Page 112 - Dynamic Loading and Design of Structures

Page 89

It will be noted that three parameters define the effective excitation: , Bs and L C. All
are presumably sensitive to motion of the structure, but in an overall physical visualization of
lock-on, the dominant effect may perhaps best be envisaged as a constraint on the phase of
shedding. This is incompatible with simple spectral visualization. The familiar action in
which phase, relative to the elastic forces, is crucial is damping, and Vickery therefore
visualized lock-on as a negative damping action superimposed on the basic excitation, as well
as the simpler effect of ensuring a uniform central frequency of shedding over a range of
height in the presence of moderate taper and mean speed profile. This negative damping is
normalized in the same way as positive structural damping, as a modified Scruton number,
which will be negative. For chimneys, presuming the critical event to lie in the transcritical
regime and turbulence levels typical of the neutral stability atmospheric boundary layer,
Vickery suggests , B =0.3 and L =1.OD to 1.5D, together with aerodynamic
C
s
damping equivalent to . The aerodynamic damping is combined with
structural damping (δ s) in the response variance equation (i.e. , with δ a negative),
so for a structure with basic Scruton number K =15, lock on would halve the effective
s
damping and increase response by a factor of 2, but if the basic value were only K =7.6, the
s
response would increase without limit. A term modelling damping forces proportional to the
cube of the response amplitude can be added to the linear damping which leads to simple
Scruton number normalization, in order to express the self-limiting nature of vortex excitation
at amplitudes of the order of D (Vickery, 1981).
The ESDU approach commences with evaluation of a so-called broadband formulation, by
which is denoted an input spectrum broad by comparison with the frequency response
function, as in Vickery’s model. The structural response will, however, be narrowband
dominated by the structural natural frequency, albeit with a broadly modulated amplitude, and
the maximum value is taken as four times the r.m.s. Another spectrally based model has been
postulated for the locked on condition, in which the force bandwidth is treated as if narrow by
comparison with the frequency response function. Although originating from the same school
as Vickery, the continuing application has been in these data items (ESDU 85038, etc.)

(ESDU, 1986a). In both formulations and L c are treated as increasing with amplitude,
but in the second model the bandwidth assumption gives reversion to the same functional
form as the basic deterministic model, with the addition of a correlation admittance. The
response in the second model is deemed to be constant amplitude sinusoidal, giving the peak
value as times the r.m.s. The outcome is presumed to be whichever model gives the greater
peak response.

3.2.6 Other vortex shedding problems: proximity, alongwind and ovalling
excitation

Any structure placed in a vortex street originating from another structure nearby is