import numpy as np
from scipy import stats
from scipy import integrate
from .jiao_moan import JiaoMoan
from ..tools import pdf_rayleigh_sum
[docs]
class GaoMoan(JiaoMoan):
"""Class for fatigue life estimation using frequency domain
method by Gao and Moan [1].
References
----------
[1] Zhen Gao and Torgeir Moan. Frequency-domain fatigue analysis of
wide-band stationary Gaussian processes using a trimodal spectral formulation.
International Journal of Fatigue, 30(10-11): 1944-1955, 2008
[2] Aleš Zorman and Janko Slavič and Miha Boltežar.
Vibration fatigue by spectral methods—A review with open-source support,
Mechanical Systems and Signal Processing, 2023,
https://doi.org/10.1016/j.ymssp.2023.110149
Example
-------
Import modules, define time- and frequency-domain data
>>> import FLife
>>> import pyExSi as es
>>> import numpy as np
>>> from matplotlib import pyplot as plt
>>> # time-domain data
>>> N = 2 ** 16 # number of data points of time signal
>>> fs = 2048 # sampling frequency [Hz]
>>> t = np.arange(0, N) / fs # time vector
>>> # frequency-domain data
>>> M = N // 2 + 1 # number of data points of frequency vector
>>> freq = np.arange(0, M, 1) * fs / N # frequency vector
>>> PSD_lower = es.get_psd(freq, 20, 60, variance = 5) # lower mode of random process
>>> PSD_middle = es.get_psd(freq, 100, 120, variance = 1) # middle mode of random process
>>> PSD_higher = es.get_psd(freq, 300, 350, variance = 2) # higher mode of random process
>>> PSD = PSD_lower + PSD_middle + PSDb_higher # trimodal one-sided flat-shaped PSD
Get Gaussian stationary signal, instantiate SpectralData object and plot PSD
>>> rg = np.random.default_rng(123) # random generator seed
>>> x = es.random_gaussian(N, PSD, fs, rg) # Gaussian stationary signal
>>> sd = FLife.SpectralData(input=x, dt=1/fs) # SpectralData instance
>>> plt.plot(sd.psd[:,0], sd.psd[:,1])
>>> plt.xlabel('Frequency [Hz]')
>>> plt.ylabel('PSD')
Define S-N curve parameters and get fatigue-life estimatate
>>> C = 1.8e+22 # S-N curve intercept [MPa**k]
>>> k = 7.3 # S-N curve inverse slope [/]
>>> gm = FLife.GaoMoan(sd, PSD_splitting=('userDefinedBands', [80,150,400])) # fatigue-life estimator
>>> print(f'Fatigue life: {gm.get_life(C,k):.3e} s.')
Plot segmentated PSD, used in Gao-Moan method
>>> lower_band_index, middle_band_index, upper_band_index= gm.band_stop_indexes
>>> plt.plot(sd.psd[:,0], sd.psd[:,1])
>>> plt.vlines(sd.psd[:,0][lower_band_index], 0, np.max(sd.psd[:,1]), 'k', linestyles='dashed', alpha=.5)
>>> # lower band
>>> plt.fill_between(sd.psd[:lower_band_index,0], sd.psd[:lower_band_index,1], 'o', label='lower band', alpha=.2, color='blue')
>>> plt.vlines(sd.psd[:,0][middle_band_index], 0, np.max(sd.psd[:,1]), 'k', linestyles='dashed', alpha=.5)
>>> # middle band
>>> plt.fill_between(sd.psd[lower_band_index:middle_band_index,0], sd.psd[lower_band_index:middle_band_index,1], 'o', label='middle band', alpha=.5, color ='orange')
>>> plt.vlines(sd.psd[:,0][upper_band_index], 0, np.max(sd.psd[:,1]), 'k', linestyles='dashed', alpha=.5)
>>> # upper band
>>> plt.fill_between(sd.psd[middle_band_index:upper_band_index,0], sd.psd[middle_band_index:upper_band_index,1], 'o', label='upper band', alpha=.5, color ='green')
>>> plt.xlabel('Frequency [Hz]')
>>> plt.ylabel('PSD')
>>> plt.xlim(0,450)
>>> plt.legend()
"""
[docs]
def __init__(self, spectral_data, PSD_splitting = ('equalAreaBands', 3)):
"""Get needed values from reference object.
:param spectral_data: Instance of class SpectralData
:param PSD_splitting: tuple
PSD_splitting[0] is PSD spliting method, PSD_splitting[1] is method argument.
Splitting methods:
- 'userDefinedBands', PSD_splitting[1] must be of type list or tupple, with N
elements specifying upper band frequencies of N random processes.
- 'equalAreaBands', PSD_splitting[1] must be of type int, specifying N random processes.
Defaults to ('equalAreaBands', 3).
"""
JiaoMoan.__init__(self, spectral_data, PSD_splitting)
[docs]
def get_life(self, C, k):
"""Calculate fatigue life with parameters C, k, as defined in [1, 2].
:param C: [int,float];
S-N curve intercept [MPa**k].
:param k: [int,float];
S-N curve inverse slope [/].
:return:
Estimated fatigue life in seconds.
:rtype: float
"""
if len(self.band_stop_indexes) == 1: # narrow-band
T = self._life_NB(C,k)
elif len(self.band_stop_indexes) == 2: # bi-modal
T = self._life_bimodal(C,k)
elif len(self.band_stop_indexes) == 3: # tri-modal
T = self._life_trimodal(C,k)
else:
raise Exception('Specify up to tri-modal random process.')
return T
def _life_bimodal(self, C, k):
# -- spectral moments for each narrowband
moments = self.spectral_data.get_spectral_moments(self.PSD_splitting, moments=[0,2])
m0L, m2L = moments[0] #spectral moments for lower band
m0H, m2H = moments[1] #spectral moments for upper band
# -- Vanmarcke bandwidth parameter
_, epsV_H = self.spectral_data.get_vanmarcke_parameter(self.PSD_splitting)
# -- positive slope zero crossing frequency
_, v0H = self.spectral_data.get_nup(self.PSD_splitting)
v0P = 1/(2*np.pi) * (np.sqrt(m0L) / (m0H + m0L)) * np.sqrt(m2H * epsV_H**2 + m2L)
#v0P1 = m0L_norm * v0L* np.sqrt(1 + m0H_norm/m0L_norm * (v0H/v0L*epsV_H)**2) #jiao-moan, izraz je ekvivalenten za bimodalen proces
# -- damage intensity
dNB_H = self.damage_intesity_NB(m0=m0H, nu=v0H, C=C, k=k)
dNB_P = self._damage_intesity_bimodal_LF(m0L=m0L, m0H=m0H, nuP=v0P, C=C, k=k) #deduje od jiao-moana
d = dNB_H + dNB_P
T = 1 / d
return T
def _life_trimodal(self, C, k):
moments = self.spectral_data.get_spectral_moments(self.PSD_splitting, moments=[0,2])
m0L, m2L = moments[0]
m0M, m2M = moments[1]
m0H, m2H = moments[2]
# -- Vanmarcke bandwidth parameter
_, epsV_M, epsV_H = self.spectral_data.get_vanmarcke_parameter(self.PSD_splitting)
# -- positive slope zero crossing frequency
_, _, v0H = self.spectral_data.get_nup(self.PSD_splitting)
# -- positive slope zero crossing frequency
# -- process HF + MF
v0P = 1/(2*np.pi) * (np.sqrt(m0M) / (m0H + m0M)) * np.sqrt(m2H * epsV_H**2 + m2M)
# -- process HF + MF + LF
v0Q = 1/(4*np.pi) * np.sqrt(m2H * epsV_H**2 + m2M * epsV_M**2 + m2L) * \
(2.0 * np.sqrt(m0L * (m0H + m0M + m0L)) - np.pi * np.sqrt(m0H * m0M) \
+ 2.0 * np.sqrt(m0H * m0M) * np.arctan(np.sqrt((m0H * m0M ) /m0L) / np.sqrt(m0H + m0M + m0L))) \
/ (np.sqrt(m0H + m0M + m0L)**3)
# -- damage intensity
dNB_H = self.damage_intesity_NB(m0=m0H, nu=v0H, C=C, k=k)
dNB_P = self._damage_intesity_bimodal_LF(m0_LF=m0M, m0_HF=m0H, nuP=v0P, C=C, k=k)
dNB_Q = self._damage_intesity_trimodal_LF(m0_LF=m0L,m0_MF=m0M, m0_HF=m0H, nu_L=v0Q, C=C, k=k)
d = dNB_H + dNB_P + dNB_Q
T = 1 / d
return T
def _damage_intesity_trimodal_LF(self, m0_LF, m0_MF, m0_HF, nu_L, C, k):
"""Calculates damage intensity for low frequency component of bimodal random process,
with parameters m0, nuP, C, k, as defined in [2].
:param m0_LF: [int,float]
Zeroth spectral moment of low-frequency component [MPa**2].
:param m0_MF: [int,float]
Zeroth spectral moment of medium-frequency component [MPa**2].
:param m0_HF: [int,float]
Zeroth spectral moment of high-frequency component [MPa**2].
:param nu_L: [int,float]
Frequency of positive slope zero crossing of low frequency component[Hz].
:param C: [int,float]
Fatigue strength coefficient [MPa**k].
:param k : [int,float]
Fatigue strength exponent [/].
:return d: float
Estimated damage intensity of low frequency component.
"""
# medium-amplitude cycle pdf
pdf_M = pdf_rayleigh_sum(m0_MF, m0_HF)
# LF component pdf - rayleigh distributed
pdf_LF = lambda s: stats.rayleigh.pdf(s/np.sqrt(m0_LF)) / np.sqrt(m0_LF)
# large-amplitude cycle pdf
pdf_L = lambda s: np.convolve(pdf_M(s), pdf_LF(s))[0] # large cycle pdf
S_L = integrate.quad(lambda s: s**k * pdf_L(s), 0, np.inf)[0]
d = nu_L * S_L / C
return d