vibration.machinery.diagnostics

Kinematic fault frequencies of rotating machinery (Norton & Karczub Ch. 8).

Condition monitoring starts with arithmetic, not with signal processing: every rolling-contact bearing, gear pair, induction motor and bladed rotor excites a family of discrete frequencies fixed by its geometry and its shaft speed. Knowing where those lines fall turns a featureless envelope spectrum into a diagnosis, because a peak is only evidence when it sits on a named line.

This module computes the families set out in M. P. Norton and D. G. Karczub, Fundamentals of Noise and Vibration Analysis for Engineers (2nd ed., CUP 2003), Section 8.4 (8.4.1 gears, 8.4.3 bearings, 8.4.4 fans and blowers, 8.4.7 pumps, 8.4.8 electrical equipment), and hands them to the signal chain that already exists in phonometry.signals: band-pass the structural resonance the defect impacts ring, detect its envelope and transform it (envelope_spectrum), average synchronously with the shaft (time_synchronous_average) or collapse the harmonic families in the cepstrum (cepstrum). The result object’s FaultFrequencyResult.plot draws the predicted lines on top of a measured envelope spectrum, which is the working view.

Rolling-contact bearings (Eqs. 8.4 to 8.14, after Shahan & Kamperman). With shaft speed N in r/min, Z rolling elements of diameter d on a pitch diameter D and a contact angle phi, writing $g=(d/D)\cos \varphi$ and $f_{\mathrm{s}}=N/60$:

$$\mathrm{FTF}=\frac{f_{\mathrm{s}}}{2}(1-g)\text{cage, stationary outer race}\text{\hspace{1em}(8.5)}$$ $$\mathrm{FT}{\mathrm{F}}_{\mathrm{rel}}=f_{\mathrm{s}}-\mathrm{FTF}\text{cage seen from the shaft}\text{\hspace{1em}(8.11)}$$ $$\mathrm{BSF}=\frac{f_{\mathrm{s}}}{2}\frac{D}{d}(1-g^2)\text{element rotation}\text{\hspace{1em}(8.7)}$$ $$\mathrm{BDF}=2\mathrm{BSF}\text{element spin (both races)}\text{\hspace{1em}(8.10)}$$ $$\mathrm{BPFO}=Z\frac{f_{\mathrm{s}}}{2}(1-g)\text{element pass, outer race}\text{\hspace{1em}(8.8)}$$ $$\mathrm{BPFI}=Z\frac{f_{\mathrm{s}}}{2}(1+g)\text{element pass, inner race}\text{\hspace{1em}(8.9)}$$

so $\mathrm{BPFO}+\mathrm{BPFI}=Z{f}_{\mathrm{s}}$ exactly, and $\mathrm{BPFO}=Z\mathrm{FTF}$. Norton notes that Eqs. (8.8) and (8.14), and (8.9) and (8.13), are identical: BPFO and BPFI do not depend on which race turns. Only the cage does, and with the outer race rotating it becomes $(f_{\mathrm{s}}/2)(1+g)$ (Eq. 8.6).

Gears (Eq. 8.3). The gear-meshing (tooth-passing) frequency of a wheel with n_teeth teeth is $\mathrm{GMF}=n_{\text{teeth}}{f}_{\mathrm{s}}$, with integer harmonics. A discrete tooth fault adds shaft-rate lines and low, flat sidebands around every mesh harmonic; distributed wear raises tall sideband groups at $k\mathrm{GMF}\pm m{f}_{\mathrm{s}}$.

Induction motors (Eqs. 8.19, 8.20). The three lines always present in a motor bearing signal are fs (mechanical unbalance), $2f_{\mathrm{s}}$ (misalignment with the driven load) and $2f_{\mathrm{e}}$ (non-uniform air gap, torque pulses and the electrical faults). Norton’s Eq. (8.19) writes the supply frequency as $f_{\mathrm{e}}=f_{\mathrm{s}}p/2$ for p magnetic poles, which is the synchronous, zero-slip form; this module uses the slip-consistent $f_{\mathrm{e}}=f_{\mathrm{s}}p/(2(1-s))$, identical at $s=0$ and the only version that makes the rotor-slot harmonics of Eq. (8.20),

$$f_{\mathrm{sh}}=f_{\mathrm{e}}\left[\frac{2R}{p}(1-s)\pm 2(n-1)\right]\text{\hspace{1em}(8.20)}$$

collapse to the physical rotor-bar passing rate $R{f}_{\mathrm{s}}$ at $n=1$ for R rotor bars. Dynamic eccentricity dresses that line with sidebands at $\pm$ the shaft rate and $\pm$ the slip frequency.

Fans, blowers and pumps (Eqs. 8.15 to 8.18). The blade-passing frequency of an impeller with N blades is $f_{\mathrm{b}}=nN{f}_{\mathrm{s}}$; a pump’s hydraulic pulsations follow the same form with N pumping events per revolution (Eq. 8.18), and a rotary positive-displacement blower repeats four times per revolution. In a ducted axial fan the blade-vane interaction sets up rotating pressure patterns with $m_{\mathrm{L}}=nN\pm kV$ lobes for V vanes (Eq. 8.16), turning at $nN{f}_{\mathrm{s}}/m_{\mathrm{L}}$ (Eq. 8.17): a pattern that spins faster than the blades themselves, and the reason a careful choice of N and V matters for radiated power.

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bearing_fault_frequencies

bearing_fault_frequencies(
    speed_rpm: float,
    n_elements: int,
    element_diameter: float,
    pitch_diameter: float,
    *,
    contact_angle_deg: float = 0.0,
    rotating_race: str = 'inner',
) -> FaultFrequencyResult

Kinematic frequencies of a rolling-contact bearing (Norton 8.4-8.14).

Returns the seven lines a bearing generates, named with the acronyms used in condition monitoring:

============ ================================================= ========== Name Meaning Norton eq. ============ ================================================= ========== shaft shaft rotational frequency fs (8.4) FTF cage (fundamental train) frequency (8.5)/(8.6) FTF_rel cage rotation relative to the rotating race (8.11)/(8.12) BSF rolling-element rotational frequency (8.7) BDF rolling-element spin frequency, 2 BSF (8.10) BPFO element pass frequency on the outer race (8.8)/(8.14) BPFI element pass frequency on the inner race (8.9)/(8.13) ============ ================================================= ==========

BPFO and BPFI are the outer- and inner-race defect lines and BDF the rolling-element/cage defect line; all three are exact kinematics of a pure-rolling contact, so a real bearing’s lines wander by 1 % to 2 % with load-dependent slip. $\mathrm{BPFO}+\mathrm{BPFI}=Z{f}_{\mathrm{s}}$ always, and both are independent of which race turns (Norton’s Eqs. 8.8 and 8.14, and 8.9 and 8.13, are identical); rotating_race only moves the cage.

Parameters

Name	Description
speed_rpm	Shaft speed N, in r/min (> 0).
n_elements	Number of rolling elements Z (integer >= 1).
element_diameter	Rolling-element diameter d, in the same unit as pitch_diameter (> 0); only the ratio d/D enters.
pitch_diameter	Bearing pitch diameter D (> d).
contact_angle_deg	Contact angle phi between element and raceway, in degrees (Default: 0, a radial ball bearing); 0 <= phi < 90.
rotating_race	Which race turns with the shaft, "inner" (Default) or "outer".

Returns: A FaultFrequencyResult (source "rolling-contact bearing").

Raises

Exception	When
ValueError	for a non-positive or inconsistent geometry.

blade_pass_frequencies

blade_pass_frequencies(
    speed_rpm: float,
    n_blades: int,
    *,
    harmonics: int = 3,
    n_vanes: int | None = None,
    lobe_orders: int = 1,
) -> FaultFrequencyResult

Blade-passing frequency and interaction patterns (Norton 8.15-8.17).

$f_{\mathrm{b}}=n{N}_{\text{blades}}\times N/60$ for n = 1 .. harmonics, the discrete tone family of any fan, blower or pump impeller; a rotary positive-displacement blower repeats four times per revolution, so pass 4 x its speed.

In a ducted axial fan the blades also interact with the stator vanes and set up rotating pressure patterns with $m_{\mathrm{L}}=n{N}_{\text{blades}}\pm k{N}_{\text{vanes}}$ lobes (Eq. 8.16) turning at $M_{\mathrm{L}}=n{N}_{\text{blades}}{f}_{\mathrm{s}}/m_{\mathrm{L}}$ (Eq. 8.17). Those patterns radiate strongly when they can drive a higher-order duct mode, so they are the lines to check against the duct cut-on frequencies. Give n_vanes to include them, named "lobe n=1 m=2", "lobe n=1 m=10", … The blade harmonic is part of the name because Eq. (8.17) carries n: the same lobe count reached from a different harmonic is a distinct pattern turning at a different speed.

Parameters

Name	Description
speed_rpm	Shaft speed N, in r/min (> 0).
n_blades	Number of blades (integer >= 1).
harmonics	Number of blade-pass harmonics (integer >= 1, Default: 3).
n_vanes	Number of stator vanes V (integer >= 1) to include the lobed interaction patterns (Default: None, blade tones only).
lobe_orders	Highest |k| in $m_{\mathrm{L}}=n{N}_{\text{blades}}\pm k{N}_{\text{vanes}}$ (integer >= 1, Default: 1).

Returns: A FaultFrequencyResult (source "bladed rotor").

Raises

Exception	When
ValueError	for a non-positive or non-integer input.

combine_fault_lines

combine_fault_lines(
    *results: FaultFrequencyResult,
    source: str | None = None,
) -> FaultFrequencyResult

Merge several fault-line families into one overlay.

A gearbox bearing carries its own bearing lines, the mesh family of the gear it supports and the shaft harmonics; this puts them on one axes. Duplicate names are disambiguated by appending the source of the family they came from.

Parameters

Name	Description
results	Two or more FaultFrequencyResult objects.
source	Label for the merged family (Default: the sources joined by " + ").

Returns: A FaultFrequencyResult holding every line.

Raises

Exception	When
ValueError	If no result is given, or the shaft rates disagree.

FaultFrequencyResult

FaultFrequencyResult(
    lines: tuple[FaultLine, ...],
    shaft_rate: float,
    source: str,
)

A family of predicted fault lines for one machine element.

Attributes

Name	Description
lines	The predicted FaultLine entries, in the order the generating function produced them.
shaft_rate	Shaft rotational frequency fs, in hertz.
source	Description of the element ("rolling-contact bearing", "gear pair", …), used as the plot title.

FaultFrequencyResult.as_dict()

FaultFrequencyResult.as_dict() -> dict[str, float]

The lines as a {name: frequency} mapping.

FaultFrequencyResult.frequencies

property

Predicted frequencies, in hertz, in order.

FaultFrequencyResult.harmonics()

FaultFrequencyResult.harmonics(name: str, count: int) -> np.ndarray

The first count integer harmonics of the line called name.

Parameters

Name	Description
name	Line name (see names).
count	Number of harmonics, $\ge 1$ (the fundamental first).

Returns: Frequencies $nf$ for n = 1 .. count, in hertz.

Raises

Exception	When
KeyError	If no line carries that name.
ValueError	If count is not a positive integer.

FaultFrequencyResult.names

property

Names of the predicted lines, in order.

FaultFrequencyResult.orders

property

Predicted frequencies in shaft orders, in order.

FaultFrequencyResult.plot()

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

Overlay the predicted lines on a measured envelope spectrum.

Pass the measurement as spectrum= (an EnvelopeSpectrumResult, or any object exposing frequencies and amplitude); without it the predicted lines are drawn alone as a labelled stem plot.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

Name	Description
ax	Existing axes, or None to create a figure.
language	Label language, "en" (default) or "es".
kwargs	spectrum, max_frequency and anything forwarded to the spectrum curve; see phonometry._plot.vibration.plot_fault_frequencies.

FaultFrequencyResult.within()

FaultFrequencyResult.within(low: float, high: float) -> FaultFrequencyResult

The lines falling in [low, high] hertz, as a new result.

Handy before plotting on an envelope spectrum whose useful span is much narrower than the highest predicted harmonic.

Parameters

Name	Description
low	Lower edge, in hertz (>= 0).
high	Upper edge, in hertz (> low).

Returns: A FaultFrequencyResult with the surviving lines.

Raises

Exception	When
ValueError	If the edges are invalid.

FaultLine

FaultLine(
    name: str,
    frequency: float,
    order: float,
    family: str,
    description: str,
)

One predicted discrete line of a machine’s kinematic signature.

Attributes

Name	Description
name	Short label, unique within a result ("BPFO", "2xGMF", "GMF-1x", …). Acronyms are language neutral and are what the FaultFrequencyResult.plot overlay annotates.
frequency	Predicted frequency, in hertz.
order	Frequency expressed in shaft orders, frequency / fs.
family	One of "shaft", "bearing", "gear", "motor" or "blade".
description	One-line English description of the mechanism.

gear_mesh_frequencies

gear_mesh_frequencies(
    speed_rpm: float,
    n_teeth: int,
    *,
    harmonics: int = 3,
    sidebands: int = 0,
    sideband_rate: float | None = None,
) -> FaultFrequencyResult

Gear-meshing frequency and its sideband family (Norton Eq. 8.3).

$\mathrm{GMF}=n_{\text{teeth}}\times N/60$, with integer harmonics k GMF (k = 1 .. harmonics) named "GMF", "2xGMF", … Each harmonic can carry a modulation family at k GMF +/- m f_mod (m = 1 .. sidebands), named "GMF-1x", "2xGMF+2x", and so on. The default modulation rate is the shaft rate of the wheel: a chipped tooth or an eccentric wheel modulates the mesh once per revolution, which is what produces those sidebands (Norton Figs. 8.23 and 8.24). Pass sideband_rate to modulate at the mating wheel’s shaft rate instead.

Only positive sideband frequencies are returned.

Parameters

Name	Description
speed_rpm	Shaft speed N of the wheel, in r/min (> 0).
n_teeth	Number of teeth on that wheel (integer >= 1).
harmonics	Number of mesh harmonics (integer >= 1, Default: 3).
sidebands	Sideband order per harmonic (integer >= 0, Default: 0, no sidebands).
sideband_rate	Modulation rate f_mod, in hertz (> 0, Default: the wheel’s own shaft rate).

Returns: A FaultFrequencyResult (source "gear pair").

Raises

Exception	When
ValueError	for a non-positive or non-integer input.

induction_motor_frequencies

induction_motor_frequencies(
    speed_rpm: float,
    poles: int,
    rotor_bars: int,
    *,
    slip: float = 0.0,
    supply_frequency: float | None = None,
    slot_harmonics: int = 1,
    sidebands: int = 0,
) -> FaultFrequencyResult

Electrical and slot lines of an induction motor (Norton 8.19, 8.20).

The three lines always present in a motor bearing vibration signal are 1x (mechanical unbalance), 2x (misalignment with the driven load) and 2fe (a non-uniform air gap, torque pulses and the winding/rotor-bar electrical faults). Rotor defects that produce static or dynamic air-gap eccentricity are read on the slot harmonics of the stator core,

$f_{\mathrm{sh}}=f_{\mathrm{e}}\left[\left(\frac{2R}{p}\right)(1-s)\pm 2(n-1)\right]=R{f}_{\mathrm{s}}\pm 2(n-1)f_{\mathrm{e}}$

for R rotor bars, p magnetic poles (not pole pairs), unit slip s and n = 1, 2, .... Dynamic eccentricity modulates the dominant slot harmonic at +/- the shaft rate and +/- the slip frequency, which is what sidebands adds around fsh.

Give the slip directly, or give the mains supply_frequency and let it be derived from fe and the measured shaft speed. The supply frequency is taken as $f_{\mathrm{e}}=f_{\mathrm{s}}p/(2(1-s))$: Norton’s Eq. (8.19) writes $f_{\mathrm{e}}=f_{\mathrm{s}}p/2$, which is the same expression at zero slip but does not reduce Eq. (8.20) to the physical rotor-bar passing rate R fs when the machine is loaded.

With a non-zero slip the pole-pass line $F_{\mathrm{P}}=p{f}_{\text{slip}}$ is included as well. That name is standard condition-monitoring practice rather than Norton’s: he gives the slip frequency itself as the sideband spacing of a broken rotor bar (his Section 8.4.8) and does not multiply it by the pole count.

Parameters

Name	Description
speed_rpm	Shaft speed N, in r/min (> 0).
poles	Number of magnetic poles p (even integer >= 2).
rotor_bars	Number of rotor bars/slots R (integer >= 1).
slip	Unit slip s (0 <= s < 1, Default: 0); typically 0,02 to 0,05 under load. Ignored when supply_frequency is given.
supply_frequency	Mains frequency fe, in hertz (> 0), from which the slip is derived (Default: None, use slip).
slot_harmonics	Number of slot-harmonic orders n (integer >= 1, Default: 1, the fundamental R fs alone).
sidebands	Sideband order around the fundamental slot harmonic at +/- the shaft rate and +/- the slip frequency (integer >= 0, Default: 0).

Returns: A FaultFrequencyResult (source "induction motor").

Raises

Exception	When
ValueError	for a non-positive, non-integer or inconsistent input.

shaft_rate

shaft_rate(speed_rpm: float) -> float

Shaft rotational frequency $f_{\mathrm{s}}=N/60$ (Norton Eq. 8.4).

Parameters

Name	Description
speed_rpm	Shaft speed N, in r/min (> 0).

Returns: The shaft rotational frequency $f_{\mathrm{s}}$, in hertz.

Raises

Exception	When
ValueError	for a non-positive speed.