Civil Engineering Reference
In-Depth Information
5.3 The random vibration or spectral approach
In some important papers in the 1960s, Davenport outlined an approach to the wind-
induced vibration of structures based on random vibration theory (Davenport, 1961,
1963, 1964). Other significant early contributions to the development of this approach
were made by Harris (1963) and Vickery (1965, 1966).
The approach uses the concept of the stationary random process to describe wind
velocities, pressures and forces. This assumes that the complexities of nature are such
that we can never describe, or predict, perfectly (or 'deterministically') the forces
generated by wind storms. However, we are able to use averaged quantities like standard
deviations, correlations and spectral densities (or 'spectra') to describe the main features
of both the exciting forces and the structural response. The
spectral density,
which has
already been introduced in Section 3.3.4 and Figure 5.1, is the most important quantity to
be considered in this approach, which primarily uses the
frequency domain
to perform
calculations and is alternatively known as the
spectral approach
.
Wind speeds, pressures and resulting structural response have generally been treated
as stationary random processes in which the time-averaged or mean component is
separated from the fluctuating component. Thus:
(5.1)
where
X(t)
denotes either a wind velocity component, a pressure (measured with respect
to a defined reference static pressure) or a structural response such as bending moment,
stress resultant, deflection, etc.; the mean or time-averaged component; and
x′(t)
the
fluctuating component such that If
x
is a response variable,
x′(t)
should include
any resonant dynamic response resulting from excitation of any natural modes of
vibration of the structure.
Figure 5.4 (after Davenport, 1963) illustrates graphically the elements of the spectral
approach. The main calculations are done in the bottom row, in which the total
meansquare fluctuating response is computed from the spectral density or 'spectrum' of
the response. The latter is calculated from the spectrum of the aerodynamic forces, which
are, in turn, calculated from the wind turbulence or gust spectrum. The frequency-
dependent
aerodynamic
and
mechanical admittance
functions form links between these
spectra. The amplification at the resonant frequency, for structures with a low
fundamental frequency, will result in a higher mean-square fluctuating and peak
response, than is the case for structures with a higher natural frequency, as previously
illustrated in Figure 5.2.
