Table 3.1: Common condition indicators.
Root Mean Square
The root mean square represents the energy of the signal. As most serious defects in gear and bearing assemblies will increase the signal energy, this is a general fault indicator.
RMSx =1 (x(n) . μx)2 (3.5)
N
n∈N
1
μx = x(n) (3.6)
N
n∈N
Residual Energy
The
residual
signal
[55]
is
given
by
(Eq.
3.7),
where
DF Cx is the DFT co-e.cients of x. This transform captures the noise .oor w(t) of the signal, by removing the signal components corresponding to the harmonics of the mesh-ing
tone.
An
alternative
de.nition
[42]
exists,
which
also
removes
the
signal
components corresponding to the .rst modulation sidebands. By calculating the rms of the residual signal, RMSxres , the energy of the signal noise .oor is estimated. Several gear failures tend to increase the noise .oor, making this an indicator both to localized and distributed damage.
xres = x . DF T .1[MDF C] (3.7)
MDF Ck = DF Ck.[modulus(z, k)! = 0] (3.8)
Residual Energy Ratio
The residual energy ratio is the ratio between the residual energy and the total signal energy. Alternatively, it can be de.ned as the ratio between the residual
energy
and
the
meshing
energy
[44].
The
former
de.nition
is
always
between zero and one, where zero indicates the perfect gear de.nition (Eq. 3.1).
RMSx
ER = (3.9)
RMSxres
Kurtosis
Kurtosis is the fourth statical moment of a dataset, and indicates how outlier-prone the dataset is. In vibration monitoring, this provides a good shock indicator, indicating if a small portion of the signal has signi.cantly higher amplitude than the rest. Kurtosis is associated with localized gear damage, as well as a cracks and corrosion for bearings.
3.3. FEATURE EXTRACTION
(x(n) . μx)4
Kurtosisx = n∈N (3.10)
RMSx
Omega
With Ω being the shaft rotation frequency, the Ωn is simply a spectral pointer de.ned relative to the shaft rotation. For synchronously sampled signals, Ωn corresponds simply to the n’th DFT coe.cient. The values 1 and 2 for n, denotes frequencies for detection of shaft unbalance and misalignment respectively. Values for n being multiples of the number of teeth extracts frequencies associated with gear damage.
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