Small displacements of the suspension bridge girder
(because of wind) interacting with the wind velocity
U result in the motion-induced forces that again can
produce violent vibration and develop into destructive
mode (flutter mode) within a short time, as mentioned
earlier in Chapter 1, [00,0]. The flutter mode can be a
combination of several modes, but is primarily in the
vertical and the torsional mode at the lowest
frequencies, where the critical frequency wcr is the
one between them.
The motion of one joint on the girdercan be described by three translations and three rotational components, which can be written in the deformation vector v(x,t), Figure 5.1 Qk(x) is the undamped mode shape vector in mode k, in which fx,k , fy,k and fz,k are the longitudinal, lateral and vertical translation component, yx,k ,yy,k , and yz,k are the rotational component about the x-, y- and z-axis, respectively. xk(t) is the corresponding modal coordinate and ndof is theoretically the total number of degrees-of-freedom of all the joints created in the bridge.
In (5.2) the physical displacement vz(x,t) in purely
vertical mode i and the physical rotation rx(x,t) in
purely torsional mode j can be written as independent
modal coordinates in the form (5.3) and (5.4), where
fi(x) and yj(x) are the vertical and torsional modes
that couple to produce flutter vibrations.
The modal force Fdeck(t) due to a coupled vertical-
torsional mode k is given by (5.5) which can be
written as the vertical modal equation Fz(t) and
torsional modal equation Fx(t).
The question of how many and what modes a
suspension bridge couples at the flutter mode (beyond
the SV1 and ST1) refers to a lot of factors depending
on each individual case. During the CAE work to
identify the natural modes, one can see that many of
the modes (in low frequency range) look like each other
with closed natural frequencies. Considering this
information, a number of important modes can be
selected for the multimode coupled flutter computation.
The total motion-induced wind loads per unit span on a girder cross-section including the leading and trailing flaps are shown in Figure 5.2. The same coordinate system and definition of the girder deformations as Figure 5.1 is used, but the flaps and their deformation directions are shown. The system is assumed to oscillate from position B to C at wind action U.
It is assumed that the generalised forces caused by
the girder-wind-interaction and the generalised forces
caused by the flap-wind-interaction (for separate flaps)
are computed on the independent flutter derivatives of
the girder and of the flaps.
When the girder vibrates in the coupled vertical/
torsional mode, the generalised moment of the flaps is
composed of two contributions. One refers to the
“purely” vertical motion of the girder and the rotation
of the flaps (that computed from the rotation of the
girder). The other refers to the vertical motion of the
girder due to the girder rotation and the same rotation
of the flaps.
A new graphical solution for solving the flutter
conditions using Maple is introduced in [00,0].
There are no needs for a reduced frequency K and a
reduced wind velocity u in the between step, since
the critical frequency wcr and the critical wind
velocity Ucr can be solved directly. Moreover, the
change of any equation to produce the flutter
conditions (e.g. the flutter derivative functions)
chapter for several reasons. Firstly, to be able to
reduce the complications related to the additional
control flaps. Secondly, it is easier to operate the
multimode coupled buffeting response from the single-
mode analysis (Section 7.4). Thirdly, since the buffeting
response occurs at a lower mean wind velocity than
flutter, the aerodynamic effects that cause modal
coupling will not produce significant differences between
the two assumptions (see e.g. [99,2]). Finally, to ensure
that the single-mode vibrations do not develop a
catastrophic vibration amplitude.
Expression (6.7) to (6.10) are the buffeting-induced lift
and moment per unit span of the deck, the leasing flap
(le) and the trailing flap (tr), respectively. u(x,t) is the
along wind turbulence component and w(x,t) is the
vertical turbulence component. CL , CM , and CD are the
non-dimensional lift and the moment and drag coefficient,
respectively.
The mean wind velocity is assumed to be smaller at buffeting conditions than at those causing coupled instabilities. Thus, the uncoupled aeroelastic forces are transferred to the left-hand side of the motion equation to modify the natural frequency and the structural damping ratio to the new quantities in the stochastic buffeting modal equation, cf. [00,0]. You can download entire PhD Thesis in Reference below 123456789_123456789_123456789_123456789_12345
Truc Huynh & Palle Thoft-Christensen, "Suspension Bridge Flutter for Girder with Separate Control Flaps”, Journal of Bridge Engineering, ASCE p.168-175, May/Jun 2001 Vol.6 No.3 or ISSN 1395-7953 R0013
2000
[0]
[3]
Truc Huynh, "Suspension Bridge Aerodynamics and Active Vibration Control", PhD Thesis, Aalborg University Denmark, ISSN 1395-7953 R0015
Truc Huynh & Palle Thoft-Christensen, "Buffeting Response of Suspension Bridge Girder with Separate Control Flaps”, Presented in Second European Conference on Structural Control, Paris July 2000, ISSN 1395-7953 R0014
1999
[1]
Nielsen, S. R. K & Truc Huynh “Vibration Theory, Vol. 7A. Special Structures: Aerodynamics of Suspension Bridges”, Aalborg University Denmark, ISSN 1395-8232 U9902
[2]
Katsuchi H., N. P. Jones & R. H. Scanlan “Multimode Coupled Flutter and Buffeting Analysis of the Akashi-Kaikyo Bridge”, Journal of Structural Engineering /January 1999, Vol. 125, No. 1
1997
[2]
Dyrbye C. & S. O. Hansen, “Wind Load on Structures”, John Wiley & Sons.
1992
[3]
Scanlan R. H. “Wind dynamic of long-span bridges”, Larsen A. editor [1992,1, pp. 47-57].
