THE BUFFETING THEORY IN A FINITE ELEMENT FORMAT - Theory of Bridge Aerodynamics - page 242

ˆ ˆ

⎡ ⎤

Co Co

ˆ

( )

11 nm 12 n m

Co Δ

s ,

ω = (9.121)

⎢ ⎥

nm nm

ˆ ˆ

⎢ Co Co ⎥

⎣ ⎦

12 nm 22 nm

where

( )

ˆ

⎡ ⎤ ⎫

Co s , 0 0

Δ ω

uu 11

nm

⎪

⎢ ⎥ ⎪

( )

⎢ ⎥

ˆ ˆ

0 Co s , 0 Co = Δ ω

⎪

11 nm

⎢

vv 11

⎥

nm

⎢ ⎥ ⎪

( )

ˆ

0 0 Co s ,

Δ ω

⎪

⎢ ⎥

ww 11

⎣ ⎦ ⎬

nm

(9.122)

( )

ˆ

⎡ ⎤ ⎪

Co s , 0 0

Δ ω

uu 22

⎢

nm

⎥ ⎪

( )

ˆ

⎢

ˆ

⎥ ⎪

0 Co s , 0 Co = Δ ω

22 nm

⎢

uu 22

⎥ ⎪

nm

⎢ ⎥ ⎪

( )

ˆ

0 0 Co s ,

Δ ω

⎢ ⎥ ⎪

uu 22

⎣ ⎦⎭

nm

and

( )

ˆ

⎡ ⎤ ⎫

Co s , 0 0

Δ ω

uu 12

nm

⎪

⎢ ⎥ ⎪

( )

⎢ ⎥

ˆ ˆ

0 Co s , 0 Co = Δ ω

⎪

12 nm

⎢

vv 12

⎥

nm

⎢ ⎥ ⎪

( )

ˆ

0 0 Co s ,

Δ ω

⎪

⎢ ⎥

ww 12

⎣ ⎦ ⎬

nm

(9.123)

( )

ˆ

⎡ ⎤ ⎪

Co s , 0 0

Δ ω

uu 21

⎢

nm

⎥ ⎪

( )

ˆ

⎢

ˆ

⎥ ⎪

0 Co s , 0 Co = Δ ω

21 nm

⎢

vv 21

⎥ ⎪

nm

⎢ ⎥ ⎪

( )

ˆ

0 0 Co s ,

Δ ω

⎢ ⎥ ⎪

ww 21

⎣ ⎦⎭

nm

()

then S ω in Eq. 9.114 is given by

vv ˆˆ

ˆ ˆ ˆ

() () () ()

S ω = S ω ⋅ Co ω ⋅ S ω

(9.124)

vv ˆˆ n nm m

Combining Eqs. 9.109, 9.114 and 9.124 then the following cross spectral density
response matrix is obtained

()

* T

S ω =⋅ H S ⋅ H

rr r RR r

NN

o S ψ BA H (9.125)

⎧ ⎫

⎪

( )

⎪

ˆ ˆ ˆ

*

∑∑

T

⎡

1 2 1 2 T T

⎤

T

=⋅ H A ⋅ B ψ S ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⎨ ⎬

r n nn n mm mm m r

⎣ ⎦

⎪ ⎪

⎩ ⎭

nm 11

==