From c388c86463574a8785c3287054989692c779f95e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Ja=C5=A1a?= <78748619+jasasonc@users.noreply.github.com> Date: Mon, 22 Jun 2026 09:52:31 +0200 Subject: [PATCH 1/4] Fix factor-of-2 error in random-vibration FDS The random Fatigue Damage Spectrum was 2x too high, from two independent causes (each a factor of 2; they are in separate code paths, so the net error is 2x, not 4x): 1. _get_random_psd_ers_fds: the cycle rate used n0 = (1/pi)*(dz_rms/z_rms), i.e. ~2*f0. The mean upward zero-crossing rate is n0+ = (1/2pi)*sqrt(M2/M0) = (1/2pi)*(dz_rms/z_rms) (Lalanne, Random Vibration, Vol.3 eq [5.76]); for a Q=10 narrow-band SDOF response n0+ = f0 (Specification Development, Vol.5 p.46 and Example 4.4). FDS scales linearly with n0 -> was 2x too high. (ERS uses the same n0 inside sqrt(2*ln(n0*T)), so the ERS effect is only ~3-4%; corrected here too for consistency with the book.) 2. _get_random_time_ers_fds (FDS): rainflow.count_cycles returns full cycles, so full-cycle Miner-Basquin damage is (p**k/C)*sum(count*(range/2)**k). The extra '*2' (full->half cycles) had no matching 1/2 from Lalanne Fatigue Damage Vol.4 eq [4.6] (K**b/(2C)), doubling the result. Removed the '*2'. Sine and sine-sweep FDS are unaffected and were already correct. Tests: - Regenerated random golden arrays in tests/test_data.py. Verified the change is exactly what was intended: the three random FDS arrays are exactly 0.5x the previous values; sine/sweep and the convolution-ERS arrays are unchanged; the two random-ERS arrays drop ~3.5%. - Added test_narrowband_crossing_rate_equals_f0: an absolute, literature- anchored check (Lalanne Vol.5 Example 4.4) that the response cycle rate of a Q=10 narrow-band oscillator equals f0, guarding against reintroducing 1/pi. BREAKING (numerical): random FDS magnitudes are halved and random ERS shifts ~3-4%. Relative comparisons / FDS inversion of random-only profiles are unchanged; absolute damage and any sine-vs-random mixing are corrected. Suggest a 0.2.0 -> 0.3.0 release. --- FatigueDS/spectrum.py | 6 +- tests/test_basic.py | 28 +- tests/test_data.py | 1071 +++++++++++++++++++++++------------------ 3 files changed, 635 insertions(+), 470 deletions(-) diff --git a/FatigueDS/spectrum.py b/FatigueDS/spectrum.py index a7f94d4..0619fcb 100644 --- a/FatigueDS/spectrum.py +++ b/FatigueDS/spectrum.py @@ -424,7 +424,7 @@ def _get_random_psd_ers_fds(self, output=None): # ERS calculation if output == 'ERS': - n0 = 1 / np.pi * dz_rms / z_rms + n0 = 1 / (2 * np.pi) * dz_rms / z_rms # n0+ = (1/2pi)*sqrt(M2/M0), Lalanne Vol.3 eq [5.76]; for a Q=10 narrow-band response n0+ = f0 (Vol.5 p.46, Ex 4.4) ers = (2 * np.pi * self.f0_range)**2 * z_rms * np.sqrt(2 * np.log(n0 * self.T)) return ers @@ -432,7 +432,7 @@ def _get_random_psd_ers_fds(self, output=None): elif output == 'FDS': z_rms *= self.unit_scale dz_rms *= self.unit_scale - n0 = 1 / np.pi * dz_rms / z_rms + n0 = 1 / (2 * np.pi) * dz_rms / z_rms # n0+ = (1/2pi)*sqrt(M2/M0), Lalanne Vol.3 eq [5.76]; for a Q=10 narrow-band response n0+ = f0 (Vol.5 p.46, Ex 4.4) fds = self.p**self.k / self.C * n0 * self.T * (z_rms * np.sqrt(2))**self.k * gamma(1 + self.k / 2) return fds @@ -457,7 +457,7 @@ def _get_random_time_ers_fds(self, output=None): rf = rainflow.count_cycles(z) rf = np.asarray(rf) - cyc_sum = np.sum(rf[:,1] * 2 * (rf[:,0] / 2)**self.k) # *2 and /2 because rainflow returns cycles and ranges, fds theory is defined for half cycles and amplitudes + cyc_sum = np.sum(rf[:,1] * (rf[:,0] / 2)**self.k) # rainflow count = full cycles, range/2 = amplitude; full-cycle Miner-Basquin damage (Lalanne Vol.4 eq [4.6], full-cycle form) D_i = self.p**self.k / (self.C) * cyc_sum fds[i] = D_i return fds diff --git a/tests/test_basic.py b/tests/test_basic.py index 4fb0575..afb2116 100644 --- a/tests/test_basic.py +++ b/tests/test_basic.py @@ -74,7 +74,7 @@ def test_random_time_psd_averaging(self): """ Test the random time history function with psd averaging""" _time_data = np.load('test_data/test_time_history.npy', allow_pickle=True) time_history_data = _time_data[:,1] - t = _time_data[:,0] + t = _time_data[:,0] dt = t[2] - t[1] load_spectrum_averaging = FatigueDS.Spectrum(freq_data=(20, 200, 5)) @@ -86,3 +86,29 @@ def test_random_time_psd_averaging(self): assert np.allclose(load_spectrum_averaging.fds, random_time_averaging_fds_true) +def test_narrowband_crossing_rate_equals_f0(): + """Absolute, literature-anchored check of the FDS/ERS cycle-rate convention. + + For a single-DOF oscillator (Q=10) excited by broadband (white) random vibration, + the response is narrow-band and its mean number of upward zero-crossings per + second equals the natural frequency f0 (Lalanne, *Specification Development*, + Vol.5 p.46: "n0 is equal to f0"; verified numerically in Vol.5 Example 4.4, where + a 10 Hz oscillator shows 50 up-crossings in 5 s -> 10 Hz). This is the quantity + n0+ used in the random ERS/FDS, so it pins down the (1/(2*pi)) factor in + n0+ = (1/(2*pi)) * (dz_rms/z_rms) and guards against the factor-of-2 error of + using 1/pi (which would yield ~2*f0). + """ + rng = np.random.default_rng(0) + fs = 2048.0 + dt = 1 / fs + acc = rng.standard_normal(int(120.0 * fs)) # broadband (white) excitation + damp = 1 / (2 * 10) # Q = 10 + + for f0 in (50.0, 100.0, 150.0, 200.0): + z = FatigueDS.tools.response_relative_displacement(acc, dt, f_0=f0, damp=damp) + z = z[len(z) // 20:] # drop start-up transient + n_up = np.count_nonzero((z[:-1] < 0) & (z[1:] >= 0)) + rate = n_up / (len(z) * dt) + assert np.isclose(rate, f0, rtol=0.02), f"f0={f0}: crossing rate {rate:.2f} != f0" + + diff --git a/tests/test_data.py b/tests/test_data.py index 5aa2e3b..d53f5f7 100644 --- a/tests/test_data.py +++ b/tests/test_data.py @@ -1,477 +1,616 @@ import numpy as np -sine_ers_true = np.array([4.00000000e-11, 1.00009951e-03, 4.00159263e-03, 9.00806669e-03, - 1.60255124e-02, 2.50623418e-02, 3.61294132e-02, 4.92400640e-02, - 6.44101506e-02, 8.16580965e-02, 1.01004948e-01, 1.22474441e-01, - 1.46093068e-01, 1.71890170e-01, 1.99898021e-01, 2.30151932e-01, - 2.62690368e-01, 2.97555069e-01, 3.34791187e-01, 3.74447440e-01, - 4.16576274e-01, 4.61234040e-01, 5.08481196e-01, 5.58382517e-01, - 6.11007327e-01, 6.66429756e-01, 7.24729014e-01, 7.85989693e-01, - 8.50302094e-01, 9.17762584e-01, 9.88473986e-01, 1.06254600e+00, - 1.14009568e+00, 1.22124792e+00, 1.30613601e+00, 1.39490226e+00, - 1.48769865e+00, 1.58468753e+00, 1.68604246e+00, 1.79194906e+00, - 1.90260598e+00, 2.01822592e+00, 2.13903688e+00, 2.26528336e+00, - 2.39722782e+00, 2.53515229e+00, 2.67936005e+00, 2.83017765e+00, - 2.98795702e+00, 3.15307790e+00, 3.32595053e+00, 3.50701868e+00, - 3.69676304e+00, 3.89570500e+00, 4.10441093e+00, 4.32349707e+00, - 4.55363492e+00, 4.79555754e+00, 5.05006654e+00, 5.31804017e+00, - 5.60044252e+00, 5.89833409e+00, 6.21288392e+00, 6.54538361e+00, - 6.89726356e+00, 7.27011180e+00, 7.66569594e+00, 8.08598889e+00, - 8.53319901e+00, 9.00980569e+00, 9.51860145e+00, 1.00627420e+01, - 1.06458062e+01, 1.12718674e+01, 1.19455804e+01, 1.26722861e+01, - 1.34581391e+01, 1.43102639e+01, 1.52369475e+01, 1.62478772e+01, - 1.73544366e+01, 1.85700754e+01, 1.99107735e+01, 2.13956264e+01, - 2.30475826e+01, 2.48943751e+01, 2.69696910e+01, 2.93146234e+01, - 3.19794311e+01, 3.50255669e+01, 3.85277710e+01, 4.25756329e+01, - 4.72731440e+01, 5.27328937e+01, 5.90578382e+01, 6.62971491e+01, - 7.43550842e+01, 8.28358146e+01, 9.08622691e+01, 9.70585875e+01, - 1.00000000e+02, 9.90574642e+01, 9.48323183e+01, 8.86617200e+01, - 8.18207684e+01, 7.51352916e+01, 6.90053276e+01, 6.35614212e+01, - 5.87974930e+01, 5.46505007e+01, 5.10407694e+01, 4.78900336e+01, - 4.51283287e+01, 4.26957508e+01, 4.05420560e+01, 3.86254883e+01, - 3.69114469e+01, 3.53712416e+01, 3.39810184e+01, 3.27208661e+01, - 3.15740887e+01, 3.05266172e+01, 2.95665364e+01, 2.86837047e+01, - 2.78694470e+01, 2.71163072e+01, 2.64178473e+01, 2.57684832e+01, - 2.51633512e+01, 2.45981977e+01, 2.40692881e+01, 2.35733313e+01, - 2.31074169e+01, 2.26689626e+01, 2.22556696e+01, 2.18654852e+01, - 2.14965716e+01, 2.11472782e+01, 2.08161189e+01, 2.05017525e+01, - 2.02029650e+01, 1.99186559e+01, 1.96478244e+01, 1.93895593e+01, - 1.91430286e+01, 1.89074716e+01, 1.86821912e+01, 1.84665477e+01, - 1.82599528e+01, 1.80618645e+01, 1.78717831e+01, 1.76892468e+01, - 1.75138282e+01, 1.73451314e+01, 1.71827891e+01, 1.70264598e+01, - 1.68758260e+01, 1.67305919e+01, 1.65904815e+01, 1.64552373e+01, - 1.63246185e+01, 1.61983997e+01, 1.60763700e+01, 1.59583314e+01, - 1.58440980e+01, 1.57334955e+01, 1.56263595e+01, 1.55225357e+01, - 1.54218784e+01, 1.53242505e+01, 1.52295223e+01, 1.51375717e+01, - 1.50482828e+01, 1.49615465e+01, 1.48772591e+01, 1.47953226e+01, - 1.47156441e+01, 1.46381353e+01, 1.45627124e+01, 1.44892961e+01, - 1.44178108e+01, 1.43481844e+01, 1.42803487e+01, 1.42142385e+01, - 1.41497918e+01, 1.40869493e+01, 1.40256547e+01, 1.39658541e+01, - 1.39074961e+01, 1.38505316e+01, 1.37949136e+01, 1.37405972e+01, - 1.36875396e+01, 1.36356995e+01, 1.35850376e+01, 1.35355163e+01, - 1.34870992e+01, 1.34397519e+01, 1.33934410e+01, 1.33481346e+01, - 1.33038021e+01, 1.32604141e+01, 1.32179422e+01, 1.31763594e+01, - 1.31356395e+01, 1.30957573e+01, 1.30566888e+01, 1.30184105e+01, - 1.29809000e+01, 1.29441359e+01, 1.29080973e+01, 1.28727640e+01, - 1.28381169e+01, 1.28041372e+01, 1.27708069e+01, 1.27381088e+01, - 1.27060259e+01, 1.26745421e+01, 1.26436419e+01, 1.26133100e+01, - 1.25835319e+01, 1.25542934e+01, 1.25255810e+01, 1.24973813e+01, - 1.24696817e+01, 1.24424697e+01, 1.24157334e+01, 1.23894612e+01, - 1.23636418e+01, 1.23382645e+01, 1.23133186e+01, 1.22887939e+01, - 1.22646806e+01, 1.22409690e+01, 1.22176498e+01, 1.21947141e+01, - 1.21721529e+01, 1.21499579e+01, 1.21281208e+01, 1.21066335e+01, - 1.20854883e+01, 1.20646775e+01, 1.20441940e+01, 1.20240305e+01, - 1.20041800e+01, 1.19846359e+01, 1.19653916e+01, 1.19464407e+01, - 1.19277770e+01, 1.19093945e+01, 1.18912874e+01, 1.18734498e+01, - 1.18558762e+01, 1.18385613e+01, 1.18214998e+01, 1.18046865e+01, - 1.17881164e+01, 1.17717847e+01, 1.17556867e+01, 1.17398176e+01, - 1.17241731e+01, 1.17087487e+01, 1.16935401e+01, 1.16785432e+01, - 1.16637539e+01, 1.16491682e+01, 1.16347823e+01, 1.16205924e+01, - 1.16065948e+01, 1.15927859e+01, 1.15791623e+01, 1.15657204e+01, - 1.15524569e+01, 1.15393686e+01, 1.15264523e+01, 1.15137049e+01, - 1.15011233e+01, 1.14887045e+01, 1.14764457e+01, 1.14643441e+01, - 1.14523968e+01, 1.14406011e+01, 1.14289544e+01, 1.14174541e+01, - 1.14060977e+01, 1.13948827e+01, 1.13838067e+01, 1.13728673e+01, - 1.13620622e+01, 1.13513891e+01, 1.13408459e+01, 1.13304303e+01, - 1.13201402e+01, 1.13099736e+01, 1.12999283e+01, 1.12900025e+01, - 1.12801942e+01, 1.12705015e+01, 1.12609224e+01, 1.12514553e+01, - 1.12420982e+01, 1.12328494e+01, 1.12237073e+01, 1.12146700e+01, - 1.12057361e+01, 1.11969038e+01, 1.11881715e+01, 1.11795378e+01, - 1.11710011e+01, 1.11625599e+01, 1.11542127e+01, 1.11459582e+01, - 1.11377948e+01, 1.11297213e+01, 1.11217362e+01, 1.11138383e+01, - 1.11060262e+01, 1.10982987e+01, 1.10906545e+01, 1.10830924e+01, - 1.10756112e+01, 1.10682096e+01, 1.10608866e+01, 1.10536410e+01, - 1.10464717e+01, 1.10393776e+01, 1.10323576e+01, 1.10254106e+01, - 1.10185357e+01, 1.10117318e+01, 1.10049978e+01, 1.09983328e+01, - 1.09917359e+01, 1.09852061e+01, 1.09787425e+01, 1.09723441e+01, - 1.09660101e+01, 1.09597395e+01, 1.09535316e+01, 1.09473853e+01, - 1.09413000e+01, 1.09352748e+01, 1.09293089e+01, 1.09234015e+01, - 1.09175517e+01, 1.09117589e+01, 1.09060223e+01, 1.09003412e+01, - 1.08947147e+01, 1.08891422e+01, 1.08836230e+01, 1.08781564e+01, - 1.08727418e+01, 1.08673783e+01, 1.08620654e+01, 1.08568025e+01, - 1.08515889e+01, 1.08464239e+01, 1.08413070e+01, 1.08362375e+01, - 1.08312148e+01, 1.08262384e+01, 1.08213077e+01, 1.08164220e+01, - 1.08115809e+01, 1.08067838e+01, 1.08020302e+01, 1.07973194e+01, - 1.07926511e+01, 1.07880246e+01, 1.07834394e+01, 1.07788951e+01, - 1.07743912e+01, 1.07699271e+01, 1.07655025e+01, 1.07611167e+01, - 1.07567694e+01, 1.07524601e+01, 1.07481884e+01, 1.07439537e+01, - 1.07397558e+01, 1.07355940e+01, 1.07314681e+01, 1.07273775e+01, - 1.07233219e+01, 1.07193009e+01, 1.07153141e+01, 1.07113611e+01, - 1.07074414e+01, 1.07035548e+01, 1.06997007e+01, 1.06958790e+01, - 1.06920891e+01, 1.06883308e+01, 1.06846036e+01, 1.06809073e+01, - 1.06772414e+01, 1.06736057e+01, 1.06699998e+01, 1.06664234e+01, - 1.06628761e+01]) +sine_ers_true = np.array([4.000000000016e-11, 1.000099509851e-03, 4.001592630672e-03, + 9.008066687294e-03, 1.602551241082e-02, 2.506234179202e-02, + 3.612941321441e-02, 4.924006400668e-02, 6.441015059645e-02, + 8.165809649059e-02, 1.010049483536e-01, 1.224744405034e-01, + 1.460930681943e-01, 1.718901701139e-01, 1.998980205755e-01, + 2.301519319541e-01, 2.626903679801e-01, 2.975550685814e-01, + 3.347911870430e-01, 3.744474403452e-01, 4.165762736367e-01, + 4.612340399092e-01, 5.084811960562e-01, 5.583825166347e-01, + 6.110073267903e-01, 6.664297559745e-01, 7.247290142609e-01, + 7.859896932718e-01, 8.503020939558e-01, 9.177625837098e-01, + 9.884739856291e-01, 1.062546002988e+00, 1.140095682426e+00, + 1.221247919714e+00, 1.306136012460e+00, 1.394902264624e+00, + 1.487698648325e+00, 1.584687529112e+00, 1.686042461632e+00, + 1.791949063558e+00, 1.902605976618e+00, 2.018225924784e+00, + 2.139036881003e+00, 2.265283355426e+00, 2.397227819879e+00, + 2.535152285416e+00, 2.679360052208e+00, 2.830177653837e+00, + 2.987957021349e+00, 3.153077896255e+00, 3.325950526189e+00, + 3.507018682187e+00, 3.696763042844e+00, 3.895704997923e+00, + 4.104410932799e+00, 4.323497065478e+00, 4.553634920379e+00, + 4.795557537921e+00, 5.050066536834e+00, 5.318040167608e+00, + 5.600442521583e+00, 5.898334091784e+00, 6.212883920160e+00, + 6.545383613089e+00, 6.897263564968e+00, 7.270111801261e+00, + 7.665695941033e+00, 8.085988889371e+00, 8.533199008044e+00, + 9.009805686141e+00, 9.518601451101e+00, 1.006274203793e+01, + 1.064580618766e+01, 1.127186739818e+01, 1.194558043171e+01, + 1.267228613331e+01, 1.345813908631e+01, 1.431026389342e+01, + 1.523694751546e+01, 1.624787723943e+01, 1.735443662549e+01, + 1.857007536575e+01, 1.991077354401e+01, 2.139562643034e+01, + 2.304758258691e+01, 2.489437507184e+01, 2.696969096810e+01, + 2.931462340616e+01, 3.197943109129e+01, 3.502556685194e+01, + 3.852777102651e+01, 4.257563289707e+01, 4.727314401161e+01, + 5.273289374822e+01, 5.905783822758e+01, 6.629714905649e+01, + 7.435508424832e+01, 8.283581463023e+01, 9.086226905418e+01, + 9.705858747377e+01, 1.000000000000e+02, 9.905746424975e+01, + 9.483231825348e+01, 8.866171995093e+01, 8.182076844088e+01, + 7.513529161748e+01, 6.900532763443e+01, 6.356142119338e+01, + 5.879749297750e+01, 5.465050070692e+01, 5.104076941376e+01, + 4.789003355516e+01, 4.512832870259e+01, 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1.086737830767e+01, + 1.086206544412e+01, 1.085680251091e+01, 1.085158886945e+01, + 1.084642389154e+01, 1.084130695917e+01, 1.083623746431e+01, + 1.083121480874e+01, 1.082623840380e+01, 1.082130767028e+01, + 1.081642203818e+01, 1.081158094654e+01, 1.080678384328e+01, + 1.080203018503e+01, 1.079731943695e+01, 1.079265107256e+01, + 1.078802457361e+01, 1.078343942991e+01, 1.077889513914e+01, + 1.077439120677e+01, 1.076992714587e+01, 1.076550247697e+01, + 1.076111672792e+01, 1.075676943378e+01, 1.075246013667e+01, + 1.074818838560e+01, 1.074395373642e+01, 1.073975575164e+01, + 1.073559400033e+01, 1.073146805797e+01, 1.072737750637e+01, + 1.072332193357e+01, 1.071930093365e+01, 1.071531410670e+01, + 1.071136105869e+01, 1.070744140134e+01, 1.070355475205e+01, + 1.069970073376e+01, 1.069587897492e+01, 1.069208910931e+01, + 1.068833077600e+01, 1.068460361926e+01, 1.068090728842e+01, + 1.067724143785e+01, 1.067360572681e+01, 1.066999981941e+01, + 1.066642338450e+01, 1.066287609559e+01]) -sine_fds_true = np.array([1.92209026e-24, 1.92304679e-24, 1.92591978e-24, 1.93071955e-24, - 1.93746332e-24, 1.94617539e-24, 1.95688730e-24, 1.96963803e-24, - 1.98447431e-24, 2.00145092e-24, 2.02063110e-24, 2.04208703e-24, - 2.06590034e-24, 2.09216273e-24, 2.12097666e-24, 2.15245614e-24, - 2.18672765e-24, 2.22393109e-24, 2.26422100e-24, 2.30776775e-24, - 2.35475905e-24, 2.40540156e-24, 2.45992267e-24, 2.51857261e-24, - 2.58162674e-24, 2.64938816e-24, 2.72219067e-24, 2.80040208e-24, - 2.88442803e-24, 2.97471620e-24, 3.07176121e-24, 3.17611010e-24, - 3.28836866e-24, 3.40920851e-24, 3.53937539e-24, 3.67969846e-24, - 3.83110110e-24, 3.99461325e-24, 4.17138573e-24, 4.36270664e-24, - 4.57002050e-24, 4.79495034e-24, 5.03932349e-24, 5.30520160e-24, - 5.59491585e-24, 5.91110811e-24, 6.25677952e-24, 6.63534772e-24, - 7.05071461e-24, 7.50734679e-24, 8.01037151e-24, 8.56569121e-24, - 9.18012111e-24, 9.86155485e-24, 1.06191648e-23, 1.14636449e-23, - 1.24075077e-23, 1.34654463e-23, 1.46547814e-23, 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1.26598371e-22, - 9.74126686e-23, 7.57379379e-23, 5.94535417e-23, 4.70867675e-23, - 3.76009383e-23, 3.02570357e-23, 2.45219754e-23, 2.00069480e-23, - 1.64254198e-23, 1.35641666e-23, 1.12630621e-23, 9.40081745e-24, - 7.88481430e-24, 6.64378548e-24, 5.62249553e-24, 4.77784099e-24, - 4.07596742e-24, 3.49012113e-24, 2.99903627e-24, 2.58571529e-24, - 2.23650041e-24, 1.94036214e-24, 1.68835075e-24, 1.47317076e-24, - 1.28884917e-24, 1.13047519e-24, 9.93995199e-25, 8.76050202e-25, - 7.73846587e-25, 6.85052715e-25, 6.07715918e-25, 5.40195556e-25, - 4.81108826e-25, 4.29286713e-25, 3.83738054e-25, 3.43620121e-25, - 3.08214448e-25, 2.76906902e-25, 2.49171210e-25, 2.24555286e-25, - 2.02669853e-25, 1.83178956e-25, 1.65792011e-25, 1.50257143e-25, - 1.36355576e-25, 1.23896898e-25, 1.12715063e-25, 1.02664998e-25, - 9.36197199e-26, 8.54678822e-26, 7.81116793e-26, 7.14650552e-26, - 6.54521677e-26, 6.00060692e-26, 5.50675723e-26, 5.05842705e-26, - 4.65096933e-26, 4.28025746e-26, 3.94262186e-26, 3.63479493e-26, - 3.35386311e-26, 3.09722518e-26, 2.86255582e-26, 2.64777377e-26, - 2.45101400e-26, 2.27060324e-26, 2.10503853e-26, 1.95296840e-26, - 1.81317624e-26, 1.68456566e-26, 1.56614760e-26, 1.45702886e-26, - 1.35640203e-26, 1.26353651e-26, 1.17777059e-26, 1.09850437e-26, - 1.02519346e-26, 9.57343454e-27, 8.94504885e-27, 8.36268812e-27, - 7.82262843e-27, 7.32147583e-27, 6.85613457e-27, 6.42377861e-27, - 6.02182611e-27, 5.64791646e-27, 5.29988972e-27, 4.97576804e-27, - 4.67373902e-27, 4.39214065e-27, 4.12944777e-27, 3.88425978e-27, - 3.65528966e-27, 3.44135391e-27, 3.24136354e-27, 3.05431582e-27, - 2.87928693e-27, 2.71542517e-27, 2.56194484e-27, 2.41812073e-27, - 2.28328300e-27, 2.15681264e-27, 2.03813727e-27, 1.92672733e-27, - 1.82209256e-27, 1.72377889e-27, 1.63136553e-27, 1.54446227e-27, - 1.46270716e-27, 1.38576418e-27, 1.31332133e-27, 1.24508868e-27, - 1.18079672e-27, 1.12019481e-27, 1.06304971e-27, 1.00914428e-27, - 9.58276303e-28, 9.10257333e-28, 8.64911702e-28, 8.22075571e-28, - 7.81596068e-28, 7.43330491e-28, 7.07145576e-28, 6.72916818e-28, - 6.40527845e-28, 6.09869848e-28, 5.80841040e-28, 5.53346170e-28, - 5.27296068e-28, 5.02607220e-28, 4.79201382e-28, 4.57005219e-28, - 4.35949972e-28, 4.15971146e-28, 3.97008225e-28, 3.79004406e-28, - 3.61906351e-28, 3.45663960e-28, 3.30230154e-28, 3.15560682e-28, - 3.01613935e-28, 2.88350774e-28, 2.75734373e-28, 2.63730069e-28, - 2.52305225e-28, 2.41429104e-28, 2.31072746e-28, 2.21208856e-28, - 2.11811706e-28, 2.02857030e-28, 1.94321940e-28, 1.86184837e-28, - 1.78425336e-28, 1.71024190e-28, 1.63963220e-28, 1.57225254e-28, - 1.50794065e-28, 1.44654313e-28, 1.38791496e-28, 1.33191897e-28, - 1.27842540e-28, 1.22731147e-28, 1.17846095e-28, 1.13176381e-28, - 1.08711584e-28, 1.04441832e-28, 1.00357773e-28, 9.64505424e-29, - 9.27117366e-29, 8.91333874e-29, 8.57079372e-29, 8.24282161e-29, - 7.92874206e-29, 7.62790932e-29, 7.33971032e-29, 7.06356290e-29, - 6.79891411e-29, 6.54523866e-29, 6.30203735e-29, 6.06883575e-29, - 5.84518282e-29, 5.63064966e-29, 5.42482837e-29, 5.22733089e-29, - 5.03778801e-29, 4.85584833e-29, 4.68117735e-29, 4.51345661e-29, - 4.35238283e-29, 4.19766713e-29, 4.04903429e-29, 3.90622209e-29, - 3.76898059e-29, 3.63707154e-29, 3.51026777e-29, 3.38835266e-29, - 3.27111958e-29, 3.15837141e-29, 3.04992007e-29, 2.94558604e-29, - 2.84519800e-29, 2.74859235e-29, 2.65561288e-29, 2.56611042e-29, - 2.47994246e-29, 2.39697285e-29, 2.31707148e-29, 2.24011401e-29, - 2.16598157e-29, 2.09456053e-29, 2.02574221e-29, 1.95942269e-29, - 1.89550256e-29, 1.83388669e-29, 1.77448407e-29, 1.71720761e-29, - 1.66197393e-29, 1.60870322e-29, 1.55731904e-29, 1.50774821e-29, - 1.45992064e-29, 1.41376915e-29, 1.36922943e-29, 1.32623980e-29, - 1.28474118e-29, 1.24467694e-29, 1.20599277e-29, 1.16863660e-29, - 1.13255850e-29, 1.09771060e-29, 1.06404694e-29, 1.03152347e-29, - 1.00009788e-29, 9.69729601e-30, 9.40379674e-30, 9.12010708e-30, - 8.84586802e-30, 8.58073487e-30, 8.32437656e-30, 8.07647513e-30, - 7.83672515e-30, 7.60483317e-30, 7.38051722e-30, 7.16350634e-30, - 6.95354009e-30, 6.75036813e-30, 6.55374979e-30, 6.36345366e-30, - 6.17925722e-30, 6.00094646e-30, 5.82831551e-30, 5.66116635e-30, - 5.49930847e-30, 5.34255851e-30, 5.19074007e-30, 5.04368334e-30, - 4.90122486e-30, 4.76320729e-30, 4.62947911e-30, 4.49989444e-30, - 4.37431279e-30, 4.25259884e-30, 4.13462223e-30, 4.02025739e-30, - 3.90938333e-30, 3.80188346e-30, 3.69764542e-30, 3.59656093e-30, - 3.49852558e-30, 3.40343875e-30, 3.31120342e-30, 3.22172602e-30, - 3.13491634e-30, 3.05068736e-30, 2.96895514e-30, 2.88963872e-30, - 2.81265999e-30, 2.73794357e-30, 2.66541672e-30, 2.59500926e-30, - 2.52665344e-30]) +sine_fds_true = np.array([1.922090260873e-24, 1.923046785764e-24, 1.925919784010e-24, + 1.930719549752e-24, 1.937463319562e-24, 1.946175392746e-24, + 1.956887301694e-24, 1.969638034159e-24, 1.984474309970e-24, + 2.001450915331e-24, 2.020631098554e-24, 2.042087031821e-24, + 2.065900344418e-24, 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1.097710598471e-29, 1.064046944052e-29, + 1.031523468064e-29, 1.000097881615e-29, 9.697296011054e-30, + 9.403796739165e-30, 9.120107075932e-30, 8.845868023222e-30, + 8.580734865550e-30, 8.324376556137e-30, 8.076475131327e-30, + 7.836725151971e-30, 7.604833170440e-30, 7.380517222006e-30, + 7.163506339417e-30, 6.953540089512e-30, 6.750368130826e-30, + 6.553749791149e-30, 6.363453664097e-30, 6.179257223760e-30, + 6.000946456576e-30, 5.828315509600e-30, 5.661166354386e-30, + 5.499308465742e-30, 5.342558514667e-30, 5.190740074770e-30, + 5.043683341576e-30, 4.901224864080e-30, 4.763207288003e-30, + 4.629479110186e-30, 4.499894443620e-30, 4.374312792609e-30, + 4.252598837601e-30, 4.134622229251e-30, 4.020257391275e-30, + 3.909383331714e-30, 3.801883462211e-30, 3.697645424937e-30, + 3.596560926829e-30, 3.498525580805e-30, 3.403438753634e-30, + 3.311203420178e-30, 3.221726023713e-30, 3.134916342049e-30, + 3.050687359212e-30, 2.968955142418e-30, 2.889638724131e-30, + 2.812659988950e-30, 2.737943565145e-30, 2.665416720611e-30, + 2.595009263071e-30, 2.526653444318e-30]) -sine_sweep_ers_true = np.array([1.25000000e-08, 3.33214878e-01, 1.66297526e+00, 6.33614307e+00, - 5.00000000e+01, 5.00000000e+01, 5.00000000e+01, 5.00000000e+01, - 5.00000000e+01, 5.00000000e+01, 5.00000000e+01, 5.00000000e+01, - 5.00000000e+01, 5.00000000e+01, 5.00000000e+01, 5.00000000e+01, - 5.00000000e+01, 5.00000000e+01, 5.00000000e+01, 6.62971491e+01, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, 1.00000000e+02, - 1.00000000e+02, 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2.93532476e+01, 2.92762705e+01, 2.92003363e+01, - 2.91254249e+01, 2.90515166e+01, 2.89785923e+01, 2.89066333e+01, - 2.88356215e+01, 2.87655391e+01, 2.86963689e+01, 2.86280938e+01, - 2.85606975e+01, 2.84941638e+01, 2.84284771e+01, 2.83636220e+01, - 2.82995835e+01, 2.82363472e+01, 2.81738986e+01, 2.81122239e+01, - 2.80513094e+01, 2.79911418e+01, 2.79317082e+01, 2.78729958e+01, - 2.78149922e+01, 2.77576853e+01, 2.77010631e+01, 2.76451142e+01, - 2.75898272e+01, 2.75351909e+01, 2.74811945e+01, 2.74278274e+01, - 2.73750791e+01, 2.73229397e+01, 2.72713990e+01, 2.72204474e+01, - 2.71700752e+01, 2.71202733e+01, 2.70710325e+01, 2.70223438e+01, - 2.69741984e+01, 2.69265879e+01, 2.68795038e+01, 2.68329378e+01, - 2.67868820e+01, 2.67413283e+01, 2.66962692e+01, 2.66516970e+01, - 2.66076042e+01]) +sine_sweep_ers_true = np.array([1.250000003109e-08, 3.332148779873e-01, 1.662975263094e+00, + 6.336143066657e+00, 5.000000000000e+01, 5.000000000000e+01, + 5.000000000000e+01, 5.000000000000e+01, 5.000000000000e+01, + 5.000000000000e+01, 5.000000000000e+01, 5.000000000000e+01, + 5.000000000000e+01, 5.000000000000e+01, 5.000000000000e+01, + 5.000000000000e+01, 5.000000000000e+01, 5.000000000000e+01, + 5.000000000000e+01, 6.629714905649e+01, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 1.000000000000e+02, + 1.000000000000e+02, 1.000000000000e+02, 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3.009656565327e+01, + 3.000920439917e+01, 2.992309297901e+01, 2.983820586965e+01, + 2.975451822951e+01, 2.967200587605e+01, 2.959064526421e+01, + 2.951041346560e+01, 2.943128814859e+01, 2.935324755910e+01, + 2.927627050216e+01, 2.920033632412e+01, 2.912542489563e+01, + 2.905151659513e+01, 2.897859229304e+01, 2.890663333657e+01, + 2.883562153495e+01, 2.876553914536e+01, 2.869636885925e+01, + 2.862809378926e+01, 2.856069745647e+01, 2.849416377831e+01, + 2.842847705665e+01, 2.836362196655e+01, 2.829958354524e+01, + 2.823634718156e+01, 2.817389860574e+01, 2.811222387955e+01, + 2.805130938680e+01, 2.799114182409e+01, 2.793170819200e+01, + 2.787299578644e+01, 2.781499219041e+01, 2.775768526597e+01, + 2.770106314645e+01, 2.764511422901e+01, 2.758982716738e+01, + 2.753519086483e+01, 2.748119446739e+01, 2.742782735730e+01, + 2.737507914666e+01, 2.732293967126e+01, 2.727139898466e+01, + 2.722044735238e+01, 2.717007524637e+01, 2.712027333955e+01, + 2.707103250062e+01, 2.702234378893e+01, 2.697419844961e+01, + 2.692658790878e+01, 2.687950376892e+01, 2.683293780439e+01, + 2.678688195709e+01, 2.674132833226e+01, 2.669626919434e+01, + 2.665169696304e+01, 2.660760420951e+01]) -sine_sweep_fds_true = np.array([1.48829368e-15, 1.93501515e-15, 4.77631432e-15, 4.08234019e-14, - 4.05729827e-11, 1.20790969e-11, 2.34361197e-12, 5.85376481e-13, - 1.76008210e-13, 6.09779979e-14, 2.36245938e-14, 1.00193219e-14, - 4.57882664e-15, 2.22805855e-15, 1.14382333e-15, 6.15196968e-16, - 3.45171664e-16, 2.02886013e-16, 1.32974225e-16, 1.63796165e-16, - 6.88062359e-16, 8.75296113e-16, 6.16454585e-16, 4.17555013e-16, - 2.85481057e-16, 1.97912487e-16, 1.39118445e-16, 9.90797586e-17, - 7.14339568e-17, 5.20941657e-17, 3.83981265e-17, 2.85869523e-17, - 2.14827352e-17, 1.62864516e-17, 1.24494916e-17, 9.59085909e-18, - 7.44310307e-18, 5.81656496e-18, 4.57545680e-18, 3.62167474e-18, - 2.88372967e-18, 2.30910085e-18, 1.85890334e-18, 1.50413516e-18, - 1.22301336e-18, 9.99067489e-19, 8.19764298e-19, 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2.762747943832e-29, 2.677425152642e-29, 2.595051258531e-29, + 2.515513157999e-29, 2.438702526851e-29, 2.364515599112e-29, + 2.292852957088e-29, 2.223619331936e-29, 2.156723414197e-29, + 2.092077673736e-29, 2.029598188601e-29, 1.969204482304e-29, + 1.910819369088e-29, 1.854368806751e-29]) -random_psd_ers_true = np.array([ 0.67091768, 0.98388194, 1.63247542, 3.29788837, - 4.75383438, 4.99785655, 6.0270407 , 7.67571032, - 9.35047492, 12.0497534 , 15.75766271, 17.61621176, - 18.93799533, 16.95103332, 16.97641796, 19.05362423, - 22.40096774, 27.10816543, 33.66999142, 41.89796133, - 50.12713593, 58.22040026, 68.31443109, 82.49658532, - 103.51585301, 135.71439027, 184.26416655, 242.8514731 , - 273.92720623, 252.32139196, 208.69281484, 168.63234503, - 138.37984191, 117.15507006, 102.17017646, 91.25217518, - 83.01077056]) +random_psd_ers_true = np.array([ 0.648090424988, 0.950662701293, 1.576695180869, 3.18135848496 , + 4.58500815002 , 4.823609898684, 5.818083049655, 7.409584972776, + 9.027029380723, 11.632304979946, 15.210364978889, 17.007492097751, + 18.287702558843, 16.377949833245, 16.408659703298, 18.41880183438 , + 21.655280368578, 26.205315034631, 32.54706760129 , 40.499247659264, + 48.454954673049, 56.282067298352, 66.044592935802, 79.759871603961, + 100.085713263054, 131.220684907453, 178.166133896886, 234.818884160492, + 264.872874687469, 243.986820927801, 201.803000178221, 163.066277658454, + 133.812182369895, 113.287616809697, 98.797113836241, 88.239452679016, + 80.270179197913]) -random_psd_fds_true = np.array([3.82709923e-15, 2.95840805e-15, 5.47767260e-15, 3.06611256e-14, - 4.83763821e-14, 2.19304752e-14, 2.03374664e-14, 2.62627689e-14, - 3.00317155e-14, 4.73934452e-14, 8.47422726e-14, 7.71568736e-14, - 6.09463324e-14, 2.15303382e-14, 1.33345238e-14, 1.42507575e-14, - 1.93008425e-14, 3.06236179e-14, 5.62393884e-14, 1.06743408e-13, - 1.71983299e-13, 2.45479841e-13, 3.74800030e-13, 6.68359831e-13, - 1.45822973e-12, 4.00066484e-12, 1.32073361e-11, 3.80002040e-11, - 5.07843483e-11, 2.48902485e-11, 7.17773588e-12, 1.85355859e-12, - 5.20252849e-13, 1.71916345e-13, 6.63776932e-14, 2.90858483e-14, - 1.40680124e-14]) +random_psd_fds_true = np.array([1.913549616983e-15, 1.479204025928e-15, 2.738836299824e-15, + 1.533056278836e-14, 2.418819106024e-14, 1.096523758199e-14, + 1.016873317891e-14, 1.313138442793e-14, 1.501585776452e-14, + 2.369672259044e-14, 4.237113628645e-14, 3.857843681479e-14, + 3.047316621633e-14, 1.076516909464e-14, 6.667261875130e-15, + 7.125378749098e-15, 9.650421232457e-15, 1.531180894821e-14, + 2.811969422259e-14, 5.337170416851e-14, 8.599164926747e-14, + 1.227399206751e-13, 1.874000147893e-13, 3.341799156894e-13, + 7.291148627469e-13, 2.000332421405e-12, 6.603668050857e-12, + 1.900010198647e-11, 2.539217417048e-11, 1.244512424006e-11, + 3.588867940773e-12, 9.267792966951e-13, 2.601264245105e-13, + 8.595817234579e-14, 3.318884660841e-14, 1.454292412754e-14, + 7.034006193421e-15]) -random_time_convolution_ers_true = np.array([ 0.6099179 , 0.90024939, 1.58842974, 3.27566123, - 3.52033832, 3.52442658, 5.06568159, 6.91651683, - 6.58642658, 9.08957127, 15.44237487, 18.6571416 , - 22.95008567, 14.64092216, 16.12775012, 18.23665893, - 21.43841285, 27.03874696, 31.89604905, 38.48662753, - 44.07566044, 51.43851906, 60.06826414, 77.19192935, - 97.55560784, 128.18880101, 177.46755538, 235.12800744, - 255.86616316, 223.11173246, 175.9850834 , 140.50045444, - 116.25702449, 101.63699583, 89.61811721, 80.55439421, - 73.42014547]) +random_time_convolution_ers_true = np.array([ 0.609917902834, 0.900249389815, 1.588429740467, 3.275661225469, + 3.520338319522, 3.524426583473, 5.065681589184, 6.916516833636, + 6.586426583431, 9.089571269482, 15.442374870328, 18.657141595561, + 22.950085666686, 14.640922156082, 16.127750116007, 18.236658925636, + 21.438412853576, 27.038746959674, 31.896049054443, 38.486627531051, + 44.075660444232, 51.438519061149, 60.068264144119, 77.191929345094, + 97.555607837718, 128.188801009866, 177.467555375057, 235.128007441727, + 255.86616316305 , 223.111732460848, 175.985083402906, 140.50045443875 , + 116.257024494942, 101.636995827753, 89.618117207621, 80.554394207243, + 73.420145467834]) -random_time_convolution_fds_true = np.array([4.44750346e-15, 3.64972875e-15, 4.81530707e-15, 3.93224792e-14, - 4.97618080e-14, 1.84276306e-14, 2.05773726e-14, 5.90468334e-14, - 2.81286302e-14, 5.18810683e-14, 1.77647730e-13, 2.25013716e-13, - 4.17914600e-13, 8.08228777e-14, 3.74689816e-14, 3.17274940e-14, - 3.87678188e-14, 5.31947296e-14, 8.18807569e-14, 1.42588064e-13, - 2.18840066e-13, 3.03394132e-13, 5.14781240e-13, 1.10101535e-12, - 2.72993636e-12, 7.99340960e-12, 2.96421138e-11, 1.00091619e-10, - 1.40830580e-10, 6.32629767e-11, 1.51780343e-11, 3.61950545e-12, - 1.00532493e-12, 3.36335237e-13, 1.32052620e-13, 5.87197129e-14, - 2.87066429e-14]) +random_time_convolution_fds_true = np.array([2.223751729714e-15, 1.824864375801e-15, 2.407653533898e-15, + 1.966123962121e-14, 2.488090402189e-14, 9.213815291773e-15, + 1.028868630564e-14, 2.952341668012e-14, 1.406431511512e-14, + 2.594053413940e-14, 8.882386511550e-14, 1.125068581595e-13, + 2.089573000984e-13, 4.041143885167e-14, 1.873449080281e-14, + 1.586374702163e-14, 1.938390941911e-14, 2.659736477947e-14, + 4.094037843490e-14, 7.129403180252e-14, 1.094200330692e-13, + 1.516970660210e-13, 2.573906200294e-13, 5.505076765620e-13, + 1.364968179386e-12, 3.996704802336e-12, 1.482105691075e-11, + 5.004580950374e-11, 7.041528999769e-11, 3.163148836192e-11, + 7.589017164135e-12, 1.809752726388e-12, 5.026624655536e-13, + 1.681676184770e-13, 6.602631020244e-14, 2.935985645443e-14, + 1.435332147165e-14]) +random_time_averaging_ers_true = np.array([ 0.688798678829, 0.972813471527, 1.548654343504, 2.940156834089, + 4.596034070523, 4.857705303958, 5.842274087815, 7.438486748778, + 9.065322540381, 11.68387655829 , 15.324564539318, 16.79554214044 , + 17.163821574784, 15.470942844752, 15.892242871032, 17.998417106553, + 21.245457501816, 25.8123281857 , 32.219659350718, 40.288489291266, + 48.224325727331, 55.841729846577, 65.307739019269, 78.590121814914, + 98.37160204807 , 128.914586890175, 174.931188343257, 229.450575991764, + 255.54387335346 , 234.053207465682, 194.488500274938, 157.708798942311, + 129.566084718982, 109.726670678954, 95.703129215536, 85.485342502239, + 77.77237799016 ]) -random_time_averaging_ers_true = np.array([ 0.7132824 , 1.00698015, 1.6034918 , 3.04733707, - 4.76565368, 5.03366731, 6.05260711, 7.70623989, - 9.39082283, 12.10395372, 15.87696591, 17.39723053, - 17.773186 , 16.01129239, 16.44190528, 18.61883622, - 21.97728329, 26.7020753 , 33.33198404, 41.68093115, - 49.88969719, 57.76600833, 67.55327577, 81.28758475, - 101.74375727, 133.33007801, 180.91936669, 237.30055765, - 264.2802926 , 242.04891168, 201.12861375, 163.09189855, - 133.98871832, 113.47250634, 98.97053004, 88.40401509, - 80.42767645]) - -random_time_averaging_fds_true = np.array([4.85734344e-15, 3.19769291e-15, 4.97511369e-15, 2.13702173e-14, - 4.81996227e-14, 2.22810029e-14, 2.04118103e-14, 2.63634910e-14, - 3.02192968e-14, 4.78197297e-14, 8.68570356e-14, 7.19924092e-14, - 4.48922136e-14, 1.64147366e-14, 1.13993314e-14, 1.26838790e-14, - 1.74972647e-14, 2.82908138e-14, 5.32163817e-14, 1.03432956e-13, - 1.67062699e-13, 2.35004268e-13, 3.53138486e-13, 6.19228167e-13, - 1.33527914e-12, 3.65649900e-12, 1.20378509e-11, 3.38169505e-11, - 4.24128165e-11, 2.02105694e-11, 5.96769262e-12, 1.56864466e-12, - 4.42850493e-13, 1.46558997e-13, 5.66188439e-14, 2.48225150e-14, - 1.20116961e-14]) - +random_time_averaging_fds_true = np.array([2.428671722244e-15, 1.598846453750e-15, 2.487556844155e-15, + 1.068510867358e-14, 2.409981134308e-14, 1.114050143555e-14, + 1.020590513724e-14, 1.318174551576e-14, 1.510964841022e-14, + 2.390986487275e-14, 4.342851779439e-14, 3.599620458225e-14, + 2.244610682047e-14, 8.207368288748e-15, 5.699665698140e-15, + 6.341939487772e-15, 8.748632333978e-15, 1.414540692292e-14, + 2.660819086063e-14, 5.171647794890e-14, 8.353134956490e-14, + 1.175021337606e-13, 1.765692428609e-13, 3.096140834934e-13, + 6.676395685341e-13, 1.828249501354e-12, 6.018925454984e-12, + 1.690847523660e-11, 2.120640824931e-11, 1.010528469127e-11, + 2.983846307715e-12, 7.843223288389e-13, 2.214252465719e-13, + 7.327949872890e-14, 2.830942194915e-14, 1.241125749202e-14, + 6.005848066016e-15]) From 79b4883bd04d1d8816518f240043aa73727f4e3f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Ja=C5=A1a?= <78748619+jasasonc@users.noreply.github.com> Date: Mon, 22 Jun 2026 10:35:13 +0200 Subject: [PATCH 2/4] Add validation doc for the random-FDS factor-of-2 fix Documents the correction and its validation against Lalanne (2009): - numeric convergence to the book's closed form eq [4.9] (-> 1.000 as Q->inf; pre-fix code -> 2.000) - side-by-side reproduction of Vol.5 Figs 4.3 and 4.4 - Example 4.3 damage ratio (15.34 vs book 16) - note on the book's own 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zRF1x-i_YN0uRmwyt??c-RCZ@s@A%@@te>i-f-xLir^TJufBuFrlRkWd$Op&*$<)Lwxzm)+3MlvH~hta zy1Ip(iY?YWte%jleST=p2ezYOLY+@>opnUCwY9O3L!;n~s3F70Ob3KUI5qqFbxhnr zV&G;Dm+XjA_Yn#%N0xhBG1SOzvT-l=05{~m=Ozp4Ld_;d+gX|tob3%DP)M~|>Oe0) zy$t6taPX76G4kPdL?Pa72A()SKm6;`L?$msv)ElYy Date: Tue, 23 Jun 2026 08:54:39 +0200 Subject: [PATCH 3/4] docs: update validation - affected releases (0.2.0 & 0.3.0) + independent verification Correct the version note (the bug is present in both 0.2.0 and 0.3.0; 0.3.0 shipped without the fix), and add an independent-verification section: direct rainflow ground truth (corrected = 1.0000x, pre-fix = 2.0000x), measured cycle rate = 1.000*f0, an adversarial review that failed to defend the pre-fix code, and the half-cycle/full-cycle root cause. --- validation/validation.md | 43 +++++++++++++++++++++++++++++++++++++++- 1 file changed, 42 insertions(+), 1 deletion(-) diff --git a/validation/validation.md b/validation/validation.md index 6641b48..a1cee50 100644 --- a/validation/validation.md +++ b/validation/validation.md @@ -1,9 +1,13 @@ -# Validation of the random-vibration FDS fix (v0.2.0 → v0.3.0) +# Validation of the random-vibration FDS fix This documents the correction of a factor-of-2 error in the random Fatigue Damage Spectrum and its validation against Christian Lalanne, *Mechanical Vibration and Shock Analysis*, 2nd ed., Wiley-ISTE (2009). +**Affected releases:** the error is present in the published releases **0.2.0 and +0.3.0** (the 0.3.0 release did not include this fix). The correction is intended for +the next release (0.3.1 / 0.4.0). + ## The correction Two independent factors of 2 made the **random FDS 2× too high** (they sit in @@ -65,6 +69,43 @@ Package: damage ratio **15.34** (book: 16 = 2^(b/2)); narrow-band contribution **6.52%** (book: ~6.25%). The small gap is the genuine finite-band correction (the package computes the exact integral; 16 is the idealized asymptote). +### 4. Independent verification (convention-free ground truth + adversarial review) + +The fix was cross-checked by four mutually independent methods, none sharing the +others' reasoning, plus the book. All converge on the corrected formulas. + +**Direct rainflow ground truth.** A Gaussian white-noise signal is passed through an +SDOF oscillator (computed independently of the package), and fatigue damage is taken +directly as `Σ count·(amplitude)^k` over rainflow-counted full cycles — the +definition of Miner–Basquin damage, with no spectral formula involved. Two separate +implementations (one using `scipy.signal.lsim`, one using an impulse-invariant IIR +filter, 5 seeds, T = 2000 s) give, to machine precision: + +| quantity | corrected (`Σ count·(range/2)^k`) | pre-fix (`Σ count·2·(range/2)^k`) | +|----------|-----------------------------------|-----------------------------------| +| rainflow damage ÷ direct ground truth | **1.0000** | **2.0000** | + +**Cycle rate, measured.** The mean upward zero-crossing rate of the SDOF response +was measured directly: **1.000·f0** (e.g. 100.0 / 200.0 / 400.0 Hz for f0 = 100 / +200 / 400). `n0 = (1/2π)·(ż_rms/z_rms)` reproduces it (≈ 0.998·f0); the pre-fix +`n0 = (1/π)·(ż_rms/z_rms)` gives ≈ 1.996·f0 — twice the true cycle rate. + +**Closed-form FDS.** Against the same ground truth, the corrected closed form lands +at 1.03–1.07× (the residual is the known narrow-band Rayleigh approximation, larger +at high `k`); the pre-fix form lands at ~2.1×. + +**Adversarial review.** An independent reviewer was instructed to build the strongest +possible case that the *pre-fix* formulas were correct. The defense (Lalanne's +half-cycle accounting, Coffin–Manson 2N reversals, the explicit `×2` in eq [4.6]) +was found citable but internally inconsistent, and collapsed against the +convention-free measurements above. + +**Root cause.** Both factors of 2 are genuine Lalanne *half-cycle* artifacts — the +`×2` in eq [4.6] and the `1/π` (= 2·f0 reversals) rate — that were applied to +**full-cycle** rainflow counts and to the **upward-crossing** rate, without the +compensating `1/2`. A half-cycle-vs-full-cycle bookkeeping slip; the factor is real +in the book, but double-counts once paired with full-cycle quantities. + ## Regression tests `tests/test_data.py` golden arrays were regenerated. The change was verified to be From a4b6f20b76cc8c9cbc0b7a3fa484722f0e137614 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Ja=C5=A1a?= <78748619+jasasonc@users.noreply.github.com> Date: Wed, 24 Jun 2026 12:32:12 +0200 Subject: [PATCH 4/4] docs: add literature references throughout the code (closes #3) Add in-code citations to the underlying theory so the references are not confined to the README. - spectrum.py / tools.py: add a module-level References bibliography (Lalanne Mechanical Vibration and Shock Analysis, Vol. 1/3/4/5, 2009, + Thomson) with consistent [LalanneN]/[Thomson] keys. - Spectrum class docstring maps each signal type to its governing equation. - Per-method/inline references with exact equation numbers for the sine, sine-sweep, random-PSD and random-time ERS/FDS formulae, the segmented-PSD response RMS (eq [8.86]), the I0/I2/I4 closed forms (App. A6), and the damped SDOF impulse response. Also corrects several pre-existing inaccurate citations: - sine-sweep ERS comments cited Vol.1 "[1.21]/[1.22]/[1.23], p.31-32"; the correct source is Vol.5 eqs [1.31]/[1.32]/[1.33], p.16 (branch tags were also swapped). - random-PSD FDS cited "Vol. 0 ... (A1-93)"; replaced with Vol.4 eq [4.41] = Vol.5 eqs [4.7]-[4.8]. - response_relative_displacement cited Thomson eq (4.2-5), the undamped kernel; clarified that the code implements the damped impulse response. Documentation only; no behaviour change. Full test suite passes. --- FatigueDS/spectrum.py | 143 ++++++++++++++++++++++++++++++++---------- FatigueDS/tools.py | 87 ++++++++++++++++--------- 2 files changed, 168 insertions(+), 62 deletions(-) diff --git a/FatigueDS/spectrum.py b/FatigueDS/spectrum.py index 0619fcb..9e9ca2b 100644 --- a/FatigueDS/spectrum.py +++ b/FatigueDS/spectrum.py @@ -1,3 +1,27 @@ +"""Extreme Response Spectrum (ERS) and Fatigue Damage Spectrum (FDS) of sine, +sine-sweep and random (PSD or time-history) signals. + +The theory implemented here follows C. Lalanne, *Mechanical Vibration and Shock +Analysis*, 2nd ed., ISTE Ltd / John Wiley & Sons, 2009 -- a five-volume set. The +equation and page numbers given in the docstrings and inline comments below refer +to the following volumes: + +References +---------- +.. [Lalanne1] C. Lalanne, *Mechanical Vibration and Shock Analysis, Vol. 1: + Sinusoidal Vibration*, 2nd ed., ISTE/Wiley, 2009. +.. [Lalanne3] C. Lalanne, *Mechanical Vibration and Shock Analysis, Vol. 3: + Random Vibration*, 2nd ed., ISTE/Wiley, 2009. +.. [Lalanne4] C. Lalanne, *Mechanical Vibration and Shock Analysis, Vol. 4: + Fatigue Damage*, 2nd ed., ISTE/Wiley, 2009. +.. [Lalanne5] C. Lalanne, *Mechanical Vibration and Shock Analysis, Vol. 5: + Specification Development*, 2nd ed., ISTE/Wiley, 2009. + +Notation note: the package renames Lalanne's material parameters ``b, K, C`` +(S-N slope, stress/relative-displacement proportionality, S-N constant) to +``k, p, C`` for consistency with the FLife package. +""" + import numpy as np import matplotlib.pyplot as plt import scipy.integrate @@ -8,6 +32,22 @@ from . import tools class Spectrum: + """Compute the ERS and FDS of a vibration signal for a range of natural + frequencies ``f0``. + + Workflow: instantiate with a frequency range and damping, set a load with one + of the ``set_*_load`` methods, then call :meth:`get_ers` and/or :meth:`get_fds`. + + The supported signal types map to the following theory (see the module-level + bibliography for the volume references): + + - sine: ERS [Lalanne5]_ eq. [1.7]; FDS [Lalanne5]_ eq. [3.19]. + - sine-sweep: ERS [Lalanne5]_ eqs. [1.31]-[1.33]; FDS [Lalanne5]_ eq. [3.65]. + - random (PSD): response RMS [Lalanne3]_ eq. [8.86]; ERS peak factor + [Lalanne3]_ ch. 7; FDS [Lalanne4]_ eq. [4.41] / [Lalanne5]_ eqs. [4.7]-[4.8]. + - random (time history): ERS from the SDOF response peak; FDS from rainflow + counting + Basquin-Miner [Lalanne4]_ eq. [4.6]. + """ def __init__(self, freq_data=(10, 2000, 5), damp=None, Q=10): """ @@ -221,18 +261,20 @@ def get_fds(self, k, C=1, p=1): before FDS calculation, because the FDS theory is based on SI base units. NOTE: - Naming of material parameters slightly differs from the notation in literature by Lalanne [1] (``b,C,K`` -> ``k,C,p``). + Naming of material parameters slightly differs from the notation in literature by Lalanne (``b,C,K`` -> ``k,C,p``). This is done due to the consistency with the established package in this ecosystem (FLife ). Alternative material parameters ------------------------------- - If you have parameters ``b`` and ``sigma_f`` from equation - ``sigma_a = sigma_f * (2*N)**b`` you can convert them to ``k`` and ``C`` using + If you have parameters ``b`` and ``sigma_f`` from equation + ``sigma_a = sigma_f * (2*N)**b`` you can convert them to ``k`` and ``C`` using the `tools.material_parameters_convert` function. References ---------- - 1. C. Lalanne, Mechanical Vibration and Shock: Specification development, London, England: ISTE Ltd and John Wiley & Sons, 2009 + The S-N (Basquin) law ``N s**k = C`` is [Lalanne4]_ eq. [1.13]. The per-signal + FDS formulae implemented by the internal ``_get_*_ers_fds`` methods are cited + on the relevant lines; see the module-level bibliography for the volumes. :param k: S-N curve slope from Basquin equation :param C: material constant from Basquin equation (default: C=1) @@ -317,6 +359,11 @@ def plot_fds(self, new_figure=True, grid=True, *args, **kwargs): def _get_sine_ers_fds(self, output=None): """ Internal function for calculating ERS and FDS of a sine signal. + + ERS: [Lalanne5]_ p.7, eq. [1.7] (general excitation type, R = E_m*Omega**a / |H|). + FDS: [Lalanne5]_ p.98, eq. [3.19] (single-frequency sinusoidal damage). + ``a`` is the excitation-type exponent (0 = acc, 1 = vel, 2 = disp) and + ``h = f/f0``. """ omega_0i = 2 * np.pi * self.f0_range @@ -324,8 +371,9 @@ def _get_sine_ers_fds(self, output=None): # Getting the ERS with self.get_ers() if output == 'ERS': + # [Lalanne5] p.7 eq [1.7]: steady-state SDOF response, R = omega_0**2 * z_m R_i = -self.amp * (omega_0i)**self.a / (np.sqrt((1 - (self.sine_freq / self.f0_range)**2)**2 + (self.sine_freq / (self.Q * self.f0_range))**2)) - return np.abs(R_i) + return np.abs(R_i) # Getting the FDS with self.get_fds() elif output == 'FDS': @@ -334,14 +382,25 @@ def _get_sine_ers_fds(self, output=None): raise ValueError('Missing parameter `t_total`.') h = self.sine_freq / self.f0_range + # [Lalanne5] p.98 eq [3.19]: D = K**b/C * f0*T * E_m**b * omega_0**(b(a-2)) * h**(ab+1) / [(1-h**2)**2 + (h/Q)**2]**(b/2) D_i = self.p**self.k / self.C * self.f0_range * self.t_total * self.amp**self.k * omega_0i**(self.k * (self.a - 2)) * h**(self.a * self.k + 1) / ((1 - h**2)**2 + (h / self.Q)**2)**(self.k / 2) return D_i def _get_sine_sweep_ers_fds(self, output=None): """ Internal function for calculating ERS and FDS of a sine sweep signal. + + ERS: piecewise peak response, [Lalanne5]_ p.16, eqs. [1.31] (resonance, + f1 <= f0 <= f2), [1.32] (f0 < f1) and [1.33] (f0 > f2). + FDS: swept-sine damage master integral, [Lalanne5]_ p.120, eq. [3.65], + with the dwell-weighting function M(h) from [Lalanne5]_ Table 3.2 + (linear: h**2/(h2-h1); logarithmic: h/ln(h2/h1)). Sweep-time and the + dh change of variable follow [Lalanne1]_ ch. 8 / [Lalanne5]_ p.103-114 + (linear rate in Hz/min, eq. [3.39]; logarithmic rate in oct./min). + Assumes the sweep is slow enough to reach steady state at each frequency + ([Lalanne5]_ p.103). """ - + R_i_all = np.zeros((len(self.f0_range), len(self.const_amp))) fds = np.zeros(len(self.f0_range)) ers = np.zeros(len(self.f0_range)) @@ -360,21 +419,23 @@ def _get_sine_sweep_ers_fds(self, output=None): if self.sweep_type is None: raise ValueError("You need to provide either ['linear','lin'] or ['logarithmic','log'] sweep_type.") elif self.sweep_type in ['lin', 'linear']: - tb = (self.const_f_range[-1] - self.const_f_range[0]) / self.sweep_rate * 60 # sinusoidal sweep time [s] -> from [Hz/min] - dh = (f2 - f1) * self.dt / (self.f0_range[i] * tb) + tb = (self.const_f_range[-1] - self.const_f_range[0]) / self.sweep_rate * 60 # sinusoidal sweep time [s] -> from [Hz/min] ([Lalanne1] ch.8) + dh = (f2 - f1) * self.dt / (self.f0_range[i] * tb) # [Lalanne5] p.104 eq [3.39] h = np.arange(h1, h2, dh) - M_h = h**2 / (h2 - h1) + M_h = h**2 / (h2 - h1) # linear dwell weighting M(h), [Lalanne5] Table 3.2 elif self.sweep_type in ['log', 'logarithmic']: - tb = 60 * np.log(self.const_f_range[-1] / self.const_f_range[0]) / (self.sweep_rate * np.log(2)) # logarithmic sweep time [s] -> from [oct./min] + tb = 60 * np.log(self.const_f_range[-1] / self.const_f_range[0]) / (self.sweep_rate * np.log(2)) # logarithmic sweep time [s] -> from [oct./min] ([Lalanne1] ch.8) t = np.arange(0, tb, self.dt) - T1 = tb / np.log(h2 / h1) + T1 = tb / np.log(h2 / h1) # log-sweep time constant, f(t)=f1*exp(t/T1) ([Lalanne5] p.114) f_t = f1 * np.exp(t / T1) dh = f1 / (T1 * self.f0_range[i]) * np.exp(t / T1) * self.dt h = f_t / self.f0_range[i] - M_h = h / (np.log(h2 / h1)) + M_h = h / (np.log(h2 / h1)) # exponential dwell weighting M(h), [Lalanne5] Table 3.2 else: raise ValueError(f"Invalid method `method`='{self.sweep_type}'. Supported sweep types: 'lin' and 'log'.") - + + # swept-sine FDS master integral, [Lalanne5] p.120 eq [3.65]: + # D = K**b/C * f0*tb * E_m**b * omega_0**(b(a-2)) * integral( M(h) h**(ab-1) / [(1-h**2)**2+(h/Q)**2]**(b/2) dh ) const = self.p**self.k / self.C * self.f0_range[i] * tb * amp**self.k * omega_0i**(self.k * (self.a - 2)) integral = scipy.integrate.trapezoid(M_h * h**(self.a * self.k - 1) / ((1 - h**2)**2 + (h / self.Q)**2)**(self.k / 2), x=h) fds[i] += const * integral @@ -382,12 +443,12 @@ def _get_sine_sweep_ers_fds(self, output=None): elif output == 'ERS': if self.f0_range[i] <= f1: Omega_1 = 2 * np.pi * f1 - R_i = Omega_1**self.a * amp / (np.sqrt((1 - h1**2)**2 + (h1 / self.Q)**2)) # page 32/501 eq. [1.22] + R_i = Omega_1**self.a * amp / (np.sqrt((1 - h1**2)**2 + (h1 / self.Q)**2)) # f0 < f1: [Lalanne5] p.16 eq [1.32] elif self.f0_range[i] >= f2: Omega_2 = 2 * np.pi * f2 - R_i = Omega_2**self.a * amp / (np.sqrt((1 - h2**2)**2 + (h2 / self.Q)**2)) # page 32/501 eq. [1.23] + R_i = Omega_2**self.a * amp / (np.sqrt((1 - h2**2)**2 + (h2 / self.Q)**2)) # f0 > f2: [Lalanne5] p.16 eq [1.33] else: - R_i = omega_0i**self.a * amp * self.Q # page 31/501 eq. [1.21] + R_i = omega_0i**self.a * amp * self.Q # f1 <= f0 <= f2 (resonance): [Lalanne5] p.16 eq [1.31] R_i_all[i, n] = R_i ers[i] = max(R_i_all[i, :]) @@ -400,16 +461,25 @@ def _get_sine_sweep_ers_fds(self, output=None): def _get_random_psd_ers_fds(self, output=None): """ Internal function for calculating ERS and FDS of a random signal in frequency domain. + + Response RMS (relative displacement / velocity / acceleration) is obtained from + the segmented-PSD closed form [Lalanne3]_ p.395, eq. [8.86] (the per-segment + I0/I2/I4 integrals are evaluated in ``tools.integrals_b``). The mean upward + zero-crossing rate ``n0`` is Rice's formula [Lalanne3]_ p.264, eq. [5.76]. + ERS uses the extreme-peak factor sqrt(2*ln(n0*T)) ([Lalanne3]_ ch. 7, + eq. [7.34]; ERS definition [Lalanne5]_ ch. 2). FDS uses the narrow-band + Rayleigh closed form [Lalanne4]_ p.144, eq. [4.41] = [Lalanne5]_ p.117, + eqs. [4.7]-[4.8]. """ - + fds = np.zeros(len(self.f0_range)) - ers = np.zeros(len(self.f0_range)) - - # constants - C0 = np.pi / (4 * self.damp) - C_disp = C0 * 1 / ((2 * np.pi)**4 * self.f0_range**3) - C_vel = C0 * 1 / ((2 * np.pi)**2 * self.f0_range) - C_acc = C0 * self.f0_range + ers = np.zeros(len(self.f0_range)) + + # constants -- f0-power prefactors of the segmented-PSD response RMS, [Lalanne3] eq [8.86] + C0 = np.pi / (4 * self.damp) # pi/(4*xi) prefactor common to all three RMS sums + C_disp = C0 * 1 / ((2 * np.pi)**4 * self.f0_range**3) # rel. displacement RMS**2 (uses I0) + C_vel = C0 * 1 / ((2 * np.pi)**2 * self.f0_range) # rel. velocity RMS**2 (uses I2) + C_acc = C0 * self.f0_range # rel. acceleration RMS**2 (uses I4) # rms sums z_rms_2 = tools.rms_sum(f_0=self.f0_range, psd_freq=self.psd_freq, psd_data=self.psd_data, damp=self.damp, motion='rel_disp') * C_disp @@ -424,28 +494,35 @@ def _get_random_psd_ers_fds(self, output=None): # ERS calculation if output == 'ERS': - n0 = 1 / (2 * np.pi) * dz_rms / z_rms # n0+ = (1/2pi)*sqrt(M2/M0), Lalanne Vol.3 eq [5.76]; for a Q=10 narrow-band response n0+ = f0 (Vol.5 p.46, Ex 4.4) + n0 = 1 / (2 * np.pi) * dz_rms / z_rms # n0+ = (1/2pi)*sqrt(M2/M0), [Lalanne3] eq [5.76]; for a Q=10 narrow-band response n0+ = f0 ([Lalanne5] p.46, Ex 4.4) + # ERS = omega_0**2 * z_rms * peak factor, with peak factor sqrt(2*ln(n0*T)) ([Lalanne3] ch.7 eq [7.34]) ers = (2 * np.pi * self.f0_range)**2 * z_rms * np.sqrt(2 * np.log(n0 * self.T)) return ers - - # FDS calculation (damage according to Vol. 0, page 89/198, equation (A1-93)) + + # FDS calculation: narrow-band Rayleigh closed form, [Lalanne4] p.144 eq [4.41] = [Lalanne5] p.117 eqs [4.7]-[4.8] elif output == 'FDS': z_rms *= self.unit_scale dz_rms *= self.unit_scale - n0 = 1 / (2 * np.pi) * dz_rms / z_rms # n0+ = (1/2pi)*sqrt(M2/M0), Lalanne Vol.3 eq [5.76]; for a Q=10 narrow-band response n0+ = f0 (Vol.5 p.46, Ex 4.4) + n0 = 1 / (2 * np.pi) * dz_rms / z_rms # n0+ = (1/2pi)*sqrt(M2/M0), [Lalanne3] eq [5.76]; for a Q=10 narrow-band response n0+ = f0 ([Lalanne5] p.46, Ex 4.4) + # D = K**b/C * n0*T * (sqrt(2)*z_rms)**b * Gamma(1 + b/2) fds = self.p**self.k / self.C * n0 * self.T * (z_rms * np.sqrt(2))**self.k * gamma(1 + self.k / 2) return fds def _get_random_time_ers_fds(self, output=None): """ - Internal function for calculating ERS and FDS of a sine random signal in time domain. + Internal function for calculating ERS and FDS of a random signal in the time domain. + + The SDOF relative-displacement response z(t) is obtained by convolution + (``tools.response_relative_displacement``). ERS is the peak response + ``max(z) * omega_0**2``. FDS sums Basquin-Miner damage over rainflow-counted + cycles, [Lalanne4]_ p.132, eq. [4.6] (full-cycle form, see below). """ if output == 'ERS': ers = np.zeros(len(self.f0_range)) - for i in tqdm(range(len(self.f0_range))): + for i in tqdm(range(len(self.f0_range))): z = tools.response_relative_displacement(self.time_data, self.dt, f_0=self.f0_range[i], damp=self.damp) - R_i = np.max(z) * (2 * np.pi * self.f0_range[i])**2 + R_i = np.max(z) * (2 * np.pi * self.f0_range[i])**2 # ERS = peak rel. displacement * omega_0**2 ers[i] = R_i return ers @@ -457,7 +534,9 @@ def _get_random_time_ers_fds(self, output=None): rf = rainflow.count_cycles(z) rf = np.asarray(rf) - cyc_sum = np.sum(rf[:,1] * (rf[:,0] / 2)**self.k) # rainflow count = full cycles, range/2 = amplitude; full-cycle Miner-Basquin damage (Lalanne Vol.4 eq [4.6], full-cycle form) + # rainflow count = full cycles, range/2 = amplitude; full-cycle Basquin-Miner damage. + # [Lalanne4] eq [4.6] is D = K**b/(2C) * sum(n_half * z_a**b); with full cycles n_full = n_half/2 this is D = K**b/C * sum(n_full * z_a**b). + cyc_sum = np.sum(rf[:,1] * (rf[:,0] / 2)**self.k) D_i = self.p**self.k / (self.C) * cyc_sum fds[i] = D_i return fds diff --git a/FatigueDS/tools.py b/FatigueDS/tools.py index 57dff2b..740ae27 100644 --- a/FatigueDS/tools.py +++ b/FatigueDS/tools.py @@ -1,3 +1,19 @@ +"""Helper functions for the FatigueDS :class:`~FatigueDS.spectrum.Spectrum` class: +frequency-range handling, the SDOF response RMS over a segmented PSD, the closed-form +spectral integrals, the time-domain SDOF response, and material-parameter conversion. + +References +---------- +.. [Lalanne3] C. Lalanne, *Mechanical Vibration and Shock Analysis, Vol. 3: + Random Vibration*, 2nd ed., ISTE/Wiley, 2009. +.. [Lalanne4] C. Lalanne, *Mechanical Vibration and Shock Analysis, Vol. 4: + Fatigue Damage*, 2nd ed., ISTE/Wiley, 2009. +.. [Lalanne5] C. Lalanne, *Mechanical Vibration and Shock Analysis, Vol. 5: + Specification Development*, 2nd ed., ISTE/Wiley, 2009. +.. [Thomson] W. T. Thomson, *Theory of Vibration with Applications*, 2nd ed., + Prentice-Hall, 1981. +""" + import numpy as np from scipy import signal from FLife.tools import basquin_to_sn @@ -40,8 +56,14 @@ def get_freq_range(self, freq_data): def rms_sum(f_0, psd_freq, psd_data, damp, motion='rel_disp'): """ - This function calculates the response RMS (either relative displacement, velocity or acceleration) for a given - natural frequency and damping ratio. + This function calculates the response RMS (either relative displacement, velocity or acceleration) for a given + natural frequency and damping ratio. + + The excitation PSD is treated as ``n`` horizontal straight-line segments and the + response mean square is accumulated segment by segment using the closed-form + integrals I0/I2/I4 (:func:`integrals_b`). This is the summation kernel of + [Lalanne3]_ p.395, eq. [8.86]; the ``pi/(4*xi)`` and ``f0``-power prefactors are + applied by the caller in ``spectrum.py``. :param f_0: system natural frequency [Hz] :param psd_freq: PSD frequency range [Hz] @@ -67,11 +89,8 @@ def rms_sum(f_0, psd_freq, psd_data, damp, motion='rel_disp'): h1 = f1[j] / f_0 h2 = f2[j] / f_0 - # Case where the excitation is defined by PSD comprising "n" straight line segments (Vol.3, equation [8.86]) - - - - + # Per-segment contribution G_j * [I_b(h2) - I_b(h1)], the summation kernel of + # [Lalanne3] p.395 eq [8.86] (rel_disp uses I0, rel_vel uses I2, rel_acc uses I4). if motion == 'rel_disp': z_rms = psd_data[j] * (integrals_b(h=h2, b=0, damp=damp) - integrals_b(h=h1, b=0, damp=damp)) rms_sum += z_rms @@ -90,42 +109,43 @@ def rms_sum(f_0, psd_freq, psd_data, damp, motion='rel_disp'): def integrals_b(h, b, damp): """ - This function calculates integrals I_b described in [3] and [4]. See equations (A1-74), (A1-75), (A1-76) in [3] - or [A6.20], [A6.22], [A6.24] in [4] or [8.52], [8.53], [8.54] [4]. + Closed-form evaluation of the dimensionless spectral integrals I0, I2 and I4 used in + the segmented-PSD response RMS (:func:`rms_sum`, [Lalanne3]_ eq. [8.86]). + + The closed forms are [Lalanne3]_ Appendix A6, p.544: I0 = eq. [A6.20], I2 = eq. + [A6.22], I4 = eq. [A6.24]. They are identical to eqs. (A1-74), (A1-75), (A1-76) in + [Lalanne5]_. ``alpha = 2*sqrt(1-xi**2)`` and ``beta = 2*(1-2*xi**2)`` ([Lalanne3]_ + eq. [8.37]). - Literature: - [3] Mechanical Environment Test Specification Development Method - Christian LALANNE - [4] Christian Lalanne(auth.) Random Vibration Mechanical Vibration and Shock Analysis, Volume 3, Second Edition - :param h: frequency ratio (frequency vs natural frequency) [/] - :param b: exponent b [/] + :param b: exponent b [/] (0, 2 or 4) :param damp: damping ratio [/] :return: I_b integral value """ - - # constants - alpha = 2 * np.sqrt(1 - damp**2) + + # constants ([Lalanne3] eq [8.37]) + alpha = 2 * np.sqrt(1 - damp**2) beta = 2 * (1 - 2 * damp**2) - + C0 = damp / (np.pi * alpha) C1 = (h**2 + alpha * h + 1)/(h**2 - alpha * h + 1) C2 = (2 * h + alpha) / (2 * damp) C3 = (2 * h - alpha) / (2 * damp) C4 = 4 * damp / np.pi C5 = np.arctan(C2) + np.arctan(C3) - + # integrals if b == 0: - Ib = C0 * np.log(C1) + 1 / np.pi * C5 # 84/198 eq. (A1-74) and 560/610 eq. [A6.20] - + Ib = C0 * np.log(C1) + 1 / np.pi * C5 # I0: [Lalanne3] p.544 eq [A6.20] (= [Lalanne5] eq (A1-74)) + elif b == 2: - Ib = C0*np.log(1 / C1) + 1 / np.pi * C5 # 84/198 eq. (A1-75) and 560/610 eq. [A6.22] - + Ib = C0*np.log(1 / C1) + 1 / np.pi * C5 # I2: [Lalanne3] p.544 eq [A6.22] (= [Lalanne5] eq (A1-75)); log argument reciprocated vs I0 + elif b == 4: - I0 = C0 * np.log(C1) + 1 / np.pi * C5 + I0 = C0 * np.log(C1) + 1 / np.pi * C5 I2 = -C0 * np.log(C1) + 1 / np.pi * C5 - Ib = C4 * h + beta * I2 - I0 # 84/198 eq. (A1-76) and 560/610 eq. [A6.24] + Ib = C4 * h + beta * I2 - I0 # I4: [Lalanne3] p.544 eq [A6.24] (= [Lalanne5] eq (A1-76)) else: raise ValueError(f"Invalid exponent ``b``='{b}'. Supported exponents: 0, 2 and 4.") @@ -135,12 +155,18 @@ def integrals_b(h, b, damp): def response_relative_displacement(time_data, dt, f_0, damp): """ - Returns relative response displacement of a linear SDOF system by performing the convolution of a signal and impulse response - function, defined in [1]. The function is used in calculation of the extreme response spectrum (ERS) of a random time signal. + Returns the relative response displacement of a linear SDOF system by convolving the + base-excitation signal with the system's unit-impulse response (Duhamel's integral). + Used to obtain the SDOF response for the time-domain ERS/FDS of a random signal. + + The impulse response used here is that of the *damped* SDOF system, + ``h(t) = -1/omega_d * exp(-xi*omega_0*t) * sin(omega_d*t)`` with + ``omega_d = omega_0*sqrt(1-xi**2)`` -- the damped generalisation of the impulse + response / convolution (Duhamel) integral in [Thomson]_ ch. 4 (impulse response and + convolution, eq. (4.2-5) gives the undamped case). The leading sign and the + ``1/omega_d`` factor give the relative displacement of the mass w.r.t. its base for a + unit base acceleration. - Literature: - [1] WILLIAM T. THOMSON, Theory of vibration with applications -> see page 111/512 equation (4.2-5) - :param time_data: signal time data [m/s^2] :param dt: time step [s] :param f_0: system natural frequency [Hz] @@ -184,6 +210,7 @@ def material_parameters_convert(sigma_f, b, range = False): """ Converts Basquin equation parameters ``sigma_f`` and ``b`` to fatigue life parameters ``C`` and ``k``, using a function from FLife package. Basic form of Basquin equation is used here: ``sigma_a = sigma_f* (2*N)**b``. The function converts to parameters from equation ``N * s**k = C`` + (the S-N / Basquin law, [Lalanne4]_ p.22, eqs. [1.13]-[1.15]). :param sigma_f: Fatigue strength coefficient [MPa**k].