vibration.immission.railway

Evaluating the vibration of a passing train (DIN 45672-2:1995-07).

A train passing a building puts vibration into the ground for twenty seconds and then leaves, and DIN 45672-2 is the procedure that turns those twenty seconds into numbers that can be compared: between two measuring points on the way from the track to the building, between two dates either side of a mitigation measure, and between one train and the next. It is measured with the meter of DIN 45669-1 in its railway working range, 4 Hz to 315 Hz, and it is an evaluation standard rather than an assessment one: it says how to reduce a record, not what the result may be.

Three stretches of one record (Clause 5). $T_1$ is about four seconds with the largest vibration in its middle, where the characteristic values of the passage are read. $T_2$ is the passage itself, from where the amplitude first reaches about a quarter of its usual maxima to where it falls back to it. $T_3$ is the whole event, and it is the one an event value is formed over. Every quantity below carries the index of the stretch it came from.

The quantities (Clause 6). The running r.m.s. ${\tilde{v}}_F(t)$ with $\tau =125$ ms (Formula (1)) and its level against $v_0=5\cdot 10^{-8}$ m/s (Formula (2)); the interval r.m.s. ${\tilde{v}}_j$ of Formula (4), or the approximation of Formula (5) formed from the running r.m.s.; and the event value of Formula (8), the interval r.m.s. of $T_3$ referred to one hour, $v_E={\tilde{v}}_3\sqrt{T_3/3600\mathrm{s}}$, whose square adds across the passages of an hour (Formula (10)). The same quantities in third-octave bands from 4 Hz to 315 Hz give the interval and maximum third-octave levels of Formulae (6) and (7).

Narrow band (Clause 7.3). The one-sided power spectral density of Formula (12), blockwise with a Hanning window and at least half a block of overlap, at the resolution of 1,25 Hz the standard recommends because it resolves the resonance of an ordinary floor. Adding its lines back into third octaves is Formula (23), and Table 1 fixes how many lines each band takes: a number recommended for comparability rather than derived, which is why it is a table here.

What this does not do. The standard is a method of reduction, so there is no verdict anywhere in it, and none here: what the numbers mean for people and buildings is DIN 4150-2 and DIN 4150-3. The calibration checks of Clause 7.3.3 test an analyser, which this module is not.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

amplitude_distribution(
    velocity_mm_s: ArrayLike,
    *,
    bins: int = 101,
) -> AmplitudeDistribution

The amplitude distribution of a stretch of velocity (Clause 6.3).

Clause 6.3 forms it over $T_2$ by DIN 55350-23, which is a histogram normalised to unit area: equal bins across the range of the stretch.

Parameters

Returns: The density and its cumulative function, as an AmplitudeDistribution.

Raises

AmplitudeDistribution(
    edges_mm_s: NDArray[np.float64],
    density_per_mm_s: NDArray[np.float64],
    cumulative: NDArray[np.float64],
)

How often each velocity occurred in a stretch, Formula (11).

Attributes

band_sum_level(levels_db: ArrayLike) -> float

The sum level $\Sigma L$ of a spectrum, Annex B, Formula (B.3).

$10\lg \sum 10^{0,1L(f_{Tn})}$, the energy sum of the bands.

Parameters

Returns: $\Sigma L$, in decibels.

Raises

centred_interval(
    velocity_mm_s: ArrayLike,
    fs_hz: float,
    *,
    duration_s: float = 4.0,
) -> tuple[float, float]

The stretch $T_1$ of Clause 5 a), centred on the largest amplitude.

Clause 5 a) asks for the largest vibration to sit about in the middle of $T_1$, and this puts it exactly there, or as near as the record allows: a stretch that would run off either end is moved back inside it rather than cut, so it keeps its length.

Parameters

Returns: (start, end), in seconds from the start of the record. A record shorter than duration_s is returned whole.

Raises

combined_event_velocity(event_velocities_mm_s: ArrayLike) -> float

The event value of every passage in one hour, Formula (10), in mm/s.

Event values add in square, because each is already the energy of its passage spread over the same hour: $v_{E,\mathrm{ges}}=\sqrt{\sum v_{En}^2}$.

Parameters

Returns: $v_{E,\mathrm{ges}}$, in millimetres per second.

Raises

elastic_insertion_loss(
    levels_before_db: ArrayLike,
    levels_after_db: ArrayLike,
) -> NDArray[np.float64]

The insertion loss $D_e(f_{Tn})$ of Annex B, Formula (B.1), in dB.

The drop in the third-octave structure-borne level at one point of a transmission path when an elastic element is built in. Annex B says it and it bears repeating: the value belongs to the point it was measured at. The difference spectrum of Formula (B.2) is the same subtraction between any two spectra.

Parameters

Returns: $L_1-L_2$ per band, positive where the element reduces the level.

Raises

evaluate_train_passage(
    velocity_mm_s: ArrayLike,
    fs_hz: float,
    *,
    t2_s: tuple[float, float],
    t1_s: tuple[float, float] | None = None,
    t3_s: tuple[float, float] | None = None,
    upper_band_hz: float = 315.0,
) -> TrainPassage

Reduce one passage to the quantities of Clauses 5 to 7.

The record goes through the band limitation of the DIN 45669-1 railway working range first, as it does inside the meter, and every quantity is read from what is left: the running r.m.s. and its maximum, the peak, the interval r.m.s. of the three stretches, the event value, the weighted severity DIN 4150-2 reads, and the third-octave levels of Figure 6.

Parameters

Returns: The reduced passage, as a TrainPassage.

Raises

Constant (float).

EVENT_REFERENCE_DURATION_S = 3600.0

event_velocity(interval_rms_mm_s: float, duration_s: float) -> float

The event value $v_E$ of Formula (8), in mm/s.

The interval r.m.s. of the whole event referred to one hour, $v_E={\tilde{v}}_3\sqrt{T_3/3600\mathrm{s}}$: the constant velocity that, held for an hour, carries the energy the passage carried.

Parameters

Returns: $v_E$, in millimetres per second.

Raises

event_velocity_level(interval_rms_mm_s: float, duration_s: float) -> float

The event level $L_{vE}$ of Formula (9), in dB.

$20\lg ({\tilde{v}}_3/v_0)+10\lg (T_3/3600\mathrm{s})$, which is event_velocity as a level against VELOCITY_LEVEL_REFERENCE_MM_S.

Parameters

Returns: $L_{vE}$, in decibels.

Raises

interval_rms(
    values: ArrayLike,
    fs_hz: float,
    *,
    interval_s: tuple[float, float],
) -> float

The interval r.m.s. of Formula (4), or of Formula (5).

Formula (4) averages the square of the velocity over the stretch. Handed the running r.m.s. of running_velocity_rms instead, the same average is Formula (5), which Clause 6.1 calls a good approximation of Formula (4) and which is what a meter reading its display produces.

Parameters

Returns: ${\tilde{v}}_j$, in the unit of values.

Raises

narrowband_psd(
    velocity_mm_s: ArrayLike,
    fs_hz: float,
    *,
    resolution_hz: float = 1.25,
) -> SpectralDensityResult

The one-sided power spectral density of Formula (12), in (mm/s)²/Hz.

Clause 7.3.1 asks for blocks of $1/\Delta f$, a Hanning window (Formula (18)), at least half a block of overlap and a linear average of the blocks, which is the Welch estimate power_spectral_density makes. Its lines add back to the mean square of the stretch, which is Formula (14).

Parameters

Returns: The spectrum, as a SpectralDensityResult.

Raises

Constant (float).

NARROWBAND_RESOLUTION_HZ = 1.25

passage_average_level(
    levels_db: ArrayLike,
    *,
    axis: int | None = None,
) -> float | NDArray[np.float64]

The energy average of levels over repeated passages (Clause 9), in dB.

$10\lg ⟨10^{L/10}⟩$, the level of passage_average_velocity.

Parameters

Returns: The average level: a float, or one level per band.

Raises

passage_average_velocity(interval_rms_mm_s: ArrayLike) -> float

The energy average of r.m.s. values over repeated passages (Clause 9).

Clause 9 averages the results of a class of trains energetically, which for r.m.s. values is the root of the mean square, $\sqrt{⟨{\tilde{v}}^2⟩}$. The same holds for narrow-band spectra, whose power spectral densities average arithmetically line by line.

Parameters

Returns: The energy average, in millimetres per second.

Raises

Constant (tuple).

PASSAGE_BANDS_HZ = (4.0, 315.0)

passage_energy_spectral_density(
    psd: ArrayLike,
    duration_s: float,
) -> NDArray[np.float64]

The energy spectral density of $T_2$, Formula (13), in (mm/s)²/Hz².

Formula (13) writes $e_2=|V_2{|}^2/4$ with the one-sided Fourier spectrum $V=2X$ of Formula (16), which is $|X{|}^2$, the two-sided density. Set beside Formula (12) that is $e_2=G_2{T}_2/2$, and it means the lines of $e_2$ over positive frequencies add up to half the energy of the stretch where the lines of $G_2$ add up to all of its mean square. This returns the formula as printed.

Parameters

Returns: $e_2$ at the same lines, in (mm/s)²/Hz².

Raises

running_acceleration_level(
    acceleration_m_s2: ArrayLike,
    fs_hz: float,
    *,
    time_constant_s: float = 0.125,
) -> NDArray[np.float64]

The running acceleration level $L_{aF}(t)$ of Formula (3), in dB.

Parameters

Returns: $20\lg ({\tilde{a}}_F/a_0)$ with $a_0=10^{-6}$ m/s², the reference DIN EN 21683 gives and the one the rest of this package’s acceleration levels use.

Raises

running_velocity_level(
    velocity_mm_s: ArrayLike,
    fs_hz: float,
    *,
    time_constant_s: float = 0.125,
) -> NDArray[np.float64]

The running velocity level $L_{vF}(t)$ of Formula (2), in dB.

Parameters

Returns: $20\lg ({\tilde{v}}_F/v_0)$ with $v_0$ = VELOCITY_LEVEL_REFERENCE_MM_S, one value per sample. A sample whose running r.m.s. is still exactly zero, before the first nonzero sample, is -inf.

Raises

running_velocity_rms(
    velocity_mm_s: ArrayLike,
    fs_hz: float,
    *,
    time_constant_s: float = 0.125,
) -> NDArray[np.float64]

The running r.m.s. ${\tilde{v}}_F(t)$ of Formula (1), in mm/s.

The exponential average the formula integrates, started from rest. Clause 4 says what that costs at the start of a record and so why the averaging has to be running before the train arrives. It puts the cost at 14 % after $2\tau$ and 2 % after $4\tau$, which are the shortfalls of the mean square, $e^{-2}$ and $e^{-4}$; the running r.m.s. itself, which is what Figure 3 draws, is 7 % and 0,9 % short.

Parameters

Returns: ${\tilde{v}}_F(t)$, one value per sample, in mm/s.

Raises

spectral_density_level(psd: ArrayLike) -> NDArray[np.float64]

The level of a power spectral density, as Figure 7 draws it, in dB.

$10\lg (G/G_0)$ with $G_0=(5\cdot 10^{-8}m/s{)}^2/\mathrm{Hz}$, the square of VELOCITY_LEVEL_REFERENCE_MM_S per hertz.

Parameters

Returns: The level at each line, in decibels; -inf where the density is exactly zero.

Raises

Constant (float).

T1_DURATION_S = 4.0

Constant (mapping).

THIRD_OCTAVE_LINES = {4.0: 1, 5.0: 1, 6.3: 1, 8.0: 2, 10.0: 2, 12.5: 2, 16.0: 3, 20.0: 3, 25.0: 5, 31.5: 6, 40.0: 7, 50.0: 9, 63.0: 12, 80.0: 14, 100.0: 18, 125.0: 23, 160.0: 29, 200.0: 37, 250.0: 46, 315.0: 58, 400.0: 74, 500.0: 92}

third_octaves_from_narrowband(
    frequencies_hz: ArrayLike,
    psd: ArrayLike,
) -> tuple[NDArray[np.float64], NDArray[np.float64]]

Add a narrow-band spectrum back into third octaves, Formula (23).

Each band takes the number of lines Table 1 gives it, and the band r.m.s. is the square root of their summed power, $\sqrt{\sum G(f_k)\Delta f}$. The table gives the counts and not the lines, so the choice is stated here: the lines nearest the nominal centre on a logarithmic axis. Wherever the count is what a sixth of an octave either side of the centre holds, which is every band but two, those are exactly the lines inside the band’s nominal edges; at 10 Hz and 12,5 Hz, where the table departs from the edges, they are 10 and 11,25 Hz, then 12,5 and 13,75 Hz.

It is a rule of nominal bands and it behaves like one. Nominal edges do not meet: six lines fall between two bands and are in neither (71,25 Hz, 141,25 Hz and 142,5 Hz, and 353,75 Hz to 356,25 Hz), and five are in two (178,75 Hz, 223,75 Hz, and 446,25 Hz to 448,75 Hz). The bands therefore do not add up to the spectrum exactly, and a tone that sits in one of the gaps is in no band at all; that is the comparability the table buys.

A band is returned only if the spectrum holds all its lines: a spectrum that starts above the lowest bands or stops inside a band leaves those bands out rather than filling them with the lines of a neighbour.

Parameters

Returns: (centres, rms): the nominal centres of the bands the spectrum holds whole, from THIRD_OCTAVE_LINES, in hertz, and the band r.m.s. in each, in mm/s.

Raises

TrainPassage(
    velocity_mm_s: NDArray[np.float64],
    running_rms_mm_s: NDArray[np.float64],
    peak_velocity_mm_s: float,
    running_rms_max_mm_s: float,
    kbf_max: float,
    interval_rms_mm_s: tuple[float, float, float],
    intervals_s: tuple[tuple[float, float], ...],
    event_velocity_mm_s: float,
    event_level_db: float,
    band_centres_hz: NDArray[np.float64],
    band_interval_levels_db: NDArray[np.float64],
    band_max_levels_db: NDArray[np.float64],
    fs_hz: float,
)

One passage reduced the way Clauses 5 to 7 reduce it.

Attributes

TrainPassage.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the running level with the three stretches marked on it.

Requires matplotlib (pip install phonometry[plot]). The spectra of Figure 6 are plot_spectrum.

Parameters

Returns: The Axes.

TrainPassage.plot_spectrum(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the interval and maximum third-octave levels, as Figure 6 does.

Parameters

Returns: The Axes.

Constant (float).

VELOCITY_LEVEL_REFERENCE_MM_S = 5e-05

velocity_psd_from_voltage(
    psd_v2_per_hz: ArrayLike,
    *,
    sensitivity_v_per_mm_s: float,
    gain: float,
) -> NDArray[np.float64]

A density measured in volts, in the unit of the velocity (Annex A).

The transducer’s transfer coefficient ${\kappa }_A$ and the amplifier’s gain ${\kappa }_V$ multiply the velocity, so they divide its density in square: $G_{vv}=G_{\mathrm{V}}/({\kappa }_A^2{\kappa }_V^2)$.

Parameters

Returns: The density of the velocity, in (mm/s)²/Hz.

Raises