centro (Imperial Valley, USA,1930); (2)
Newhall (Northridge, USA, 1994); (3) Melipilla (Chile, 1985);
and (4) SCT (Michoacán, México, 1985). The corresponding
5%-damped pseudo-acceleration response spectra are shown in
Fig. 9. In order to get more insight into the dynamics of TMDΨ2 = 1 −
equipped structures, low-intensity harmonic excitations were
considered as well, which also provide some information about
the response for wind loads.
Frequency Response Functions (FRFs) for the displacement
response at the top of model M1 are shown in Fig. 10 (top).
The FRF for the uncontrolled structure and the one for the
structure with the BLV-TMD were obtained through Fourier
Analysis. The FRF for the structure with the BH-TMD is
actually an empirical FRF given by the ratio of the steadystate non-linear response amplitude to the amplitude of the
harmonic excitation. The values shown in Fig. 10 (top) were
obtained for ẍ g = 0.01g sin(ω̄t) and for the range of frequency
values of interest ω̄. As expected, the TMD devices are very
effective in reducing the response of low damping systems in
near-resonance conditions (ω̄/ω1 ≈ 1). It is apparent that the
efficiency of the proposed BH-TMD is essentially equivalent
J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
1555
Fig. 10. Response to harmonic excitations (PGA = 0.01g) of model M1, with and without TMDs: (a) Frequency Response Functions (FRFs) for the displacement
at the top of the chimney; (b) response histories under resonance condition of normalized interaction force (left) and normalized displacement at the top of the
chimney (right).
to that of the BLV-TMD, which is slightly more efficient for
ω̄ < ω1 , and slightly less efficient for ω̄ > ω1 . Response
time histories, obtained by considering resonance conditions,
can be seen at the bottom of Fig. 10. The left plot shows
the history of the normalized interaction force λ̂x = λx /Wd ,
while the right plot shows the history of the displacement
response at the top of the chimney, normalized by the peak
uncontrolled response. It can be seen that the displacement
response histories of the structure with the TMD devices is only
7% of that for the uncontrolled case, and that both responses
are essentially identical to each other. Please note that the
displacement response history for the structure with the BHTMD is essentially a perfect harmonic function, even though
the interaction force λx is clearly nonlinear.
A comparison between results obtained using the exact
formulation for the BH-TMD (Eq. (12)) and results obtained
using the approximate formulation (Eq. (13)) is shown in
Fig. 11. These results were obtained considering the M2
model subjected to the E-W component of the SCT record
scaled to: (a) 25% (left plots); and (b) 50% (right plots). The
normalized hysteresis loops (λx /Wd vs. u/L x ) show that the
actual constitutive relationship of the BH-TMD is essentially
triangular for displacements less than 0.3L x , and that changes
in stiffness are noticeable only for larger displacements. Such
changes occur as a result of two actions in the tensile forces in
the cables: (i) an initial increase due to centripetal accelerations
an = u̇ 2 /L x (velocity hardening); and (ii) a decrease at large
deformations due to a lesser influence of the weight of the
pendular mass Wd (deformation softening). The response of the
primary structure, however, does not seem to be affected by
these effects, as shown by the corresponding normalized base
shear response history Vx (t)/Ws (bottom plot of Fig. 11(a)).
On the other hand, the right-side normalized hysteresis loops
indicate that the deformation capacity of the friction damper
is reached in this case (s = slim ), which creates a sudden
increase in stiffness due to engagement of the restrainer.
Some differences between the exact and approximate responses
(bottom plot of Fig. 11(b)) are now observed in the 60–80 s
time window. However, the response of the structure with
the BH-TMD is still very satisfactory because model M2 is
nearly in resonance with the quasi-harmonic seismic excitation
considered in this case.
Shown in Fig. 12 is the bidirectional response of model M1,
with and without TMDs, to the Melipilla and El Centro records.
The left-side plots show displacement paths at the top of the
chimney, dx (t) vs. d y (t); while the right-sideqplots show the
response history of total displacement d(t) = dx2 (t) + d y2 (t).
The uncontrolled response shows greater displacements in
the direction for which ground accelerations are larger; these
displacements are the ones most effectively reduced by the
TMDs, leading to a balance in the plus and minus direction.
Please observe that the response for the BH-TMD is very
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
(a) 0.25 × SCT.
(b) 0.50 × SCT.
Fig. 11. Unidirectional response of model M2, with and without BH-TMD subjected to the E-W component of the SCT record scaled down to (a) 25% and (b)
50%: normalized hysteresis loops of the BH-TMD (top), and response history of normalized base shear Vx (t)/Ws (bottom).
similar to that for the BLV-TMD, especially for peak response
values.
Analogously, the bidirectional response of model M2 to the
Newhall (scaled to 50%) and El Centro records is shown in
Fig. 13. Again, results show that the BH-TMD is effective in
reducing the response of the primary structure and is similar
to the BLV-TMD. Moreover, Tables 2 and 3 summarize values
of response reduction factors Ψ1 and Ψ2 (Eqs. (20) and (21))
for models M1 and M2 under unidirectional and bidirectional
excitations. In the case of model M2 and BH-TMD, results are
presented for the records scaled to both 50% and 100%.
Some observations on the performance of the TMDs for
the records considered in this study are worth mentioning at
this point. Consider first the response to the quasi-harmonic
SCT record. FRFs of Fig. 10 clearly show that very significant
response reductions can be achieved when the natural frequency
of the primary structure is close to the resonance frequency,
while even response amplifications might occur away from
resonance. Indeed, values shown in Tables 2 and 3 for the
SCT record indicate that reductions of up to 45% are possible
in model M2, while the response of model M1 is actually
amplified by approximately 20%. Results, however, are very
different for the Melipilla record, which has wider band
characteristics. Response reductions reach 60% for model M1
and just 20% for model M2. In the case of the Newhall record,
which shows impulsive characteristics, factor Ψ1 reveals that
important reductions along the whole response history are
achieved with the TMDs, although the reduction of peak
responses (indicated by factor Ψ2 ) is small.
All values of Ψ1 and Ψ2 obtained for both unidirectional and
bidirectional excitations indicate that the efficiencies of the BHTMD and BLV-TMD are essentially the same. An exception
occurs when the intensity of the excitation is large enough as to
induce deformations in the friction damper that are larger than
its deformation capacity slim . In these cases, a sudden increment
of the interaction force takes place as the restrainer engages,
which untunes the pendular mass.
6. Conclusions
A new passive frictional and homogeneous TMD vibration
reduction device (BH-TMD) has been studied and proposed.
Based on the analytical and experimental results obtained in
this investigation, we conclude that:
(1) The advantages of the proposed BH-TMD are its simplicity,
well-known dynamic pendular behavior, stable energy
dissipation by friction, versatility in tuning the two lateral
frequencies of the building independently, and energy
dissipation that is proportional to the square of the motion
amplitude.
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
(a) Melipilla.
(b) El Centro.
Fig. 12. Bidirectional response of chimney (model M1) with and without TMDs, to two earthquake records: (a) Melipilla; and (b) El Centro. Displacement paths
are shown at left and response history of total displacement d(t) at the top of the chimney at right.
Table 2
Maximum uncontrolled response (in % of total height H ), and reduction factors Ψ1 and Ψ2 (in brackets) for chimney (model M1)
-X Y
Record
Melipilla
-X
-X Y
SCT
-X
-X Y
Mean
-X
-X Y
0.98
1.05
0.44
0.45
0.48
0.49
0.53
0.57
0.32
(0.12)
0.50
(0.32)
0.50
(0.30)
0.63
(0.58)
0.57
(0.44)
−0.21
(−0.20)
−0.21
(−0.23)
0.28
(0.16)
0.29
(0.16)
0.29
(0.13)
0.53
(0.28)
0.41
(0.27)
0.60
(0.53)
0.50
(0.45)
−0.21
(−0.26)
−0.26
(−0.24)
0.28
(0.12)
0.23
(0.15)
-X Y
Newhall
-X
0.22
0.30
BLV-TMD
0.22
(−0.04)
BH-TMD
0.22
(−0.07)
Maximum uncontrolled response
(%H )
Reduction factors
El Centro
-X
(2) The evaluation of the response of two different structural
models subjected to different unidirectional and bidirectional ground excitations shows that the level of response
reduction that can be achieved by the BH-TMD is similar
to that of an ideal linear viscous TMD; the BH-TMD became less effective only when the deformation capacity of
the friction damper is reached and the restraint engages due
to excessive displacement of the pendular mass.
few percentage points to 60%.
(4) Experimental results obtained through shaking table tests
of a scaled model of an isotropic BH-TMD demonstrate
that the proposed device is a realization of an homogeneous
device, i.e., its fundamental period and equivalent damping
ratio are essentially independent of the vibration amplitude.
(3) Depending on the excitation and structure, the BH-TMD
may reach displacement reduction factors that vary from a
This investigation has been supported by the Pontificia
Universidad Católica de Chile under Grant DIPEI 2002/09E,
Acknowledgements
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
(a) 0.5 × Newhall.
(b) El Centro.
Fig. 13. Bidirectional response of 25-story building (model M2) with and without TMDs, to two earthquake records: (a) Newhall scaled to 50%; and (b) El Centro.
Displacement paths are shown at left and response history of total roof displacement d(t) at right.
Table 3
Maximum uncontrolled response (in % of total height H ), and reduction factors Ψ1 and Ψ2 (in brackets) for 25-story building (model M2)
El Centro
-X
-X Y
Newhall
-X
-X Y
Record
Melipilla
-X
-X Y
SCT
-X
-X Y
Mean
-X
-X Y
0.33
0.33
0.82
0.82
0.19
0.26
1.42
1.43
0.69
0.71
0.42
(0.05)
0.40
(0.04)
0.53
(0.13)
0.51
(0.13)
0.24
(0.00)
0.26
(0.09)
0.45
(0.37)
0.43
(0.37)
0.41
(0.14)
0.40
(0.16)
scaled to 50%
0.41
(0.03)
0.37
(0.05)
0.54
(0.13)
0.50
(0.13)
0.22
(−0.02)
0.23
(0.10)
0.44
(0.39)
0.42
(0.38)
0.40
(0.13)
0.38
(0.17)
scaled to 100%
0.40
(0.03)
0.36
(0.03)
0.52
(0.12)
0.35
(0.13)
0.22
(−0.03)
0.22
(0.10)
0.17
(0.08)
0.15
(0.06)
0.33
(0.05)
0.27
(0.08)
Maximum uncontrolled response (%H )
BLV-TMD
Reduction factors
BH-TMD
and the Chilean National Fund for Research and Technology,
FONDECYT under Grant No 1050691. The authors are
grateful for this support.
case be expressed by:
Appendix A. Differential equation of motion of the proposed BH-TMD
where T (q, q̇) = 12 m d u̇ 2 + v̇ 2 + ẇ 2 = 12 ṙT M(r ) ṙ is the
kinetic energy of the pendular mass, M(r ) = m d I the local
mass matrix (I is a 3 × 3 identity matrix); Vg (q) = m d gw is
the gravitational potential energy, g the acceleration of gravity;
The equation of motion of the proposed BH-TMD (Eq. (5))
is derived as follows. The Euler–Lagrange equation can in this
∂ Vg
d ∂T
∂T
−
+
+ Qi + Qe = 0
dt ∂ q̇
∂q
∂q
(A.1)
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
∂s
Qi = ∂q
f µ + fr + Cq q̇ is the generalized internal force
(dissipative forces), Cq = ξo I the assumed intrinsic viscous
damping matrix of the device that takes into account the energy
dissipated in the connections (I is a 2 × 2 identity matrix in this
case); and Qe = JT m d a is the generalized external force (input
force), a the
vector of
total accelerations at the supports and
∂r ∂r
∂r
is the Jacobian matrix.
J = ∂q =
∂u ∂v
Equivalent expressions for the five terms of Eq. (A.1) can be
obtained as shown below:
(1) First term, dtd ∂∂Tq̇
d ∂T
d (q) (q)
=
M q̇ = M(q) q̈ + Ṁ q̇
dt ∂ q̇
dt
∂ ṙ T ∂ T
∂T
=
=
∂q j
∂q j ∂ ṙ
= q̇T
(A.2)
(3) Third term,
X ∂m i(q)
j
∂qk
q̇k .
(B.3)
Substitution of (B.2) and (B.3) into (B.1) gives:


X X ∂m i(q)
X (q) d ∂T
j 2


m i j q̈ j +
=
qk + ϑ (q) q̈ j
0
dt ∂ q̇i
∂q
k
j
k
j
XX
j
(A.3)
(q)
∂m i j
∂qk
k
q̇ j q̇k .
(B.4)
Clearly, only the first term of (B.4) is linear, i.e.:
X (q) d ∂T
≈
m i j q̈ j
0
dt ∂ q̇i
j
(B.5)
or
(A.4)
(4) Fourth term, Qi
d ∂T
(q)
≈ M̄ q̈;
dt ∂ q̇
(q)
∂s
∂rT ∂s
Qi =
f µ + fr + Cq q̇ =
f µ + fr
∂q
∂q ∂r
∂s
+ Cq q̇ = JT
f µ + fr + Cq q̇
∂r
where M̄
(ii) Term
T = 1/2
Appendix B. First-order approximation of Eq. (5)
This appendix shows the derivation of the first-order
approximation of Eq. (5), which has four nonlinear terms: (i)
d ∂T
∂T
∂w
∂s
dt ∂ q̇ ; (ii) ∂q ; (iii) ∂q ; y (iv) ∂q .
X (q)
d ∂T
(q)
=
m i j q̈ j + ṁ i j q̇ j
dt ∂ q̇i
j
0
1
(B.6)
is the local mass matrix evaluated at q = 0.
XX
(q)
m i j q̇i q̇ j
(B.7)
j
(B.8)
Obviously, expression (B.8) does not include linear terms,
i.e.:
Substituting Eqs. (A.2)–(A.6) into Eq. (A.1) gives Eq. (5).
(q)
1
= md
0
(q)
(A.6)
= M(q) q̈ + Ṁ
M̄
(q)
X X ∂m i j
∂T
=
q̇i q̇ j .
∂qk
∂qk
i
j
where can be obtained using Eq. (2).
(5) Fifth term, Qe
∂w
a
md h
Qe = JT m d a = I
az
∂q
∂w
az .
= m d ah + m d
∂q
∂T
∂q
i
(A.5)
∂s
∂r
d ∂T
dt ∂ q̇
(q)
ṁ i j =
+
∂ Vg
∂q
∂ Vg
∂w
=
m d g.
∂q
∂q
(i) Term
where ()|0 denotes the function () evaluated at q = 0; and ϑ 2 (q)
(q)
(q)
represents the nonlinear terms. In addition, ṁ i j = dtd (ṁ i j )
can be written as:
0
T
∂J
q̇ M(r ) ṙ
∂q j
!
∂J T (r )
M J q̇ = q̇ T H j q̇.
∂q j
0
k
where M(q) = JT M(r ) J is the generalized mass matrix.
(2) Second term, ∂∂qT
(q)
where m i j = M(q) (i, j) can be expressed by a Taylor series
as:
(q) X
∂m
i
j
(q)
(q)
qk + ϑ 2 (q)
(B.2)
m i j = m i j |0 +
∂q
k
k
q̇ (Eq. (A.2))
(B.1)
∂T
≈ 0.
∂q
(B.9)
(iii) Term ∂w
∂q
The ith component of vector ∂w
∂q can be expressed by a
Taylor series as:
X ∂ 2 w ∂w ∂w
q j + ϑ 2 (q)
+
(B.10)
=
∂qi
∂qi 0
∂q
∂q
i
j
0
j
∂w where ∂q
= 0. Hence, the linear approximation of ∂w
∂q can be
i 0
expressed by:

 1
0
∂w


≈ H̄w q;
H̄w =  L x
(B.11)
1 
∂q
0
Ly
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J.L. Almazán et al. / Engineering Structures 29 (2007) 1548–1560
2w where H̄w (i, j) = ∂q∂i ∂q
is the Hessian matrix of w(q)
j 0
evaluated at q = 0.
∂s
(iv) Term ∂q
In this case, the procedure followed to linearize terms (i) and
(iii) leads to:

 1
1
+
0
∂s


lo
(B.12)
≈ H̄s q;
H̄s =  L x

1
1
∂q
0
+
Ly
lo
2s where H̄s (i, j) = ∂q∂i ∂q
is the Hessian matrix of s(q)
j 0
evaluated at q = 0.
Substituting Eqs. (B.6), (B.9), (B.11) and (B.12) into Eq. (5)
gives Eq. (6).
[4]
[5]
[6]
[7]
[8]
[9]
[10]
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