vibration.immission.prediction

Predicting vibration before it is measured (DIN 4150-1:2001-06).

Part 1 of DIN 4150 is the first question of the series: before a blast is fired, a pile driven or a line built, how much vibration will reach the building, and how much of it will the floors feel. It gives no recipe, and says so in its foreword; what it gives is the shape of every answer, the handful of constants experience has fixed, and twenty-seven figures of measured cases to show what the shapes look like in the ground. The shapes and the constants are here.

Propagation (Clause 4.2). Beyond the reference distance $R_1=a/2+{\lambda }_R$ of Formula (1), half the source’s extent plus a surface wavelength, the velocity amplitude decays as

(Formula (2)): geometric spreading with an exponent $n$ that Figure 1 fixes at 0, 0,5, 1 or 1,5 by whether the source is a line or a point, harmonic or impulsive, and the wave a surface or a body wave; and material damping with $\alpha \approx 2\pi D/\lambda$, the damping ratio of the ground over the wavelength, which for loose ground may be taken as 0,01 at most in a preliminary estimate. Nearer than $R_1$ the formula does not hold. A train is a chain of point sources and decays with an exponent between 0,3 and 0,5.

Into the building (Clause 4.3). A building on the ground is a mass on a spring with the natural frequency of Formula (3), about 15 Hz for one or two storeys and under 8 Hz above six; the foundation passes at most $1/(2D_0)$ of the ground’s amplitude at that frequency, 2 for the 0,25 of loose ground, and a mean of 0,5 above it, or all of it on rock; a floor amplifies by at most $1/(2D_1)$, 10 to 25 for a concrete floor; and the lowest horizontal natural frequency of a building of five storeys or more is about $10/n$ Hz (Formula (4)).

Sources (Clause 5). A blast in the far field follows $v_{\max }=k(L/L_0{)}^b(R/R_0{)}^{-m}$ (Formula (5)) with the charge per delay $L$ and constants from trial blasts; a falling mass follows the same with the root of its fall energy (Formula (6)); a hall of $N$ similar machines gives $\chi v_B\sqrt{N}$ at a point where $N_B$ of them were measured (Formula (7)), with the correction $\chi$ of Figure 3, which is printed as a nomogram and is here as the nomogram read at a five-hundredth. Rail traffic excites at the speed over the spacing of whatever repeats along the track, sleepers first, and its vehicles have natural frequencies of their own.

What is not here. The measured cases of Annex A print their inputs and their peaks and say themselves that they are not a basis for a prediction; two of their figures are drawn from the formulas above with every parameter printed, and those are the conformance rows. Two symbol lists print a distance in millimetres, one working frequency has its sign the wrong way and one legend swaps two line styles; all in docs/ERRATA.md.

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

attenuation_coefficient_per_m(
    damping_ratio: float,
    *,
    wavelength_m: float,
) -> float

The material damping of the ground, $\alpha$ of Formula (2).

$\alpha \approx 2\pi D/\lambda$ with $\lambda =c/f$ the wavelength that matters. Figure A.19 prints 0,005 for a damping ratio of 0,01 and a wavelength of 12,5 m.

Parameters

Returns: $\alpha$, in reciprocal metres.

Raises

blast_peak_velocity_mm_s(
    charge_kg: float,
    distance_m: ArrayLike,
    *,
    coefficient_mm_s: float,
    charge_exponent: float,
    distance_exponent: float,
) -> NDArray[np.float64]

The peak velocity of a blast in the far field, Formula (5).

$v_{\max }=k(L/L_0{)}^b(R/R_0{)}^{-m}$ with the charge per delay $L$ against 1 kg, the distance $R$ against 1 m, and the constants $k$, $b$ and $m$ from trial blasts or comparable cases in ground, method and distance, with allowance for scatter. The standard prints no values for them. Its symbol list prints the distance in millimetres, which the reference metre says is a slip.

Parameters

Returns: $v_{\max }$, one per distance, in millimetres per second.

Raises

Constant (mapping).

BLASTING_RELEVANT_DISTANCE_M = {'quarry': 1500.0, 'construction': 400.0}

fall_energy_kj(weight_kn: float, *, drop_height_m: float) -> float

The energy of a falling mass, $E=Gh$ of Clause 5.1.3.

A weight in kilonewtons through a height in metres is that many kilojoules, the unit Formula (6) wants.

Parameters

Returns: $E$, in kilojoules.

Raises

far_field_velocity_mm_s(
    reference_velocity_mm_s: float,
    distance_m: ArrayLike,
    *,
    reference_distance_m: float,
    exponent: float,
    attenuation_per_m: float = 0.0,
) -> NDArray[np.float64]

The velocity amplitude at a distance in the far field, Formula (2).

$\bar{v}={\bar{v}}_1(R/R_1{)}^{-n}\exp [-\alpha (R-R_1)]$: the amplitude at the reference distance, spread with the exponent of Figure 1 and damped with $\alpha$. Figure A.19 draws it for 0,44 mm/s at 13 m with $\alpha$ = 0,005 for the three exponents 0, 0,5 and 1.

Parameters

Returns: $\bar{v}$, one per distance, in millimetres per second.

Raises

Constant (tuple).

FLOOR_DAMPING_RATIO_RANGE = (0.02, 0.05)

floor_transfer_max(floor_damping_ratio: float) -> float

The most a floor amplifies at its resonance, Clause 4.3.

$V_D=1/(2D_1)$, from the foundation through the walls to the floor, for a building excited in phase over its whole footprint by predominantly harmonic vibration: 25 for a concrete floor with a damping ratio of 0,02 and 10 for one with 0,05. Short spans, partitions and a foundation on loose ground raise the damping.

Parameters

Returns: $V_D$.

Raises

Constant (float).

FOUNDATION_TRANSFER_ABOVE_RESONANCE = 0.5

foundation_transfer_max(system_damping_ratio: float = 0.25) -> float

The most a foundation passes at the building’s resonance, Clause 4.3.

$V_F=1/(2D_0)$ with the system damping of the building on its ground, 0,25 for loose ground, which gives 2. Above the resonance a mean of FOUNDATION_TRANSFER_ABOVE_RESONANCE may be assumed, and on rock there is no reduction.

Parameters

Returns: $V_F$.

Raises

geometric_exponent(*, geometry: str, character: str, wave: str) -> float

The exponent $n$ of Formula (2) for a source and a wave, Figure 1.

Parameters

Returns: $n$: 0, 0,5, 1 or 1,5.

Raises

impact_peak_velocity_mm_s(
    fall_energy_kj: float,
    distance_m: ArrayLike,
    *,
    coefficient_mm_s: float,
    distance_exponent: float,
) -> NDArray[np.float64]

The peak velocity of a falling mass, Formula (6).

$v_{\max }=k(E/E_0{)}^{0,5}(R/R_0{)}^{-m}$ with the fall energy $E$ against 1 kJ, the distance against 1 m, and $k$ and $m$ from comparable cases. The blast that fells a chimney is usually the smaller source; the impact is this one.

Parameters

Returns: $v_{\max }$, one per distance, in millimetres per second.

Raises

Constant (float).

LOOSE_GROUND_DAMPING_RATIO = 0.01

Constant (float).

LOOSE_GROUND_SYSTEM_DAMPING = 0.25

Constant (tuple).

MACHINE_COUNT_AXIS = (4, 5, 6, 8, 10, 15, 20, 25, 30, 35, 40, 50, 60, 70, 80, 90, 100)

Constant (mapping).

MACHINE_COUNT_CORRECTION = {3: (0.551, 0.537, 0.519, 0.496, 0.471, 0.423, 0.383, 0.36, 0.336, 0.323, 0.309, 0.3, 0.292, 0.289, 0.289, 0.289, 0.286), 5: (0.455, 0.444, 0.427, 0.406, 0.386, 0.342, 0.308, 0.292, 0.276, 0.264, 0.256, 0.245, 0.239, 0.238, 0.236, 0.234, 0.234), 10: (0.365, 0.355, 0.342, 0.327, 0.308, 0.28, 0.252, 0.239, 0.225, 0.216, 0.207, 0.2, 0.193, 0.192, 0.192, 0.191, 0.189), 30: (0.292, 0.283, 0.273, 0.258, 0.243, 0.22, 0.2, 0.188, 0.176, 0.169, 0.164, 0.157, 0.153, 0.153, 0.151, 0.15, 0.149), 60: (0.242, 0.239, 0.23, 0.22, 0.207, 0.189, 0.17, 0.158, 0.148, 0.142, 0.136, 0.13, 0.127, 0.127, 0.126, 0.125, 0.125), 100: (0.189, 0.188, 0.18, 0.173, 0.161, 0.147, 0.134, 0.125, 0.117, 0.111, 0.108, 0.1, 0.1, 0.1, 0.1, 0.1, 0.099)}

machine_count_correction(
    machine_count: ArrayLike,
    *,
    reference_count: int,
) -> NDArray[np.float64]

The correction $\chi$ of Formula (7), Figure 3.

The figure draws $\chi$ against the number of machines running, from 4 to 100, for measurements made with 3, 5, 10, 30, 60 or 100 of them, and prints no closed form; the curve is read off the page at a five-hundredth and interpolated linearly between the readings.

Parameters

Returns: $\chi$, one per count.

Raises

Constant (mapping).

MACHINE_FREQUENCY_BANDS_HZ = {'counter_blow_hammer': (4.0, 8.0), 'forging_press_horizontal': (5.0, 15.0), 'frame_saw': (4.0, 8.0)}

machine_hall_velocity_mm_s(
    reference_velocity_mm_s: float,
    machine_count: ArrayLike,
    *,
    reference_count: int,
) -> NDArray[np.float64]

The peak velocity outside a hall of similar machines, Formula (7).

$v_N=\chi v_B\sqrt{N}$: the velocity measured at the point with $N_B$ machines running, scaled to $N$ of them with the correction of Figure 3. Figure A.18 draws it for 0,44 mm/s measured with three machines and finds the measurements on the curve up to about sixty, the nearest group; beyond that the added groups are further off.

Parameters

Returns: $v_N$, one per count, in millimetres per second.

Raises

material_damping_factor(
    distance_m: ArrayLike,
    *,
    damping_ratio: float,
    frequency_hz: float,
    wave_speed_m_s: float,
) -> NDArray[np.float64]

The share the ground absorbs over a distance, Figure 2.

$\exp [-2\pi Df(R-R_1)/c]$, the damping factor of Formula (2) alone, which Figure 2 draws for a damping ratio of 0,01 and a wave speed of 200 m/s from 10 Hz to 50 Hz: at 100 m the ground has taken 27 % of the amplitude at 10 Hz and 79 % at 50 Hz.

Parameters

Returns: The factor, one per distance, between 0 and 1.

Raises

Constant (tuple).

MEDIUM_SOIL_SHEAR_WAVE_SPEED_M_S = (150.0, 200.0)

Constant (float).

RAIL_INFLUENCE_RANGE_M = 80.0

Constant (tuple).

RAIL_SUPPORT_SPACING_M = (0.6, 0.9)

reference_distance_m(
    source_extent_m: float,
    *,
    rayleigh_wavelength_m: float,
) -> float

The distance the far field begins at, Formula (1).

$R_1=a/2+{\lambda }_R$: half the extent of the source along the direction of propagation plus a wavelength of the surface wave. Nearer than it Formula (2) does not hold.

Parameters

Returns: $R_1$, in metres.

Raises

soil_building_frequency_guide_hz(storeys: int) -> tuple[float, float]

The natural frequency Clause 4.3 gives a building on medium ground.

About 15 Hz for one or two storeys, 8 Hz to 12 Hz for two to six, under 8 Hz above six, for a ground with a shear wave speed of 150 m/s to 200 m/s. The clause prints two storeys in both ranges, so a two-storey building gets the union, 8 Hz to 15 Hz.

Parameters

Returns: (low, high) in hertz; the low bound is 0 above six storeys and both are 15 for a single storey.

Raises

soil_building_natural_frequency_hz(
    stiffness_n_per_m: float,
    *,
    mass_kg: float,
) -> float

The natural frequency of a building on its ground, Formula (3).

$f_B=\frac{1}{2\pi }\sqrt{k_B/m_B}$: the ground as a spring under the mass of the building that moves in phase, for the vertical direction and predominantly harmonic vibration in the lower frequency range.

Parameters

Returns: $f_B$, in hertz.

Raises

Constant (mapping).

SOURCE_EXPONENTS = {('line', 'harmonic', 'surface'): 0.0, ('line', 'harmonic', 'body'): 0.5, ('point', 'harmonic', 'surface'): 0.5, ('line', 'impulsive', 'surface'): 0.5, ('point', 'harmonic', 'body'): 1.0, ('line', 'impulsive', 'body'): 1.0, ('point', 'impulsive', 'surface'): 1.0, ('point', 'impulsive', 'body'): 1.5}

Constant (int).

STOREY_FORMULA_MIN_STOREYS = 5

storey_frequency_hz(storeys: int) -> float

The lowest horizontal natural frequency of a building, Formula (4).

$f_1\approx 10/n$ Hz for a building of $n$ storeys, meant for five and more, where a tall slender building meets a low excitation frequency.

Parameters

Returns: $f_1$, in hertz.

Raises

track_excitation_frequency_hz(
    train_speed_m_s: float,
    *,
    spacing_m: float,
    harmonics: int = 1,
) -> NDArray[np.float64]

The frequencies a repeating feature of the track excites, Clause 5.3.2.

$f_A=v_Z/d$ and its multiples, for the spacing $d$ of whatever repeats along the track or the wheel: the sleepers, the axles, the bogies, a flat spot once per turn of the wheel. The natural frequencies of the vehicle itself do not move with the speed.

Parameters

Returns: $f_A$ and its multiples, in hertz.

Raises

Constant (mapping).

TRACK_TRANSMITTED_BANDS_HZ = {'ballast': (40.0, 80.0), 'under_ballast_mat': (15.0, 40.0), 'mass_spring': (5.0, 20.0)}

Constant (tuple).

TRAIN_CHAIN_EXPONENT_RANGE = (0.3, 0.5)

Constant (mapping).

VEHICLE_NATURAL_FREQUENCIES_HZ = {'car_body': (1.0, 3.0), 'bogie': (6.0, 10.0)}