## Dynamic analysis: Ramping |

Simulation time can be reduced by the use of a build-up period at the beginning of the simulation. During the build-up period the wave dynamics, vessel motions and, optionally, current and line feeding payout rate are increased smoothly from zero to their full value. This gives a gentle start to the simulation which reduces transient responses and avoids the need for long simulation runs. The duration of the build-up period should normally be set to at least one wave period. OrcaFlex models default to having a build-up period in their first simulation stage, but this can be customised, or turned off altogether, by setting the data that defines the ramp. For the default build-up, negative time indicates the build-up period; so time before time zero is build-up time, time after time zero is normal simulation with the full specified excitation.

When using a time domain VIV model, ramping is also used to smooth the handover from the standard Morison drag force applied in statics to the force given by the VIV model.

In certain modes of operation, documented individually for each, the ramp can be utilised in the models for wind speed, turbine shaft rotation and constraint imposed motion.

The ramping factor, $R$, is calculated as \begin{equation} R = r^3 (6r^2 -15r + 10) \end{equation} where $r$ is the proportion of the ramp completed at time $t$, given by \begin{equation} r = \begin{cases} 0 &{\text{ for } t \leq t_s} \\ \dfrac{t - t_s}{t_e - t_s} &{\text{ for } t_s \lt t \lt t_e} \\ 1 &{\text{ for } t \geq t_e} \end{cases} \end{equation} with $t_s$ the ramp start time and $t_e$ the ramp finish time.

Note: | The form of $R$ has been chosen specifically to arrange that it increases monotonically from 0 to 1, as $r$ varies between 0 and 1, and that its first and second derivatives are both zero when $r{=}0$ and $r{=}1$. The zero derivatives are important to ensure smooth transitions between statics and dynamics, and between the build-up period and the rest of the simulation. |

Figure: | The OrcaFlex ramping function |

The above definition of $r$ applies in the usual case where $t_s \leq t_e$. It is also possible to have $t_s \gt t_e$, which corresponds to *ramping down* instead of up. In this case, $r$ is given by

\begin{equation} r = \begin{cases} 1 &{\text{ for } t \leq t_e} \\ \dfrac{t - t_s}{t_e - t_s} &{\text{ for } t_e \lt t \lt t_s} \\ 0 &{\text{ for } t \geq t_s} \end{cases} \end{equation}