Forced Vessel Motion

In many cases, the slender structure will be connected to a vessel at one end. Motions of this vessel must therefore be known during a dynamic riser analysis. Generation of motion time series will be consistent with generated time series for wave-induced water particle velocities and accelerations. The rigid body motion responses consist of 6 degrees of freedom: surge, sway, heave, roll, pitch and yaw referred to the global \(\mathrm {(X,Y,Z)}\) coordinate system. The motions are in this chapter denoted by \(\mathrm {\boldsymbol{x}}\) .

The motion model consists of a set of high-frequency (wave frequency) motions in all 6 degrees of freedom and a set of low-frequency motions in the 3 horizontal degrees of freedom: surge, sway and yaw. These two sets of motions are referred to as HF-motions and LF-motions, respectively. For most dynamic line problems it is sufficient to include only the HF-motions. The effects of typical LF-motions, with periods of 60-180 s, can often be covered by suitable selection of static (mean) position.

1. HF Vessel Motion Model

The wave frequency-, or HF motions are treated as linear responses to the waves. Thus, the HF motions are described by a set of complex transfer functions \(\mathrm {j=1,2.....6}\)
where:

\[H_{HFj}(\beta ,\omega )=\frac{x_j(\beta ,\omega )}{\zeta_a(\beta ,\omega )}\]
\[S_{xj}(\beta ,\omega )=|H_{HFj}(\beta ,\omega )|^2S_\zeta(\beta ,\omega )\]

2. LF Vessel Motion Model

The low-frequency-, or LF-motions, if included, are represented by directly input response spectra for surge, sway and yaw. The LF motions modelled this way are uncorrelated with the wave, and with each other.

Both the transfer functions of HF motions and spectra for LF motions refer to a specified vessel origin point. They are transformed to the structural attachment points on the vessel, (i.e. upper end terminal point for standard systems). This information together with the wave spectrum description, is sufficient to generate spectra and the sample time series of forced vessel motions.

3. Frequency-domain Vessel Motion Analysis

3.1. HF Responses

In the linearized, frequency-domain analysis the wave frequency motions are completely described by their autospectra, \(\mathrm {S_x}\) .

Response spectra are generated for all 6 rigid-body motions.

\[S_{HFxj}(\omega )=\int_{\beta _1-\frac{\pi }{2}}^{\beta _1+\frac{\pi }{2}}\!{|H_{xj}'(\beta ,\omega )|^2S_{\zeta,\textrm{TOT}}(\beta ,\omega )}\textrm{d}{\beta },\quad j=1,2...6\]

where \(\mathrm {H_{xj}'(\beta ,\omega )}\) is the transfer function referring to the top terminal point.

3.2. LF-Responses

\(\mathrm {S_{LFxj}(\omega )}\) is given as direct input and subsequently used for time series generation.

3.3. HF time series generation

Input to the HF time series generation is the complex motion transfer function \(\mathrm {H_j(\beta _k,\omega _l)}\) , and the harmonic wave components, \(\mathrm {Z_{kl}}\) .

The harmonic components of the motion responses are simply:

\[X_{j,l}=\sum_{\textrm{k}=1}^{N_\textrm{k}}\,H_j(\beta _\textrm{k},\omega _l)Z_{\textrm{k}l},\quad j=1,2...6\]

The transfer functions \(\mathrm {H}\) are stored on a tabular form for given arrays of directions and frequencies different from \(\mathrm {\beta _\textrm{k}}\) and \(\mathrm {\omega _l}\) , intermediate values are obtained by interpolation (3rd order spline interpolation). Symmetry properties of the platform are utilized.

Time series are obtained by adding the harmonic components by means of an FFT algorithm.

For regular wave analysis, the HF-time series may optionally be given as direct input of amplitudes and phase angles of the top terminal point of the structure.

3.4. LF time series generation

Input to the LF time series generation are the complex motion response spectra\(\mathrm {S_{LF,xj}(\omega ),j=1,2,6}\)

The harmonic components of the motion responses are:

\[\begin{array}{l}|x_{j\textrm{k}}|=\sqrt{2S_{LF,xj}(\omega _\textrm{k})\Delta \omega }\\\\\textrm{arg}(x_{j\textrm{k}})=\phi _{xj\textrm{k}}\end{array}\]

The random phase angles, \(\mathrm {\phi _{xjk}}\) , are sampled from a uniform distribution over \(\mathrm {[-\pi ,\pi ]}\) .

Addition of the harmonic components to obtain time series is performed by a Fourier Transform algorithm.

The relations between time increment, \(\mathrm {\Delta t_{LF}}\) , frequency increment \(\mathrm {\Delta \omega _{LF}}\) and number of time steps for the LF-simulations is equivalent to those described in Eq. 7.20 except that, \(\mathrm {t_{\textrm{lim}}}\) , the minimum interesting low frequency response period is greater, typically 20-30 sec.

3.5. Total Responses

Time series of total responses for the surge, sway and yaw motions are obtained by adding the HF and LF contributions directly.

If LF-responses have been generated with larger time increment than the HF-responses, a 3rd order spline interpolation is applied to obtain identical time increments.