Civil Engineering Reference
In-Depth Information
situations in which large oscillations occur, and the shedding has 'locked-in' to the cross-
wind motion of the structure (Section 5.5.4).
Sinusoidal excitation models were also proposed by Rumman (1970) and Ruscheweyh
(1990).
Unlike other loading models in wind engineering, sinusoidal excitation models are
deterministic,
rather than random. The assumption of sinusoidal excitation leads to
responses which are also sinusoidal.
To derive a simple formula for the maximum amplitude of vibration of a structure
undergoing cross-wind vibration due to vortex shedding, the following assumptions will
be made:
• sinusoidal cross-wind force variation with time;
• full correlation of the forces over the height over which they act; and
• constant amplitude of fluctuating cross-wind force coefficient.
None of these assumptions are very accurate for structures vibrating in the turbulent
natural wind. However, they are useful for simple initial calculations to determine
whether vortex-induced vibrations are a potential problem.
The structure is assumed to vibrate in the jth mode of vibration (in practice
j
will be
equal to 1 or 2), so that Equation (5.17) applies:
where
Gj
is the generalized mass equal to the mass per unit length along
the structure;
h
the height of the structure;
Cj
the modal damping;
Kj
the modal stiffness;
ωj the natural undamped circular frequency for the jth mode
Qj(t)
the
generalized force, equal to where
f(z, t)
is the fluctuating force per unit
height; Z
1
and Z
2
the lower and upper limits of the height range over which the vortex
shedding forces act.
In this case, the applied force is assumed to be harmonic (sinusoidal) with a frequency
equal to the vortex-shedding frequency,
n
s
. The maximum amplitude of vibration will
occur at resonance, when
n
s
is equal to the natural frequency of the structure,
n
j
.
Thus the generalized force (Section 5.3.6) is given by:
where
Q
j,
max
is the amplitude of the applied generalized force, given by,
(11.9)
where,
C
ℓ
is the amplitude of the sinusoidal lift (cross-wind force) per unit length
coefficient and
ρ
a
the density of air.
The result for the maximum amplitude at resonance for a single-degree-of-freedom
system can be applied:
