Skip to content

Home > Theory > Hydrodynamics > Uncoupled Analysis

Uncoupled analysis

Wave frequencies motions

The motion of floaters at the wave frequency are defined through RAOs. With respect to the wave amplitude, the phases and all motions are defined as follows :

\[ X = \sum_{i=1}^{N_h} X_0^i cos( \vec{k_i}. \vec x - \omega_it - \Phi_{RAO}^i - \varphi_i) \]

where:

  • \(X_{0}^i\) is the motions (translations or rotations) amplitude of the ith wave component. It comes : \(X_{0}^i = X_{RAO}(\omega_i)*A_{wave}(\omega_i)\), where
  • \(X_{RAO}(\omega_i)\) is the RAO amplitude at the wave component pulsation \(\omega_i\)
  • \(A_{wave}(\omega_i)\) is the wave amplitude of i-th component
  • \(\Phi_{RAO}^i\) is phase angle of the motion RAO for the of ith wave component, at the pulsation \(\omega_i\)
\[ \vec X_{RAO} = (X_{RAO}, \Phi_{RAO}) \]

is the solution of the following complex equation expressed for a one meter height wave and for a wave heading \(\theta\):

\((-(M_a(\omega)+M_p)\omega^2 + i\omega(B(\omega)+B_{sup}) +K_{hys} +K_{Lia})\vec X_{RAO}(\omega, \theta)=\vec F_{wave}(\omega, \theta) ~~\) (1)

with :

  • \(X_{RAO}(\omega_i)\) first order wave loads,

  • \(A_{wave}(\omega_i)\) added mass

  • \(\omega_i\) radiation damping

  • \(K_{hys}\) hydrostatic stiffness

  • \(K_{Lia}\) stiffness of the mooring system

  • \(B_{sup}\) additional damping which may be composed of a linear and a quadratic part. For the quadratic part, a linearization is required and the RAOs theoritically depend on the wave height. They are usually given for a wave height of one meter.

The RAOs are input data to DeepLines. They result from hydrodynamic calculations performed with a dedicated diffraciton/radiation software such as Diodore.

Tip

The tremendous importance of the consistency in conventions between DeepLines and the hydrodynamic software used is highlighted. Phases have to be adjusted most of the time.

For example, in Diodore, the wave elevation is given as :

\[ \eta = - \eta_0 sin \left [\omega t - \vec{k}.\vec{x} - \varphi \right] = \eta_0 cos \left [ \omega t - \vec k. \vec x - \varphi + \frac {\pi} 2 \right] = \eta_0 cos \left [ \vec k. \vec x - \omega t + \varphi - \frac {\pi} 2 \right] \]

Therefore, a set of RAOs \((X_{RAO}, \Phi_{RAO})\) calculated by Diodore wrt to its own wave elevation must be entered into DeepLines as \((X_{RAO}, -\Phi_{RAO}- \pi/2)\)

There exist two different ways to define the RAOs in DeepLines :

  • RAOs as input data: in that case you have to complete series of arrays for each heading of interest. During the calculation, for a given sea-state, the RAOs will be linearly interpolated in periods and headings. In case the wave heading is outside the range of the defined headings, it will not be possible to launch the analysis.

  • RAO read from an hydrodynamic database: the structure and content of hydrodynamic database files (.HDB) are described here.

Low frequencies

It may be useful to superimpose low frequencies motions to wave frequencies. These low motions may be defined as a series of periodic movements with an amplitude and a phase on the floaters degrees of freedom. In that case, the phases of the WF motions are adjusted at each time step wrt the LF offset.