Journal
of Applied
AppliedMechanics
Mechanics
Vol.13,
pp.587-594 Journal of
Vol.
13 (August
2010) （August 2010）
JSCE
JSCE
Vibration Control Effects of Tuned Cradle Damped Mass Damper
ࠢ࡟ࠗ࠼࡞ဳ೙ᝄⵝ⟎ߩ⥄↱ᝄേߦ߅ߌࠆ೙ᝄലᨐ
Hiromitsu TAKEI* and Yoji SHIMAZAKI**
ᱞ੗໪ల࡮ፉ㦮ᵗᴦ
* MS Dept. Civil Eng. Tokai University (Kitakaname, Hiratsuka, Kanagawa 259-1292)
** Member PhD Prof. Dept. Civil Eng. Tokai University (Kitakaname, Hiratsuka, Kanagawa 259-1292)
This research introduces a new mechanical device for dissipating vibration, called the tuned cradle mass damper
(TCMD). This device relies on the motion of a swing mass on a curved surface to dissipate structural vibration
energy. The objectives of this study are to develop a model of TCMD and verify its performance through both
experiments and numerical analysis when the structure is under free vibrations. The proposed device was
developed by using simple pendulum dynamics that are applied to a structure with a frequency of approximately
1 Hz. For this study, the damper was installed in a one-story simple rigid frame model and excited in free
vibration.
Key Words : vibration control, tuned mass damper, passive damper, free vibration
1.
Introduction
The Great Hanshin-Awaji Earthquake of 1995 served as a
reminder to Japanese citizens of how important it is to enforce
seismic safety in buildings. In recent years, mechanisms for
absorbing the large-amplitude rocking motions of earthquakes
have been installed not only in numerous high-rises, but also in
smaller buildings. However, it is difficult to ensure that these
mechanisms will function properly as designs have become
increasingly complicated and larger in scale, thus raising their
manufacturing and maintenance costs and posing other
problems. If less-expensive versions of these mechanisms can
be made, building owners will become more motivated about
improving the safety of their buildings, which in turn will further
protect our cities from earthquakes.
Vibration control systems for buildings can be classified into
two types: active controllers, which require an exterior energy
source to absorb the vibrational energy, and passive controllers,
which do not require an energy source. Several kinds of passive
controllers have been developed into practical products,
including a type using laminated rubber or a coil spring to
support the weight of a damper, a hanging type based on the
principle of the pendulum1), an impact damper using a steel ball2),
a tuned liquid damper (TLD) using liquid materials3,4,5) and a
tuned rotary-mass damper (TRMD) consisting of a rolling mass
and container allowing free movement of the mass along its
inner arc6,7).
This research introduces a new mechanical
vibration-dissipating device: the tuned cradle mass damper
(TCMD). This device relies on the movement of the swing
mass on a curved surface to change the dynamic characteristics
of a structure by dissipating its vibration energy. The TCMD
utilizes simple driving force, which is developed in response to
the structural motion. The benefits of TCMD are its simplicity,
compactness and ease of maintenance. Small wheels, attached
to the swing mass, are seated and they move along the curved
surface. This configuration enables the TCMD to sustain a
natural frequency closely in tune with the engineered structure.
The objectives of this study are to develop a model of
TCMD and verify its performance through both experiments
and numerical analysis when the structure is under free
vibrations. There are more than 20,000 ten-story slender
buildings in the Tokyo metropolitan area. The natural frequency
of these buildings is approximately 1 Hz. The proposed device
was developed by using simple pendulum dynamics that are
applied to a structure with a frequency of approximately 1 Hz.
For this study, the damper was installed in a one-story simple
rigid frame model and excited in free vibration.
2.
Device configuration
2.1 Experimental model
As stated above, to examine TCMD experimentally, a
model of a one-story structure is used. Figure 1 shows the model,
which is made of steel columns 1250 mm in length, 60 mm in
width and 4.5 mm in thickness. Four columns support a floor.
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The lateral spring constant k1 of the structure is 0.175 kgf/mm.
Figure 2 shows the force-displacement relationship of the
structure for k1. The effective mass of the structure is
approximately 50 kg. The natural frequency of the structure is
approximately 1 Hz.
2.2 Modeling of TCMD
Figure 3 shows the TCMD model. The swing mass is
made of two steel plates with three small wheels. The size of the
plate is 175 × 60 × 3 mm and the diameter of the wheel is 22
mm. The swing mass including the wheels is about 600 g. The
mass moves along three curved surfaces (radii of 300 mm) as
the structure moves. To obtain magnetic damping 6,7), an
aluminum plate is placed on the side of the middle curved
surface. The damping strength of the mass can be adjusted by
varying the number of magnets attached to the swing mass.
Neodymium magnets (size: 10 × 10 × 4 mm) are used to obtain
the damping. Figure 4 and 5 show the free vibration of TCMD
when zero and seven magnets are used for the TCMD
respectively.
Fig.4 Free vibration of TCMD (0-magnet)
a)Front view
b) Side view
Fig.1 Simple structure model
Fig.5 Free vibration of TCMD (7-magnet)
h2 (%)
Fig.2 Force-displacement line of the structure
Figure 6 shows the relationship between the number of
magnets used for the TCMD and the corresponding its damping
ratio h2. Here, h2. is the damping ratio obtained from the free
vibration of TCMD. The natural frequency of the TCMD is
0.92 Hz when the amplitude of the swing is small. The swing
speed can be adjusted by modifying the radii of the curved
surfaces and/or the size of three wheels.
Number of magnets
Fig.6 Number of magnets versus damping constant h2
Fig.3 TCMD model
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3.
Numerical analysis model
Only the viscous damping
Figure 7 shows the free body diagram of TCMD. Here,
m2 is the mass of the swing. Because the mass of the wheels is
c2 is considered here. We do not
take the frictional damping between the wheel and the curved
surface into account for simplicity. Then, the equation of the
cradle in the horizontal direction becomes
small compared to the swing mass itself, we do not take the
rotational energy by the wheels into account. In addition, Ft
m2 x2 Ft
is the horizontal force given by TCMD, x2 is the horizontal
where
Ft
N ˜ sinT Fd ˜ cosT ,
(4)
1
AT 2 ˜ sinT c2 AT ˜ cosT
m2
x2
or
0,
Fd
g ˜ cosT sinT
(5)
Ͱ
ǰ
Figure 8 shows the schematic figure of the structure. Here,
m1 is the mass of the structure, c1 is the viscous damping
m2
coefficient, k1 is the spring constant of the structure, x1 is the
horizontal displacement of the structure. We then obtain the
following equation of motion for the structure.
N
v
u
m2g
Ft c1 x k1 x1
Ft
m1
x1
(6)
||
m2 AT
2
m2 AT
or x1
c1
k
m
x1 1 x1 2 AT ˜ sinT
m1
m1
m1
m2
c AT
g ˜ cosT sinT 2 cosT
m1
m1
(7)
We can now calculate the coupled equations (5) and (7)
numerically.
x2
Ft
Fig.7 Free body diagram of TCMD
m1
acceleration supplied by the structure, Fd is the damping force,
k1 x1
c1 x1
and N is the normal force. The equation of motion of the mass
m2 in the u and v directions can be written as
u direction: m2 g ˜ sinT Fd
m2 AT ,
Fig.8 Free body diagram of structure
(1)
4.
v direction: m2 g ˜ cosT N
where
Fd = c2 AT .
x1
m2 AT 2 , (2)
(3)
Results of the experiment
Experimental measurements are made to clarify the
dissipation of vibration energy of the simple model
structure by TCMD action. The laser displacement
sensors are used to measure both the time-displacement
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responses of the structure and the TCMD. In response to
the vibration stimuli, TCMD shows that the free
vibration motion of the model structure can be
controlled to within a frequency range of approximately
1 Hz. Therefore, good energy dissipation is obtained.
Figure 9 shows the displacement responses of the
experimental structure in the uncontrolled condition (i.e.,
without a damper). The initial lateral displacement
given to the structure is 24 mm and the frequency of the
structure f1=0.92 Hz. The damping ratio h1 of the
structure is 0.12%.
Figure 10 shows the displacement responses of the
experimental structure in the controlled condition when
no magnet is installed on the swing mass of the TCMD.
The beats are observed because of the weak damping of
TCMD.
Fig.9 Free vibration of frame model (uncontrolled)
a)
When the swing mass with magnets moves on the
curved surface, the damping ratio becomes much
greater than that without the magnet. Figure 11 shows
the ratio obtained by the experiment. The initial
displacements given to the free vibration of the structure
are 12 mm, 24 mm and 34 mm. It is observed that the
structural damping h1 becomes more than 5% when an
appropriate number of magnets is attached to the swing
mass. Figures 12 through 17 show the experimental
wave form of the structure with TCMD when f1 = 0.90,
0.92 and 0.95 Hz. In the figures, (a) and (b) show the
experimental wave form when 3 and 7 magnets are used
for TCMD, respectively.
5.
Results of Analysis
To solve equations (5) and (7), a fourth-order
Runge-Kutta method is applied. Figures 18 through 23
show comparisons of the results obtained by the experiments
and those by the analyses. The initial lateral displacement and
the natural frequency of the structure are 24 mm and 0.92 Hz
respectively. In the figures, both the structure and the swing
mass movement are shown. It is shown that the results
obtained by the numerical analysis agree well with
those of experiment. Figures 20 and 21 suggest the
swing mass with seven magnets for TCMD gives the
best damping effect on the model structure.
Structure response for TCMD with no magnet
(a) Initial displacement = 12 mm
b) TCMD response with on magnet
(b)Initial displacement = 24 mm
Fig.10 Structure-TCMD response (controlled)
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(a) Three magnets
(c) Initial displacement = 34 mm
Fig.11 Damping ratio obtained by the experiment
(a) Seven magnets
Fig.13 Structure responses
( f1 = 0.9 Hz , initial displacement = 34 mm)
(a) Three magnets
Time ( sec )
(a) Three magnets
(b) Seven magnets
Fig.12 Structure responses
( f1 = 0.90 Hz , initial displacement = 12 mm)
(b) Seven magnets
Fig.14 Structure responses
( f1 = 0.92 Hz , initial displacement = 12 mm)
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(a) Three magnets
(a) Three magnets
(b) Seven magnets
(a) Seven magnets
Fig.17 Structure responses
( f1 = 0.95 Hz , initial displacement = 34 mm)
Displacement (mm)
Fig.15 Structure responses
( f1 = 0.92 Hz , initial displacement = 34 mm)
Time (sec)
(a) Experiment
Displacement (mm)
(a) Three magnets
Time (sec)
(b) Analysis
(b) Seven magnets
Fig. 18 Structure responses for h2=5.3% (three magnets)
( f1 = 0.92 Hz , initial displacement = 24 mm)
Fig.16 Structure responses
( f1 = 0.95 Hz , initial displacement = 12 mm)
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Displacement (mm)
Displacement (mm)
Time (sec)
Time (sec)
(a) Experiment
Displacement (mm)
Displacement (mm)
(a) Experiment
Time (sec)
Time (sec)
(c) Aanalysis
(b) Analysis
Displacement (mm)
Fig. 21 Cradle responses for h2=8.9% (seven magnets)
( f1 = 0.92 Hz , initial displacement = 24 mm)
Displacement (mm)
Fig. 19 Cradle responses for h2=5.3% (three magnets)
( f1 = 0.92 Hz , initial displacement = 24 mm)
Time
Time(sec)
(sec)
Time (sec)
(a) Experiment
Displacement (mm)
Displacement (mm)
(a) Experiment
Time (sec)
Time (sec)
(b) Analysis
(b) Analysis
Fig. 20 Structure responses for h2=8.9 % (seven magnets)
( f1 = 0.92 Hz , initial displacement = 24 mm)
Fig. 22 Structure responses for h2=12.7% (11 magnets)
( f1 = 0.92 Hz , initial displacement = 24 mm)
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Displacement (mm)
2)
3)
Time (sec)
(a) Experiment
Displacement (mm)
4)
5)
Time (sec)
(b) Analysis
6)
Fig. 23 Cradle responses for h2=12.7% (11 magnets)
( f1 = 0.92 Hz , initial displacement = 24 mm)
7)
6.
Conclusions
A new mechanical vibration absorber, Tuned
Cradle Mass Damper (TCMC) is suggested. This
TCMD has a simple construction and is applicable to
the structure which may vibrate in lower frequency
modes. The equation of reciprocal motion between
TCMD and the structure is derived using simple
pendulum theory. It is shown both experimentally and
numerically that the new tuned cradle mass damper
can efficiently dissipate undesirable vibration energy
in a structure within the range of elastic deformation.
Acknowledgement
The financial support of the “Research for Promoting
Technological Seeds” is gratefully acknowledged.
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(Received March 9, 2010)