Geology Reference
In-Depth Information
This section mainly reviews the interaction
of the in-structure damping models with the
optimal distribution of dampers and attempts to
qualitatively evaluate the influence of the various
models on optimality in terms of response. Though
no specific conclusions are drawn, our main in-
tention here is to highlight the issues associated
with certain prevalent assumptions regarding the
in-structure damping and its effect on the optimal
distribution of dampers.
longer exists. So using the classical in-structure
damping model (Rayleigh model) does not add
any benefit. Moreover, from a realistic perspective
there are a lot of issues associated with this
model, some of which are discussed briefly here-
after. One of the main issues is the proportional-
ity phenomenon exhibited by the Rayleigh
model. In reality, the test results indicate complex
nature of the eigen modes, which implies non-
orthogonality of the mode shapes and indicates
the presence of non-classical damping (Adhikari
2000). The other main issue with this model is
the strong dependence on the frequency of the
structure as the constants
α
Discussions on the Realism of
Classical Viscous Damping
and
( )
are evalu-
ated as a function of the frequency. There have
been a large number of studies investigating the
frequency dependence of the analytical model
used in practice, and interestingly the majority of
these studies emphasize that material level damp-
ing is a strong function of
x
n
and a frail function
of
w
, where
x
refers to the displacement and
w
refers to the angular frequency (Adhikari 2000,
Bandstra 1983, Baburaj and Matsukai 1994). The
facts highlighted above raise a huge concern re-
garding the optimality criterion achieved in terms
of response reduction when the classical Rayleigh
model is used for the in-structure damping. So
here we give a brief overview of other models of
damping reported in literature and perform a
numerical sensitivity study to see the effect of
different models on the response of optimally
controlled frame.
β
The sensitivity studies by Takewaki (2009) im-
plicitly pose a big question as to what is the cor-
rect model of in-structure damping that represents
the true nature of the system. Common practice
is to use the classical viscous damping model
originated by Rayleigh, through his famous 'Ray-
leigh dissipation function', in which only the
instantaneous velocities are considered as the
relevant state variables and on employing Taylor's
expansion results in a model which captures the
damping through the formation of a 'dissipation
matrix' (Adhikari 2000). In strict mathematical
sense, Rayleigh's matrix is actually representative
of a system which is mainly driven by fluid damp-
ing due to its inherent dependence on the instan-
taneous velocity. This model is commonly used
to model damping in MDOF systems and its
popularity is mainly due to the fact that it uses
the already computed mass
(M)
and stiffness
(K)
matrices
C
(
)
α β
and demands only the
calculation of the constants
α
and
β
(Carr 2007).
The main advantage of this model is that the or-
thogonality of the modes is preserved; thereby
facilitating the classical modal analysis to be
performed more or less similar to the un-damped
vibration.
In the case of a controlled frame, due to the
addition of dampers, damping becomes non-
classical and the orthogonality of the modes no
=
M
+
K
Brief Overview of the
Models of Damping
A full detail review on all models of damping is
beyond the scope of this chapter and interested
readers should refer to Banks and Inman (1991),
Woodhouse (1998), Adhikari (2000), Muravski
(2004), Puthanpurayil et al (2011), Smyrou et al
(2011). In this section, we restrict our discussion
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