Civil Engineering Reference
In-Depth Information
The equations of motion (per unit mass or moment of inertia) for the two degrees of
freedom of a bluff body can be written (Scanlan and Tomko, 1971; Scanlan and Gade,
1977; Matsumoto, 1996) as:
(5.46)
(5.47)
The terms
A
i
and
H
i
are linear aeroelastic coefficients or
flutter derivatives
which are
usually determined experimentally for particular cross-sections. They are functions of
non-dimensional or
reduced
frequency.
F
z
(t)
and
M(t)
are forces and moments due to
other mechanisms which act on a static body (e.g. turbulent buffeting or vortex
shedding).
ω
z
(=2πn
z
)
and
ω
θ
(=2πn
θ
)
are the undamped circular frequencies in still air for
vertical motion and rotation, respectively.
Note that Equations (5.46) and (5.47) have been 'linearized', i.e. they only contain
terms in ż,
θ,
etc. There could be smaller terms in ż
2
, θ
2
, etc. The two equations are
'coupled' second-order linear differential equations. The coupling arises from the
ocurrence of terms in
z
and
θ,
or their derivatives in both equations. This can result in
coupled aeroelastic instabilities, which are a combination of vertical (bending) and
rotational (torsion) motions, depending on the signs and magnitudes of the
A
i
and
H
i
derivatives. All bridge decks will reach this state at a high enough wind speed.
Several particular types of instability for bluff bodies have been defined. Three of
these are summarized in Table 5.1.
Coupled aeroelastic instabilities in relation to long-span bridge decks and flutter
derivatives are further discussed in Chapter 12—Bridges.
5.5.4
Lock-in
Motion-induced forces can occur during vibration produced by vortex shedding (Section
4.6.3). Through a feedback mechanism, the frequency of the shedding of vortices can
'lock-in' to the frequency of motion of the body. The strength of the vortices shed and the
resulting fluctuating forces are also enhanced.
Lock-in
has been observed many times
during the vibration of lightly damped cylindrical structures such as steel chimneys and
occasionally during the vortex-induced vibration of long-span bridges.
Table 5.1
Types of aerodynamic instabilities
Name
Conditions
Type of motion
Type of section
Galloping
H
1
>0
Translational
Square section
'Stall' flutter
A
2
>0
Rotational
Rectangle, H-section
'Classical' flutter
H
2
>0,
A
1
>0
Coupled
Flat plate, airfoil
