Source: https://jmrplens.github.io/phonometry/vibration/immission/vibration-prediction/ # Predicting vibration before measuring (DIN 4150-1) Part 1 of DIN 4150 is the first question of the series. Before a quarry is blasted, a chimney felled, a pile driven or a line built, someone has to say how much vibration will reach the houses and how much of it the floors will feel, and Part 1 gives the shapes of every answer. It gives no recipe, and says so in its foreword: the constants of a real case come from trial blasts, from comparable ground, from a measurement with a few machines running. What it does fix, it fixes plainly, and its Annex A shows the shapes in the ground twenty-seven times over. Section through the ground from a vibration source to a five-storey building. On the left the source sits on the surface, its extent a dimensioned along the path, and a dashed red line marks the far-field boundary R1; under the ground a chain of dimensions counts it from the centre of the source as a over 2 plus the Rayleigh wavelength, and a longer dimension gives the distance R to a point further out. The band above the ground between the source centre and R1 is shaded as the near field, where the approximations do not hold. Beyond R1, in the far field of Formula (2), three blue arrows on the surface shrink along a dashed decay curve from v1 at R1 to v at R, a source near the surface sending mostly a Rayleigh wave, while dashed fronts under the surface stand for the body waves, compression and shear. On the right the foundation of the building rests on two springs and a damper, the stiffness kB of the ground and the system damping D0, under the building mass mB; the foundation is marked VF, one floor is drawn bending and marked VD, and a double arrow at the roof marks the horizontal frequency f1. Three panels below give the exponent n of Figure 1, 0 for a harmonic line source on a surface wave with 0.5 added for a point source, for impulsive excitation and for a body wave, up to 1.5, and 0.3 to 0.5 for a train of point sources; the attenuation coefficient alpha of about 2 pi D over lambda, with D at most 0.01 on loose ground in a first estimate, higher frequencies damped more, and 0.005 per metre for D of 0.01 and a wavelength of 12.5 metres; and the building: about 15 hertz for one to two storeys, 8 to 12 hertz for two to six and under 8 hertz above six on ground of medium stiffness, a foundation passing at most 2 at that frequency on loose ground, a mean of 0.5 above it and no reduction on rock, and a floor amplifying 10 to 25 at its resonance in reinforced concrete with a damping ratio between 0.02 and 0.05. Two boxes at the foot carry Formulae (1) to (4) and the two maximum transfer values, and a closing note says the transfer values assume the whole building excited in phase by predominantly harmonic vibration, and that for a source close by, moving or impulsive they lie on the safe side *The path clause 4 of Part 1 lays down for propagation and transfer: a near field where its approximations do not hold, the decay of Formula (2) beyond it, and a building that sits on its ground as a mass on a spring and hands the motion on through its foundation to its floors.* ## 1. How far vibration carries Every source has a near field and a far field, and the boundary is `R₁ = a/2 + λ_R` (Formula (1)): half the extent of the source along the direction of propagation plus a wavelength of the surface wave. Nearer than that nothing here holds. Beyond it the velocity amplitude decays as ```text v = v₁ · (R / R₁)^-n · exp[-α (R - R₁)] ``` (Formula (2)): geometric spreading with an exponent `n` that Figure 1 fixes by three yes-or-no questions, and material damping with `α ≈ 2π D / λ`, the damping ratio of the ground over the wavelength that matters. The exponent is 0 for a harmonic line source carried by a surface wave and gains 0,5 for each of point instead of line, impulsive instead of harmonic, and body wave instead of surface wave, up to 1,5 for an impulsive point source in a body wave. A train is a chain of point sources not excited in phase and decays with something between 0,3 and 0,5. For loose ground a first estimate may take a damping ratio of 0,01 at most; more has to be proven. ```python from phonometry import vibration print(vibration.geometric_exponent(geometry="point", character="impulsive", wave="surface")) # 1.0 print(vibration.geometric_exponent(geometry="line", character="harmonic", wave="surface")) # 0.0 print(vibration.reference_distance_m(6.0, rayleigh_wavelength_m=12.5)) # 15.5 # Annex A, Figure A.19: a machine hall measured at 0,44 mm/s 13 m away, # with D = 0,01 and a wavelength of 12,5 m. alpha = vibration.attenuation_coefficient_per_m(0.01, wavelength_m=12.5) print(f"{alpha:.4f} 1/m") # 0.0050 1/m decay = vibration.far_field_velocity_mm_s( 0.44, [20.0, 40.0, 80.0], reference_distance_m=13.0, exponent=1.0, attenuation_per_m=alpha ) print(decay.round(3)) # [0.276 0.125 0.051] ``` Vertical velocity in millimetres per second against distance from 0 to 80 metres, on an axis from 0 to 0.5. Three curves start together at 0.44 at 13 metres: a green solid line for an exponent of 0 that falls gently to 0.31 at 80 metres from the damping alone, a blue dashed line for 0.5 that reaches 0.13, and a red dotted line for 1 that reaches 0.05. Four black crosses are the measured points, at 13, 23, 43 and 73 metres, and they fall between the two lower curves and then below them
Figure code ```python import matplotlib.pyplot as plt import numpy as np from phonometry import vibration distances = np.linspace(13.0, 80.0, 200) fig, ax = plt.subplots(figsize=(10, 6.2)) for exponent in (0.0, 0.5, 1.0): ax.plot( distances, vibration.far_field_velocity_mm_s( 0.44, distances, reference_distance_m=13.0, exponent=exponent, attenuation_per_m=0.005 ), label=f"$n$ = {exponent:g}", ) ax.plot([13.0, 23.0, 43.0, 73.0], [0.44, 0.27, 0.09, 0.015], "k+", markersize=11, label="measured") ax.set_xlabel("Distance [m]") ax.set_ylabel("Vertical velocity [mm/s]") ax.legend() ```
The damping alone is worth drawing, and Figure 2 draws it for a damping ratio of 0,01 and a wave speed of 200 m/s: over 100 m the ground takes 27 % of the amplitude at 10 Hz and 79 % at 50 Hz, which is why the low frequencies are what arrives at a distance. ```python from phonometry import vibration for frequency in (10.0, 50.0): left = vibration.material_damping_factor( [100.0], damping_ratio=0.01, frequency_hz=frequency, wave_speed_m_s=200.0 )[0] print(f"{frequency:g} Hz: {left:.3f} of the amplitude left after 100 m") # 10 Hz: 0.730 of the amplitude left after 100 m # 50 Hz: 0.208 of the amplitude left after 100 m ``` ## 2. Into the building, and up the floors A building on the ground is a mass on a spring, with the natural frequency of Formula (3), `f_B = √(k_B / m_B) / 2π`, for the vertical direction and predominantly harmonic vibration: about 15 Hz for one or two storeys, 8 Hz to 12 Hz for two to six, under 8 Hz above that, on a ground of medium stiffness with a shear wave speed of 150 m/s to 200 m/s. At that frequency the foundation passes at most `1 / (2 D₀)` of the ground's amplitude, and for loose ground `D₀` may be taken as 0,25, so the foundation amplifies by 2 at most; above it a mean transfer of 0,5 may be assumed, and on rock there is no reduction at all. A floor then amplifies by at most `1 / (2 D₁)` at its own resonance, 10 to 25 for a concrete floor with a damping ratio between 0,05 and 0,02, on the assumption that the building is excited in phase over its whole footprint, which is the safe side for a source that is close, moving or impulsive. And the lowest horizontal natural frequency of a building of five storeys or more is about `10 / n` Hz (Formula (4)), which matters where a tall slender building meets a low excitation frequency. ```python from phonometry import vibration print(f"{vibration.soil_building_natural_frequency_hz(2.5e9, mass_kg=4.0e5):.1f} Hz") # 12.6 Hz print(vibration.soil_building_frequency_guide_hz(4)) # (8.0, 12.0) print(vibration.foundation_transfer_max()) # 2.0 print(f"{vibration.floor_transfer_max(0.03):.1f}") # 16.7 print(vibration.storey_frequency_hz(8)) # 1.25 ``` ## 3. Single events: a blast and a falling mass In the far field a blast follows `v_max = k (L/L₀)^b (R/R₀)^-m` (Formula (5)) with the charge per delay `L` against 1 kg, the distance against 1 m, and three constants that come from trial blasts or from comparable cases in ground, method and range of distance, with allowance for scatter; the standard prints none of them. A falling mass follows the same with the root of its fall energy `G · h` (Formula (6)), which for a felled chimney is usually the larger source, the demolition blast the smaller. Quarry blasting is very rarely relevant beyond 1500 m, construction blasting beyond 400 m. ```python from phonometry import vibration # Constants of the user's own trial blasts; the standard prints none. peak = vibration.blast_peak_velocity_mm_s( 50.0, [300.0, 600.0], coefficient_mm_s=1200.0, charge_exponent=0.6, distance_exponent=1.7 ) print(peak.round(2)) # [0.77 0.24] # A 2 600 t chimney, its centre of mass 70 m up: 25 506 kN through 70 m. energy = vibration.fall_energy_kj(25506.0, drop_height_m=70.0) print(f"{energy:.0f} kJ") # 1785420 kJ impact = vibration.impact_peak_velocity_mm_s(energy, [50.0, 150.0], coefficient_mm_s=0.02, distance_exponent=1.0) print(impact.round(1)) # [0.5 0.2] ``` ## 4. What a track and a hall of machines excite at A train excites the ground at its speed over the spacing of whatever repeats along the track or around the wheel, the sleepers first at 0,6 m to 0,9 m, and at the multiples of that, while the parts of the vehicle keep their own frequencies whatever the speed: the car body on its secondary suspension at 1 Hz to 3 Hz, the bogie on its primary at 6 Hz to 10 Hz. Ballasted track passes 40 Hz to 80 Hz on preferentially, a tunnel with under-ballast mats 15 Hz to 40 Hz, a mass-spring system 5 Hz to 20 Hz, and rail vibration reaches about 80 m, further on soft layers. ```python from phonometry import vibration print(vibration.track_excitation_frequency_hz(80 / 3.6, spacing_m=0.6, harmonics=2).round(1)) # [37. 74.1] print(vibration.TRACK_TRANSMITTED_BANDS_HZ["ballast"]) # (40.0, 80.0) ``` A hall of similar machines running together gives, at a point outside, `v_N = χ · v_B · √N` (Formula (7)): the velocity measured with `N_B` of them running, scaled to `N` with a correction `χ` that the standard prints only as a nomogram, Figure 3, for measurements made with 3, 5, 10, 30, 60 or 100 machines. The nomogram is here as its six curves read off the page at a five-hundredth, the width of their stroke, and the standard's own check of it, Figure A.18, drawn for a hall measured at 0,44 mm/s with three machines running, is reproduced within 6 %: the measurements follow the curve up to about sixty machines, the nearest group, and stay flat beyond, because the groups added after that are further off. ```python from phonometry import vibration print(vibration.machine_count_correction([12.0], reference_count=3).round(3)) # [0.448] print(vibration.machine_hall_velocity_mm_s(0.44, [12.0, 44.0, 56.0], reference_count=3).round(2)) # [0.68 0.89 0.97], against 0.57, 0.95 and 1.04 measured ``` Six decreasing curves of the correction chi against the number of machines running from 4 to 100, on an axis from 0 to 0.6, one curve per reference count of 3, 5, 10, 30, 60 and 100 machines, the top curve starting at 0.55 and flattening to 0.29 and the bottom one starting at 0.19 and flattening to 0.10. Small dots mark the points read off the printed nomogram, and a note says they were read at a five-hundredth
Figure code ```python import matplotlib.pyplot as plt from phonometry import vibration fig, ax = plt.subplots(figsize=(10, 6.2)) for reference, chi in vibration.MACHINE_COUNT_CORRECTION.items(): ax.plot(vibration.MACHINE_COUNT_AXIS, chi, marker="o", markersize=3, label=f"$N_B$ = {reference}") ax.set_xlabel("Machines running") ax.set_ylabel(r"Correction $\chi$") ax.legend(ncol=2) ```
## 5. Four things printed wrong The symbol lists of Formulae (5) and (6) give the distance in millimetres against a reference of one metre, which Formula (2) and every axis of Annex A say is metres. Clause 5.2.3 speaks of vibratory drivers with a low working frequency and writes `f > 30 Hz`, the sign the wrong way round after a paragraph that calls the high frequencies above 35 Hz the favourable ones. Figure A.2's legend gives the continuous line to the vertical component and the dash-dot line to the radial one, and the drawing has them the other way about. A.5.1 prints an eccentric moment in newtons. And the sixth measured point of Figure A.18, labelled all groups of a hall of 252 machines, sits at 110, the count of the groups of the hall nearest the measuring point. The [errata page](https://jmrplens.github.io/phonometry/reference/errata/) has all five with their pages. ## What this guide covers **Propagation**: the far-field boundary of Formula (1), the decay of Formula (2) with the exponents of Figure 1 and the damping of the ground, and the damping factor of Figure 2 on its own. **The building**: the natural frequency of Formula (3) and its guide values by storeys, the transfer of a foundation and a floor at resonance, and the storey formula of Formula (4). **The sources**: the blast and the falling mass of Formulae (5) and (6), the machine hall of Formula (7) with the nomogram of Figure 3 read off the page, the excitation frequencies of a track, and the frequency bands, ranges and constants Clause 5 gives its sources. **No constants for a real case.** The `k`, `b` and `m` of a blast and the `k` and `m` of an impact are the user's; the standard prints none and the library carries none. The measured cases of Annex A are described in the standard and not reproduced here, and its two figures drawn from the formulas are held as conformance rows, not offered as data. **No near field.** Nearer than `R₁` the standard asks for a numerical or experimental investigation of its own, and the library refuses the distance. ## See also - [Predicting railway vibration (E DIN 45672-3)](https://jmrplens.github.io/phonometry/vibration/immission/railway-prediction/): the same decay law as a third-octave chain, with the building's transfer as tables. - [Vibration and people in buildings (DIN 4150-2)](https://jmrplens.github.io/phonometry/vibration/immission/people-in-buildings/) and [Vibration damage to structures (DIN 4150-3)](https://jmrplens.github.io/phonometry/vibration/structural/structural-damage/): what the predicted numbers are judged against. - [Vibration next to a railway (DIN 45672)](https://jmrplens.github.io/phonometry/vibration/immission/railway-vibration/): the wave speeds of the ground that the wavelength here comes from. - API reference: [`vibration.immission.prediction`](https://jmrplens.github.io/phonometry/reference/api/vibration/prediction/). ## References - Deutsches Institut für Normung. (2001). *Erschütterungen im Bauwesen — Teil 1: Vorermittlung von Schwingungsgrößen* (DIN 4150-1:2001-06). Formulae (1) to (7), the exponents of Figure 1, the damping curves of Figure 2, the nomogram of Figure 3 read off the page, and the guide values and frequency bands Clauses 4 and 5 give for the ground, the building and each kind of source. The measured cases of Annex A are described, not implemented; Figures A.18 and A.19, which the standard draws from Formulae (7) and (2) with every parameter printed, are the conformance rows.