Geology Reference
In-Depth Information
*
ζ
=
ζ
(44)
harmonic vertical excitation can be given by,
Reiterer (2004),
A opt
,
A opt
,
Again parametric excitation remains ineffec-
tive for pure torsional motion,
w
=0 in Eq. (32),
if the equivalent viscous damping is above the
critical cut-off damping
ζ
ϑ
A
,
(
)
max
w x
=
ξ
4
U
δ
π
(
w
)
ζ
=
0
L
>
ζ
=
κ
A opt
,
A
,
0
1
3
H
(42)
( )
0
for time harmonic
torsional vibrations,
If the bridge vibration is dominated by tor-
sional vibrations, oblique bending is assumed
zero,
φ
≈ 0
,
χ
≈ 0
and the equivalent mass ratio
changes accordingly to
ζ
=
A opt
,
4
U
δ
π
d
H
1
8
0
L
>
ζ
( )
ϑ
=
κ
A
−
κ
maxangle
ϑ
2
(
x
=
ξ
)
3
A
,
0
1
2
0
(45)
(
)
(
)
µ κ
B
+
κ
d
κ
B
+
κ
d
ψ
2
2
2
A
A
µ
*
=
1
1
Due to the different type of excitation, the
critical damping differs from Eq.(42).
To demonstrate the increase of effective
bridge damping by TLCGD, a scaled model of
a cable-stayed bridge (see Figure 7a) is tested
dynamically under laboratory conditions. This
model refers to the cantilever method of bridge
construction. Harmonic horizontal forcing at the
position of the cantilever carriage was used to
excite the dynamic laboratory model, and vibration
measurements rendered the dynamic magnifica-
tion factor given in Figure 7b). The substantial
reduction of displacements proves that the TLCGD
2
d
e
2
2
H
e
1
4
(
)
(
)
e
2
+
µψ κ
2
+
A
−
κ
B
+
κ
d
κ
B
+
κ
d
4
1
2
2
A
A
3
2
e
2
1
1
x
=
ξ
2
µ
=
m e
I
f
(43)
e
with the optimal tuning parameter determined by
*
ω
δ
δ
=
A opt
,
=
opt
opt
Ω
2
2
d
e
2
2
H
e
1
4
(
)
(
)
2
1
+
µψ κ
+
A
−
κ
B
+
2
κ
d
κ
B
+
2
κ
d
3
e
2
1
A
1
A
x
=
ξ
Figure 7. a) Scaled model of the bridge with TLCGD and harmonic excitation at the position of the
cantilever carriage, b) Dynamic magnification factor of predominant horizontal motion (Adapted from
Ziegler, 2008)
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