Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 82

APPENDIX A

Finite Element Matrices of Different VE Damper Models

The explicit form of the matrices used to describe the generalized Kelvin model is given by:

K K
K K

C C
C

 

T zz zw T zz zw

K = T K T = , C = T C T = (A.1)

d d d d d d d d

C

wz ww wz ww

c k

2

csk 0 0 0 0 0 0

0 0

0 0 0 0

csk s k

2

0 0

, C zz

=

K zz =

0 0

(A.2)

0 0 c c

2

csc

2

0 0 c k csk m m

m m

0 0 csc csc

2

0 0 csk s k m m

m m

0 0 ..... 0 ....... 0

−
−

ck
sk

0 ..... 0 ....... 0

0

0 0 ..... 0 ....... 0

0 ..... 0 ....... 0

T

T 0

, C C = =

K K
zw

= = (A.3) zw wz

0 0 ..... .....

..... .....

0 −

−

cc
sc

wz

0 0 ..... .....

..... ......

0 −

−

ck
sk

m

m

0 0 0

0 0 0

m

m

( ) ( )

 

K K
ww

= = diag k , k , .........., k , , C C

ww

= = diag c , c , .........., c , (A.4)

ww 1 2 m ww 1 2 m

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

The explicit form of the matrices used to describe the generalized Maxwell model is given by:

K K
K K

C C
C

 

T zz zw T zz zw

K = T K T = , C = T C T = (A.5)

d d d d d d d d

C

wz ww wz ww

m m

∑ ∑

2 2

c k + k cs k + k − c k − csk

0 i 0 i 0 0

i = 1 i = 1

m m

∑ ∑

2 2

cs k + k s k + k − csk − s k

K zz = (A.6)

0 i 0 i 0 0

i = 1 i = 1

2 2

− c k − c sk c k csk

0 0 0 0

2 2

− csk − s k csk s k

0 0 0 0