Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 58

Figure 4. A schematic of the generalized Maxwell
model

A typical transformation of nodal parameters
to the global coordinate system is used. The dis-
placements of the damper's external nodes are
transformed as usual but the internal variables of
the damper are still defined in the local coordi-
nate system. This means that the transformation
matrix is:



T 0
0

d

T = (14)

d

I

where



T 0
0 T

c s
s c

 T = −

T

d =

, (15)



c = cos β , s = sin β , β is the angle between the
global and the local coordinate systems and I is
the (

damping matrices given in the local coordinate
system, respectively.
The equilibrium conditions of the internal
nodes, i.e., u

m × identity matrix.
The equation of the considered model, written
in the global coordinate system, has the form:

)

( ) t − u t ( ) = 0 for i = 1, .., lead m

i − 1 1

to the following matrix equation:

R t K q t

+ 

C q t ( )

=

( ) ( )

(16)

d d d d d

=
(11)

K q t + K q t + C q t + C q t 0

( ) ( ) ( ) ( )

wz z ww w wz z ww w

where



where   ,  

T T

R ( ) t = [ R ( ), t 0 ] = T R T ,

= .
The equation of the generalized Kelvin model
written in the local coordinate system can be finally
presented in the form:

T

K K
wz

= C C

wz

d z d d d

zw zw

T

R z ( ) t = [ R t R t R t R t ( ), ( ), ( ), ( )] ,

1 2 3 4



T T



q ( ) t = [ q ( ), t q ( ) t = q ( )] t = T q T ,

d z w w d d d

T

q z ( ) t = [ ( ), q t q t ( ), q t ( ), q t ( )]

1 2 3 4

R t = K q t + C q t (12)

( ) ( ) ( )

d d d d d

are the vector of nodal reactions and the vector
of nodal parameters, respectively, written in the
global coordinate system. The explicit forms of
matrices K d and C d are given in Appendix A.
The equation of the generalized Maxwell
model could be derived in a similar way. In the
global coordinate system the equation mentioned
above has the form of Equation (16) though with
the matrices K d and C d as given in Appendix A.

 

T

w h e r e R ( ) t = [ R ( ), t 0 ] ,

d z

  

T

q ( ) t = [ q ( ), t q ( )] t ,

d z w

 
 

 
 

K K
K K

C C
C C

 

zz zw zz zw

K = , C = (13)

d d

wz ww wz ww