residual part contains information of non-uniformity, including shaft eccentricity, misalignment,
and uneven loading, as well as irregularities in individual teeth or local tooth defects. Furthermore,
shaft problems and misalignment usually manifest as low-frequency oscillations, so they can be
easily filtered out, leaving only a signal that reflects the tooth irregularities, and thus defects.
Narrow Band Demodulation (NBD). This method assumes that a gear signal bandpass
filtered about the mth meshing harmonic can be expressed approximately by
q m ( t ) = (1 + a m (t ))X m cos(2 mTf s t + m + b m (t )) (5)
where a and b are amplitude and phase modulations and T is number of teeth. Mcfadden (1986)
employed the Hilbert transform to obtain an approximation of the amplitude and phase modulations
of qm(t)from gear signal band pass filtered about the dominant meshing harmonic. His experimental
study showed that the phase modulation revealed incipient gear tooth crack better than the
amplitude modulation.
Energy Operator (EO). For continuous time signal y(t), the energy operator Ψ y t is
given by (Maragos, et. al., 1993)
2
⎛ dy t ⎞ ⎛d2y t ⎞
Ψ y t =⎜ ⎟ −y t ⎜⎜ 2
⎟⎟ (6)
⎝ dt ⎠ ⎝ dt ⎠
Ma (1995) employed the energy operator for gear local fault detection. If the signal has the form of
a simple modulated cosine wave
s t = a t cos t (7)
where a(t) is the time-varying amplitude and t is the time-varying frequency, then, in the
discrete form, the energy operator Ψ s n of the signal is given by
Ψ s n = s2 n − s n −1 s n +1 (8)
5
= a2 t 2
t
which includes both amplitude and/or frequency modulations.
Model Prediction Error (MPE). This algorithm involves making a parametric model of
the gear vibration and watching the prediction error of the model over the life of the gear and
around the circumference of the gear. Wear and pitting on gear teeth will gradually change the tooth
meshing stiffness, which, in turn, will change the gear vibration signal. For a digital gear signal
y(n), Li et al. (1996) made the following linear autoregressive (AR) model
y n = a 0 + a1 y n − k + 1 + a 2 y n − k + 2 + e n (9)
where k is the delay, y(n) is the signal and e(n) is the model prediction error. The coefficients of a0,
a1, and a2 were calculated by minimizing e(n) over the duration of the signal in a least-squares
sense. When the model is applied to a signal, e(n), the prediction error quantifies how well the
model describes the data. By formulating an adequate model at the beginning of gear life and using
the model to calculate the prediction errors over the life of the gear, the amount of overall damage
present in the gear teeth can be estimated (Li, et al, 1996). Their test results showed that MPE
produced an upward trend as the gear teeth wore off.
Correlation Coefficient Discriminant (CCD). Li and Yoo (1998) developed a method to
detect the localized helical gear tooth fault by employing correlation coefficients. The gear
vibration was segmented into as many non-overlapping pieces as the number of teeth, with each
segment centered about a tooth. The correlation coefficient was then calculated between any two
segments. The segment corresponding to a cracked tooth is expected to be different from the others
and therefore a lower correlation level.
Polynomial Discriminant Function (PDF). Li and Yoo (1998) developed a method to
detect a localized tooth defect by employing polynomial regression between time series. Given two
