Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 63

D λ Q λ + D λ Q λ = 0 (46)

1 T

( ) ( ) ( ) ( )

where z ( ) t = [ q ( ), t D q ( )] , t ds s dd d

s t s

T

s ( ) t = [ p ( ), t 0 ] ,

s

where

C M
M 0

C 0

ss ss dd

A = , A

1 =

,

2

λ α

d d

D ( ) λ = − λ M + i λ C + ( i ) C + K + K ,

0 0

ss ss ss ss ss ss

ss

λ α

d d

D ( ) λ = ( i ) C − K ,

sd sd sd

d d

K + K 0 D ( ) λ = ( i λ α ) C − K ,

ss dd ds ds ds

B = (43)

(47)

0 − M

D ( ) λ = ( i λ α ) C

d

+ K

d

ss

dd dd dd

The eigenvalue problem, which can be solved
to determine the eigenvalue s and the eigenvector
a , is nonlinear and has the following form:

Finally, it is possible to write the relationships

Q ( ) λ = H ( ) , λ P Q λ = H λ P (48) ( ) ( )

s ss d ds

( s s A +

α

A + B a ) = 0 (44)

1

where the frequency response functions H ss ( λ
and H rd ( λ could be written in the following form:

The above nonlinear eigenvalue problem can
be solved using the continuation method described
by Lewandowski and Pawlak (2010). Moreover,
relationships (28) can be used to determine the
natural frequencies and non-dimensional damp-
ing ratios.
For the fractional derivative Kelvin model of
dampers, the frequency response functions are
defined as:

− 1

− 1

H H ≡ ( ) λ = D ( ) λ − D ( ) λ D D ( ) λ ,

ss ss sd dd ds

− 1

H ( ) λ = − D ( ) λ D ( ) λ H ( ) λ =

ds dd ds ss

− 1

− 1 − 1

− D ( ) λ D ( ) λ D ( ) λ − D ( ) λ D D ( λ

dd ds ss sd dd ds

(49)

The vector H d ( λ of the frequency transfer
functions of interstorey drifts can be calculated
from the following formula:

− 1

2

λ α H ( ) λ = − λ M + i λ C + ( i ) C + K + K

ss ss dd ss dd

(45)

The shear frame model is also used as a struc-
ture model in this chapter. The detailed derivation
of the equation of motion is given by Lewandowski
and Pawlak (2010). The final form of the equation
of motion, the eigenvalue problem and the matrix
of transfer functions can be written in the form of
relationships (41), (42), (44) and (45), respectively.
In the case of a structure with the fractional
derivative Maxwell dampers, after substituting
relationships (29) and (30) into Equations (36)
and (37), the following relationships are obtained:

H

d ( )

=

T H ( )

λ λ (50)

where T is the transformation matrix. In the case
of a share frame, the transformation matrix is:

...
...
........................
...

1 0 0 0 0 0

T = −

1 1 0 0 0 0

(51)

0 0 0 0 − 1 1

D ( ) λ Q ( ) λ + D ( ) λ Q ( ) λ = P ,

ss s sd d