Page 222 - Dynamic Loading and Design of Structures

Page 195

wave frequencies enhanced by the current, resulting in force components with frequencies
equal to a difference, sum and double frequencies of the wave components.
Components containing the difference frequencies lead to long period forces which may be
critical for rigid body modes of behaviour. The terms with ω+ω lead to high frequency
1
2
forces, which may cause dynamic effects in bottom supported platforms.
The non-linearity in Morison’s equation may be linearized to facilitate efficient response
analysis. Linearization may, for instance, be achieved:
●deterministically by requiring that the same energy be dissipated per wave cycle for the
linear and non-linear model;
●stochastically by assuming that the particle velocity follows a Gaussian distribution and
finding the linearization that minimizes the expected mean square error.

Linearization by consideration of energy dissipation for a single wave-component
corresponds to taking the first term in the Fourier expansion, eqn (5.32), and ignoring high-
frequency terms (see e.g. chapter 2, Almar-Næss, 1985). Stochastic linearization of

yields where σis the standard deviation of v(t) (see e.g. Leira,
v
1987). Stochastic linearization yields accurate estimates of loads when used to determine
response variance, which is relevant for fatigue analysis, but needs to be used with caution in
estimation of extreme values.
As mentioned in section 5.2.3, particle velocities and accelerations are Gaussian processes
in the time domain. In the frequency domain the kinematics is described by spectral densities
(e.g. eqn (5.23)). The forces (eqn (5.28)) on slender members may also be expressed in the
frequency domain by the cross-spectral density of the load intensity at two locations m and n.
This topic is thoroughly treated by Borgman (1972). It is seen that the spectral density has
peaks at the wave frequency as well as at multiples of the wave frequency as displayed by the
Fourier expansion, eqns (5.34), (5.35).
Clearly, a linearization of eqn (5.36), which yields:

(5.37)

where c is a constant, ignores the higher order components.

Large volume structures
Higher order terms in the potential theory to account for finite wave elevation also cause time-
variant sum and frequency forces on large volume structures in (irregular) waves. For instance,
the second order term of the surface elevation (in e.g. eqn (5.16) for the deterministic wave)
and the quadratic velocity terms in Bernoulli’s equation (eqn (5.2e)) based on the first order
potential will contribute second order force components. The term in Bernoulli’s equation is
somewhat analogous to the velocity squared term in the Morison equation.