## Results: Time history and XY graphs |

Time history graphs are of a single variable against time. XY graphs are of one time-dependent variable against another. These graphs are a natural result of time domain simulation, but it is also possible to synthesise them following frequency domain simulation.

The period of simulation covered by the graph is chosen from a list.

- Select the
**time history**or**XY graph**result type. - Select the object.
- Select the period.
- Select the variable. More than one variable can be selected for time histories.
- Click the
**show**button.

For XY graphs the steps 2 and 3 need to be done for both axes. Do this by clicking on one of the options labelled **X-axis** or **Y-axis**, which are located at the bottom of the results form, and then repeating steps 2 and 3.

For time histories of angles, OrcaFlex chooses the angle's range so that the time history is continuous.

For example consider vessel heading, which is normally reported in the range -180° to +180°. If the vessel's heading passes through 180° then without range jump suppression the time history would be:

…, 179°, 180°, -179°, …

i.e. with a 360° jump. To avoid this jump OrcaFlex adds or subtracts multiples of 360° to give the best continuation of the previous value. So in this example it adds 360° to the -179° value and hence reports:

…, 179°, 180°, 181°, …

This addition is valid since 181° and -179° are of course identical headings.

Note that this means that angle time history results can go outside the range -360° to +360°.

From any time history graph generated from time domain simulation, you can use the popup menu to obtain the spectral density graph for that time history. The curve shown on the graph is the one-sided power spectral density (PSD) per unit time of the sampled time history, obtained using the Fourier transform. The fundamental frequency is specified on the general data form.

Notes: | Using the Fourier transform to estimate the PSD inevitably introduces 'noise' or 'leakage' to the spectrum. To reduce the leakage the time history is partitioned into a number of overlapping periods. The PSDs are calculated for each period and then averaged to give the reported PSD which has the effect of smoothing the resulting PSD. |

This smoothing technique is only applied if the time history duration and the sample count are of sufficient size. |

From any time history graph you can use the popup menu to obtain the empirical cumulative distribution graph for that time history. This graph shows what proportion of the samples in the time history are less than or equal to a given value.

These graphs are sometimes referred to as **exceedence plots**, since they can sometimes be used to estimate the probability that the variable will exceed a given value.

Warning: | The samples in a time history are not independent. They have what is called 'serial correlation', which often affects the accuracy of statistical results based on them. |

From any time history graph you can use the popup menu to obtain the rainflow half-cycle empirical cumulative distribution graph for that time history. The curve on this graph is produced in the following way:

- The time history is analysed using the rainflow cycle-counting algorithm. For details of this algorithm see the paper by Rychlik.
- The rainflow algorithm produces a list of half-cycles associated with the time history. The empirical cumulative distribution of these half-cycles is then plotted.

From any time history graph you can use the popup menu to obtain a table listing each rainflow half-cycle range, $\Yrange$, its associated mean value, $\Ymean$, and the corresponding R ratio, $R$.

The associated mean value is defined to be the mean of the two local turning points, $Y_1$ and $Y_2$, that were used to define the half-cycle range \begin{equation} \begin{aligned} \Yrange &= Y_2 - Y_1 \\ \Ymean &= \frac{Y_1 + Y_2}{2} \\ \end{aligned} \end{equation} The R ratio is defined to be \begin{equation} R = \begin{cases} \cfrac{\Ymin}{\Ymax} & \text{if $\Ymax \neq 0$} \\ 0 & \text{otherwise} \\ \end{cases} \end{equation} where \begin{equation} \begin{aligned} \Ymin &= \begin{cases} Y_1 & \text{if $|Y_1| \lt |Y_2|$} \\ Y_2 & \text{otherwise} \\ \end{cases} \\ \Ymax &= \begin{cases} Y_2 & \text{if $|Y_1| \lt |Y_2|$} \\ Y_1 & \text{otherwise} \\ \end{cases} \\ \end{aligned} \end{equation}