Geology Reference
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bridge model is coupled with the TLCGD and
analyzed as MDOF system. Alternatively, a more
analytic approach is to describe the long span
bridge of length
l
as a vibrating beam with coupled
oblique bending torsional vibrations. The dynamic
equation can be solved for a selected isolated
mode using the single term Ritz approximations
of the deformations,
coefficients
κ
1
,
κ
2
and
κ
3
renders the interaction
forces in the relative coordinates, Reiterer et al.
(2006),
κ
(
)
(
)
+
(
)
+
2
2
2
F
=
m a
−
κ
u
−
u
ϑ
1
H
+
u
ϑ
4
uu
ϑ
y
'
f
y
'
2
H
F
=
z
'
κ
(
)
(
)
+
(
)
−
(
)
m a
−
κ ϑ
u
+
2
u
ϑ
1
H
2
+
u
2
ϑ
2
2
u
2
+
uu
v x t
( , )
=
Y t
( ) ( ),
χ
x w x t
( , )
=
Y t
( ) ( )
φ
x
f
z
'
2
H
u x t
T
( , )
=
ϑ
( , )
x t e Y t
=
( ) ( )
ψ
x
(34)
M
=
Ax
κ
B
u
κ
(
)
+
(
)
2
2
2
2
m
κ
H
ϑ
+
1
−
κ
ua
+
H
H
1
+
u a
κ
u
ϑ
+
2
uu
ϑ
f
3
z
'
y
'
2
2
2
Together with the Ritz-Galerkin approximation
the coupled dynamic bending torsional differen-
tial beam equations can be solved. The linearized
equation of motion of a mode number
j
is obtained
as a function of the generalized coordinate
Y t
+
M
AG
κ
(
)
M
=
m g
κ
u
+
H
H
1
2
+
u
2
ϑ
AG
f
2
( )
=
( )
, light modal structural damping has
been added, see Reiterer et al. (2006)
q t
j
B
+
2
H
L
cos
β
2
H
sin
β
B
cos
β
+
2
H
κ
=
,
κ
=
,
κ
=
1
2
L
L
1
1
1
Y
+
2
ζ
Ω
Y
+
Ω
2
Y
=
2
A
A
3
d
e
M
e
2
3
H
L
3
4
B
H
3
B
cos
β
B
H
1
Y
e
Y
e
−
−
−
F
χ φψ
+
−
A
ψ
F
φ χψ
−
ψ
Ax
κ
=
1
+
+
+
B
'
y
'
z
M
3
2
2
H
3
8
x
=
ξ
1
H
F t
M
( )
(33)
+
(35)
From Eq.(32) a parametric excitation by ver-
tical flexural accelerations
w
as well as by angu-
lar velocities
ϑ
2
is apparent. Reiterer (2004) has
shown theoretically and by experiments that the
parametric excitation of the bridge remains inef-
fective if the linearized damping coefficient is
sufficiently large and thus can be omitted.
To obtain a dynamic model of the bridge, two
straightforward approaches are possible. Most
accurate and flexible is a numeric approach using
the finite element method with subsequent modal
analysis. Having calculated the mode shape vec-
tor and the natural frequency a modal truncation
becomes necessary to reduce the dynamic degrees
of freedom substantially. The resulting low order
e
=
2
is the effective modal mass
and
Ω =
ω
Sj
the natural circular frequency of the
mode considered.
e
where
M I
e
= + +
denotes
the radius of gyration with respect to the center
of stiffness, see Figure 6b, and
ξ
the position of
the TLCGD on the span. The effective external
load
F(t)
is obtained from the projection of the
distributed bridge load
p
y
,
p
z
and when including
a distributed moment
m
x
onto the Ritz ap-
proximations of the modal shape
2
c
2
d
2
I A
0
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