MOTION INDUCED INSTABILITIES - Theory of Bridge Aerodynamics - page 203

where

2

0

⎫

⎡ ⎤ ⎡ − κ ⎤

( )

1 / 0

− κ − ω ω

ae z

ˆ ˆ

θ

ae r z

⎪

⎢

zz

⎥ ⎢ ⎥

E = E =

1 2

2

( )

⎪

⎢ ⎥ ⎢ ⎥

0 0 1 /

− κ − κ − ω ω

⎣ ⎦ ⎣ ⎦ ⎪ ⎬

ae ae r θ

θ z θθ

(8.35)

( )

0 /

⎡ ⎤ ⎡ − ζ ⋅ ωω ⎤ ⎪

/ 0

ζζ − ⋅ ωω

ae r z

z ae r z z θ

ˆ ˆ

⎢

zz

⎥ ⎢ ⎥

E = E =

⎪

( )

3 4

⎢ ⎥ ⎢

0 /

⎥

/ 0

ζζ − ⋅ ωω

− ζ ⋅ ωω ⎪

ae r

⎣

ae r θ

⎦ ⎣

θ θ

⎦ ⎭

θ z θθ

The stability limit is then defined by the following two conditions
()

( )

( ) ( )

ˆ ˆ ˆ ˆ ˆ

Re det det 4 det 0

E = E + E − ⋅ E + E = (8.36)

η 1 2 3 4

( )

() ( ) ( )

ˆ ˆ ˆ ˆ ˆ ⎡ ⎤

Im det 2 det det 0 E =⋅ E + E + E + E = (8.37)

1 4 2 3

η

⎣ ⎦

Fully expanded these equations become
()

( )

ˆ

Re det 1 E = − κ − κ + κ ⋅ κ − κ ⋅ κ −

ae ae ae ae ae ae

η

zz zz z z

θθ θθ θ θ

( ) ( )

⎡ ⎤

( ) ( )

4

⋅ ζζ − ⋅ ζζ − − ζ ⋅ ζ ⋅ ωω ωω

/

⋅

/

−

z ae ae ae ae r z r

⎣

θ

⎦

θ

zz z z

θθ θ θ

( ) ( )

2 2 2 2

( ) ( ) ( ) ( )

1 / 1 / / /

− κ ⋅ ωω − − κ ⋅ ωω + ωω ⋅ ωω

ae r z ae r θ r z r θ

θθ zz

=

0

(8.38)

( )

{

()

( )

ˆ

⎡

( )

⎤

Im det 2 1 / E =⋅ − κ ⋅ ζ − ζ − κ ⋅ ζ ⋅ ω ω +

η

⎣

ae z zz ae ae

⎦

r z

θθ θ z z θ

( ) ( )

⎡ ⎤

1

− κ ⋅ ζ − ζ − κ ⋅ ζ ⋅ ω

/

ω −

ae ae ae ae r

⎣

θ

⎦

θ

zz z z

θθ θ θ

) }

( ) ( )

2 2

( ) ( ) ( ) (

/ / / /

ζ ζ − ⋅ ωω ⋅ ωω − ζ ζ − ⋅ ωω ⋅ ωω

θ ae r θ r z z ae r z r θ

θθ zz

0

=

(8.39)

where (see Eqs. 8.12 - 8.15)

2

∫

dx

∫

dx

φ φφ

z z θ

2 2

2 3

B

⎛⎞

B

⎛⎞

ρ ω ρ ω

L L

exp exp

r *

4

r *

3

κ =

H

κ =

H

⎜⎟
⎝⎠

⎜⎟
⎝⎠

ae zz ae z

2 m

2

2 m

2

ω

∫

dx

θ

ω

∫

dx

φ φ

z z z z

z z

L L

2

∫

dx

∫

dx

φ φφ

z

θ θ

2 2

4 3

B

⎛ ⎞

B

⎛ ⎞

ρ ω ρ ω

L L

exp exp

*
3

*
4

r r

A A

κ = κ =

⎜ ⎟
⎝ ⎠

⎜ ⎟
⎝ ⎠

ae ae z

2 2

θθ

2 m

ω

∫

θ

2 m

ω

∫

φ

dx

φ

dx

θ θ θ θ

θ θ

L L