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(Eq.
3.1),
with
a
harmonic
distribution Pn given by the geometry of the gear, over a noise .oor w(n). The variables z, Ω and Φn symbolized shaft rotation frequency, the number of gear teeth, and phase o.set for each harmonic.
∞ 
xperfect(t)= Pncos(ntzΩ+Φn)+ w(t) (3.1) n=0
Due to the imperfect nature of any physically gear implementation, each gear mesh harmonic is subject to amplitude and phase modulation by any multiple
of
the
shaft
rotating
frequency
(Eq.
3.2).

∞ 
xrealistic(t)= an(t)cos(ntzΩ+ bn(t)) + w(t) (3.2)
n=0
∞ 
an(t)= Ak,ncos(ntΩ+ αk,n) (3.3)
k=0
∞ 
bn(t)= Bk,ncos(ntΩ+ βk,n) (3.4) k=0
Consequently, a gear vibration signature becomes a function of the am-plitude modulation amplitude matrix Ak,n, the amplitude modulation phase matrix αk,n, the phase modulation amplitude matrix Bk,n, and the phase modulation phase matrix βk,n. As the coe.cient values tend to drop o. quickly for increasing values of n and k, simpli.ed .nite-size approximations of these matrices can provide a good approximation of a gear vibration sig-nature.
According
to
[42],
any
presence
of
gear
failures
tends
to
increase
the
modulation between the meshing tone harmonics and low multiples of the shaft rotation. This corresponds to a value increase in the coe.cient matrix Ak,n for low values of k. Traditional condition indicators are designed to capture this phenomenon. Indicators do also exist which capture changes in the noise .oor w(t), which also is associated with certain types of damage.

3.3. FEATURE EXTRACTION
Overview
The indicator de.nitions presented here assume that the input signal is .nite, which is the case for all commercial HUMS. It is indeed possible to create indicator algorithms working on in.nite signals, but this topic is not treated in this study due to lack of relevance in the context of HUMS. The indicators explained here are only few examples of the total number existing in the literature, and only an extract of those are given an in-depth explanation.
Indicator  Damage Detected  Ref 
IR  Bearing inner race crack  [35]
 
OR  Bearing outer race crack  [35]
 
BS  Bearing roller crack  [35]
 
Crest Factor  General gear  [10]
 
Energy Operator  Localized gear  [26]
 
Energy Ratio  General gear  [44]
 
FM0  General gear  [42]
 
FM4  Localized gear  [42]
 
Kurtosis  Localized gear / bearing  [39]
 
M6A  Localized gear / bearing  [28]
　
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