A passive control device (Tuned liquid damper (TLD)) is widely used in the
structural vibration control (Tanaka and Mak, 1983;
Fulin et al., 1993; Fulin,
1997). However, the damping of TLD used in the past days is not large. So
that it leads to the worse control performance compared with another widely-used
control device-tuned mass damper (TMD). Thus, increasing the sloshing damping
is a key task in TLD study (Soong and Dargush, 1997).
The energy method to study the damping characteristics when liquid flows through
the vertical damping devices and obtained the formulation for additional sloshing
damping of sloshing liquid is studied by Warnitchai and
Pinkaew (1998). In view of this, the damping characteristics of a new liquid
damper-TLD embedded a transverse cylinder (TLDETC) is studied in this paper.
Meanwhile, the fluid mechanics principle and energy method will be combined
to derived the optimal sloshing damping of TLDETC. Xu
et al. (1992) indicated that a U-shaped water tank filled with water
could be used as the mass of a TMD, namely, tuned liquid column/mass damper
(TLCMD) (Lan, 2001). presented a control device combined
tuned liquid column damper (TLCD) and tuned liquid damper (TLD), namely, hybrid
tuned liquid dampers system (MTLDs, or HTLD), which could be used to reduce
the structural vibration. This paper presents two new tuned-type damper systems-tuned
hybrid-tank/mass damper (THMD) and tuned double liquid columns/mass damper (TDLCMD),
which make fully use of the extra space in the backside of TLCMD and the control
effectiveness of HTLD to obtain better practicability and control performance
in structural control. However, excess water motion in both systems may reduce
the effectiveness of these two damper systems. This paper presents a phasic
difference analysis method to investigate synthetical effect of TLCD, TLD and
TMD simultaneously. A proposed method is also discussed to give design suggestions
to these two systems in the real engineering.
MODEL OF THE TLDETC
The cylinder is embedded in a certain location in the middle (half of the container
length) of the container and the surface of the cylinder is assumed to be smooth.
Figure 1 shows the schematic of an ordinary TLD model and
a TLDETC model with the same container dimension and liquid mass.
As shown in Fig. 1, M is the total liquid mass and A is the
container length. H is the liquid height of TLD. For TLDETC, H' is the liquid
height, R is the radius of the cylinder, while h is the vertical distance from
the center of the cylinder to the free surface of liquid. Moreover, the container
width is presented as B. It can be observed from Fig. 1 that
the liquid height in a container is increased from H to H' due to the embedded
||Schematic of damper models (a) TLD and (b) TLDETC
FORMULAS OF TLDETC
According to the assumptions of hydrodynamics theory used in TLD calculation
(Zhenan et al., 1993) the basic parameters (sloshing
mass and frequency) of TLD are given as follows:
The first sloshing mode of liquid sloshing is the most effective in TLD research.
Hence, only the first sloshing mode is considered when the dynamic characteristics
of TLD or TLDETC is discussed in this paper.
When the liquid flows around the cylinder, the flow-induced force on the cylinder
along liquid flowing direction can be obtained by Morisons Eq.
where, fm(x, z, t) and fd(x, z, t) are the inertial and
drag components of the flow-induced force f(x, z, t), respectively; Cm
and Cd are the coefficients of inertia and drag of the cylinder,
respectively. Moreover, Cm = 2 and Cd = 1 (Xu
et al., 1992) are considered in this discussion. The additional sloshing
damping ratio ξa can be obtained by using the method proposed
by Lazan and Goodman (Lan, 2001).
where, ΔE is the energy loss of the sloshing due to the cylinder in one
sloshing cycle, the expression of which was given by Keulegan (Housner,
1957; Haroun and Pires, 1994) Eq.
where, A1 is the sloshing amplitude of the first sloshing mode of
TLDETC. And E can be obtained through the gravitational potential energy of
the sloshing liquid.
Hence, substitute Eq. 5 and 6 into Eq.
4, additional sloshing damping ratio is then obtained:
where, it can be observed from Eq. 7 that ξa
is proportional to the sloshing amplitude A1 whose expression can
be obtained by using D' Alemberts principle:
where, X0 is the amplitude of external excitation. Hence, A1
is positive real solution of Eq. 8, while ωT
is the sloshing frequency of TLDETC which can be obtained by hydrodynamics theory:
where, in order to determine the optimal additional sloshing damping of TLDETC
with definite container dimension and liquid mass, h and R are differentiated
in Eq. 7.
Hence, the relationships between h and R can be obtained by solving Eq.
where, Eq. 11 presents the relationships between h and R
for the optimal additional sloshing damping ratio of TLDETC. When the definite
h and R meet Eq. 11, the optimal additional sloshing damping
ratio can be obtained in theory; however, both h and R can not be designed too
great to disturb the stability of the liquid flowing around the cylinder. Thus,
in this paper, the ranges of h and R are set to be H/3 = h = 2H/3 and 0 = R
Two schemes are conducted to compare the damping characteristics of TLDETC
and TLD in this paper. External excitation is assumed to be Χ (t) and the
liquid masses of TLDETC and TLD are the same in the two schemes.
Scheme 1: The container dimensions of TLDETC and TLD are the same while
the liquid depths in TLDETC and TLD are different: ATLD = ATLDETC
= 5 m, BTLD = BTLDETC = 5 m, while HTLD = 4
m. The curve diagram for the relationship between R and h can be obtained from
Eq. (11) (Fig. 2).
As shown in Fig. 2, R is not in the range of≤R≤0.4
m when h is in the range of 1.333≤h≤2.666 m for the optimal sloshing mass,
but it can be observed that the curve goes closer to Rs value range as
h goes lower. Hence, h can be ascertained as h = 1.333 m, then R can be ascertained
as R = 0.4 m to obtained the optimal additional sloshing damping ratio. Meanwhile,
HTLDETC = 4.100 m can be obtained. Then the increasing amplitude
of sloshing damping ratio between TLDETC and TLD under different excitation
frequency in this scheme is presented in Fig. 3.
As shown in Fig. 3, the additional damping ratio appears
when TLDETC is excited by the external excitation, which means TLDETC runs with
a larger sloshing damping compared with the ordinary TLD. In this scheme, the
maximal increasing amplitude of damping ratio reaches up to about 1.27% which
happens when the external excitation ω≈2.095 rad sec-1.
Scheme 2: The liquid depths in TLDETC and TLD and the sloshing frequencies
of TLDETC and TLD are the same while the container dimensions of TLDETC and
TLD are different: HTLD = HTLDETC = 4 m while ATLDETC
= 5 m, BTLDETC = 5 m, then h = 1.333 m and R = 0.4 m are ascertained
to obtain the optimal sloshing mass of TLDETC by means of the method used in
Scheme 1, which yields ωTLDETC = 2.446 rad sec-1;
then ωTLD = ωTLDETC = 2.446 rad sec-1
is given. In this case, ATLD = 5.073 m, BTLD = 4.804 m
are obtained. The increasing amplitude of sloshing damping ratio between TLDETC
and TLD under different excitation frequencies in this scheme is presented in
As shown in Fig. 4, the damping characteristics of TLDETC
is similar to the case of Scheme 1. In this scheme, the maximal increasing amplitude
of damping ratio reaches up to about 1.29% which happens when the external excitation
ω≈2.093 rad sec-1.
The results of the numerical simulation indicate that the sloshing damping
ratio produced by TLDETC is greater than that produced by TLD (Tanaka
and Mak, 1983; Fulin et al., 1993; Fulin,
|| Relationship between R and h (Scheme 1)
|| Additional damping ratio of TLDETC (Scheme 1)
|| Additional damping ratio of TLDETC (Scheme 2)
||A new type of TLD device-TLDETC is presented in this paper.
The additional sloshing damping ratio of TLDETC when liquid flows through
the transverse cylinder is obtained by combining the hydrodynamics theory
and energy method, then the optimal additional sloshing damping ratio is
||The comparison analysis of damping characteristics between
TLDETC and ordinary TLD are made through two different setting schemes.
Analysis results indicate that the sloshing damping ratio produced by TLDETC
is greater than that produced by TLD, which means the damping performance
of TLDETC outperforms that of ordinary TLD