Page 172 - Pipelines and Risers
P. 172
Force Model and Wave Fatigue 145
p water density
D pipe diameter
CD drag coefficient
CM= (C,+l) is the inertia coefficient and C, is the added mass coefficient
CD and CM are functions of the Keulegan-Carpenter (KC) number and the ratio between
current velocity and wave velocity a. The added mass coefficient is taken from the
Figure 9-1 of the DNV (1998) guideline, multiplied by a factor due to the gap between
the span and the seabed.
The motion of the beam as a function of time and position along the beam is obtained by
solving Equation (10.7) with appropriate boundary conditions.
Equation (10.7) is a non-linear partial differential equation that can not be solved analytically.
The dependency of the position along the pipe axis can be eliminated from the equation by
applying modal analysis. The modal analysis is based upon the assumption that the vibration
mode shape of the beam is represented by a summation of beam eigen modes, whereby
increasing the number of modes improves the accuracy. Modal analysis reduces the non-
linear partial differential equation to a set of non-linear ordinary differential equations.
The non-linear ordinary differential equations can either be solved numerically or linearised
and then solved analytically. The first approach is called a ‘Time Domain Solution’ and the
latter a ‘Frequency Domain Solution’.
The time domain approach demands more computing power than the frequency domain
approach, but the latter will in some cases give erroneous results. In this context it shall also
be mentioned that the Morison force representation is empirical, and originally intended to be
used on stationary vertical piles. Since the first presentation of the formula it has been verified
to cover other scenarios. The relative velocity model, used to describe the wave forces on a
vibrating cylinder. The force coefficients are empirical and probably obtained from
experiments with regular waves.
16.3.2 Modal Analysis
The modal analysis method reduces the partial differential equation to a set of ordinary
differential equations. The key assumption is that the vibration mode of the beam can be
described by a superposition of the eigen-modes.
Eigen-frequencies and modes are determined from the equation of motion describing free
vibrations.
Solutions to the above equation are expressed as:
Z(tA = V(X)X@)
