Evaluating machine vibration

Machine fault frequencies answers where a fault would show. This page answers the question a plant asks first: is this machine acceptable at all. The two are different trades. A kinematic line tells you what is wrong; a severity grade tells you whether to keep running. It comes from one broad-band number, and 4.3.3 says which one: the largest of the magnitudes measured over the agreed positions and directions.

ISO 20816-1:2016 is the basis document of the ISO 20816 series, which merged and replaced ISO 10816-1 and ISO 7919-1. It fixes the shape of the judgement and leaves the numbers to the machine-specific parts: Part 2 for large gas turbines, Part 3 for industrial machines, Part 4 for gas turbines, Part 5 for hydraulic plant, Part 8 for reciprocating compressors, Part 9 for gear units.

Two kinds of measurement, and a machine may carry both.

On non-rotating parts, an accelerometer or velocity transducer on the bearing, its support housing, or another structural part that responds to the dynamic forces coming through the bearing. Clause 4.4.1 asks for three mutually perpendicular directions at each position for acceptance testing; operational monitoring is usually met by one or both radial directions, and the axial one is evaluated only on thrust bearings, where direct axial dynamic force is transmitted.

On the rotating shaft, a pair of non-contacting probes reading the shaft displacement directly. Clause 4.4.2.1 places them radially in one transverse plane, their axes within 5° of a radial line, preferably 90° ± 5° apart on the same bearing half, and at the same positions on every bearing of the machine. A single probe per plane is allowed only where it is known to give adequate information.

The two measurement families of Clause 4.4, on one train. The axial arrow is drawn as a foreshortened diagonal because it leaves the plane of an elevation.

The measured quantity is broad-band, and the vibration severity is the largest magnitude found at any bearing at rated speed under steady operation (4.3.3). It is that single number the zones grade.

Clause 6.3.2.3 grades rather than judges:

Three boundaries separate them, and evaluation_zone is the comparison itself. It is deliberately blind to the quantity: the parts of the series set boundaries on shaft displacement, housing velocity or housing acceleration, and the grading is the same in all three. A magnitude exactly on a boundary is the top of the zone below it, which is how the tables print their limits.

from phonometry import vibration

# Three boundaries from whichever part of ISO 20816 applies, in mm/s r.m.s.
zones = vibration.ZoneBoundaries(2.8, 7.1, 11.2)
print(vibration.evaluation_zone([0.9, 3.4, 8.0, 20.0], zones))   # ['A' 'B' 'C' 'D']
print(vibration.evaluation_zone(2.8, zones))                     # A, the limit of A

The boundaries have to rise through the zones, and ZoneBoundaries refuses a set that does not: out of order they would grade a good machine as a bad one and say nothing.

Velocity carries a severity criterion over a wide speed range, which is why the tables are written in it. But a single velocity limit regardless of frequency allows unacceptable displacement at low frequency and unacceptable acceleration at high frequency (6.3.2.1). So the criterion of Figure 9 is flat only between two corner frequencies, and slopes outside them. Annex C.2 writes it as Formula (C.1), and allowable_velocity is that formula:

One curve, moved onto the three boundaries by the factor Zbound. Below the lower corner the criterion holds displacement constant, above the upper one it holds acceleration constant.

import matplotlib.pyplot as plt
import numpy as np
from phonometry import vibration

freq = np.logspace(np.log10(2.0), np.log10(3000.0), 600)
common = dict(constant_velocity_mm_s=1.12, corner_low_hz=10.0,
              corner_high_hz=1000.0)
for zone, factor in vibration.ZONE_LIMIT_FACTORS.items():
    plt.plot(freq, vibration.allowable_velocity(freq, zone_factor=factor,
                                                **common),
             label=f"limit of zone {zone}")
plt.xscale("log"); plt.yscale("log"); plt.legend()
plt.xlabel("Frequency [Hz]"); plt.ylabel("Allowable r.m.s. velocity (mm/s)")
plt.show()

The factor Zbound moves the one curve onto the three limits, and Annex C.2 prints what it takes: 1 for the limit of zone A, 2.56 for zone B and 6.4 for zone C. Those are 1.6 squared and 1.6 to the fourth, near enough, and the ladder of Table C.1 steps by about 1.6 as well, so the three limits land close to its rungs without falling on them.

from phonometry import vibration

common = dict(constant_velocity_mm_s=1.12, corner_low_hz=10.0,
              corner_high_hz=1000.0)
for zone, factor in vibration.ZONE_LIMIT_FACTORS.items():
    limit = vibration.allowable_velocity(100.0, zone_factor=factor, **common)
    print(zone, round(float(limit), 2))       # A 1.12 / B 2.87 / C 7.17

The corner frequencies and the exponents belong to the machine, and the specific parts state them. The defaults of 1 are the physical reading of the shape rather than a value taken from anywhere: constant displacement below the lower corner, which makes the allowable velocity rise at 6 dB per octave, and constant acceleration above the upper one, which makes it fall at the same rate. The curve is lowest at both ends and flat in between.

Criterion I judges a magnitude. Criterion II (6.3.3) judges a change from an established baseline, and it exists because a machine can go wrong inside zone B: a significant increase or decrease can occur, and require action, long before zone C is reached.

The trap is that a broad-band magnitude cannot express a change. Annex D makes the point with a machine whose vibration fell from 3 mm/s to 2.5 mm/s while its phase swung from 40° to 180°:

The magnitude fell by half a millimetre per second. The vibration moved by 5.2, ten times as much.

from phonometry import vibration

change = vibration.vibration_vector_change(3.0, 40.0, 2.5, 180.0)
print(round(change.magnitude, 2))          # 5.17 mm/s: the real change
print(round(change.magnitude_change, 2))   # -0.5 mm/s: what magnitudes say

This is why phase belongs in a trend record. A component that swings while its contribution falls changes the machine, and a magnitude-only criterion sees a small improvement.

For a machine covered by a part of the series, from that part. For one that is not, and where no experience is available, Annex C.1 offers Table C.1: a ladder of preferred magnitudes and the range each boundary is typically drawn from, with small machines (electric motors up to 15 kW) at the low end and large ones on flexible supports at the high end.

from phonometry import vibration

print(vibration.TYPICAL_ZONE_BOUNDARY_RANGES_MM_S["B/C"])   # (1.8, 9.3)
print(vibration.TYPICAL_BOUNDARY_LADDER_MM_S[:5])           # the first rungs

The ranges overlap on purpose: a large machine’s A/B boundary can sit above a small machine’s B/C one, which is what makes them ranges rather than limits. They are a starting point for agreement between supplier and customer, and the standard says plainly that the values assigned to zone boundaries are not themselves intended to serve as acceptance specifications (6.3.2.5).

Part 1 fixes the shape of the judgement and declines to give boundaries. ISO 10816-3 gives them, for the machines an industrial site is actually made of: rotary compressors, generators, electric motors of any type, blowers and fans, steam turbines up to 50 MW and industrial gas turbines up to 3 MW, all measured in situ on the non-rotating parts.

Two questions place a machine in one of four classes.

How big it is (4.2). Group 1 is a large machine with rated power above 300 kW and not more than 50 MW, or an electrical machine of shaft height 315 mm or more. Group 2 is a medium-sized machine above 15 kW and up to and including 300 kW, or an electrical machine of shaft height from 160 mm up to but not including 315 mm.

What it stands on (4.3). If the lowest natural frequency of the combined machine and support system, in the direction being measured, is at least 25 % above the main excitation frequency, which is usually the running speed, the support is rigid in that direction. Every other support is flexible. The clause is explicit that the two can differ within one machine: a foundation stiff vertically and soft horizontally is judged by the rigid table in one direction and the flexible table in the other.

Each of the four classes is then stated twice, once in displacement and once in velocity, and 5.2.3 settles the disagreement: the more restrictive severity zone applies.

The same machine, read two ways. Velocity says zone B; displacement says zone C; the grade is C.

from phonometry import vibration

print(vibration.industrial_machine_zone("group_2", "rigid", velocity_mm_s=2.0))
# B
print(vibration.industrial_machine_zone("group_2", "rigid", displacement_um=50.0))
# C
print(
    vibration.industrial_machine_zone(
        "group_2", "rigid", displacement_um=50.0, velocity_mm_s=2.0
    )
)  # C

Give whichever quantities were measured. Annex A says velocity alone is enough in most cases, and that a machine whose spectrum is expected to carry low-frequency components should be judged on both. The limits are broad-band r.m.s. values between 10 Hz and 1 kHz, or from 2 Hz for a machine running below 600 r/min.

The edition implemented here is ISO 10816-3:2009, which ISO 20816-3:2022 has since replaced. Part 3 is the one part of the series not held here, so the boundaries come from its direct predecessor, and this page says so rather than implying a currency it does not have.

Part 1 describes the two operational limits and leaves their values to the parts. Part 3 gives them, and the two are set from different things.

A significant change (5.3) is an increase or a decrease of more than 25 % of the upper limit of zone B, measured at the same transducer location and orientation and under the same operating conditions. That is Criterion II made arithmetic.

An ALARM (5.4.1) sits 25 % of the upper limit of zone B above the established baseline for that position, and should not normally exceed 1.25 times that limit. Both halves matter. Set from the baseline, an alarm on a quiet machine can fall well inside zone B, which is the point of it. Capped at 1.25 times the limit, an alarm on a machine whose baseline has crept up cannot follow it indefinitely.

A TRIP (5.4.2) relates to the mechanical integrity of the machine, so it is generally the same for every machine of a design and is not tied to the baseline. It will normally lie within zone C or D, and should not exceed 1.25 times the upper limit of zone C.

The quiet machine gets an alarm inside zone B. The one that has drifted runs into the cap.

from phonometry import vibration

limits = vibration.INDUSTRIAL_MACHINE_ZONES["group_2", "rigid"].velocity_mm_s

print(vibration.is_significant_change(0.8, limits.b_c))   # True
print(vibration.is_significant_change(0.6, limits.b_c))   # False
print(round(vibration.alarm_limit(0.9, limits.b_c), 3))   # 1.6
print(round(vibration.alarm_limit(2.9, limits.b_c), 3))   # 3.5, at the cap
print(round(vibration.trip_limit(limits.c_d), 3))         # 5.625

Part 3 asks what the machine is and what it stands on. Part 9 asks a different question: it grades a gear unit against a rating number, chosen for the unit at the start of negotiation, and the rating indexes a row of boundaries.

There are three ratings, one per quantity, and Table 1 fixes the vocabulary of each: DR for shaft relative peak-to-peak displacement in micrometres, VR for housing r.m.s. velocity in millimetres per second, and AR for housing true peak acceleration in metres per second squared. Tables 2, 3 and 4 give the boundaries.

Those three tables are one ladder seen three times. Every printed row is three consecutive rungs of 2, 3.15, 5, 8, 12.5, 20, 31.5, 50, 80, 125, 200, 315, with the rating itself as the B/C boundary and its neighbours as the other two. So choosing a rating is choosing a rung, and the library refuses a rating between two rows rather than interpolating one: the ladder is an agreement, not a continuum.

Table 5 is where a unit gets its ratings. Class I is special-purpose precision parallel-shaft units, class II general-purpose parallel-shaft, helical and spiral-bevel units, class III epicyclic units, and class IV straight-cut units; subclass a) covers any power and subclass b) splits by it.

from phonometry import vibration

ratings = vibration.GEAR_UNIT_CLASSES["III", "a"]
print(ratings.displacement, ratings.velocity, ratings.acceleration)
# 80.0 8.0 125.0

boundaries = vibration.gear_unit_zone_boundaries("velocity", ratings.velocity)
print(boundaries.as_tuple)                          # (5.0, 8.0, 12.5)
print(vibration.evaluation_zone(6.0, boundaries))   # B

Only the subclass a) rows carry an acceleration rating. Every b) row prints “no information available at this time”, and the library carries that as None rather than filling the gap with a plausible number.

A rating is not only a set of three boundaries: Annex A draws it as a curve against frequency, so a filtered measurement can be judged line by line rather than as one broad-band value. The rating of a shaft or a position is then the lowest curve that encloses its whole spectrum.

Left, Figure A.1; right, Figure A.2. The right-hand family is the criterion of section 3 again, with the corners and the slope Part 9 states.

from phonometry import vibration

print(round(float(vibration.gear_housing_velocity_limit(3000.0, rating=8.0)), 2))
# 5.13 mm/s: a 3 kHz mesh line is judged against less than the rating
print(round(float(vibration.gear_shaft_displacement_limit(500.0, rating=80.0)), 1))
# 25.3 µm

The velocity curve is worth a second look. Flat between two corners and falling outside them at one rate is exactly Formula (C.1) of Part 1, and this is that formula with $f_x$ = 45 Hz, $f_y$ = 1590 Hz and $k=m=0.7$, which is 14 dB per decade. The library computes it by calling allowable_velocity, so the two parts of the series share one curve rather than two implementations of it.

ALARM and TRIP: 8.2 sends the reader to Part 1 for them, which is section 7 of this page. Acceptance criteria: 8.3 says they are agreed between manufacturer and customer, are historically specified inside zone A or B, and would normally not exceed 1.25 times the A/B boundary, which for a class II unit judged on velocity is 3.94 mm/s.

Measurement is Clause 6. Shaft displacement is read relative to the housing by non-contacting probes in orthogonal pairs through the journal bearing housing, and the combined mechanical and electrical runout should not exceed a quarter of the allowable displacement at shaft rotational frequency, or 6 µm, whichever is greater. Housing vibration is read on a rigid section such as a bearing block, in up to three orthogonal directions, two of them perpendicular to the gear axis; a housing panel that supports no bearing does not give a true indication of the unit.

• Covered

  The four evaluation zones and the grading of 6.3.2.3, the frequency-shaped criterion of Figure 9 as Formula (C.1) with the zone factors of Annex C.2, the typical ranges and ladder of Table C.1, and the vector reading of a change from Annex D, through ZoneBoundaries, evaluation_zone, allowable_velocity and vibration_vector_change. For industrial machines, the boundaries of Tables A.1 and A.2 in both quantities, the most-restrictive rule of 5.2.3, the significant change of 5.3 and the ALARM and TRIP settings of 5.4, through INDUSTRIAL_MACHINE_ZONES, industrial_machine_zone, is_significant_change, alarm_limit and trip_limit. For gear units, the three rating tables and the classification of ISO 20816-9, and the two rating curves of its Annex A, through GEAR_UNIT_ZONES, GEAR_UNIT_CLASSES, gear_unit_zone_boundaries, gear_shaft_displacement_limit and gear_housing_velocity_limit.

• Not covered

  The zone boundaries of the machine-specific parts this page does not name, which it takes as inputs; the shaft criterion of Figure 10, whose values Annex C.1 declines to give for machines no part covers, on the grounds that such machines are not normally fitted with shaft transducers; and the current third edition of Part 3, ISO 20816-3:2022, which is not held here, so its predecessor supplies the industrial boundaries.