Optimal Passive Damper Positioning Techniques: State-of-the-Art - Design Optimization of Active and Passive Structural Control Systems - page 90

each floor. Parametric studies were conducted to
determine the effect of structural damping and
the inherent structural flexibility on the control
parameters.
Cheng and Pantelides (1988) pioneered an
approach in which the locations of active control-
lers were optimized in terms of a controllability
index. This controllability index as defined by
them is a measure associated with the structure's
response to a specific earthquake. The basic idea
underlying the controllability index method is that
a controller is optimally placed when it is located
at a position where the displacement or relative
displacement response of the uncontrolled system
is the maximum. Though it was done in the context
of active control, the philosophy was very much
applicable for addressing the positioning issues
in passive control.
Zhang and Soong (1992) pioneered an exten-
sion to the above described controllability index
method to address the issue of locating passive
dampers. They developed a sequential procedure
for the optimal placement of the damper devices.
This procedure is called the Sequential Search
Algorithm (SSA), and it determines the optimal
location index by evaluating the random seismic
response of a structure using the transfer matrix
method. The mean square values of the inter-story
drifts are used as optimal location indices. The
procedure starts with determining the best location
for the first damper. It was shown that the best
position for the first damper is the location where
the inter-story drift of the uncontrolled frame is
the maximum (Cheng and Pantelides 1988). After
determining this location, the damper is added and
the procedure is repeated incorporating the added
stiffness and damping and the optimal location for
the second damper is determined. This procedure
is repeated till all dampers are placed. In this
method, the earthquake excitation is modeled as
a stationary stochastic process.
Hahn and Sathiavageeswaran (1992) proved
through a series of sensitivity analyses that in order
to get an optimum response for a shear building

with uniform stiffness during an earthquake, the
dampers should be placed in the lower half of
the building. This study mainly focused on as-
sessing the effect of distribution of visco-elastic
dampers. They also proved that tall buildings are
more sensitive to changes in the distribution of
dampers as compared to short buildings. Gurgoze
and Muller (1992) came up with a numerical
method for optimally placing the dampers and
to determine their capacities based on an energy
criterion. One common observation that could be
made in these works is that all of them considered
shear buildings with either uniform story stiffness
or with specified story stiffness. In other words,
in the optimality problem considered, stiffness
of the parent frame was never considered as a
design variable.
Tsuji and Nakamura (1996) made a significant
advancement by pioneering an algorithm to derive
an optimum set of stiffness of a shear building
frame along with the optimum set of viscous
damping devices, imposing necessary behavioral
constraints. The constraints imposed were on
maximum inter-story drifts due to a set of spectrum
compatible ground motions, on upper bounds of
the damping coefficient of each damper and on the
sum of the damping coefficients of all dampers.
Optimum problem addressed in this study was to
find a minimum cost design. The method proposed
by Tsuji and Nakamura was more efficient in the
sense that it produces an ordered set of optimum
design of shear buildings with viscous dampers by
minimizing the sum of the story stiffness subjected
to the current constraints, and each design in the
ordered set could be considered to be a 'candidate
design' corresponding to various upper bound lev-
els of damper damping coefficients. On the other
hand, the method developed by Zhang and Soong
(1992) was more intuitive as their ultimate solu-
tion only approximately optimizes the objective
function. Connor and Klink (1996) and Connor
et al. (1997) introduced the concept of a quasi-
optimal distribution in which the damper devices
are proportional to the stiffness distribution.