Vibration and people in buildings (DIN 4150-2)

A vibration meter turns a record into two numbers, the maximum weighted vibration severity KB_Fmax and the clock maximum r.m.s. KB_FTm, and stops. DIN 4150-2 is where those numbers are judged for the people who live or work where the vibration arrives: a table of guide values by kind of area and time of day, a procedure that reads them in a fixed order, and a set of rules for the sources that most often shake a house, from a hammer in the workshop next door to the railway at the end of the street.

It is not a standard of limits, and it says so. The guide values are “not to be applied mechanically”, the annex that explains them calls them recommended values rather than binding ones, and one of its own worked examples finds a reading 13 % above the lower value and calls the requirement met, because a measurement of KB_F is uncertain by about that much. What the standard does fix is the procedure, and the procedure is what this page implements.

KB_Fmax says how bad the vibration was at its worst. The assessment vibration severity KB_FTr says what it added up to over the whole assessment period, 16 h by day (6:00 to 22:00) and 8 h by night. Both are formed for the three directions and the largest is assessed.

Clause 6.2 reads the guide values of Table 1 in a fixed order, which its Figure 2 draws as a flowchart:

• KB_Fmax at or below the lower value A_u: the requirement is met, and nothing else is asked.
• KB_Fmax above the upper value A_o: not met, however short the exposure.
• In between: up to three short events a day are met as they are; anything else is decided by KB_FTr against A_r.

from phonometry import vibration

guide = vibration.guide_values("commercial")
print(guide)  # GuideValues(a_u=0.3, a_o=6.0, a_r=0.15, time_of_day='day', edition='1999')

# Annex C, Example 1: a sawmill's stationary 5 Hz vibration at KB_Fmax = 0.25.
verdict = vibration.assess_people_in_buildings(0.25, guide)
print(verdict.complies, verdict.criterion)  # True A_u

# Above A_o the question ends the other way, unless the events are rare.
print(vibration.assess_people_in_buildings(6.5, guide).criterion)  # A_o
blast = vibration.assess_people_in_buildings(2.0, guide, rare_short_events=True)
print(blast.complies)  # True

import matplotlib.pyplot as plt
import numpy as np

from phonometry import vibration

areas = list(vibration.GUIDE_VALUES)
x = np.arange(len(areas))
fig, axes = plt.subplots(1, 2, figsize=(11.5, 5.6), sharey=True)
for ax, time_of_day in zip(axes, ("day", "night"), strict=True):
    values = [vibration.guide_values(a, time_of_day=time_of_day) for a in areas]
    ax.bar(x - 0.19, [v.a_u for v in values], 0.38, label="$A_u$")
    ax.bar(x + 0.19, [v.a_r for v in values], 0.38, label="$A_r$")
    ax.plot(x, [v.a_o for v in values], "v", label="$A_o$")
    ax.set_yscale("log")
    ax.set_xticks(x, areas)
    ax.set_title(time_of_day)
axes[0].legend()

The verdict compares at the decimals the guide value is printed with, half up as a hand rounds, because that is what the standard does: its Example 4 forms a KB_FTr of 0,154, writes it as 0,15 and finds it at the A_r of 0,15, met. And it follows Example 3 on a KB_Fmax above A_u by less than the 15 % a measurement of KB_F is uncertain by: the requirement “can as a rule still be regarded as met”, the standard concludes, and the verdict says so, with within_uncertainty set for anyone who wants to read it more strictly.

Use a meter to DIN 45669-1, working from 1 Hz to 80 Hz, and check the chain before and after measuring, with a calibrator, a tap test or the vibration already there. Measure on the floor of the room itself, where the strongest vibration is expected; for the vertical that is usually the middle of the floor panel. Take z and two horizontals along the outer walls, x towards the source where possible; the horizontals may also go by a wall or in a door or window recess. Record the three at once, or one after another only when the vibration is steady, with a fourth channel near the source to tell it from other disturbances. Prefer a hard surface: set down loose, a transducer holds while the peaks stay at or below 3 m/s², its horizontals only to 40 Hz, and on a carpet it stands on the spiked device, about 2,5 kg with the transducer, pressed and tapped through the covering. Measure long enough to capture the characteristic exposure, mark an untypical state as such, and form KB_Fmax and, where it is needed, KB_FTr separately by day and by night, recording the rest hours apart. Clock maxima of 0,1 or less count as zero, because vibration that weak is as a rule not felt, and the clock still counts in N. A disturbance is kept out by interrupting the clock maxima, erasing at most the running clock and the one before it, and is judged, where possible, with the source running and stopped. The report names the institution and the person responsible, the purpose, the date and measuring times, the sources and how they ran, a site plan and the propagation conditions, the measuring and immission point, the directions and the coupling, the meter with make, type and number, its settings and the rest of the equipment, the quantities measured, the disturbances and the subjective observations.

KB_FTr is the clock maximum r.m.s. of each stretch of exposure, weighted by the share of the assessment period it lasts for (Formula (4a); with one stretch, Formula (4b)):

KB_FTr = √( (1 / T_r) · Σ_j T_e,j · KB²_FTm,j )

A stretch that falls in the rest hours of the day, 6:00 to 7:00 and 19:00 to 22:00 on working days and the whole day on Sundays and public holidays, carries the weight 2 (Formula (5)). Annex C works the same two forging hammers through both formulas: hammer a) for 6 h at a KB_FTm of 0,16 and hammer b) for 1,5 h at 0,39, first with both outside the rest hours and then with hammer b) moved into them.

from phonometry import vibration

hour = 3600.0
hammers = [0.16, 0.39]
hours = [6 * hour, 1.5 * hour]

# Annex C, Example 4: Formula (4a), both hammers outside the rest hours.
kb_ftr = vibration.assessment_vibration_severity(hammers, hours)
print(f"KB_FTr = {kb_ftr:.3f}")  # 0.154, which the standard writes as 0.15

# Example 5: hammer b) runs 19:00 to 20:30, in the rest hours, Formula (5).
evening = vibration.assessment_vibration_severity(
    hammers, hours, in_rest_time=[False, True]
)
print(f"KB_FTr = {evening:.2f}")  # 0.20

guide = vibration.guide_values("commercial")
print(vibration.assess_people_in_buildings(0.47, guide, kb_ftr=kb_ftr).complies)   # True
print(vibration.assess_people_in_buildings(0.47, guide, kb_ftr=evening).complies)  # False

The same formula turned around says how long a source may act before KB_FTr reaches A_r, which is what Example 2 asks: the sawmill of Example 1 in a residential area instead, where A_r is 0,07, may run for 1,48 h of the 16 h day.

from phonometry import vibration

exposure = vibration.admissible_exposure_s(0.23, 0.07)
print(f"{exposure / 3600:.2f} h")  # 1.48 h

Clause 6.5 adds a rule per kind of source, and each is one argument:

• Rare short events, up to three a day, blasting among them: KB_Fmax is compared with A_o alone, which rare_short_events=True asks for. Quarry blasting has more: blasts in immediate succession may count as one event, at most fifteen a week if they do; and blasts on working days with the neighbours warned, between 7:00 and 13:00 or 15:00 and 19:00, one a day, are held in a mixed or residential area to the A_o of an industrial one, 6, which source="quarry_blasting" asks for, with a KB_Fmax of 8 allowed a few times a year in exceptional cases.
• Road traffic uses the procedure as it stands, without the rest-time weighting.
• A railway is judged on A_u and A_r only, with the rest-time weighting not applied either, which source="railway" asks for; A_o is not a verdict for it, and 6.5.3.5 sets its own thresholds instead, a night-time clock maximum above 0,6 on a surface line or 0,3 underground being a reason to look into the cause, flat spots on wheels for one, and to put it right. An urban surface line, a tram, light rail or S-Bahn, gets A_u and A_r raised by the factor 1,5 with source="urban_railway". A new line is held to Table 1; an existing one often exceeds it, and the standard leaves that case to judgement.
• A construction site has its own Table 2, by how many working days it shakes the neighbours and by the stage the operator is held to, stage I below which no considerable annoyance is expected, stage II which needs the measures of 6.5.4.3, and stage III above which the exposure is unreasonable. The values for two to six days are interpolated between the one-day column and the column that starts at seven, as Figure 3 draws them; at night Table 1 applies; and the site’s blasting is held to an A_o of 8.

from phonometry import vibration

for days in (1, 3, 6, 7, 30):
    guide = vibration.construction_guide_values(days, stage="I")
    print(f"{days:2d} working days: A_u {guide.a_u:.2f}, A_r {guide.a_r:.2f}")
#  1 working days: A_u 0.80, A_r 0.40
#  3 working days: A_u 0.67, A_r 0.37
#  6 working days: A_u 0.47, A_r 0.32
#  7 working days: A_u 0.40, A_r 0.30
# 30 working days: A_u 0.30, A_r 0.20

urban = vibration.guide_values("residential", time_of_day="night", source="urban_railway")
print(urban)  # GuideValues(a_u=0.15, a_o=0.2, a_r=0.075, time_of_day='night', edition='1999')

The trains of a railway occupy a few clock intervals each and leave the rest quiet, so Annex A forms KB_FTm for each class of train over the intervals its trains occupied (Formula (A.1)), puts a standard deviation on its square (Formula (A.2)), and weights each class in KB_FTr by the intervals it occupies in the period, 1920 by day and 960 by night (Formula (A.3)). Example 8 does it for a ten-minute record with three passages, the third of which takes three intervals: the peak interval of each train is one class and the two flank intervals of the third train another, extrapolated to a day of 288 and 192 occupied intervals:

from phonometry import vibration

class_1 = [0.92, 0.6, 0.9]  # the clock maxima trains of class 1 occupied
class_2 = [0.2, 0.24]

kb_ftm = [vibration.railway_takt_maximum_rms(c) for c in (class_1, class_2)]
spread = [vibration.railway_takt_spread(c) for c in (class_1, class_2)]
print([round(v, 2) for v in kb_ftm])  # [0.82, 0.22]

railway = vibration.railway_assessment_severity(kb_ftm, [288, 192], spread=spread)
print(f"KB_FTr = {railway.kb_ftr:.3f}")  # 0.325
print(f"  from {railway.lower:.3f} to {railway.upper:.3f}")  # 0.253 to 0.384

guide = vibration.guide_values("residential")
verdict = vibration.assess_people_in_buildings(
    0.92, guide, kb_ftr=railway.kb_ftr, source="railway"
)
print(verdict.complies, verdict.criterion)  # False A_r

Annex D turns Formula (A.3) around into a figure: with one class of train and each train occupying one clock interval, how many an hour keep KB_FTr at A_r. It reads off 7 trains for the 0,05 of a dwelling at night and 14 for the 0,07 of a mixed area, both at a KB_FTm of 0,2, and the formula behind the figure is 120 (A_r / KB_FTm)².

import matplotlib.pyplot as plt
import numpy as np

from phonometry import vibration

trains = np.geomspace(0.5, 100.0, 200)
fig, ax = plt.subplots(figsize=(10, 6.2))
for a_r in (0.2, 0.15, 0.1, 0.07, 0.05):
    ax.loglog(trains, a_r * np.sqrt(120.0 / trains), label=f"$A_r$ = {a_r:g}")
for a_r in (0.05, 0.07):
    n = vibration.admissible_trains_per_hour(0.2, a_r)
    ax.plot([n], [0.2], "o")
ax.set_xlabel("Trains an hour")
ax.set_ylabel("$KB_{FTm}$")
ax.legend()

from phonometry import vibration

print(int(vibration.admissible_trains_per_hour(0.2, 0.05)))  # 7
print(int(vibration.admissible_trains_per_hour(0.2, 0.07)))  # 14

Where only an unweighted velocity record exists, Clause 7 estimates KB_Fmax from its peak and its frequency: Formula (6) is what the KB weighting would leave of a sine of that peak, with the 5,6 Hz corner of the meter, and Formula (7) scales it by an empirical factor of Table 3 for the kind of vibration, 0,9 for a clean harmonic signal down to 0,6 for a single short event with no resonance in the floor. The standard marks the result with an asterisk because it is an estimate, and puts the factors at about 15 % either way.

from phonometry import vibration

# Annex C, Example 7: a blast on a ceiling, 4 mm/s peak at 14 Hz.
estimate = vibration.kb_fmax_from_peak_velocity(4.0, 14.0, kind="single_event_resonant")
print(f"KB*_Fmax = {estimate:.1f}")  # 2.1, against an A_o of 3

Formula (A.1b) prints KB_FTm,j equal to a mean of squares with no root over it, while Formula (A.1a) beside it takes the root, Formula (A.2) beneath them uses KB²_FTm,j for the mean square, and Example 8 applies (A.1b) with the root. The library takes the root, the tests hold it to the example, and the errata page has the reading.

E DIN 4150-2:2023-08 is to replace this edition, and the library reads it with edition="2023" on guide_values and assess_people_in_buildings. It changes one cell of Table 1, the night A_u of a mixed area down to 0,1; no longer excuses a KB_Fmax within the 15 % above A_u; compares a railway with A_o and forms its numbers by category of train, with a weighting factor per kind of train, and a road by night not; and adds an existing road whose neighbours must put up with 50 % more, and an induced seismic event held to the daytime A_o. The guide values carry the edition they were read for and the verdict follows it. The railway and the rest of the changes have their own page.

from phonometry import vibration

print(vibration.guide_values("mixed", time_of_day="night", edition="2023"))
# GuideValues(a_u=0.1, a_o=0.3, a_r=0.07, time_of_day='night', edition='2023')

What this guide covers

Section titled “What this guide covers”

• The guide values of Table 1 by area, time of day and kind of source, the procedure of Clause 6.2 as a verdict with the criterion that decided it, and the assessment vibration severity of Formulae (4a), (4b) and (5) with the exposure A_r allows.

• The source rules of Clause 6.5: rare short events on A_o alone, the railway on A_u and A_r with the factor 1,5 of an urban surface line, and the construction site’s Table 2 by duration and stage with the interpolation of Figure 3.

• Annex A in full: the clock maximum r.m.s. of a class of train, the spread of its square, the assessment severity by classes and the interval the spread puts on it; and Figure D.1, trains an hour against A_r. And Clause 7, the estimate of KB_Fmax from a peak velocity with Table 3.

• Not covered

  No judgement of the cases the standard leaves open. An existing railway line, a construction site beyond 78 working days, a hospital next to one, and whatever 6.2 sends to an individual assessment are decided case by case in the standard, and are not decided here.

• The measurement is described, not checked. Where the transducers go, how they are coupled and how long a measurement runs are Clause 5 and DIN 45669-2, set out above for the person measuring. Of all that, the library puts a number only to the loose-mounting limits, the wax limit of Table 1 and the mass loading of 7.2.4, on the meter’s page; the quantities themselves come from the meter of DIN 45669-1, which forms them from a record.