Fix/random fds factor 2#6
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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.
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 table-vs-figure inconsistency in Example 4.2
…dent 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.
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.
docs: add literature references throughout the code (closes #3)
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