A mixed-mode test as one specification

A sine-on-random test drives one signal made of parts: a broadband random floor written as breakpoints, and tones written in amplitude (peak), each at a fixed frequency or swept. A random-on-random test lays narrow bands of random over the floor instead, each a few percent of its center wide, fixed or swept. The parts are judged by different rules, and a report that quotes one rms figure for the test has said nothing, because there are three: the floor's rms with the tone lines left out, each tone in amplitude, and the total rms of the whole drive.

Take four tones of 0.12, 1.944, 2.5 and 1.5 g peak over a 1.0 grms floor. The tones' mean squares, \(A^2/2\), are 0.0072, 1.89, 3.13 and 1.13 g², 2.48 g rms together. The parts are independent, so their mean squares add, and the total is

\[ g_{\text{rms,total}} = \sqrt{g_{\text{rms,floor}}^2 + \sum_i \frac{A_i^2}{2} + \sum_j W_j B_j} = \sqrt{1.0 + 6.15} = 2.67 \]

with \(W_j\) a narrowband's level and \(B_j\) its width. All three are right, and each misleads when quoted alone as "the test level": 2.67 read as the floor is nearly three times too high, and a tone's g peak set beside the floor's grms compares two different quantities.

The specification

MixedSpecification holds the parts together: the floor (a random Specification), the tones (a SineSweepSpecification), the narrowbands (Narrowband) and the width left out of the floor either side of each tone.

from visualdynamics.core.mixed_mode import (MixedSpecification,
                                            Narrowband, Tone)

g = 9.80665
spec = MixedSpecification(
    floor,                                    # the random specification
    [Tone(4.86, 0.12 * g), Tone(19.44, 1.944 * g),
     Tone(38.88, 2.50 * g), Tone(58.32, 1.50 * g)],
    seconds=60.0,                             # how long a fixed tone holds
    exclusion=2.0)                            # Hz left out either side
spec.reference_rms()                          # floor, tones, bands, total

The tone table is entered as Tone rows. A fixed tone is a row with a frequency and an amplitude; a swept tone adds sweep_to, a rate and a mode (LINEAR in Hz/s, LOG in octaves a minute). A fixed tone is written into the sine specification as the dwell a controller's file holds (SineTone.dwell). A tone's written mean square is its level squared over two, averaged over its run, so a tone held at two levels in turn counts each for the time it is held. spec.tone_table() reads any sine specification back as rows, fixed where every segment holds one frequency.

A narrowband is written the way a standard writes it, or by its edges:

Narrowband.plus_minus(400.0, 5.0, 0.01 * g**2)    # 380 to 420 Hz
Narrowband.between(380.0, 420.0, 0.01 * g**2)     # the same band
Narrowband(56.0, 0.05 * g**2, width=4.0,          # swept 56 to 93 Hz
           sweep_to=93.0, rate=2.0)               # and back, at 2 Hz/s

Its level is the density it adds over the floor between its edges, so its mean square is the level times its width. A band given as a share of its center (plus_minus) widens with the center as it sweeps, the way a harmonic's band does; one given in hertz keeps its width. A "5" typed into a field that wants the full width makes a band half as wide, with half the mean square, which is why a band is checked by its edges in hertz: band.edges(), and on the plot below. tolerance_db is the tolerance the plan states for a band, since a swept band cannot have the floor's statistics.

Nothing new in the file

A mixed test adds no object of its own to a project file. The floor is saved as the random specification it is and the tones as the sine specification they are, the two objects a random and sine project already holds; MixedSpecification(floor, sweep) puts them back together. A fixed tone is a dwell, so no segment type is added. The narrowbands and the width left out are attributes of the floor, Specification.narrowbands and Specification.tone_exclusion, saved with it the way its band constraints are, so a file written with them opens in an older version as the plain random specification. Given to MixedSpecification, they are written onto a copy of the floor, which spec.floor then holds; add that to the project to keep them.

The data sheet

judge_mixed(history, spec) reads one channel of a record against it, part by part:

1. Where the tones begin in the record is found by matched filter, as the extraction finds it, unless onset is given. Each tone is read through a tracking band locked to its frequency (0.5 Hz wide by default), the way a controller reads it (How a tracking filter reads a sweep). The band settles in about one over its width, so nothing before that is read; the tone's level is the median of the readings after it, compared with the table at plus or minus 10 percent, an example of what test documents ask for. A small tone near the floor scatters through its band from reading to reading, and the median reads its level through that scatter.
2. What each band passed is taken out of the record. What is left, the residual, is the floor and the narrowbands.
3. The floor is judged on the residual's averaged spectrum with the lines that held a tone, and every narrowband's lines, left out. A line held a tone when the tone spent a tenth of its run or more within the left-out width of it: every line beside a fixed tone, and few or none of the lines a long sweep passes, since it is near each only briefly (a tone that dwells at two frequencies in turn holds the lines beside each, not the ones its hop between them crosses) and what the band took with it is a small share of that line's power. The floor is judged as a comparison is (compliance.judge): its rms against the reference's over the same lines (plus or minus 10 percent), and its lines against the reference moved 3 dB either way. An averaged line scatters, so a correctly controlled floor has a few lines outside 3 dB on a short record: on a 10 s record read from 2 s, about 28 degrees of freedom a line, 1.4 percent of lines, five of 384. The lines fail only when more are outside than chance would put there one time in a thousand; the sheet gives the count beside the number chance expects and the number allowed. A floor with no line left to judge is not judged, and the reading says so.
4. Each narrowband is judged at the degrees of freedom of a narrow band. A fixed band's level is the mean of its lines, worth what their correlation under the window leaves it, about half their count under Hann (Combining lines). A swept band stays over any one frequency only as long as it takes to move its own width, so it is read in slices half that long, each worth only the few frames it holds, and its level is the slices pooled. Its tolerance is the stated one, or the floor's, widened to the chi-square scatter at those degrees of freedom (90 percent by default), so a band of few degrees of freedom is not failed for scatter wider than its tolerance. A band with no line under it, or a swept band with no slice, is not judged.
5. The total is every line of the record's spectrum. It has no tolerance of its own: it is the figure the amplifier, the coil and the fixture see, and the first one another lab computes.

from visualdynamics.core.mixed_mode import judge_mixed

reading = judge_mixed(history, spec, onset=0.0)  # tones from the start
reading.floor.residual / g        # 1.003, the residual over every line
reading.tones_measured / g        # 2.477, the tracked tones' rms
reading.total.measured / g        # 2.671, the whole drive
print(reading.describe('g', 1 / g))

visualdynamics.plot.plot_mixed_mode(history, spec, onset=0.0,
                                    path='sheet.png')

The figures are for one minute of the composite above, the guide composite of Reading a tone off an ASD. The tracked tones read 0.122, 1.945, 2.496 and 1.498 g peak, 2.477 g rms together against 2.479 written; the residual 1.003 grms over every line, and 0.986 against 0.986 written with the tone lines left out (2.8 percent of the floor's lines); and the whole drive 2.671. Every value is stored in SI, as every record is; describe and the plot take the display unit.

plot_mixed_mode draws the residual's spectrum over the reference, the record with its tones dotted, a tone marker on each fixed tone (the height its spike stands at for the spacing and window, with its amplitude read off the spectrum in words), the lines left out hatched and labeled "left out", and the data sheet beside it: the floor, the tones together and each tone, each narrowband and the total, each with its written and measured figure, its tolerance and its verdict in words, and the settings that decide what the numbers mean under them.

In the second figure the swept band is read in 37 slices of about 9 degrees of freedom each, where a floor line has 112; one slice's scatter alone is wider than 3 dB either way, and the band's level pooled over the slices reads 0.2 dB over its written level. Its edges are drawn dash-dotted and its travel cross-hatched, so the plot shows where the band went as well as how strong it was. The figures are made by tools/make_concept_figures.py.

In the app

In a random and sine project, select the record and choose the sine view on the bar. When the project also holds the floor's random specification at a channel the record has, a Mixed Mode readout sits under the extraction's settings: the channel, the width left out either side of a tone, the tracking band and the tones' tolerance. Reading a long record takes seconds, so the first reading waits for Read Data Sheet, and runs with the progress bar up; once the sheet is up, an edit reads again at once. A refusal (a dead or constant channel, a sample that is not finite, a record of another quantity) is shown in place of the sheet and on the status line. The status line also carries the sheet in one line once it is read, with its notes, so a part not judged or a band that may read low says so there as well as on the sheet. The sheet shows the same rows as the plot, the settings under them, and a small spectrum of the residual over the reference with the left-out lines hatched. The narrowbands are the floor's own (Specification.narrowbands). The reading is journaled as its plot_mixed_mode call.

What to watch

• A tone's spike on the spectrum is not its level. The tone is judged in amplitude, read through its band; the marker on the plot shows where the spike stands for this spacing, and is a picture.
• The width left out is part of the result. A floor feature inside it, a fixture resonance at a tone, is not judged against the floor's tolerance at all. The sheet gives the width and the share of the floor's lines it took, and says so in a note when more than half the floor was left out.
• A steady tone has nothing in time for the matched filter to lock onto but where it starts, so with no onset given it is placed somewhere it plays, never before it, and the floor is read from there. Give onset when the tones' start is known. A swept narrowband needs it: where the band is at a moment is counted from the start, so with steady tones and no onset it is not judged, and a note says why.
• A channel holding a sample that is not finite, zero throughout or one value throughout is refused, naming the cause: one sample that is not finite spreads over every line of its frame's spectrum, and a dead or constant channel has no floor or tones to judge. A part with nothing left to judge reads "not judged", never "passes", and shows no count of lines outside and no degrees of freedom; a sheet that judged nothing has not passed.
• The channel is read by its full identity. Where a record holds two quantities at the control DOF (a force and an acceleration at 1Z+), the one read is the quantity the floor is the density of; two records of that one quantity at the DOF are refused, since the sheet reads one.
• A record of another quantity than the floor or the tones (velocity against an acceleration floor) is refused, by the rule every comparison uses.
• A tone at the floor's edge has no floor under it on one side. The reading says so in a note.
• A swept band moving faster than a frame fine enough to resolve it reads low; the note says so and gives the line spacing used.