Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 68

sion of the Newmark method. The chosen param-
eters of the Newmark method are: γ = 0 . and
β = 0 2. (see topic by Chopra (2000)).
It is the main objective of this example to
show that some dynamic characteristics of a
structure with dampers, modelled using the above
mentioned models is practically identical if both
models possess approximately equal possibilities
to dissipate energy. This conclusion is supported
by the results presented in Table 4. This table
contains logarithmic decrements of damping and
the peak values of displacements and accelerations
of the fourth-floor, calculated from the obtained
solutions to the equations of motion for both
damper models.

As the result of the optimization procedure, the
following distributions of dampers are obtained:
there are six dampers on the first storey, four
dampers on the second storey, one damper on
the sixth storey, and two dampers on the seventh
storey. The sequence of the successive optimal
positions of dampers was obtained as follows:
the first damper on the sixth storey, the second
damper on the seventh storey, the third damper on
the first storey, the fourth damper on the second
storey, and the remaining five dampers are located
on the first storey.
In Figure 9, the peak values of the bending
moments for columns on all storeys are presented.
The results of calculation for three cases are pre-
sented: i) the dampers in the optimal positions,
ii) all dampers are on the first storey, and iii) the
dampers are uniformly distributed on the frame.
In the last case, there are eight dampers within the
structure but the parameters of a single damper are
1.25 times as great as in the other cases. Results
for all of the considered cases are shown by the
bar with forward slashes, the bar with diagonal
crosses, and the bar with backward slashes, re-
spectively. It is easy to notice that the peak values
of the bending moments for all storeys are greater
in the case 2. Comparing the results for cases 1
and 3, it can be seen that the peak values of the
bending moment for the first storey are smaller
in the case 1, but on the higher storeys, these
peak values of the bending moments are smaller
in the case 3.
Because the algorithm of sequential optimiza-
tion method is very simple in Example 1 and 3
all steps of this algorithm were done using Excel
or manually except the calculation of the values
of objective function.
A similar comparison between the peak val-
ues of the horizontal displacements of floors is
presented, in the same manner, in Figure 10. The
peak values of horizontal displacement obtained
for the frame with the optimal distribution of
dampers are greater than those for the frame with
uniformly distributed dampers and smaller than

Example 3: An Eight-Storey Frame

A frame very similar to the one presented in Ex-
ample 1 is considered. Here, the frame with
granulated masses is taken into account, with
masses concentrated at the floor levels. The
value of every mass is 225.0 Mg. The first two
non-dimensional damping ratios of frame without
dampers, used to calculate the damping matrix of
the frame, are γ = = . . The generalized

Kelvin model is used as the model of VE damp-
er. The model parameters are sixteen times as
great as the ones shown in Table 3. It is assumed
that ten dampers are located on the frame. The
chevron braces used are as the ones described in
Example 1.
The structure is loaded with forces caused by
the horizontal North South component of the El
Centro earthquake (Peknold Version). The com-
mercial program Autodesk® Robot™ Structural
Analysis (2010) is used to solve the equations of
motion with the help of the average acceleration
version of the Newmark method.
The sequential optimization method is used
to solve the optimization problem. The maximal
peak value of the bending moments in columns
is chosen as the objective function.

γ 0 02

1 2