Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 65

Table 1. Dimensions of the eight-storey frame
elements

Table 2. Unit mass of the eight-storey frame ele-
ments

Lateral
column
[cm]

Central
column
[cm]

Unit lateral
column mass
[kg/m]

Unit central
column mass
[kg/m]

Unit beam
mass
[kg/m]

Beams
[cm]

Storey Storey

7, 8 35×35 40×40 30×40 7, 8 306.2 400.0 15000.0

5, 6 40×40 45×45 30×45 5, 6 400.0 506.2 15000.0

3, 4 45×45 53×53 30×50 3, 4 506.2 702.2 15000.0

1, 2 50×50 60×60 30×50 1, 2 625.0 900.0 15000.0

Table 3. Parameters of generalized Kelvin and Maxwell models

Stiffness [MN/m] Damping factor [MN sec/m]

Kelvin model Maxwell model Kelvin model Maxwell model

k 0 57.650 0.1065 - -

k 1 18.350 33.385 c 1 2.729 1.478

k 2 6.160 3.310 c 2 6.190 1.732

k 3 0.5545 1.443 c 3 8.675 8.305

α

K '( ) λ = + k c λ cos( απ / ), 2

The energy dissipated by the damper is calcu-
lated by assuming that a damper executes har-
monically varying vibrations. This energy can be
calculated using the following formula:

α

K "( ) λ = c λ sin( απ / ) 2 (53)

The expressions (53) are analytical formulae
for the fractional-derivative Kelvin model of
dampers. The chosen parameters of the fraction-
al-derivative Kelvin model are α = 0 6. ,
k =

T
= ∫



E u t x t dt ( ) ( ) (54)

d

0

0 4 10 6 . × N/m , and

. Nsec /m α The value of the param-
eter α is similar to the one used by Chang and
Singh (2009) and by Singh and Chang (2009)
except that the original values of k and c are di-
vided by 2.0.
In the paper by Chang and Singh (2009), the
parameters of generalized models are obtained by
minimizing the mean square norm of the differ-
ences between the targeted modules and analytical
modules of the considered model. The parameters
of the generalized Kelvin model (which are used
in this example) and the generalized Maxwell
model (which will be used in Example 2), both
with seven parameters, are given in Table 3.

3 6 10 6 c = ×

where T is the period of excitation and x(t) is the
relative displacement of damper, i.e., the differ-
ence between displacements of the right and the
left end of damper, respectively.
The amplitude of displacements is equal to
0.01 m in all of the considered cases. A com-
parison of dissipated energy calculated for the
considered models of a VE damper is shown in
Figure 7. From this calculation, it may be con-
cluded that the dissipation energies of the frac-
tional-derivative Kelvin model and both general-
ized models are approximately equal in the range
0 - 15.0 rad/sec of excitation frequency. This
range of frequency covers the range of the first
three natural frequencies of vibration of the con-