def test_srsmap():
from pyyeti import ytools, dsp
sig, ts, fs = ytools.gensweep(10, 1, 50, 4)
sr = 1 / ts[1]
frq = np.arange(1.0, 50.1)
Q = 20
mp, t, f = srs.srsmap(2, 0.5, sig, sr, frq, Q, 0.02, eqsine=1)
assert np.all(f == frq)
dt = 1.0
segments = int(np.floor(ts[-1] / dt)) - 1
assert len(t) == segments
t_should_be = np.arange(segments) * dt + dt
assert np.allclose(t, t_should_be)
# test segment 63:
seg = 63
pv = np.logical_and(ts >= t[seg] - dt, ts < t[seg] + dt)
sh = srs.srs(dsp.windowends(sig[pv], 0.02), sr, frq, Q, eqsine=1)
assert np.allclose(sh, mp[:, seg])
def test_windowends():
s = dsp.windowends(np.ones(8), 4)
assert np.allclose([0.0, 0.25, 0.75, 1.0, 1.0, 1.0, 1.0, 1.0], s)
s = dsp.windowends(np.ones(8), 0.7, ends="back")
assert np.allclose([
1.0, 1.0, 1.0, 1.0, 0.85355339059327373, 0.5, 0.14644660940672621, 0.0
], s)
s = dsp.windowends(np.ones(8), 0.5, ends="both")
assert np.allclose([0.0, 0.25, 0.75, 1.0, 1.0, 0.75, 0.25, 0.0], s)
s = dsp.windowends(np.ones(8), 0.5, ends="none")
assert np.allclose([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], s)
s = dsp.windowends(np.ones(8), 3)
assert np.allclose([0.0, 0.5, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], s)
s = dsp.windowends(np.ones(8), 2)
assert np.allclose([0.0, 0.5, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], s)
def fdepsd(
sig,
sr,
freq,
Q,
resp="absacce",
hpfilter=5.0,
winends="auto",
nbins=300,
T0=60.0,
rolloff="lanczos",
ppc=12,
parallel="auto",
maxcpu=14,
verbose=False,
):
r"""
Compute a fatigue damage equivalent PSD from a signal.
Parameters
----------
sig : 1d array_like
Base acceleration signal.
sr : scalar
Sample rate.
freq : array_like
Frequency vector in Hz. This defines the single DOF (SDOF)
systems to use.
Q : scalar > 0.5
Dynamic amplification factor :math:`Q = 1/(2\zeta)` where
:math:`\zeta` is the fraction of critical damping.
resp : string; optional
The type of response to base the damage calculations on:
========= =======================================
`resp` Damage is based on
========= =======================================
'absacce' absolute acceleration [#fde1]_
'pvelo' pseudo velocity [#fde2]_
========= =======================================
hpfilter : scalar or None; optional
High pass filter frequency; if None, no filtering is done.
If filtering is done, it is a 3rd order butterworth via
:func:`scipy.signal.lfilter`.
winends : None or 'auto' or dictionary; optional
If None, :func:`pyyeti.dsp.windowends` is not called. If
'auto', :func:`pyyeti.dsp.windowends` is called to apply a
0.25 second window or a 50 point window (whichever is smaller)
to the front. Otherwise, `winends` must be a dictionary of
arguments that will be passed to :func:`pyyeti.dsp.windowends`
(not including `signal`).
nbins : integer; optional
The number of amplitude levels at which to count cycles
T0 : scalar; optional
Specifies test duration in seconds
rolloff : string or function or None; optional
Indicate which method to use to account for the SRS roll off
when the minimum `ppc` value is not met. Either 'fft' or
'lanczos' seem the best. If a string, it must be one of these
values:
=========== ==========================================
`rolloff` Notes
=========== ==========================================
'fft' Use FFT to upsample data as needed. See
:func:`scipy.signal.resample`.
'lanczos' Use Lanczos resampling to upsample as
needed. See :func:`pyyeti.dsp.resample`.
'prefilter' Apply a high freq. gain filter to account
for the SRS roll-off. See
:func:`pyyeti.srs.preroll` for more
information. This option ignores `ppc`.
'linear' Use linear interpolation to increase the
points per cycle (this is not recommended;
method; it's only here as a test case).
'none' Don't do anything to enforce the minimum
`ppc`. Note error bounds listed above.
None Same as 'none'.
=========== ==========================================
If a function, the call signature is:
``sig_new, sr_new = rollfunc(sig, sr, ppc, frq)``. Here, `sig`
is 1d, len(time). The last three inputs are scalars. For
example, the 'fft' function is (trimmed of documentation)::
def fftroll(sig, sr, ppc, frq):
N = sig.shape[0]
if N > 1:
curppc = sr/frq
factor = int(np.ceil(ppc/curppc))
sig = signal.resample(sig, factor*N, axis=0)
sr *= factor
return sig, sr
ppc : scalar; optional
Specifies the minimum points per cycle for SRS calculations.
See also `rolloff`.
====== ==================================
`ppc` Maximum error at highest frequency
====== ==================================
3 81.61%
4 48.23%
5 31.58%
10 8.14% (minimum recommended `ppc`)
12 5.67%
15 3.64%
20 2.05%
25 1.31%
50 0.33%
====== ==================================
parallel : string; optional
Controls the parallelization of the calculations:
========== ============================================
`parallel` Notes
========== ============================================
'auto' Routine determines whether or not to run
parallel.
'no' Do not use parallel processing.
'yes' Use parallel processing. Beware, depending
on the particular problem, using parallel
processing can be slower than not using it.
On Windows, be sure the :func:`fdepsd` call
is contained within:
``if __name__ == "__main__":``
========== ============================================
maxcpu : integer or None; optional
Specifies maximum number of CPUs to use. If None, it is
internally set to 4/5 of available CPUs (as determined from
:func:`multiprocessing.cpu_count`).
verbose : bool; optional
If True, routine will print some status information.
Returns
-------
A SimpleNamespace with the members:
freq : 1d ndarray
Same as input `freq`.
psd : pandas DataFrame; ``len(freq) x 5``
The amplitude and damage based PSDs. The index is `freq` and
the five columns are: [G1, G2, G4, G8, G12]
=========== ===============================================
Name Description
=========== ===============================================
G1 The "G1" PSD (Mile's or similar equivalent from
SRS); uses the maximum cycle amplitude instead
of the raw SRS peak for each frequency. G1 is
not a damage-based PSD.
G2 The "G2" PSD of reference [#fde1]_; G2 >= G1 by
bounding lower amplitude counts down to 1/3 of
the maximum cycle amplitude. G2 is not a
damage-based PSD.
G4, G8, G12 The damage-based PSDs with fatigue exponents of
4, 8, and 12
=========== ===============================================
peakamp : pandas DataFrame; ``len(freq) x 5``
The peak response of SDOFs (single DOF oscillators) using each
PSD as a base input. The index and the five columns are the
same as for `psd`. The peaks are computed from the Mile's
equation (or similar if using ``resp='pvelo'``). The peak
factor used is ``sqrt(2*log(f*T0))``. Note that the first
column is, by definition, the maximum cycle amplitude for each
SDOF from the rainflow count (G1 was calculated from
this). Typically, this should be very close to the raw SRS
peaks contained in the `srs` output but a little lower since
SRS just grabs peaks without consideration of the opposite
peak.
binamps : pandas DataFrame; ``len(freq) x nbins``
A DataFrame of linearly spaced amplitude values defining the
cycle counting bins. The index is `freq` and the columns are
integers 0 to ``nbins - 1``. The values in each row (for a
specific frequency SDOF), range from 0.0 up to
``peakamp.loc[freq, "G1"] * (nbins - 1) / nbins``. In other
words, each value is the left-side amplitude boundary for that
bin. The next column for this matrix would be ``peakamp.loc[:,
"G1"]``.
count : pandas DataFrame; ``len(freq) x nbins``
Summary matrix of the rainflow cycle counts. Size corresponds
with `binamps` and the count is cumulative; that is, the count
in each entry includes cycles at the `binamps` amplitude and
above. Therefore, first column has total cycles for the SDOF.
bincount : pandas DataFrame; ``len(freq) x nbins``
Non-cumulative version of `count`. In other words, the values
are the number of cycles in the bin, left-side inclusive. The
last bin includes the count of maximum amplitude cycles.
di_sig : pandas DataFrame; ``len(freq) x 3``
Damage indicators computed from SDOF responses to the `sig`
signal. Index is `freq` and columns are ['b=4', 'b=8',
'b=12']. The value for each frequency is the sum of the cycle
count for a bin times its amplitude to the b power. That is,
for the j-th frequency, the indicator is::
amps = binamps.loc[freq[j]]
counts = bincount.loc[freq[j]]
di = (amps ** b) @ counts # dot product of two vectors
Note that this definition is slightly different than equation
14 from [#fde1]_ (would have to divide by the frequency), but
the same as equation 10 of [#fde2]_ without the constant.
di_test_part : pandas DataFrame; ``len(freq) x 3``
Test damage indicator without including the variance factor
(see note). Same size as `di_sig`. Each value depends only on
the frequency, `T0`, and the fatigue exponent ``b``. The ratio
of a signal damage indicator to the corresponding partial test
damage indicator is equal to the variance of the single DOF
response to the test raised to the ``b / 2`` power::
var_test ** (b / 2) = di_sig / di_test_part
.. note::
If the variance vactor (`var_test`) were included, then
the test damage indicator would be the same as
`di_sig`. This relationship is the basis of determining
the amplitude of the test signal.
var_test : pandas DataFrame; ``len(freq) x 3``
The required SDOF test response variances (see `di_test_part`
description). Same size as `di_sig`. The amplitude of the G4,
G8, and G12 columns of `psd` are computed from Mile's equation
(or similar) and `var_test`.
sig : 1d ndarray
The version of the input `sig` that is fed into the fatique
damage algorithm. This would be after any filtering,
windowing, and upsampling.
sr : scalar
The sample rate of the output `sig`.
srs : pandas Series; length = ``len(freq)``
The raw SRS peaks version of the first column in `amp`. See
`amp`. Index is `freq`.
var : pandas Series; length = ``len(freq)``
Vector of the SDOF response variances. Index is `freq`.
parallel : string
Either 'yes' or 'no' depending on whether parallel processing
was used or not.
ncpu : integer
Specifies the number of CPUs used.
resp : string
Same as the input `resp`.
Notes
-----
Steps (see [#fde1]_, [#fde2]_):
1. Resample signal to higher rate if highest frequency would
have less than `ppc` points-per-cycle. Method of increasing
the sample rate is controlled by the `rolloff` input.
2. For each frequency:
a. Compute the SDOF base-drive response
b. Calculate `srs` and `var` outputs
c. Use :func:`pyyeti.cyclecount.findap` to find cycle peaks
d. Use :func:`pyyeti.cyclecount.rainflow` to count cycles
and amplitudes
e. Put counts into amplitude bins
3. Calculate `g1` based on cycle amplitudes from maximum
amplitude (step 2d) and Mile's (or similar) equation.
4. Calculate `g2` to bound `g1` & lower amplitude cycles with
high counts. Ignore amplitudes < ``Amax/3``.
5. Calculate damage indicators from data with b = 4, 8, 12
where b is the fatigue exponent.
6. By equating the theoretical damage from a `T0` second random
vibration test to the damage from the input signal (step 5),
solve for the required test response variances for b = 4, 8,
12.
7. Solve for `g4`, `g8`, `g12` from the results of step 6 using
the Mile's equation (or similar); equations are shown below.
No checks are done regarding the suitability of this method for
the input data. It is recommended to read the references [#fde1]_
[#fde2]_ and do those checks (such as plotting Count or Time
vs. Amp**2 and comparing to theoretical).
The Mile's equation (or similar) is used in this methodology to
relate acceleration PSDs to peak responses. If `resp` is
'absacce', it is the Mile's equation:
.. math::
\sigma_{absacce}(f) = \sqrt{\frac{\pi}{2} \cdot f \cdot Q
\cdot PSD(f)}
If `resp` is 'pvelo', the similar equation is:
.. math::
\sigma_{pvelo}(f) = \sqrt{\frac{Q \cdot PSD(f)}{8 \pi f}}
Those two equations assume a flat acceleration PSD. Therefore, it
is recommended to compare SDOF responses from flight data (SRS) to
SDOF VRS responses from the developed specification (see
:func:`pyyeti.srs.vrs` to compute the VRS response in the
absolute-acceleration case). This is to check for conservatism.
Instead of using 3 for peak factor (for 3-rms or 3-sigma), use
:math:`\sqrt{2 \ln(f \cdot T_0)}` for the peak factor (derived
below). Also, enveloping multiple specifications from multiple Q's
is worth considering.
Note that this analysis can be time consuming; the time is
proportional to the number of frequencies multiplied by the number
of time steps in the signal.
The derivation of the peak factor is as follows. For the special
case of narrow band noise where the instantaneous amplitudes
follow the Gaussian distribution, the resulting probability
density function for the peak amplitudes follow the Rayleigh
distribution [#fde3]_. The single DOF response to Gaussian input
is reasonably estimated as Gaussian narrow band. Let this response
have the standard deviation :math:`\sigma`. From the Rayleigh
distribution, the probability of a peak being greater than
:math:`A` is:
.. math::
Prob[peak > A] = e ^ {\frac{-A^2}{2 \sigma^2}}
To estimate the maximum peak for the response of a single DOF
system with frequency :math:`f`, find the amplitude that would be
expected to occur once within the allotted time
(:math:`T_0`). That is, set the product of the probability of a
cycle amplitude being greater than :math:`A` and the number of
cycles equal to 1.0, and then solve for :math:`A`.
The number of cycles of :math:`f` Hz is :math:`N = f \cdot T_0`.
Therefore:
.. math::
\begin{aligned}
Prob[peak > A] \cdot N &= 1.0
e ^ {\frac{-A^2}{2 \sigma^2}} f \cdot T_0 &= 1.0
\frac{-A^2}{2 \sigma^2} &= \ln(1.0) - \ln(f \cdot T_0)
\frac{A^2}{2 \sigma^2} &= \ln(f \cdot T_0)
A &= \sqrt{2 \ln(f \cdot T_0)} \sigma
\end{aligned}
.. note::
In addition to the example shown below, this routine is
demonstrated in the pyYeti :ref:`tutorial`:
:doc:`/tutorials/fatigue`. There is also a link to the source
Jupyter notebook at the top of the tutorial.
References
----------
.. [#fde1] "Analysis of Nonstationary Vibroacoustic Flight Data
Using a Damage-Potential Basis"; S. J. DiMaggio, B. H. Sako,
S. Rubin; Journal of Spacecraft and Rockets, Vol 40, No. 5,
September-October 2003.
.. [#fde2] "Implementing the Fatigue Damage Spectrum and Fatigue
Damage Equivalent Vibration Testing"; Scot I. McNeill; 79th
Shock and Vibration Symposium, October 26 – 30, 2008.
.. [#fde3] Bendat, Julius S., "Probability Functions for Random
Responses: Prediction of Peaks, Fatigue Damage, and
Catastrophic Failures", NASA Contractor Report 33 (NASA
CR-33), 1964.
See also
--------
:func:`scipy.signal.welch`, :func:`pyyeti.psd.psdmod`,
:func:`pyyeti.cyclecount.rainflow`, :func:`pyyeti.srs.srs`.
Examples
--------
Generate 60 second random signal to a pre-defined spec level,
compute the PSD several different ways and compare. Since it's 60
seconds, the damage-based PSDs should be fairly close to the input
spec level. The damage-based PSDs will be calculated with several
Qs and enveloped.
In this example, G2 envelopes G1, G4, G8, G12. This is not always
the case. For example, try TF=120; the damage-based curves go up
in order to fit equal damage in 60s.
One Count vs. Amp**2 plot is done for illustration.
.. plot::
:context: close-figs
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from pyyeti import psd, fdepsd
>>> import scipy.signal as signal
>>>
>>> TF = 60 # make a 60 second signal
>>> spec = np.array([[20, 1], [50, 1]])
>>> sig, sr, t = psd.psd2time(
... spec, ppc=10, fstart=20, fstop=50, df=1 / TF,
... winends=dict(portion=10), gettime=True)
>>>
>>> fig = plt.figure('Example', figsize=[9, 6])
>>> fig.clf()
>>> _ = plt.subplot(211)
>>> _ = plt.plot(t, sig)
>>> _ = plt.title(r'Input Signal - Specification Level = '
... '1.0 $g^{2}$/Hz')
>>> _ = plt.xlabel('Time (sec)')
>>> _ = plt.ylabel('Acceleration (g)')
>>> ax = plt.subplot(212)
>>> f, p = signal.welch(sig, sr, nperseg=sr)
>>> f2, p2 = psd.psdmod(sig, sr, nperseg=sr, timeslice=4,
... tsoverlap=0.5)
Calculate G1, G2, and the damage potential PSDs:
>>> psd_ = 0
>>> freq = np.arange(20., 50.1)
>>> for q in (10, 25, 50):
... fde = fdepsd.fdepsd(sig, sr, freq, q)
... psd_ = np.fmax(psd_, fde.psd)
>>> #
>>> _ = plt.plot(*spec.T, 'k--', lw=2.5, label='Spec')
>>> _ = plt.plot(f, p, label='Welch PSD')
>>> _ = plt.plot(f2, p2, label='PSDmod')
>>>
>>> # For plot, rename columns in DataFrame to include "Env":
>>> psd_ = (psd_
... .rename(columns={i: i + ' Env'
... for i in psd_.columns}))
>>> _ = psd_.plot.line(ax=ax)
>>> _ = plt.xlim(20, 50)
>>> _ = plt.title('PSD Comparison')
>>> _ = plt.xlabel('Freq (Hz)')
>>> _ = plt.ylabel(r'PSD ($g^{2}$/Hz)')
>>> _ = plt.legend(loc='upper left',
... bbox_to_anchor=(1.02, 1.),
... borderaxespad=0.)
>>> plt.tight_layout()
>>> fig.subplots_adjust(right=0.78)
.. plot::
:context: close-figs
Compare to theoretical bin counts @ 30 Hz:
>>> _ = plt.figure('Example 2')
>>> plt.clf()
>>> Frq = freq[np.searchsorted(freq, 30)]
>>> _ = plt.semilogy(fde.binamps.loc[Frq]**2,
... fde.count.loc[Frq],
... label='Data')
>>> # use flight time here (TF), not test time (T0)
>>> Amax2 = 2 * fde.var.loc[Frq] * np.log(Frq * TF)
>>> _ = plt.plot([0, Amax2], [Frq * TF, 1], label='Theory')
>>> y1 = fde.count.loc[Frq, 0]
>>> peakamp = fde.peakamp.loc[Frq]
>>> for j, lbl in enumerate(fde.peakamp.columns):
... _ = plt.plot([0, peakamp[j]**2], [y1, 1], label=lbl)
>>> _ = plt.title('Bin Count Check for Q=50, Freq=30 Hz')
>>> _ = plt.xlabel(r'$Amp^2$')
>>> _ = plt.ylabel('Count')
>>> _ = plt.legend(loc='best')
"""
sig, freq = np.atleast_1d(sig, freq)
if sig.ndim > 1 or freq.ndim > 1:
raise ValueError("`sig` and `freq` must both be 1d arrays")
if resp not in ("absacce", "pvelo"):
raise ValueError("`resp` must be 'absacce' or 'pvelo'")
(coeffunc, methfunc, rollfunc,
ptr) = srs._process_inputs(resp, "abs", rolloff, "primary")
if hpfilter is not None:
if verbose:
print(f"High pass filtering @ {hpfilter} Hz")
b, a = signal.butter(3, hpfilter / (sr / 2), "high")
# to try to get rid of filter transient at the beginning:
# - put a 0.25 second buffer on the front (length from
# looking at impulse response)
# - filter
# - chop off buffer
n = int(0.25 * sr)
sig2 = np.empty(n + sig.size)
sig2[:n] = sig[0]
sig2[n:] = sig
sig = signal.lfilter(b, a, sig2)[n:]
if winends == "auto":
sig = dsp.windowends(sig, min(int(0.25 * sr), 50))
elif winends is not None:
sig = dsp.windowends(sig, **winends)
mxfrq = freq.max()
curppc = sr / mxfrq
if rolloff == "prefilter":
sig, sr = rollfunc(sig, sr, ppc, mxfrq)
rollfunc = None
if curppc < ppc and rollfunc:
if verbose:
print(f"Using {rolloff} method to increase sample rate (have "
f"only {curppc} pts/cycle @ {mxfrq} Hz")
sig, sr = rollfunc(sig, sr, ppc, mxfrq)
ppc = sr / mxfrq
if verbose:
print(f"After interpolation, have {ppc} pts/cycle @ {mxfrq} Hz\n")
LF = freq.size
dT = 1 / sr
pi = np.pi
Wn = 2 * pi * freq
parallel, ncpu = srs._process_parallel(parallel,
LF,
sig.size,
maxcpu,
getresp=False)
# allocate RAM:
if parallel == "yes":
# global shared vars will be: WN, SIG, ASV, BinAmps, Count
WN = (srs.copyToSharedArray(Wn), Wn.shape)
SIG = (srs.copyToSharedArray(sig), sig.shape)
ASV = (srs.createSharedArray((3, LF)), (3, LF))
BinAmps = (srs.createSharedArray((LF, nbins)), (LF, nbins))
a = _to_np_array(BinAmps)
a += np.arange(nbins, dtype=float) / nbins
Count = (srs.createSharedArray((LF, nbins)), (LF, nbins))
args = (coeffunc, Q, dT, verbose)
gvars = (WN, SIG, ASV, BinAmps, Count)
func = _dofde
with mp.Pool(processes=ncpu,
initializer=_mk_par_globals,
initargs=gvars) as pool:
for _ in pool.imap_unordered(func,
zip(range(LF), it.repeat(args, LF))):
pass
ASV = _to_np_array(ASV)
Amax = ASV[0]
SRSmax = ASV[1]
Var = ASV[2]
Count = _to_np_array(Count)
BinAmps = a
else:
Amax = np.zeros(LF)
SRSmax = np.zeros(LF)
Var = np.zeros(LF)
BinAmps = np.zeros((LF, nbins))
BinAmps += np.arange(nbins, dtype=float) / nbins
Count = np.zeros((LF, nbins))
# loop over frequencies, calculating responses & counting
# cycles
for j, wn in enumerate(Wn):
if verbose:
print(f"Processing frequency {wn / 2 / pi:8.2f} Hz", end="\r")
b, a = coeffunc(Q, dT, wn)
resphist = signal.lfilter(b, a, sig)
SRSmax[j] = abs(resphist).max()
Var[j] = np.var(resphist, ddof=1)
# use rainflow to count cycles:
ind = cyclecount.findap(resphist)
rf = cyclecount.rainflow(resphist[ind])
amp = rf["amp"]
count = rf["count"]
Amax[j] = amp.max()
BinAmps[j] *= Amax[j]
# cumulative bin count:
for jj in range(nbins):
pv = amp >= BinAmps[j, jj]
Count[j, jj] = np.sum(count[pv])
if verbose:
print()
print("Computing outputs G1, G2, etc.")
# calculate non-cumulative counts per bin:
BinCount = np.hstack((Count[:, :-1] - Count[:, 1:], Count[:, -1:]))
# for calculating G2:
G2max = Amax**2
for j in range(LF):
pv = BinAmps[j] >= Amax[j] / 3 # ignore small amp cycles
if np.any(pv):
x = BinAmps[j, pv]**2
x2 = G2max[j]
y = np.log(Count[j, pv])
y1 = np.log(Count[j, 0])
g1y = np.interp(x, [0, x2], [y1, 0])
tantheta = (y - g1y) / x
k = np.argmax(tantheta)
if tantheta[k] > 0:
# g2 line is higher than g1 line, so find BinAmps**2
# where log(count) = 0; ie, solve for x-intercept in
# y = m x + b; (x, y) pts are: (0, y1), (x[k], y[k]):
G2max[j] = x[k] * y1 / (y1 - y[k])
# calculate flight-damage indicators for b = 4, 8 and 12:
b4 = 4
b8 = 8
b12 = 12
Df4 = np.zeros(LF)
Df8 = np.zeros(LF)
Df12 = np.zeros(LF)
for j in range(LF):
Df4[j] = (BinAmps[j]**b4).dot(BinCount[j])
Df8[j] = (BinAmps[j]**b8).dot(BinCount[j])
Df12[j] = (BinAmps[j]**b12).dot(BinCount[j])
N0 = freq * T0
lnN0 = np.log(N0)
if resp == "absacce":
G1 = Amax**2 / (Q * pi * freq * lnN0)
G2 = G2max / (Q * pi * freq * lnN0)
# calculate test-damage indicators for b = 4, 8 and 12:
Abar = 2 * lnN0
Abar2 = Abar**2
Dt4 = N0 * 8 - (Abar2 + 4 * Abar + 8)
sig2_4 = np.sqrt(Df4 / Dt4)
G4 = sig2_4 / ((Q * pi / 2) * freq)
Abar3 = Abar2 * Abar
Abar4 = Abar2 * Abar2
Dt8 = N0 * 384 - (Abar4 + 8 * Abar3 + 48 * Abar2 + 192 * Abar + 384)
sig2_8 = (Df8 / Dt8)**(1 / 4)
G8 = sig2_8 / ((Q * pi / 2) * freq)
Abar5 = Abar4 * Abar
Abar6 = Abar4 * Abar2
Dt12 = N0 * 46080 - (Abar6 + 12 * Abar5 + 120 * Abar4 + 960 * Abar3 +
5760 * Abar2 + 23040 * Abar + 46080)
sig2_12 = (Df12 / Dt12)**(1 / 6)
G12 = sig2_12 / ((Q * pi / 2) * freq)
Gmax = np.sqrt(np.vstack((G4, G8, G12)) * (Q * pi * freq * lnN0))
else:
G1 = (Amax**2 * 4 * pi * freq) / (Q * lnN0)
G2 = (G2max * 4 * pi * freq) / (Q * lnN0)
Dt4 = 2 * N0
sig2_4 = np.sqrt(Df4 / Dt4)
G4 = sig2_4 * ((4 * pi / Q) * freq)
Dt8 = 24 * N0
sig2_8 = (Df8 / Dt8)**(1 / 4)
G8 = sig2_8 * ((4 * pi / Q) * freq)
Dt12 = 720 * N0
sig2_12 = (Df12 / Dt12)**(1 / 6)
G12 = sig2_12 * ((4 * pi / Q) * freq)
Gmax = np.sqrt(np.vstack((G4, G8, G12)) * (Q * lnN0) / (4 * pi * freq))
# for output, scale the damage indicators:
Dt4 *= 4 # 2 ** (b/2)
Dt8 *= 16
Dt12 *= 64
# assemble outputs:
columns = ["G1", "G2", "G4", "G8", "G12"]
lcls = locals()
dct = {k: lcls[k] for k in columns}
Gpsd = pd.DataFrame(dct, columns=columns, index=freq)
Gpsd.index.name = "Frequency"
index = Gpsd.index
G2max = np.sqrt(G2max)
Gmax = pd.DataFrame(np.vstack((Amax, G2max, Gmax)).T,
columns=columns,
index=index)
BinAmps = pd.DataFrame(BinAmps, index=index)
Count = pd.DataFrame(Count, index=index)
BinCount = pd.DataFrame(BinCount, index=index)
Var = pd.Series(Var, index=index)
SRSmax = pd.Series(SRSmax, index=index)
di_sig = pd.DataFrame(np.column_stack((Df4, Df8, Df12)),
columns=["b=4", "b=8", "b=12"],
index=index)
di_test = pd.DataFrame(np.column_stack((Dt4, Dt8, Dt12)),
columns=["b=4", "b=8", "b=12"],
index=index)
var_test = pd.DataFrame(
np.column_stack((sig2_4, sig2_8, sig2_12)),
columns=["b=4", "b=8", "b=12"],
index=index,
)
return SimpleNamespace(
freq=freq,
psd=Gpsd,
peakamp=Gmax,
binamps=BinAmps,
count=Count,
bincount=BinCount,
var=Var,
srs=SRSmax,
parallel=parallel,
ncpu=ncpu,
di_sig=di_sig,
di_test=di_test,
var_test=var_test,
resp=resp,
sig=sig,
sr=sr,
)
def psd2time(
spec,
fstart,
fstop,
*,
ppc=10,
df=None,
winends=None,
gettime=False,
expand_method="interp",
):
r"""
Generate a 'random' time domain signal given a PSD specification.
Parameters
----------
spec : 2d ndarray or 2-element tuple/list
If ndarray, it has two columns: ``[Freq, PSD]``.
Otherwise, it must be a 2-element tuple or list, eg:
``(Freq, PSD)``.
fstart : scalar
Starting frequency in Hz
fstop : scalar
Stopping frequency in Hz
ppc : scalar; optional
Points per cycle at highest (`fstop`) frequency; if < 2, it is
internally reset to 2. With `fstop`, determines the sample
rate: ``sr = ppc * fstop``.
df : scalar or None; optional
Serves two purposes: it is the frequency step between
sinusoids included in the time signal, and it also determines
the duration of the signal. If None, it is set to ``fstart /
100`` in order to have 100 cycles at the lowest frequency. If
input as a scalar value, it is taken as a hint and will be
internally adjusted lower as needed; see Notes section. If
routine gives poor results, try refining `df`. If `df` is
greater than `fstart`, it is reset internally to `fstart` (in
that case though, you'll only get 1 cycle at the lowest
frequency). Total duration of the signal: ``T = 1 / df`` (see
equations below for more details).
.. note::
`df` can be used to indirectly specify the number of
cycles desired at the lowest frequency (`fstart`). For
example, if ``fstart=5.0`` and you want to have 500 cycles
of the 5.0 Hz content, then use ``df=0.01``: ``fstart / df
== 500``.
winends : None or dictionary; optional
If None, :func:`pyyeti.dsp.windowends` is not
called. Otherwise, `winends` must be a dictionary of arguments
that will be passed to :func:`pyyeti.dsp.windowends` (not
including `signal`).
gettime : bool; optional
If True, a time vector is output.
expand_method : str; optional
Either 'interp' or 'rescale', referring to which function in
this module will be used to expand input `spec` to all needed
frequencies. Use 'interp' if `spec` is a specification with
constant dB/octave slopes. Use 'rescale' if `spec` provides
the PSD levels on a center-band scale. See :func:`interp` and
:func:`rescale` for more information.
Returns
-------
sig : 1d ndarray
The time domain signal with properties set to match input PSD
spectrum. Duration of signal: ``1 / df``.
sr : scalar
The sample rate of the signal (``ppc * fstop``)
time : 1d ndarray; optional
Time vector for the signal starting at zero with step of
``1.0 / sr``: ``time = np.arange(len(sig)) / sr``.
Notes
-----
The following outlines the equations used in this routine.
If :math:`df` is None: :math:`df = fstart / 100`.
If :math:`df` is greater than :math:`fstart`, it is reset to
:math:`fstart`:
.. math::
df = \min(df, fstart)
The total time of the signal :math:`T` is determined by the lowest
frequency cycle (at frequency :math:`df`):
.. math::
T = 1 / df
The number of points needed is determined by the number of cycles
at the highest frequency multiplied by the points/cycle
:math:`ppc`:
.. math::
N = \lceil {fstop \cdot ppc \cdot T} \rceil
The frequency step :math:`df` is reset to match :math:`N` (because
of the possible round-up in the last equation):
.. math::
df = fstop \cdot ppc / N
The final downward adjustment to :math:`df` is to make sure we hit
the :math:`fstart` frequency exactly:
.. math::
df = fstart / \lfloor fstart / df \rfloor
The frequency vector is defined by:
.. math::
f = [fstart, fstart+df, ...,
\lceil fstop / df \rceil \cdot df]
A sinusoid is to be defined at each of those frequencies, so we
need to compute an amplitude and phase for each.
The amplitude is determined by the magnitude of the PSD. Since a
"power spectral density" is really mean-square spectral density,
and since the mean-square value of a sinusoid is the amplitude
squared over 2, the amplitude of each sinusoid is computed by:
.. math::
amp(f) = \sqrt { 2 \cdot PSD(f) \cdot df }
The phase is determined by using pseudo-random deviates from a
uniform distribution:
.. math::
phase(f) = U(0, 2\pi)
The amplitude and phase are assembled together in a complex array
so that an inverse-FFT will give a real time signal matching the
desired mean-square profile defined by the input PSD.
Raises
------
ValueError
If more than one PSD specification is input.
ValueError
On invalid setting for `expand_method`.
See also
--------
:func:`interp`, :func:`rescale`, :func:`pyyeti.dsp.windowends`
Examples
--------
.. plot::
:context: close-figs
>>> import numpy as np
>>> from pyyeti import psd
>>> spec = np.array([[20, .0768],
... [50, .48],
... [100, .48]])
>>> sig, sr = psd.psd2time(spec, ppc=10, fstart=35,
... fstop=70, df=.01,
... winends=dict(portion=.01))
>>> sr # the sample rate should be 70*10 = 700
700.0
Compare the resulting psd to the spec from 37 to 68:
>>> import matplotlib.pyplot as plt
>>> import scipy.signal as signal
>>> f, p = signal.welch(sig, sr, nperseg=sr)
>>> pv = np.logical_and(f >= 37, f <= 68)
>>> fi = f[pv]
>>> psdi = p[pv]
>>> speci = psd.interp(spec, fi).ravel()
>>> abs(speci - psdi).max() < .05
True
>>> abs(np.trapz(psdi, fi) - np.trapz(speci, fi)) < .25
True
>>> fig = plt.figure('Example')
>>> fig.clf()
>>> a = plt.subplot(211)
>>> line = plt.plot(np.arange(len(sig))/sr, sig)
>>> plt.grid(True)
>>> a = plt.subplot(212)
>>> line = plt.loglog(spec[:, 0], spec[:, 1], label='spec')
>>> line = plt.loglog(f, p, label='PSD of time signal')
>>> leg = plt.legend(loc='best')
>>> x = plt.xlim(20, 100)
>>> y = plt.ylim(.05, 1)
>>> plt.grid(True)
>>> xticks = np.arange(20, 105, 10)
>>> x = plt.xticks(xticks, xticks)
>>> yticks = (.05, .1, .2, .5, 1)
>>> y = plt.yticks(yticks, yticks)
>>> v = plt.axvline(35, color='black', linestyle='--')
>>> v = plt.axvline(70, color='black', linestyle='--')
"""
_freq, _psd, npsds = proc_psd_spec(spec)
if npsds > 1:
raise ValueError(
"only a single PSD specification is currently allowed in "
f"`psd2time`, but {npsds} were provided")
if df is None:
df = fstart / 100
if df > fstart:
df = fstart
if ppc < 2:
ppc = 2
# compute parameters
# 1 cycle of lowest frequency defines length of signal:
T = 1 / df # number of seconds for lowest frequency cycle
N = int(np.ceil(fstop * ppc * T)) # total number of pts
df = fstop * ppc / N
# adjust df to make sure fstart is an even multiple of df
df = fstart / np.floor(fstart / df)
sr = N * df # sr = N/T = N/(1/df)
odd = N & 1
# define constants
# freq = np.arange(fstart, fstop + df / 2, df)
freq = np.arange(fstart, fstop + df, df) # 4/9/22
# generate amp(f) vector
if expand_method == "interp":
speclevel = interp(spec, freq).ravel()
elif expand_method == "rescale":
speclevel, *_ = rescale(_psd.ravel(), _freq, freq=freq)
else:
raise ValueError(
'`expand_method` must be either "interp" or "rescale", '
f"not {expand_method!r}")
amp = np.sqrt(2 * speclevel * df)
m = (N + 1) // 2 if odd else N // 2 + 1
# build up amp to include zero frequency to fstart and from fstop
# to fhighest:
ntop = int(np.floor((fstart - df / 2) / df) + 1)
nbot = m - ntop - len(amp)
if nbot > 0:
amp = np.hstack((np.zeros(ntop), amp, np.zeros(nbot)))
else:
amp = np.hstack((np.zeros(ntop), amp))
# generate F(t)
phi = np.random.rand(m) * np.pi * 2 # random phase angle
# force these terms to be pure cosine
phi[0] = 0.0
if not odd:
phi[m - 1] = 0
# coefficients:
a = amp * np.cos(phi)
b = -amp * np.sin(phi)
# form matrix ready for ifft:
if odd:
a2 = a[1:m] / 2
b2 = b[1:m] / 2
r = N * np.hstack((a[0], a2, a2[::-1])) # real part
i = N * np.hstack((0, b2, -b2[::-1])) # imag part
else:
a2 = a[1:m - 1] / 2
b2 = b[1:m - 1] / 2
r = N * np.hstack((a[0], a2, a[m - 1], a2[::-1])) # real part
i = N * np.hstack((0, b2, 0, -b2[::-1])) # imag part
F_time = np.fft.ifft(r + 1j * i)
mxi = abs(F_time.imag).max()
mxr = abs(F_time.real).max()
if mxi > 1e-7 * mxr: # pragma: no cover
# bug in the code if this ever happens
warn(
f"method failed accuracy test; (max imag)/(max real) = {mxi / mxr}",
RuntimeWarning,
)
F_time = F_time.real
if winends is not None:
F_time = dsp.windowends(F_time, **winends)
if gettime:
return F_time, sr, np.arange(N) / sr
return F_time, sr
def psd2time(spec,
ppc,
fstart,
fstop,
df,
winends=None,
gettime=False,
expand_method="interp"):
"""
Generate a 'random' time domain signal given a PSD specification.
Parameters
----------
spec : 2d ndarray or 2-element tuple/list
If ndarray, its columns are ``[Freq, PSD1, PSD2, ... PSDn]``.
Otherwise, it must be a 2-element tuple or list, eg:
``(Freq, PSD)`` where PSD is: ``[PSD1, PSD2, ... PSDn]``. In
the second usage, PSD can be 1d; in the first usage, PSD is
always considered 2d.
ppc : scalar
Points per cycle at highest (`fstop`) frequency; if < 2, it is
internally reset to 2.
fstart : scalar
Starting frequency in Hz
fstop : scalar
Stopping frequency in Hz
df : scalar
Frequency step to be represented in the time signal. If
routine gives poor results, try refining `df`. If `df` is
greater than `fstart`, it is reset internally to `fstart`.
winends : None or dictionary; optional
If None, :func:`pyyeti.dsp.windowends` is not
called. Otherwise, `winends` must be a dictionary of arguments
that will be passed to :func:`pyyeti.dsp.windowends` (not
including `signal`).
gettime : bool; optional
If True, a time vector is output.
expand_method : str; optional
Either 'interp' or 'rescale', referring to which function in
this module will be used to expand input `spec` to all needed
frequencies. Use 'interp' if `spec` is a specification with
constant dB/octave slopes. Use 'rescale' if `spec` provides
the PSD levels on a center-band scale. See :func:`interp` and
:func:`rescale` for more information.
Returns
-------
sig : 1d ndarray
The time domain signal with properties set to match input PSD
spectrum.
sr : scalar
The sample rate of the signal.
time : 1d ndarray; optional
Time vector for the signal starting at zero with step of
``1/sr``: ``time = np.arange(len(sig))/sr``
Raises
------
ValueError
On invalid setting for `expand_method`.
See also
--------
:func:`interp`, :func:`rescale`, :func:`pyyeti.dsp.windowends`
Examples
--------
.. plot::
:context: close-figs
>>> import numpy as np
>>> from pyyeti import psd
>>> spec = np.array([[20, .0768],
... [50, .48],
... [100, .48]])
>>> sig, sr = psd.psd2time(spec, ppc=10, fstart=35,
... fstop=70, df=.01,
... winends=dict(portion=.01))
>>> sr # the sample rate should be 70*10 = 700
700.0
Compare the resulting psd to the spec from 37 to 68:
>>> import matplotlib.pyplot as plt
>>> import scipy.signal as signal
>>> f, p = signal.welch(sig, sr, nperseg=sr)
>>> pv = np.logical_and(f >= 37, f <= 68)
>>> fi = f[pv]
>>> psdi = p[pv]
>>> speci = psd.interp(spec, fi).ravel()
>>> abs(speci - psdi).max() < .05
True
>>> abs(np.trapz(psdi, fi) - np.trapz(speci, fi)) < .25
True
>>> fig = plt.figure('Example')
>>> fig.clf()
>>> a = plt.subplot(211)
>>> line = plt.plot(np.arange(len(sig))/sr, sig)
>>> plt.grid(True)
>>> a = plt.subplot(212)
>>> line = plt.loglog(spec[:, 0], spec[:, 1], label='spec')
>>> line = plt.loglog(f, p, label='PSD of time signal')
>>> leg = plt.legend(loc='best')
>>> x = plt.xlim(20, 100)
>>> y = plt.ylim(.05, 1)
>>> plt.grid(True)
>>> xticks = np.arange(20, 105, 10)
>>> x = plt.xticks(xticks, xticks)
>>> yticks = (.05, .1, .2, .5, 1)
>>> y = plt.yticks(yticks, yticks)
>>> v = plt.axvline(35, color='black', linestyle='--')
>>> v = plt.axvline(70, color='black', linestyle='--')
"""
if df > fstart:
df = fstart
if ppc < 2:
ppc = 2
# compute parameters
# 1 cycle of lowest frequency defines length of signal:
T = 1 / df # number of seconds for lowest frequency cycle
N = int(np.ceil(fstop * ppc * T)) # total number of pts
df = fstop * ppc / N
# adjust df to make sure fstart is an even multiple of df
df = fstart / np.floor(fstart / df)
sr = N * df # sr = N/T = N/(1/df)
odd = N & 1
# define constants
freq = np.arange(fstart, fstop + df / 2, df)
# generate amp(f) vector
if expand_method == "interp":
speclevel = interp(spec, freq).ravel()
elif expand_method == "rescale":
_freq, _psd, npsds = proc_psd_spec(spec)
speclevel, *_ = rescale(_psd.ravel(), _freq, freq=freq)
else:
raise ValueError(
'`expand_method` must be either "interp" or "rescale", '
f"not {expand_method!r}")
amp = np.sqrt(2 * speclevel * df)
m = (N + 1) // 2 if odd else N // 2 + 1
# build up amp to include zero frequency to fstart and from fstop
# to fhighest:
ntop = int(np.floor((fstart - df / 2) / df) + 1)
nbot = m - ntop - len(amp)
if nbot > 0:
amp = np.hstack((np.zeros(ntop), amp, np.zeros(nbot)))
else:
amp = np.hstack((np.zeros(ntop), amp))
# generate F(t)
phi = np.random.rand(m) * np.pi * 2 # random phase angle
# force these terms to be pure cosine
phi[0] = 0.0
if not odd:
phi[m - 1] = 0
# coefficients:
a = amp * np.cos(phi)
b = -amp * np.sin(phi)
# form matrix ready for ifft:
if odd:
a2 = a[1:m] / 2
b2 = b[1:m] / 2
r = N * np.hstack((a[0], a2, a2[::-1])) # real part
i = N * np.hstack((0, b2, -b2[::-1])) # imag part
else:
a2 = a[1:m - 1] / 2
b2 = b[1:m - 1] / 2
r = N * np.hstack((a[0], a2, a[m - 1], a2[::-1])) # real part
i = N * np.hstack((0, b2, 0, -b2[::-1])) # imag part
F_time = np.fft.ifft(r + 1j * i)
mxi = abs(F_time.imag).max()
mxr = abs(F_time.real).max()
if mxi > 1e-7 * mxr: # pragma: no cover
# bug in the code if this ever happens
warn(
f"method failed accuracy test; (max imag)/(max real) = {mxi / mxr}",
RuntimeWarning,
)
F_time = F_time.real
if winends is not None:
F_time = dsp.windowends(F_time, **winends)
if gettime:
return F_time, sr, np.arange(N) / sr
return F_time, sr