Geology Reference
In-Depth Information
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to non-viscous damping models and frequency-
independent damping models.
Models in which the damping force is a function
of past history of motion via convolution integrals
over a suitable Kernel function constitutes non-
viscous damping. They are called non-viscous
because the force depends on state variables other
than just the instantaneous velocity (Adhikari et
al 2003). The most generic form of linear non-
viscous damping given in the form of modified
dissipation function is as follows (Woodhouse
1998, Adhikari 2000):
F
=
k x
+
η
| |
| |
x
(3)
This model could be in general a better rep-
resentation of the boundary/structural damping
occurring at structural joints. As material damp-
ing is negligible in comparison to the boundary
damping, it could well be assumed that the use
of this model in dynamic analysis would give
a better representation of the overall damping
phenomenon.
It should also be noted that, in this section,
we are discussing damping models suited only
for linear dynamic analysis. Detail literature on
damping models suited for nonlinear dynamic
analysis is available and interested readers should
refer to Brenal (1994), Leger and Dussault (1992),
Carr (1997, 2005), Hall (2006), Charney (2008),
and Zareian and Medina (2010). These studies
mainly propose modeling approaches to overcome
the limitations of Rayleigh damping based on
initial stiffness.
t
1
2
(
)
∫
F
==
q
'
g t
−
τ
q
( )
τ τ
d
(1)
0
where
g
( )
represents the Kernel function and
q
(
τ
represents system velocity. This could also
be looked as a time hysteresis model applied to
discrete systems. The generality of this model is
evident from the aspect that the Kernel function
g
( )
could adopt any causal model where the
energy functional is non-negative (Adhikari et al
2003). Incorporating this model, the equation of
motion of the system can be expressed as
Numerical Study
A numerical study is carried out for illustrating
the effect of in-structure damping models on the
optimal distribution of dampers. Models used for
the study are the classical Rayleigh model and the
non-viscous model given by Equation (1) as it
represents the most general damping model within
the scope of a linear analysis (Woodhouse 1998).
t
(
) ( )
+
∫
0
Mu
+
g t
−
τ
u
τ
d
τ
Ku
=
f t
( )
(2)
where
M
is the mass,
K
the stiffness,
f
( )
the ap-
plied force,
u
the acceleration,
u
the velocity and
u
the displacement of the system.
The other most popular model is the frequency
independent damping model. The concept of
frequency independent damping arose when in
1927 Kimball and Lovell claimed that hysteretic
damping is universal in nature. Since then there has
been several studies which further strengthened
their claim. One of the most popular models in
this category is the linear Coulomb friction force
model given as (Reid 1956, Muravski 2004)
Description of the Frame
The optimal distribution of dampers derived by
Takewaki (1997) in a six storey shear building
model is used for the study. The shear building
model is shown in Figure 5. All masses are assumed
to be lumped at storey levels with m
1
=m
2
…..=m
6
=
0.8x10
5
kg. A uniform storey stiffness is assumed
with k
1
=k
2
=………=k
6
= 4.0x10
7
N/m. The optimal
damper locations are indicated in Figure 5.
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