Loads Module

The loads module contains a set of functions to calculate quantities of interest for mechanical loads assessments.

General

The loads general submodule contains general loads calculations that can be applied to most MRE devices.

bin_statistics	Bins calculated statistics against data signal (or channel) according to IEC TS 62600-3:2020 ED1.
blade_moments	Transfer function for deriving blade flap and edge moments using blade matrix.
damage_equivalent_load	Calculates the damage equivalent load of a single data signal (or channel) based on IEC TS 62600-3:2020 ED1.

mhkit.loads.general.bin_statistics(data, bin_against, bin_edges, data_signal=[])

Bins calculated statistics against data signal (or channel) according to IEC TS 62600-3:2020 ED1.

Parameters

• data (pandas DataFrame) – Time-series statistics of data signal(s)

• bin_against (array) – Data signal to bin data against (e.g. wind speed)

• bin_edges (array) – Bin edges with consistent step size

• data_signal (list, optional) – List of data signal(s) to bin, default = all data signals

Returns

• bin_mean (pandas DataFrame) – Mean of each bin

• bin_std (pandas DataFrame) – Standard deviation of each bim

mhkit.loads.general.blade_moments(blade_coefficients, flap_offset, flap_raw, edge_offset, edge_raw)

Transfer function for deriving blade flap and edge moments using blade matrix.

Parameters

• blade_coefficients (numpy array) – Derived blade calibration coefficients listed in order of D1, D2, D3, D4

• flap_offset (float) – Derived offset of raw flap signal obtained during calibration process

• flap_raw (numpy array) – Raw strain signal of blade in the flapwise direction

• edge_offset (float) – Derived offset of raw edge signal obtained during calibration process

• edge_raw (numpy array) – Raw strain signal of blade in the edgewise direction

Returns

• M_flap (numpy array) – Blade flapwise moment in SI units

• M_edge (numpy array) – Blade edgewise moment in SI units

mhkit.loads.general.damage_equivalent_load(data_signal, m, bin_num=100, data_length=600)

Calculates the damage equivalent load of a single data signal (or channel) based on IEC TS 62600-3:2020 ED1. 4-point rainflow counting algorithm from fatpack module is based on the following resources:

• C. Amzallag et. al. Standardization of the rainflow counting method for fatigue analysis. International Journal of Fatigue, 16 (1994) 287-293

• ISO 12110-2, Metallic materials - Fatigue testing - Variable amplitude fatigue testing.

• G. Marsh et. al. Review and application of Rainflow residue processing techniques for accurate fatigue damage estimation. International Journal of Fatigue, 82 (2016) 757-765

Parameters:

data_signalarray

Data signal being analyzed

mfloat/int

Fatigue slope factor of material

bin_numint

Number of bins for rainflow counting method (minimum=100)

data_lengthfloat/int

Length of measured data (seconds)

returns

DEL (float) – Damage equivalent load (DEL) of single data signal

Graphics

The graphics submodule contains functions to plot loads metrics.

plot_statistics	Plot showing standard raw statistics of variable
plot_bin_statistics	Plot showing standard binned statistics of single variable

mhkit.loads.graphics.plot_statistics(x, y_mean, y_max, y_min, y_stdev=[], **kwargs)

Plot showing standard raw statistics of variable

Parameters

• x (numpy array) – Array of x-axis values

• y_mean (numpy array) – Array of mean statistical values of variable

• y_max (numpy array) – Array of max statistical values of variable

• y_min (numpy array) – Array of min statistical values of variable

• y_stdev (numpy array, optional) – Array of standard deviation statistical values of variable

• **kwargs (optional) –

  x_labelstring

  x axis label for plot

  y_labelstring

  y axis label for plot

  titlestring, optional

  Title for plot

  save_pathstring

  Path and filename to save figure.

Returns

ax (matplotlib pyplot axes)

mhkit.loads.graphics.plot_bin_statistics(bin_centers, bin_mean, bin_max, bin_min, bin_mean_std, bin_max_std, bin_min_std, **kwargs)

Plot showing standard binned statistics of single variable

Parameters

• bin_centers (numpy array) – x-axis bin center values

• bin_mean (numpy array) – Binned mean statistical values of variable

• bin_max (numpy array) – Binned max statistical values of variable

• bin_min (numpy array) – Binned min statistical values of variable

• bin_mean_std (numpy array) – Standard deviations of mean binned statistics

• bin_max_std (numpy array) – Standard deviations of max binned statistics

• bin_min_std (numpy array) – Standard deviations of min binned statistics

• **kwargs (optional) –

  x_labelstring

  x axis label for plot

  y_labelstring

  y axis label for plot

  titlestring, optional

  Title for plot

  save_pathstring

  Path and filename to save figure.

Returns

ax (matplotlib pyplot axes)

Extreme

The extreme submodule contains functions to calculate peak distribution.

global_peaks	Find the global peaks of a zero-centered response time-series.
number_of_short_term_peaks	Estimate the number of peaks in a specified period.
peaks_distribution_weibull	Estimate the peaks distribution by fitting a Weibull distribution to the peaks of the response.
peaks_distribution_weibull_tail_fit	Estimate the peaks distribution using the Weibull tail fit method.
peaks_distribution_peaks_over_threshold	Estimate the peaks distribution using the peaks over threshold method.
ste_peaks	Estimate the short-term extreme distribution from the peaks distribution.
block_maxima	Find the block maxima of a time-series.
ste_block_maxima_gev	Approximate the short-term extreme distribution using the block maxima method and the Generalized Extreme Value distribution.
ste_block_maxima_gumbel	Approximate the short-term extreme distribution using the block maxima method and the Gumbel (right) distribution.
short_term_extreme	Approximate the short-term extreme distribution from a timeseries of the response using chosen method.
full_seastate_long_term_extreme	Return the long-term extreme distribution of a response of interest using the full sea state approach.
mler_coefficients	Calculate MLER (most likely extreme response) coefficients from a sea state spectrum and a response RAO.
mler_simulation	Define the simulation parameters that are used in various MLER functionalities.
mler_wave_amp_normalize	Function that renormalizes the incoming amplitude of the MLER wave to the desired peak height (peak to MSL).
mler_export_time_series	Generate the wave amplitude time series at X0 from the calculated MLER coefficients

mhkit.loads.extreme.global_peaks(t, data)

Find the global peaks of a zero-centered response time-series.

The global peaks are the maxima between consecutive zero up-crossings.

Parameters

• t (np.array) – Time array.

• data (np.array) – Response time-series.

Returns

• t_peaks (np.array) – Time array for peaks

• peaks (np.array) – Peak values of the response time-series

mhkit.loads.extreme.number_of_short_term_peaks(n, t, t_st)

Estimate the number of peaks in a specified period.

Parameters

• n (int) – Number of peaks in analyzed timeseries.

• t (float) – Length of time of analyzed timeseries.

• t_st (float) – Short-term period for which to estimate the number of peaks.

Returns

n_st (float) – Number of peaks in short term period.

mhkit.loads.extreme.peaks_distribution_weibull(x)

Estimate the peaks distribution by fitting a Weibull distribution to the peaks of the response.

The fitted parameters can be accessed through the params field of the returned distribution.

Parameters

x (np.array) – Global peaks.

Returns

peaks (scipy.stats.rv_frozen) – Probability distribution of the peaks.

mhkit.loads.extreme.peaks_distribution_weibull_tail_fit(x)

Estimate the peaks distribution using the Weibull tail fit method.

The fitted parameters can be accessed through the params field of the returned distribution.

Parameters

x (np.array) – Global peaks.

Returns

peaks (scipy.stats.rv_frozen) – Probability distribution of the peaks.

mhkit.loads.extreme.peaks_distribution_peaks_over_threshold(x, threshold=None)

Estimate the peaks distribution using the peaks over threshold method.

This fits a generalized Pareto distribution to all the peaks above the specified threshold. The distribution is only defined for values above the threshold and therefore cannot be used to obtain integral metrics such as the expected value. A typical choice of threshold is 1.4 standard deviations above the mean. The peaks over threshold distribution can be accessed through the pot field of the returned peaks distribution.

Parameters

• x (np.array) – Global peaks.

• threshold (float) – Threshold value. Only peaks above this value will be used. Default value calculated as: np.mean(x) + 1.4 * np.std(x)

Returns

peaks (scipy.stats.rv_frozen) – Probability distribution of the peaks.

mhkit.loads.extreme.ste_peaks(peaks_distribution, npeaks)

Estimate the short-term extreme distribution from the peaks distribution.

Parameters

• peaks_distribution (scipy.stats.rv_frozen) – Probability distribution of the peaks.

• npeaks (float) – Number of peaks in short term period.

Returns

ste (scipy.stats.rv_frozen) – Short-term extreme distribution.

mhkit.loads.extreme.block_maxima(t, x, t_st)

Find the block maxima of a time-series.

The timeseries (t,x) is divided into blocks of length t_st, and the maxima of each bloock is returned.

Parameters

• t (np.array) – Time array.

• x (np.array) – global peaks timeseries.

• t_st (float) – Short-term period.

Returns

block_maxima (np.array) – Block maxima (i.e. largest peak in each block).

mhkit.loads.extreme.ste_block_maxima_gev(block_maxima)

Approximate the short-term extreme distribution using the block maxima method and the Generalized Extreme Value distribution.

Parameters

block_maxima (np.array) – Block maxima (i.e. largest peak in each block).

Returns

ste (scipy.stats.rv_frozen) – Short-term extreme distribution.

mhkit.loads.extreme.ste_block_maxima_gumbel(block_maxima)

Approximate the short-term extreme distribution using the block maxima method and the Gumbel (right) distribution.

Parameters

block_maxima (np.array) – Block maxima (i.e. largest peak in each block).

Returns

ste (scipy.stats.rv_frozen) – Short-term extreme distribution.

mhkit.loads.extreme.ste(t, data, t_st, method)

Alias for short_term_extreme.

mhkit.loads.extreme.short_term_extreme(t, data, t_st, method)

Approximate the short-term extreme distribution from a timeseries of the response using chosen method.

The availabe methods are: ‘peaks_weibull’, ‘peaks_weibull_tail_fit’, ‘peaks_over_threshold’, ‘block_maxima_gev’, and ‘block_maxima_gumbel’. For the block maxima methods the timeseries needs to be many times longer than the short-term period. For the peak-fitting methods the timeseries can be of arbitrary length.

Parameters

• t (np.array) – Time array.

• data (np.array) – Response timeseries.

• t_st (float) – Short-term period.

• method (string) – Method for estimating the short-term extreme distribution.

Returns

ste (scipy.stats.rv_frozen) – Short-term extreme distribution.

mhkit.loads.extreme.full_seastate_long_term_extreme(ste, weights)

Return the long-term extreme distribution of a response of interest using the full sea state approach.

Parameters

• ste (list[scipy.stats.rv_frozen]) – Short-term extreme distribution of the quantity of interest for each sample sea state.

• weights (list[floats]) – The weights from the full sea state sampling

Returns

ste (scipy.stats.rv_frozen) – Short-term extreme distribution.

mhkit.loads.extreme.mler_coefficients(rao, wave_spectrum, response_desired)

Calculate MLER (most likely extreme response) coefficients from a sea state spectrum and a response RAO.

Parameters

• rao (numpy ndarray) – Response amplitude operator.

• wave_spectrum (pd.DataFrame) – Wave spectral density [m^2/Hz] indexed by frequency [Hz].

• response_desired (int or float) – Desired response, units should correspond to a motion RAO or units of force for a force RAO.

Returns

mler (pd.DataFrame) – DataFrame containing conditioned wave spectral amplitude coefficient [m^2-s], and Phase [rad] indexed by freq [Hz].

mhkit.loads.extreme.mler_simulation(parameters=None)

Define the simulation parameters that are used in various MLER functionalities.

See extreme_response_contour_example.ipynb example for how this is useful. If no input is given, then default values are returned.

Parameters

parameters (dict (optional)) – Simulation parameters. Keys: —– ‘startTime’: starting time [s] ‘endTime’: ending time [s] ‘dT’: time-step size [s] ‘T0’: time of maximum event [s] ‘startx’: start of simulation space [m] ‘endX’: end of simulation space [m] ‘dX’: horizontal spacing [m] ‘X’: position of maximum event [m]

Returns

sim (dict) – Simulation parameters including spatial and time calculated arrays.

mhkit.loads.extreme.mler_wave_amp_normalize(wave_amp, mler, sim, k)

Function that renormalizes the incoming amplitude of the MLER wave to the desired peak height (peak to MSL).

Parameters

• wave_amp (float) – Desired wave amplitude (peak to MSL).

• mler (pd.DataFrame) – MLER coefficients generated by ‘mler_coefficients’ function.

• sim (dict) – Simulation parameters formatted by output from ‘mler_simulation’.

• k (numpy ndarray) – Wave number.

Returns

mler_norm (pd.DataFrame) – MLER coefficients

mhkit.loads.extreme.mler_export_time_series(rao, mler, sim, k)

Generate the wave amplitude time series at X0 from the calculated MLER coefficients

Parameters

• rao (numpy ndarray) – Response amplitude operator.

• mler (pd.DataFrame) – MLER coefficients dataframe generated from an MLER function.

• sim (dict) – Simulation parameters formatted by output from ‘mler_simulation’.

• k (numpy ndarray) – Wave number.

Returns

mler_ts (pd.DataFrame) – Time series of wave height [m] and linear response [*] indexed by time [s].

mhkit.loads.extreme.return_year_value(ppf, return_year, short_term_period_hr)

Calculate the value from a given distribution corresponding to a particular return year.

Parameters

• ppf (callable function of 1 argument) – Percentage Point Function (inverse CDF) of short term distribution.

• return_year (int, float) – Return period in years.

• short_term_period_hr (int, float) – Short term period the distribution is created from in hours.

Returns

value (float) – The value corresponding to the return period from the distribution.