from ast import Raise
import numpy as np
from scipy import stats
from scipy import integrate
from scipy import special
import warnings
[docs]
class Low(object):
"""Class for fatigue life estimation using frequency domain
method by Low[1].
Notes
-----
Numerical implementation supports only integer values of
S-N curve parameter k (inverse slope). Due to approximation of
large stress cycles through McLaurin series, sufficient engineering
precision is up to k=6 [1].
References
----------
[1] Y.M.Low. A method for accurate estimation of the fatigue damage
induced by bimodal processes. Probabilistic Engineering Mechanics, 25(1):75-85, 2010
[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_higher = es.get_psd(freq, 100, 120, variance = 2) # higher mode of random process
>>> PSD = PSD_lower + PSD_higher # bimodal 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 [/]
>>> low = FLife.Low(sd, PSD_splitting=('userDefinedBands', [80,150]))
>>> print(f'Fatigue life: {low.get_life(C,int(k)):.3e} s.')
Plot segmentated PSD, used in Low's method
>>> lower_band_index, upper_band_index= low.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)
>>> 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][upper_band_index], 0, np.max(sd.psd[:,1]), 'k', linestyles='dashed', alpha=.5)
>>> plt.fill_between(sd.psd[lower_band_index:upper_band_index,0], sd.psd[lower_band_index:upper_band_index,1], 'o', label='upper band', alpha=.5, color ='orange')
>>> plt.xlabel('Frequency [Hz]')
>>> plt.ylabel('PSD')
>>> plt.xlim(0,300)
>>> plt.legend()
"""
[docs]
def __init__(self, spectral_data, PSD_splitting = ('equalAreaBands', 2)):
"""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', 2).
"""
self.spectral_data = spectral_data
self.PSD_splitting = PSD_splitting
self.band_stop_indexes = self.spectral_data._get_band_stop_frequency(self.PSD_splitting)
[docs]
def get_life(self, C, k):
"""Calculate fatigue life with parameters C, k, as defined in [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
"""
#Data type check
if not isinstance(k, int):
raise Exception('Parameter `k` must be integer. Sufficient engineering precision is up to k=6.')
if k > 6:
warnings.warn(f'Sufficient engineering precision is up to k=6. Results should be evaluated carefully.')
# -- spectral moments for each narrowband
moments = self.spectral_data.get_spectral_moments(self.PSD_splitting, moments=[0])
m0L, = moments[0] #spectral moments for lower band
m0H, = moments[1] #spectral moments for upper band
# -- positive slope zero crossing frequency
v0L, v0H = self.spectral_data.get_nup(self.PSD_splitting)
v0Small = v0H - v0L #frequency of small cycless
#band's zero crossing frequency ratio
beta = v0H/v0L
#Damage from small cycles
#stress discrepancy
eps = lambda r_lf, phi, beta: np.pi / (2*beta) * r_lf * np.abs(np.sin(phi))
#peak PDF
pdf_r = lambda r, var: stats.rayleigh.pdf(r, scale=np.sqrt(var))
#phase angle
pdf_phi = lambda phi, beta: stats.uniform.pdf(phi, loc=np.pi/4/beta, scale=np.pi/2-np.pi/4/beta)
#damage
int_func_small = lambda phi, r_lf: self._inner_integral_small(k, eps(r_lf, phi, beta), m0H) * pdf_phi(phi, beta) * pdf_r(r_lf, m0L)
I_small = integrate.dblquad(int_func_small, 0, np.inf, lambda r_lf: np.pi/4/beta, lambda r_lf: np.pi/2)[0]
d_small = v0Small / C * I_small
#Damage from large cycles
int_func_large = lambda r_hf, r_lf: self._inner_integral_large(k, r_lf, r_hf, beta) * pdf_r(r_hf, m0H) * pdf_r(r_lf, m0L)
I_large = 1/np.pi * integrate.dblquad(int_func_large, 0, np.inf, lambda r_lf: 0, lambda r_lf: np.inf)[0]
d_large = v0L / C * I_large
#Agregated damage
d = d_small + d_large
T = 1 / d
return T
def _Ik(self, eps, var, K):
"""Calculates coefficients for binomial series expansion for small cycles damage estimation[1].
"""
Ik_array = np.zeros(K)
Ik_array[0] = np.exp(-eps**2/(2*var))
Ik_array[1] = eps * Ik_array[0] + np.sqrt(2*np.pi) * np.sqrt(var) * stats.norm.cdf(-eps/np.sqrt(var))
for i in range(K-2):
term1 = eps**(i+2) * Ik_array[0]
term2 = (i+2) * var * Ik_array[i]
Ik_array[i+2] = term1 + term2
return Ik_array
def _inner_integral_small(self, k, eps, var):
"""Returns analytical expression of innermost integral for small cycles damage estimation,
based on binomial series expansion[1].
"""
out = 0
Ik_array = self._Ik(eps,var,int(k)+1)
for i in range(len(Ik_array)):
out += special.binom(k,i) * (-eps)**(k-i) * Ik_array[i]
return out
def _ro_j(self, r_lf, r_hf, beta, j):
"""Calculates coefficients for MacLaurin series expansion for large cycles damage estimation[1].
"""
c = beta * r_hf / (r_lf + beta**2 * r_hf)
out = r_lf * c**j + r_hf * (beta*c -1)**j
out /= special.factorial(j)
return out
def _inner_integral_large(self, k, r_lf, r_hf, beta):
"""Returns analytical approximation of innermost integral for large cycles damage estimation,
based on MacLaurin series expansion. The approximation gives sufficient engineering
precision in the damage estimate for up to k = 6 [1].
"""
r_sum = r_lf + r_hf
ro_2 = self._ro_j(r_lf, r_hf, beta, 2)
ro_4 = self._ro_j(r_lf, r_hf, beta, 4)
ro_6 = self._ro_j(r_lf, r_hf, beta, 6)
out = r_sum**k * (np.pi - 1/3/r_sum * k * ro_2 * np.pi**3 \
+ k/5/r_sum * (ro_4 + (k-1) * ro_2**2 /2/r_sum) * np.pi**5 \
- k/7/r_sum * (ro_6 + (k-1) * ro_2 * ro_4 /r_sum + (k-1) * (k-2) * ro_2**3 /(6*r_sum**2)) * np.pi**7)
return out