Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 71

Table 6. Non-dimensional damping ratios γ i

Uniform distribution of dampers Optimal distribution of dampers

Mode
Number

Frame without
dampers

Kelvin model Maxwell model Kelvin model Maxwell model

1 0.0008 0.0038 0.0036 0.0043 0.0049

2 0.0022 0.0099 0.0087 0.0085 0.0139

3 0.0035 0.0137 0.0113 0.0107 0.0148

4 0.0047 0.0162 0.0129 0.0118 0.0094

5 0.0061 0.0231 0.0177 0.0181 0.0251

6 0.0066 0.0195 0.0143 0.0200 0.0150

7 0.0073 0.0219 0.0157 0.0261 0.0218

8 0.0085 0.0208 0.0151 0.0237 0.0230

9 0.0097 0.0209 0.0154 0.0228 0.0111

10 0.0112 0.0212 0.0160 0.0214 0.0112

12a). The assumed values of the sum of the
damping coefficients and the sum of the stiffness
parameters are: C d = 500 kNsec / α , and
K d = 25000 kNm 2 , respectively. If dampers
are uniformly distributed within a structure, then
the data for every single damper are:
k d = 2500 kNm 2 , c d = 50 kNsec /m

= = kN/m ,
but the mass value is the same for every floor:
m s = 2 07

k k 16450

9 10

. Mg . The structure's damping factors

are: c = = . c 4 76 kNsec/m

1 2

c c
3

= = . 3 73 kNsec/m

4

c = = . c 2 91 kNsec/m

α

,

5 6

c = = . c 1 98 kNsec/m

c = / 0 02 . The values of fractional
parameters for all dampers are identical, i.e.,
α = 0 . . The above values of damper param-
eters are used for both of the considered frac-
tional models. Using the suggested procedure,
the dynamic properties of the considered system
were computed (see Table 6). These results show
that a frame with uniformly distributed dampers
is less damped by the fractional Maxwell damp-
er than by the fractional Kelvin damper if both
dampers have identical values of parameters.
The optimization procedures provide lower
values of damping ratios in the case of the Kel-
vin model of damper for the first, second, third
and fifth modes of vibration in comparison with
the Maxwell model. But the results of optimiza-
tion correspond to the different dampers distribu-
tion at frame structure (see Table 6 and Figure
12).

τ d

7 8

d d

c c
9

= = . kNsec/m .
The data is taken from Zhang and Soong (1992).
Two rheological models describing the dynamic
behaviour of dampers were applied in the calcu-
lations; the Kelvin fractional model and the
Maxwell fractional model.
Firstly, the calculations were carried out for a
frame without dampers, only the damping proper-
ties of the structure were taken into account. The
solution to Equation (44), where A

1 44

10

1 = and
K dd = 0 , leads to the eigenvalues s i which enables
determination of the dynamic properties of the
structure described by Equation (28). The results,
the natural frequencies of the structure and the
values of non-dimensional damping ratios are
presented in Table 5 and in the second column of
Table 6, respectively.
Next, the authors investigated a structure with
one damper mounted on every storey (see Figure

0