Optimal Placement of Viscoelastic Dampers Represented by the Classical and Fractional Rheological Models - Design Optimization of Active and Passive Structural Control Systems - page 83

0 0 0 0

0 0 0 0

m m

C zz =

∑ ∑

(A.7)

0 0 c

2

c cs c

i i

i 1 i 1

= =

m m

∑ ∑

0 0 cs c s

2

c i

i

i = 1 i = 1

− ck − ck ..... − ck ....... − ck

1 2 i m

− sk − sk ..... − sk ....... − sk

T 1 2 i m

K K
zw

= = (A.8)

zw

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

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

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

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

C C
zw

=

T

= (A.9)

zw

− cc − cc ..... − cc i ....... − cc

1 2 m

2 .....

.......

− sc − sc − sc − sc

1 i m



( )



( )

diag k , k , .........., k , , diag c , c , .........., c ,

K K
ww

= = C C

ww

= = (A.10)

ww 1 2 m ww 1 2 m

The explicit form of the vectors and matrices used to describe the simple Kelvin model and the
fractional-derivative Kelvin model of a VE damper is:

T

q ( ) t = q ( ) t = [ ( ), q t q t ( ), q t ( ), q t ( )] (A.11)

d z 1 2 3 4

2 2

2 2

c cs − c − cs

c cs − c − cs

2 2

2 2

cs s − cs − s

cs s − cs − s

C C
d

= = c

K K
d

= = k ,

zz 1

− c

2

− cs c

2

cs

zz 1

− c

2

− cs c

2

cs

cs s

2

cs s

2

cs s

2

cs s

2

− −

− −

(A.12)

The explicit form of the matrices used to describe the simple Maxwell model and the fractional
Maxwell model of a VE damper is:

T

q ( ) t = [ q ( ), t q ( )] t (A.13)

d z w

T T

q z ( ) t = [ ( ), q t q t ( ), q t ( ), q t ( )] , q w ( ) t = [ q ( )] t (A.14)

1 2 3 4 w , 1