BASIC THEORY OF STOCHASTIC DYNAMIC RESPONSE CALCULATIONS - Theory of Bridge Aerodynamics - page 86

ˆ

() () ()

a ω = H ω ⋅ a ω

(4.66)

ˆ

η η

Q

where

1

−

⎧ ⎫

⎛ ⎞

⎡⎤ ⎛

2

⎞

1

⎡ ⎤ ζ

⎪ ⎪

ˆ

K C

()

1 1 2 1

− − i −

diag diag i

H ω =− I K K − ⎜ − K M ⎟ ω + − ω

⎨ ⎢⎥ ⎜ ⎟ ⎬

⎢ ⎥ η 0 ae 0 ae

⎜

0 ae

⎟

⎜

2

⎟

ω

ω

⎢⎥

⎣ ⎦

⎪ ⎣⎦ ⎝ ⎠ ⎪

⎝ ⎠

i

⎩

i

⎭

(4.67)

is the non-dimensional frequency-response-matrix, and I is the identity matrix
( N by N ). It is convenient to define the following N by N

mod mod mod mod

matrices

( )

⎫

2 1

⎡⎤

−

diag

μ = ω ⋅ KM ⋅

⎪

⎣⎦

ae i 0 ae

⎪
⎪ ⎪

− 1

κ KK =⋅

(4.68)

⎬
⎪
⎪
⎪

ae 0 ae

1
2

( )

− 1

diag

ζ =⋅ ⎡⎤ ω ⋅ KC ⋅

⎣⎦ ⎪

ae i 0 ae

⎭

]

[

diag

ζ = ζ

as well as introducing . The non-dimensional frequency-response-matrix

i

is then given by

1

−

2

⎧ ⎫

⎛ ⎞

⎡ ⎤

1

⎡ ⎤

1

⎪ ⎪

ˆ

() ( ) ( )

diag 2 i diag

H ω =− I κ − ⋅ ω ⋅ I μ − + ω ⋅ ⋅ ζ ζ −

⎜ ⎟

⎨ ⎢⎥ ⎢⎥ ⎬

η

ae

⎜ ⎟

⎣⎦

ae ae

ω ω

⎣⎦

⎪ ⎝ ⎠ ⎪

i i

⎩ ⎭

(4.69)

By combination of Eqs. 4.60, 4.61 and 4.64, then the content of

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

%$ %$ %$

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

μ = μ κ = κ ζ = ζ

(4.70)

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

ae ae ij ae ae ij ae ae ij

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

$% $% $%

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦