Tuned Liquid Column Gas Damper in Structural Control - Design Optimization of Active and Passive Structural Control Systems - page 170

2

l

( ) ( ) + ( ) ( )

ψ d

e

d
e

H
d

2

2

∫

−

F t ( ) = χ x p x t , φ x p x t ,

a = χ

2

+ + φ

2 A

+ κ

A

ψχ

1

y z

0

11 3 2

+ ( )

( )

ψ x m x t , e dx

(36)

A

x

The linearized interaction forces determined
by Eq.(34) together with Eq.(33) render

d
e

B
e

a = − + χκ ψκ +

A

ψκ

12 1

2

d
e

 =

F

' ≈

m χ −

A

ψ Y − κ u  d

e

B
e

y f

a = − + χκ ψκ +

A

ψκ (39)

21 1

2

x ξ



If the bridge is modeled as an MDOF system
using a finite series Ritz approximation, and
multiple TLCGD are installed, Eq.(37) still holds
in a hypermatrix notation, Reiterer et al. (2006).
In case of a low order bridge model, the equa-
tions of motion of a TMD and a TLCGD can be
compared to get the optimal tuning values. For
dominating lateral vibrations the vertical and
torsional mode shapes φ and ψ are set to zero in
Eq.(38), and the mass ratio µ * of the equivalent
TMD results

F m Y

' =

φ

z f

x = ξ

κ B u

H
e

2





M ≈ m κ ψ Y +

1
2

+ g u κ (37)

Ax f 3

Neglecting the TLCGD parametric excitation,

( )





wu H d H A , in Eq.(32)
renders, together with the bridge dynamics (rep-
resented by a generalized SDOF system), a linear
coupled matrix equation of motion of the two
DOF isolated system

− κ + κ − κ ϑ

2

1 1 2

m
M

2

µχ κκ
µχ

f

*

µ = , µ = (40)

( )

1 +

2

1 − κκ






Y Y Y

u

M 1

x = ξ

( )

M + C + K = F t

 u

S S

u

S

0

According to the optimal Den Hartog tuning,
see Eq. (9), the optimal frequency tuning becomes

+ µ a µ a

1

=

M S =

11 21

a

1

*

ω δ

12

x ξ

A opt , opt

δ = = ,

opt

Ω

( )

1

2

1

+ µχ − κκ

x = ξ

ζ Ω

2 0

*

ζ = ζ (41)

C S = A opt , A opt ,

ζ ω 0 2

A A

The active TLCGD absorber mass is defined
by m

* =

m f κκχ

2

and thus determines the

Ω 2 µψκ g e

=

x = ξ

K S = (38)

optimal absorber placement at the position of
maximum horizontal modal displacement χ . The
possible parametric excitation from vertical bend-
ing remains ineffective for sufficiently large
damping, because the cut-off value ζ A ,

2

ψκ g e ω

A

x ξ

with constant coefficients, position ξ , see Eqs.
(32) and (33) for the geometry factors,

(

0 for time

)